Correction of
Geometric Perceptual Distortions
in Pictures
Thesis by
Denis N Zorin
In Partial Ful llment of the Requirements
for the Degree of
Master of Science
California Institute of Technology
Pasadena California
i
Abstract
We suggest an approach for correcting several typ es of p erceived geometric distortions in computer
generated and photographic images The approach is based on a mathematical formalization of
desirable prop erties of pictures
Weprovide a review of p erception of pictures and identify p erceptually imp ortant geometric
prop erties of images
From a small set of simple assumptions we obtain p erceptually preferable pro jections of three
dimensional space into the plane and show that these pro jections can b e decomp osed into a p er
sp ective or parallel pro jection followed by a planar transformation The decomp osition is easily
implemented and provides a convenient framework for further analysis of the image mapping
Weprove that two p erceptually imp ortant prop erties are incompatible and cannot b e satis ed
simultaneously It is imp ossible to construct a pro jection such that the images of all lines are straight
and the images of all spheres are exact circles Perceptually preferable tradeo s b etween these two
typ es of distortions can dep end on the content of the picture We construct parametric families of
pro jections with parameters representing the relative imp ortance of the p erceptual characteristics
By adjusting the settings of the parameters we can minimize the overall distortion of the picture
It turns out that a simple family of transformations pro duces results that are su ciently close to
optimal We implement the prop osed transformations and apply them to computer generated and
photographic p ersp ective pro jection images Our transformations can considerably reduce distortion in wide angle motion pictures and computer generated animations
ii
Contents
Abstract i
Intro duction
Motivation
Contributions
Persp ective Pro jection and Viewing Position Nomenclature
Summary
Perception of Pictures
Assumptions and De nitions
Perceptual Nomenclature
De ning Pictures
Assumptions ab out Perception
Robustness of Pictures
Perception of Ob jects in Pictures
Rectangular Ob jects
Spheres Cylinders Humans and Animals
Foreshortening
Other Prop erties of Visual Perception
Visual Field
Perception of Straightness and Curvature
Perception of Verticality
Linear Persp ective
Contents iii
Rectangular Ob jects in Linear Persp ective
Spheres Cylinders Humans
Straightness Verticality Texture Density Movement
Other Traditional Persp ective Systems
Summary
Formalization of Perceptual Requirements
Technical Requirements
Structural Requirements
De nitions and Preliminary Lemmas
Holes in Images
Lo ops Folds and Twists
Lo ops and Twists
Decomp osition
Nonstructural Requirements
De nitions
Zero Curvature Condition
Direct View Condition
Error Functionals
Optimization Problem
Construction of Pro jections
Previous Work
Perceptually based systems
Artistic systems
Technical Pro jections
Axially Symmetric Pro jection
Reformulation of the Problem
Limiting Cases
Lower Bound of the Optimal Values of F
Second Bound
Contents iv
Interp olations
Extentions
Implementation and Applications
Implementation
Deep and Shallow Scenes When Do We Need a Wide Angle Pro jection
Applications of Correcting Transformations
Other Applications
Summary
Conclusions and Future Work
Conclusions
Future Work
App endices
A Review of the Exp erimental Studies of Perception of Pictures
A Robustness and Perception of the Picture Surface and Frame
A Rigidity in Motion Pictures
A Size and Distance in Pictures
A Orientation and Layout
A Rectangular Corners
A Foreshortening
A Visual eld
A Straightness and Curvature
A Verticality
ter of pro jection from a pro jection of a rectangle B Reconstruction of the cen
Bibliography
Chapter
Intro duction
Motivation
The pinhole photograph from Pirenne Figure demonstrates the main problem that is
considered in this thesis The sphere in the picture app ears to b e deformed and only the fact that
ellipsoidal architectural decorations are very uncommon in ancient buildings makes us assume that
the ob ject in the picture is indeed a sphere If we strip the picture of additional cues we are likely
to p erceive the ob ject in the picture as an ellipsoid like shap e rather than a sphere
The shap e of two dimensional images in a photograph is determined by the pro jection of three
dimensional space into the plane pro duced by the camera For a pinhole camera and for most
standard lenses this pro jection is the perspective projection The distortion in Figure is one of
several typ es of distortion that are quite common for wide angle p ersp ective images
There are several questions that we can ask ab out these distortions Why do es p ersp ective
pro jection result in distorted images Whydowe p erceive p ersp ective pro jection images as realistic
in most cases Howdowe obtain a picture similar to Figure which lo oks less distorted The
p ossibility of decreasing destortion is demonstrated in Figure
These questions are imp ortant for image synthesis b ecause most existing rendering systems
Contributions
Figure Left Wide angle pinhole photograph taken on the ro of of the Church of St Ignatzio in Rome classical
example of p ersp ective distortions from Pirenne
Figure Right Corrected version of the picture with transformation describ ed in Section applied
mo del some asp ects of photographic pro cess practically all of them use p ersp ective pro jection as
the viewing transformation
Answering the questions that wehave p osed requires analysis of some features of human p er
ception of pictures for example the p erception of shap e Such analysis can b e used to evaluate
commonly used pro jections and create new ones
The main goals of the work describ ed in this thesis were
to collect and systematize available data on the p erception of pictures of three dimensional
scenes
todevelop a framework for evaluation and construction of viewing transformation based on p er
ceptual considerations
to construct sample families of transformations in this framework
Contributions
We formalize a numb er of p erceptual requirements as mathematical constraints imp osed on
the viewing transformation
Weprove that a viewing transformation can pro duce p erceptually acceptable images for arbi
Viewing transformation is the computer graphics term for the pro jection of three dimensional space into the plane
Persp ective Projection and Viewing Position Nomenclature
principal ray projection(picture) projection(picture) plane center plane of projection picture picture α α center of center of the picture the picture
projection angle
direction of projection
direction of projection
Figure Left Central pro jection
Figure Right Parallel pro jection
trary scene geometry only if it can b e represented as a comp osition of a linear pro jection and
planar transformation for each disjoint part of the scene
Weintro duce quantitative measures for twotyp es of p erceptual distortion that can b e used
for evaluation of viewing transformations
We demonstrate that two imp ortanttyp es of distortion cannot b e eliminated simultaneously
We construct a useful family of axially symmetric viewing transformations whichallow the
reduction p erceptual distortion
We describ e a non symmetric extension of this family which allows further reduction of geo
metric distortion
Persp ective Projection and Viewing Position Nomenclature
We shall use a numb er of terms describing various characteristics of p ersp ective pro jections and
viewing p ositions There is a lot of variation in usage of some of these terms we try to use a
consistent nomenclature whenever p ossible even when describing results of authors who originally
used a di erentone Terminology more sp eci c to perception can b e found in Section and
de nitions related to more general typ es of pro jection are given in Section and Section
central pro jection Let C be a pointinR letp b e a plane such that C p For anypoint
l to b e the line through C and x Thenacentral projection with the x R x C de ne
x
center C to the plane p is the mapping R n C p which maps anypoint x R to l p x
Persp ective Projection and Viewing Position Nomenclature
parallel pro jection Let p a plane v avector which is not parallel to the plane p For any p oint
x R de ne l to b e the line through x parallel to v Then a paral lel projection in the
x
direction v to the plane p is the mapping R p wqhich maps any p oint x R to l p
x
perspective pro jection Either central or paral lel projection of a part of the three dimensional
space into a plane the projection or picture plane
orthogonal pro jection A p ersp ective pro jection is orthogonal if the direction of pro jection is
p erep endicular to the plane of pro jection this de nition applies b oth to the parallel and
central pro jections
principal ray In central pro jection the principal ray is the ray through the center of pro jection
hintersects the pro jection plane p erp endicularly whic
direction of pro jection For the parallel pro jection the direction of projection is a part of the
standard de nition For the central pro jection we de ne the direction of projection to b e the
direction from the center of pro jection to the center of the picture This de nition is precise
for ob jects with central symmetry such as a rectangle For a picture of arbitrary shap e the
center can b e de ned to b e the geometric center the center of mass
When wesay direction of pro jection we mean that the angle between the pro jection
plane and the direction of pro jection Figure Figure is
pro jection angle For a central pro jection with the center C wecho ose a characteristic line segment
AB in the picture passing through the center of the picture and endp oints A B on the
AC B For rectangular pictures b oundary of the picture De ne the projection angle to b e
we can use the horizontal or vertical line segment going through the center of the picture The
projection angle is not de ned uniquely for a given picture even if it is rectangular If the ratio
of the minimal linear dimension of the picture to the maximal is not to o small the projection
angle do esn t vary much with the choice of AB
For the parallel pro jection we de ne the pro jection angle to b e zero
pro jection distance For the central pro jection the distance from the center of pro jection to the
center of the picture is the projection distance
Persp ective Projection and Viewing Position Nomenclature
projection(picture) viewing point plane picture α center of the picture
visual angle
viewing direction
Figure
viewing p oint For mono cular viewing the viewing point is de ned as the center of the lens of the
eye For bino cular viewing the viewing point is the middle of the line segmentbetween the
eyes
viewing angle The viewing angle is equal to the projection angle ofacentral pro jection from the
viewing p oint to the picture plane
viewing direction The viewing direction is the direction from the viewing p oint to the center of
the picture
viewing distance The distance from the viewing p oint to the center of the picture
visual eld visual eld is the solid angle with the ap ex at at the viewing p ointsuchthelightfrom
all directions within this angle reaches the eye s
eld of view If an ap erture is used for viewing a large part of the visual eld is o ccupied by the
ap erture T he remaining part is called the eld of view
Summary
Summary
We start with an analysis of the p erception of geometry in pictures Chapter There is no general
theory that explains all the asp ects of p erception of pictures Instead of using any particular theory
we describ e some minimal requirements of p erceptual realism that arise from everyday exp erience
and some exp erimental work in psychology This part of the thesis together with App endix A
attempts to provide a concise but su cient description of facts that are known ab out the p erception
of pictures The most imp ortant topics that are discussed in Chapter are the robustness of pictures
the p erception of common geometric shap es such as rectangular corners and spheres the p erception
of curvature and verticality and the p erception of rigidity in motion pictures
Perceptual requirements provide us with a set of constraints that we imp ose on the pro jection
Violation of some of these constraints results in unacceptable artifacts in the images suchasloops
in the images of lines or twisted images of planes Section Other constraints are more exible
and some deviation from the exact satisfaction of these constraints can b e tolerated An example
of such a constraint is the curvature of the image of a straight line it can b e nonzero for relatively
short line segments
The rst group of p erceptual constraints structural constraints allows us to limit the set of
viewing transformations that we consider Section Using additional practical considerations
we prove that any pro jection satisfying these constraints is a comp osition of a p ersp ective pro jection
and a two dimensional transformation
Some constraints in the second group cannot b e satis ed simultaneously the curvature of lines
and the distortion of shap e cannot b e eliminated completely at the same time Section We
intro duce error functions for each constraint whichcharacterize the quality of a pro jection with
resp ect to this constraintatapoint We can use the average or the maximum error to obtain an
overall characteristic of a pro jection
Using global error functionals for the direct view constraint and the zero curvature constraint
we formulate a parametric optimization problem which allows one to nd pro jections with minimal
global errors for a given tradeo b etween the twotyp es of the error Section
We consider a simple parametric family of pro jections Section and show that this family
approximates exact solutions of the optimization problem in the axially symmetric case well enough
for practical purp oses The resulting family of transformations has a very simple form
Summary
If weallow di erent tradeo s b etween twotyp es of constraints in di erent parts of the picture
we can pro duce non symmetric pro jections which can b e b etter adjusted to the sp eci c scene con
guration The resulting extended family of transformations is still quite easy to use but is much
more exible
Our families of pro jections havetwo principal comp onents that can b e chosen indep endently
the p ersp ective pro jection and the two dimensional transformation We describ e how the choice of
appropriate pro jection is in uenced by the geometry of the scene Section
The fact that the rst part of our decomp osition of pro jections is the standard p ersp ective
pro jection makes it p ossible to implementthetwo dimensional transformation as a p ostpro cessing
step of rendering It is also p ossible to apply this transformation to photographs if the angle of view
is known or if it can b e computed from the information in the picture Section App endix B
In the wide angle motion pictures varying distortion of shap e results in an impression of nonrigidity
whichmay b e undesirable our transformations allow to reduce this e ect
Chapter
Perception of Pictures
One of the goals of image synthesis is to achievetherealism in pictures Realism is hard to de ne
although most p eople haveanintuitive notion of realism Wewillcallthisintuitive notion perceptual
realism For computer graphics applications a precise de nition is required The most common
solution is to de ne realism as equivalent to photorealism Photographs are considered to b e a
standard of truth in images and the mo deling of an idealized photographic pro cess b ecomes the
main task of the image synthesis This solution has a numb er of advantages but inevitably su ers
from a fundamental aw the image synthesis algorithms are constructed and evaluated as mo dels
of a photographic pro cess while the output of these algorithms pictures is evaluated primarily by
a p erceptual pro cess If we use this approach the p erceptual quality of computer generated images
is inherently limited by that of photographs
It is well known that an arbitrary photograph do es not necessarily lo ok go o d Although photog
raphers use some typ es of p erceptual distortions o ccurring in photographs to achievevarious artistic
e ects in many cases distortions are undesirable and must b e avoided Avoiding these distortions by
purely photographic means choice of the viewing p oint or of the eld of view results in restrictions
on the scenes that can b e realistically repro duced in photographs Computer generation of images
is more exible and instead of faithfully imitating the photographic pro cess wecantrytouse
Assumptions and De nitions
p erceptual principles directly instead of mo deling a camera we can rede ne our working concept
of realism formalizing some facts ab out picture p erception Moreover photographic images can b e
pro cessed digitally and we can eliminate distortion in already exiting images
These goals require a more precise de nition of the class of pictures that we are going to consider
after all any pattern on a at surface can b e a picture Section describ es the class of pictures
in whichwe are interested
There are three main sources of information ab out picture p erception available to us psy
chophysical research art history and theory andoureveryday exp erience Quite a few facts ab out
p erception app ear to b e so obvious that nob o dy ever states them explicitly or b others to test them
exp erimentally We will see that some of these obvious facts are quite imp ortant Section
There is a considerable di erence b etween our approach to the problems of p erception and the
approach that is typically used in psychophysical research Our main goal is not to prove or disprove
some particular theory of p erception but to try to collect facts ab out p erception that can b e used
for making b etter pictures Wewishtoavoid relying on any particular theory as much as p ossible
We cannot avoid making some theoretical assumptions altogether we describ e our assumptions in
Section
Most exp erimental studies consider linear p ersp ective images but their results apply to anypro
jection pro ducing similar images Whenever p ossible we will emphasize two dimensional character
istics of images rather than the characteristics of the central pro jection used for their construction
Then we can extend our conclusions to any mappings from three dimensional space into the plane
Linear p ersp ectivenevertheless is extremely imp ortant it do esn t pro duce any distortion in most
cases and our main practical application is correction of distortions in the cases in whichitdoes In
Section we discuss linear p ersp ective in greater detail
Assumptions and De nitions
Perceptual Nomenclature
One of the problems with using data from the psychological literature is the lack of consistent
and clear terminology We attempted to de ne and use consistently a limited numb er of terms for
frequently used concepts and quantities Some of our de nitions are nonstandard such as parallel
Assumptions and De nitions
and p erp endicular foreshortening Our motivation in these cases was to create an unambiguous and
concise way to describ e a particular concept
The word quantity here refers to numerical characteristics of three dimensional ob jects and scenes
represented in a picture suchasslant size distance or orientation The numerical value of a quantity
can b e
depicted A depicted quantity size slant distance etc is the actual value for the ob ject or scene
represented in the picture
apparent An apparent quantity is the value as p erceived or deduced from the picture Apparent
quantity cannot b e measureed directly it can b e estimated from the judged quantity or from
some other resp onse such as ball throwing in Smith
judged A judged quantity is the value as rep orted by a sub ject in a go o d exp eriment it is deter
mined by the apparent measure but can b e biased bythemechanism of rep orting or by the
sp eci ed task This bias however should b e the same for identical apparentquantities
The totality of apparentquantities can b e thoughtofastheperceptual space Theperceptual space
tity need not p ossess a consistent geometry in mathematical sense the value of an apparentquan
can dep end on the task attention fo cus and many other factors In most cases the perceptual space
of a picture is not connected to the real space around the observer A p ossible reason for this is
the scalability of the pictures in most cases familiar ob jects in the pictures are smaller or larger
than their real size and they cannot b e directly placed in the real space There are several notable
exceptions the p erceptual space of trompe l oeil pictures Milman is merged with the real
space the orientation of ob jects in the p erceptual space can b e judged with resp ect to the viewer
Most names of quantities e g size slant distance are unambiguous The de nition of orienta
tion requires some comments
orientation We shall distinguish b etween twotyp e of orientation relative to the viewer and relative
to the other ob jects in the picture We can estimate the angle b etween the viewing direction
and some direction sp eci ed by an elongated ob ject in a picture We shall call this angle
orientation with resp ect to the viewer It is also p ossible to estimate the angle b etween two
directions in a picture We shall call this angle relative orientation
Assumptions and De nitions
Information ab out the geometry of three dimensional ob jects is partly contained in the two
dimensional geometry of images Most terms that we use to sp ecify the two dimensional geometry of
images are just general geometric terms e g angles lines curves areas etc A sp eci c characteristic
of images which is inherently two dimensional but cannot b e describ ed without referring to the three
dimensional ob ject is foreshortening For p ersp ective pro jections we shall distinguish twotyp es of
foreshortening paral lel and perpendicular
foreshortening If the size of the image of an ob ject dep ends on its p osition in the scene we shall
call this e ect foreshortening
parallel foreshortening The decrease in the size of the image of an ob ject when it is placed at
increasing distances from the picture plane It is also called perspective convergence
p erp endicular foreshortening Supp ose it is p ossible to destinguish a feature of the ob ject p er
p endicular to the pro jection plane and another feature which is parallel to the pro jection plane
such as faces in a cub e or the lateral and frontal side of the head The di erence b etween the
ratio of the sizes of these features and the ratio of the sizes of their images is the perpendicular
foreshortening
It is p ossible to extend these de nitions to the case of nonp ersp ective pro jections if wecan
somehow de ne the distance to the pro jection plane which need not coincide with the physical
distance
It is imp ortanttomake a clear distinction b etween several related p erceptual phenomena
distortion Some images of familiar ob jects can b e identi ed as distorted if p erceptual information
in the picture is su cienttoidentify the ob ject with a high degree of con dence yet some
tradict exp erience part of this information results in conclusions ab out the ob ject that con
Returning to the example in Figure one is reasonably con dent that the depicted ob ject is
a sphere b ecause a sphere is more likely to b e a part of architectural decoration than a tilted
ellipsoid At the same time the shap e of the image suggests that the ob ject is not spherical
which results in a contradiction
misp erception Some images of ob jects can convey incorrect information ab out the ob jects without
causing distortion If an ob ject is part of a class of ob jects whichvary in size some images
Assumptions and De nitions
result in an apparent size which is close to the actual size or shap e while others will result
in a di erent apparent size We will call the later case misperception For example the right
monitor in Figure app ears to have an asp ect ratio signi cantly greater than the actual
one no more than and than that of the left monitor the monitors are known to b e
identical
illusion Some pictures under sp ecial conditions can b e mistaken for the real scene that they repre
sent We shall call this condition il lusion
Figure Photo from the article Navigating Close to Shore byDave Do oling IEEE Sp ectrum Dec
c
IEEE photo byIntergraph Corp viewing angle The two monitors have the same size But the left one
app ears to b e wider misp erception
De ning Pictures
This section de nes more precisely what wemeanby a picture It is also imp ortant to describ e
assumptions ab out viewing conditions the p erception of a picture dep ends on the p osition of the
viewer with resp ect to the picture presence or absence of ap ertures frames etc
In this thesis our main ob ject of study will b e at or nearly at pictures on pap er a pro jection
screen or any other at surface observed bino cularly without any restrictions on the p osition of
the head and without any sp ecial devices Pictures of this typ e include b o ok illustrations photos
p osters screen pro jections of slides motion pictures and pictures on computer displays Excluded
are stereograms of all typ es pictures that are designed for observation through a xed small ap erture
pictures in head mounted displays and anamorphic pictures
Assumptions and De nitions
Further we are interested in pictures that are representations of three dimensional ob jects There
are many di erenttyp es of representation from purely symb olic likeaverbal description or a kanji
character to a highly realistic photographic image which when viewed from a correct p osition
mono cularly can b e confused for a real ob ject Linear p ersp ective is used to some extent in most
pictures of the latter typ e
Some authors argued that essentially all forms of pictorial representation are based on convention
Arnheim Go o dman For example it has sometimes b een claimed that linear p ersp ective
is a matter of convention and p ersp ective images can b e understo o d only within the context of
a particular culture It app ears that such radical approach cannot b e accepted Cross cultural
Deregowski developmental Hagen a Hagen and Jones studies contain indications
that no culture dep endent learning is required for adequate p erception of p ersp ective images The
existence of various phenomena in p erception of pictures that we will discuss b elow also demonstrate
that there is a fundamental di erence b etween the p erception of most pictures and the p erception
of purely conventional representations such as text
Figure Examples of highly conventional pictures a road sign a map a carto on
The presence or absence of a conventional comp onent in a picture is di cult to determine In
some cases Figure a it is easy to classify a drawing as almost purely conventional We will b e
most interested in drawings that don t use a symb olic metho d of representation or use it in a very
limited way This includes a wide range of pictures Figure
Assumptions ab out Perception
We will try to make as few assumptions ab out p erception as p ossible Wehave however to assume
that pictorial representation of information is mostly nonsymb olic Our assumptions can b e most
clearly formulated in terms of retinal images and internal representations
Assumptions and De nitions
a c
b d
Figure Examples of pictures that use symb ols in a limited way or don t use them line drawings a painting
Piano lesson by A Renoir a photograph Beckman Institute at Caltech
Our ability to recognize ob jects and textures that wehave seen b efore is crucial b oth to p erception
of the three dimensional world and pictures Wehave to assume
existenceofsomeinternalrepresentation of objects and a process that compares information
extractedfrom the retinal image with this representation
All the visual information ab out our environmentiscontained in the images formed on the retina
of the eye
Therefore
our internal representation of three dimensional objects should bebased on the information con
tained in these images
We don t know what part of the information contained in retinal images is utilized in this internal
representation
As retinal images are two dimensional central pro jections there is one to one corresp ondence
between a retinal image of an ob ject and a pro jection of the ob ject onto a plane a linear p ersp ective
Assumptions and De nitions
picture of an ob ject Therefore we can consider geometry of p ersp ective pro jections instead of
geometry of retinal images
We can distinguish at least twotyp es of geometric information in retinal images structural and
nonstructural
Structural features have more qualitative nature and are the easiest to detect Examples of
structural information are the numb er of holes in an image of an ob ject dimension of the image or
its parts the numb er of edges meeting at a vertex Figure Mathematically this information
corresp onds to the top ological prop erties of the image We assume that
structural information about projections of an object is present in the internal representation of
the object If an image of an object has structural features that don t match those of any perspective
projection it is perceivedascontradictory distortedoritisnotrecognizedatall
a
b
c
d
e
Figure Examples of structural di erences in images a the numb er of holes b numb er of edges meeting at a
vertex c dimension d numb er of self intersections lo cal structure e number of intersections of edges lo cal structure
Weshouldpoint out that our assumptions ab out the imp ortance of structural features are mostly
motivated by common sense and intuitive ideas ab out p erception due to the lack of exp erimental
Robustness of Pictures
data It is also imp ortant to note that the precision of the visual system is nite Finke and Kurtzman
and the ab ove statement is an idealization for example if a hole in an image is to o small it
is not detected by the visual system
Nonstructural features typically can vary without causing any signi cantchange in p erception
Examples of nonstructural features are the degree of convergence foreshortening gradients of tex
ture angles and curvature of edges Figure The imp ortant di erence from structural features
is that variation in nonstructural features can b e registered by the visual system without pro ducing
considerable change in p erception
a
b
c
d
Figure Examples of nonstructural di erences in images a texture gradients b parallel foreshortening c p er
p endicular foreshortening d curvature
Robustness of Pictures
The term robustness of pictures was intro duced by Kub ovy The p erception of a picture
typically do esn t dep end on the viewing p oint wecanwalk past a painting tilt a b o ok moveour
head away closer or to the side of a computer display and the ob jects in the picture are unlikely
Robustness of Pictures
to change their shap e or p osition This is one of the most imp ortant prop erties of pictures were
it not true a viewing apparatus or correct choice of viewing p osition would b e required for correct
p erception Robustness is not total some p erceptual variables are less robust then others
Anumb er of particular p erceptual variables were con rmed to b e indep endent of the viewing
p oint
Previous studies Rosinski and Fab er Rosinski et al showed that for normal viewing
conditions p erception of slant do esn t dep end on viewing p osition This study was extended by
Halloran who showed that for extremely oblique viewing directions with the angle b etween the
viewing direction and the picture surface less then robustness of the p erception of slant breaks
down
There is some evidence Hagen b Smith b Rosinski and Fab er Exp eriment that
relative size judgements are robust as well only two viewing directions and were used in
the rst study and only the viewing distance was varied in the last two
Robustness is very imp ortant for moving pictures for example inverse p ersp ective constructions
such as describ ed in Deregowski and La Gournerie applied to the images of rotating b o dies
might result in non rigid ob jects if the pro jection p oint used for reconstruction is di erent from the
actual pro jection p oint However this do esn t happ en in most cases As it was shown by Cutting
for images of rotating near rectangular solids there is a general tendency to ignore small
deformations esp ecially if they preserve angles b etween edges and that rigidity is preserved at least
up to a viewing direction
Relative orientations and layout are quite robust Halloran Exp eriment Goldstein
Exp eriment Goldstein Exp eriment although relative distances b etween ob jects in the
direction p erp endicular to the picture plane dep end somewhat on the viewing distance see discussion
b elow
Not all p erceptual variables are robust and the range of viewing p oints within which robustness
is preserved might dep end on the depicted ob ject
Wehave already mentioned that relative distance b etween ob jects in the direction p erp endic
ular to the picture plane is not very robust Exp eriments show that the viewing p osition a ects
judgements of relative distance Smith b Smith a although the e ect is less than predicted
by geometric reconstruction It should b e noted however that reduced viewing conditions were
Robustness of Pictures
used in these studies mono cular viewing through a p eephole and the absence of robustness can b e
attributed to the lack of information ab out the surface of the picture see b elow
The orientation of the ob jects with resp ect to the observer is not robust at all in some cases
It is the most apparent nonrobust p erceptual variable it was noted bymany authors that ob jects
p ointing p erp endicular to the plane of the picture such as gun barrels or ngers Figure app ear
to b e following the observer as he moves past the picture Portraits often app ear to b e following
the observer with their eyes It turns out Goldstein Goldstein Halloran that this
e ect is more pronounced for orientations close to the p erp endicular to the picture and decreases
with the angle of orientation This results in the following p erceptual paradox while spatial layout
is p erceived as more or less invariant orientations of di erent ob jects change in di erentways for
example the rut in the road in Figure A do esn t rotate much while the direction of the road
rotates considerably Thus the di erence b etween apparent orientations changes while the relative
orientation of the rut and the road do esn t change This is an example of a more general phenomenon
the structure of p erceived space need not b e consistent di erent p erceptual mechanisms might
pro duce contradictory information without causing any p erceptual problems
Robustness of p erception may dep end on the contents of the picture For example parallel pro
jections of rotating rigid parallelepip eds are p erceived as rigid in a wider range of viewing directions
than central pro jections Cutting Exp eriments and
Robustness of pictures is related to the dual nature of pictures weperceive b oth the picture
surface and the three dimensional scene represented in this surface Surface texture atness and
visible frame are imp ortant factors in p erception of pictures Numerous exp eriments provide evidence
that the absence of these factors results in a decrease in the robustness
Perception of slant b ecomes completely unrobust when all the information ab out the picture
surface is removed Rosinski and Fab er Rosinski et al In unpublished exp eriments by
W Purdy Lumsden only the picture frame was absent and the information ab out the picture
surface was not completely removed Still for the simple pictures used in the exp eriments slanted
strip ed surfaces picture frame removal was su cient for almost total suppression of robustness
Perception of Objects in Pictures
Figure A World War I p oster from Taylor The nger app ears to b e following the viewer when he
walks past the picture Similar p osters were created in the US Uncle Sam wants you and in Russia Did YOU
volunteer during times when the government felt it necessary to p oint ngers at each citizen
Perception of Objects in Pictures
The distortions that wementioned in Chapter are asso ciated with images of particular ob jects
Two groups of ob jects which frequently app ear to b e distorted in photographs Kub ovy The
rst group included ob jects with rectangular three dimensional corners i e corners formed by
three edges with all angles b etween them equal to The second group included spherical and
cylindrical ob jects and humans
Rectangular Objects
Perception of pictures of rectangular corners is well describ ed by simple rules Perkins Perkins
Shepard
These basic facts ab out p erception of rectangular corners are usually called Perkins laws We
will call three line segments meeting at a p ointathree star Perkins We will call an image of
Perception of Objects in Pictures
a rectangular corner two faced if only two out of three b oundary surfaces are visible and three faced if all three are visible Figure
Three-faced image Two-faced image
Figure Twotyp es of images of rectangular corners
Perkins rst law A three star is acceptable as a two faced image of a rectangular
corner if and only if it contains two angles less than or equal to whose sum is greater
then
Perkins second law A three star is acceptable as a three faced image of a rectangular
corner if and only if all three angles are greater than
Small deviations from Perkins laws can o ccur but in general there is a go o d agreementbetween
the exp eriment and theory Perkins Shepard
It is imp ortanttonotethatPerkins laws haveavery simple geometric interpretation acceptable
pro jections coincide with orthogonal p ersp ective pro jections There is no di erence b etween central
and parallel pro jections in this case b ecause only angles b etween lines are imp ortant For the
central orthogonal pro jection wehave to assume that the principal ray go es through the center of
the three star
From Perkins data Figure A we can make a rough estimate of the deviations from Perkins
laws that don t result in considerable distortion the three stars deviating from Perkins laws by less
then still can b e p erceived as rectangular corners Of course this is a rough estimate We will
discuss Perkins laws in greater detail in Section
Perception of Objects in Pictures
Spheres Cylinders Humans and Animals
Perceptual requirements on the images of spheres are remarkably restrictive only disks are gen
erally accepted as go o d images of spheres Pirenne Kub ovy No data on detection of
non circularity were available to us It app ears to b e safe to assume that asp ect ratio is de
tectable Figure and accept it as a rough upp er b oundary Spheres are not that common in real
environments although quite p opular in computer graphics images They are also convenient test ob jects b ecause the distortion of shap e in the image of a sphere is easy to detect and describ e
1.00 1.01 1.02
1.03 1.04 1.05
1.06 1.07 1.08
1.10 1.11 1.12
Figure Ellipses with di erent asp ect ratios the asp ect ratio is shown in the center of each ellipse
There are two distinct problems asso ciated with images of cylinders such as columns the one that
is most often mentioned arises when a row of cylinders is depicted It is not a problem asso ciated
with the image of any particular cylinder but rather a problem of unacceptable relative sizes of
images We will discuss it in the Section
The second problem can b e considered a separate case of a more general class of problems that
is asso ciated with axially symmetric ob jects
The image of the foundation of the column or the upp er part of a cup or a b owl is generally
acceptable if it is an ellipse with ma jor axis oriented p erp endicular to the axis of symmetry cf
Section Figure
Our p erception of humans is likelytobevery sp ecialized Humans come in many shap es colors
and sizes so the result of p erception of an inadequate image is more often a misp erception a
false conclusion ab out the p erson in the picture rather than direct p erception of distortion as with
spheres If we stretch or shrink the image vertically or horizontally within a wide range these
changes are likely to pro duce acceptable although sometimes misleading pictures For example the
Perception of Objects in Pictures
men in Figure are all identical but those close to the edges of the picture app ear to b e quite
di erent from those in the middle The part of the b o dy which is most a ected by deformation is
the head due to less variation in the shap e of the head b etween individuals compared to other b o dy
parts
Figure Deformationsofhuman gures in a wide angle p ersp ective picture horizontally
There are typ es of distortion that are more likely to pro duce deformed not just misleading
images An imp ortant feature of acceptable frontal pictures of humans is their axial symmetry
When this symmetry is broken the image b ecomes unacceptable
Similar conclusions can b e reached ab out images of animals although larger deformations are
tolerated
Foreshortening
We consider twotyp es of foreshortening parallel and p erp endicular Our terms here are not quite
standard we b elieve it is useful to destinguish b etweeen these twotyp es of foreshortening
For rectangular solids the preferred amount of p erp endicular foreshortening dep ends on the
asp ect ratio Nicholls and Kennedy b it is almost constant for cub es the preferred amountwas
close to for all viewing conditions It should b e noted that the use of line drawings in this
study could considerably a ect the results p erception of the whole ob ject could b e imp ortantin
this study the absence of any distinctive features in the drawing sp ecifying absolute size markings
Perception of Objects in Pictures
Figure Left pro jections of cub es from Pirenne
Figure Right pro jections of cylinders from Pirenne
on a matchbox or windows on a building could a ect the preferred amount of foreshortening in an
unpredictable way
The amount of p erp endicular foreshortening deviating signi cantly from the preferred one results
in distortion if it is known that the depicted ob jects are supp osed to b e cub es and in misp erception
if no such information is available Figure
Perp endicular foreshortening is easy to de ne only for ob jects with distinct edges parallel and
p erp endicular to the picture plane it can b e also de ned for familiar ob jects with distinctive features
parallel or p erp endicular to the picture plane Some distortions of the human gures can b e describ ed
in terms of the p erp endicular foreshortening ratio
Parallel foreshortening across ob jects can vary in a wide range without causing p erceptual prob
lems In p ersp ective images the ratio of parallel foreshortening is directly related to the viewing
angle the greater the viewing angle the greater the ratio for an ob ject of a given size Nicholls and
Kennedy a studied the e ects of the change in the viewing angle on the p erception of cub es it
was found that a mo derate degree of parallel foreshortening approximately was consistently
preferred indep endent of the viewing conditions
Other Prop erties of Visual Perception
Other Prop erties of Visual Perception
Visual Field
The total size of the visual eld is quite large it extends more than horizontally and
vertically for eacheye Carterette and Friedman However the density of the receptors is very
nonuniform and maximal resolution is achieved only in the fovea of the eyewhich has an angular
size of only ab out Lateral vision is very limited in resolution its main function is presumably
motion detection
As the resolution decreases less and less detail and precision is available to the rest of the visual
pro cessing system Finke and Kurtzman found that the size of the eld where gratings apart
can b e resolved is approximately For example the angular size of the mo on is Outside
eld of view we cannot detect any details of the lunar surface and cannot even tell if the mo on
is round or not
This also means that only low resolution information in the retinal pro jections of the ob jects
that are far enough o the viewing direction can b e incorp orated into the internal representation
The same study Finke and Kurtzman provides some evidence that indeed the resolution of the
internal imagery corresp onds to the resolution of the actual visual eld
Perception of Straightness and Curvature
Intuitively it is clear that the images of straight edges and lines should b e straight It is imp ortant
to know the accuracy of the p erception of straightness Exp erimental studies that we knowabout
considered the p erception of straightness only for very short lines with angular size close to Itwas
found that for small p erturbations p erceptual threshold is determined by the solid angle subtended
y the maximal bump with resp ect to the least squares straight line Figure Watt et al b
The threshold value was determined to b e close to sq arc min
In Ogilvie and Daicar a similar measure was used although a chord rather than a least
squares line was considered and there was only one bump Resulting thresholds had the same order
of magnitude sq arc min These values are remarkably small Watt et al p oints out
that the receptor density is not su cient for the detection of p erturbations of that size and some
typ e of higher level pro cessing should b e involved to account for these thresholds hyp eracuity It
LinearPersp ective
app ears that this threshold is likely to increase with the angular size of the line However it is b ound
to stay quite lowandany considerable deviation from straightness will cause p erceptual distortion
Least-squares straight line Maximal bump
Curved line
Figure Watt s measure of p erceptual curvature
Perception of Verticality
The p erception of verticality is in uenced bytwo main factors vestibular p erception of the force
of gravity and visual p erception Ob jects that we assume to b e vertical walls trees etc or
horizontal the surface of the ground determine the visual vertical A numberofexperiments Ong
and Kessinger DiLorenzo and Ro ck show that when there is a con ict b etween vestibular
and visual information there is no clear preference for the ob jectivevertical determined by the
vestibular system The ro d and frame e ect DiLorenzo and Ro ck indicates that a visible
frame a ects p erception of actual vertical when a vertical ro d is viewed inside a tilted frame it
app ears to b e tilted in the direction opp osite to the direction of the frame
In pictures lines that are not parallel to the edges of a rectangular frame are p erceived as non
vertical If these lines representvertical or horizontal edges of ob jects buildings or furniture the
picture creates a feeling of instabilitywhich is sometimes undesirable The ro d and frame e ect
explains to some extent the origin of this feeling
LinearPersp ective
Linear p ersp ectivewas intro duced as a systematic to ol during the Renaissance To some extent
linear p ersp ectivewas known to the Romans fresco es of Pomp eii and Chinese see discussion of
other p ersp ective systems in Section but during the Renaissance it was intro duced as a rigorous
system whichwas considered to b e the foundation of painting and drawing till the second half
LinearPersp ective
of the nineteenth century The rst artist to use p ersp ective during the Renaissance was Filipp o
Bruneleschi but the rst written description and analysis was done by Leon Alb erti in Alb erti
and Leonardo da Vinci in da Vinci He was also the rst to observe the limitations of the
metho d As it is discussed in Kub ovy Leonardo was not convinced that p ersp ective pictures
are robust hence his overly stringent requirements to the application of p ersp ective Leonardo s
rule
do not trouble yourself ab out representing anything unless you takeyour view p oint
at a distance of at least twenty times the maximum width and height of the thing that
you represent and this will satisfy every b eholder who places himself in frontofthework
at any angle whatso ever
This means that angular size of any ob ject in a picture should b e less than clearly an
overkill Leonardo o ers the following example of p ersp ective distortion Figure in the image
ofarow of columns depicted from a close pro jection p oint the width of the images of columns closer
to the edges of a picture would increase which is quite counterintuitive
Figure Leonardo s example of a p ersp ective distortion
Statements similar to Leonardo s rule are quite common For example Glaeser doesnot
recommend using viewing angles ab ove Olmer recommends viewing angles no more
than horizontally and vertically
Thus the limitation of linear p ersp ectivewere recognized simultaneously with its discovery As
it is p ointed out by Kub ovy it is not correct to identify artistic linear p ersp ective with central
pro jection of a halfspace onto a plane only a limited range of viewing angles is used and as we
will see the images of many ob jects are often painted with deviations from linear p ersp ective
LinearPersp ective
In photographs and computer generated images linear p ersp ective is practically identical to
central pro jection exact linear p ersp ective wide angle images are not uncommon and due to the
physics of the pro cess or nature of the algorithms resulting images follow the rules of the central
pro jection exactly Intuitive comp ensation used by the artists is not available in this case Two
metho ds are helpful in the analysis of exact linear p ersp ective rst we can checkhowwell linear
p ersp ective images satisfy various p erceptually desirable requirements describ ed in the previous
sections second we can examine delib erate deviations from linear p ersp ective that are common in
painting
Structural features Linear p ersp ective images have no undesirable structural features see
Section
Robustness The studies of robustness were done using p ersp ective images Summarizing their
results wecansay that under normal viewing conditions p ersp ective images are robust when the
viewing angle is su ciently small or the viewing direction is not to o oblique In general robustness
is a concern only for moving images
Rectangular Objects in LinearPersp ective
In p ersp ective images Perkins laws are always violated to some extent with the only exception of
orthogonal parallel pro jection The extent of this violation dep ends for the parallel pro jection on
the direction of pro jection and for the central pro jection on the direction from the center to the
ap ex of the rectangular corner We will call b oth directions directions of pro jection
Wehave selected two parameters to characterize the violation of Perkins laws fraction of
rectangular corners with pro jections violating Perkins laws for a given direction of pro jection and
maximal deviation of the o ending angle from
There are in nitely many p ossible p ositions of rectangular corners in space and each of them
o angles and determining the orientation of one of the can b e represented by three angles tw
edges and the angle of rotation of the remaining pair of edges around the rst one
De ne a triple to b e go o d if the pro jection of the corner in the given direction satis es
Perkins laws
Then we can de ne the fraction of go o d pro jections to b e
LinearPersp ective
A B A B C
C
Figure Three sided and two sided images of a rectangular corner
R
Z
d d d
all go o d
R
d d d
d d d
all good
all
We de ne the o ending angle in the following way LetA B C b e the angles b etween the edges
of the three star always less then Let A B C Then there are two p ossible violations of
Perkins laws either all three angles are less than or only one of them is less than C In
the rst case A B C and A should b e greater than while B and C can b e less than
In this case we de ne A the largest angle to b e o ending angle In the second case there are two
p ossibilities A B C or A B C In the rst case B should b e less than second
Perkins law applies and we de ne it to b e the o ending angle In the second case all angles should
b e greater than and we de ne C to b e the o ending angle
It can b e easily shown that our de nition is equivalent to the following simpler one
the o ending angle is the angle b etween edges of the three star which is the closest to
Figure shows the fraction of go o d corners as a function of the angle b etween the pro jection
direction and the picture plane Monte Carlo evaluation It also shows the fractions of almost
go o d corners those with deviations less than degrees
Figure shows the maximal o ending angle as a function of the angle b etween the pro jection
direction and the picture plane
We can see that all factors are negligibly small for small angles b etween the pro jection direction
and the pro jection plane From Perkins we can see that deviations of approximately degrees
are still tolerated in almost of the cases Therefore in most cases pictures with pro jection
angles that are small enough not to create distortions of more than can b e considered acceptable
LinearPersp ective
1 90
0.9 80
0.8 70
0.7 60
0.6 50
0.5 40 0.4
fraction of rectangular corners 30 0.3 less then 1 deg violation of Perkins’ laws less then 3 deg violation of Perkins’ laws less then 5 deg violation of Perkins’ laws 20
0.2 Maximal deviation from Perkins’ laws, degrees
0.1 10
0 0 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90
Projection direction, degrees Projection direction, degrees
Figure Left Fraction of go o d rectangular corners vs pro jection angle
Figure Right Maximal o ending angle vs pro jection angle
Hagen and Jones and Nicholls and Kennedy a examined preferred parallel foreshortening
for parallelepip eds and other typ es of prisms Again the general trend of their results is in the
direction of lesser pro jection angles although Nicholls and Kennedy a argues that a mo derate
degree of parallel foreshortening is preferred to parallel pro jection while Hagen and Jones
claims that parallel pro jection has the highest rating For reasons explained in our comments to
Hagen and Jones in App endix A webelieve that the former pap er is more reliable
The p erp endicular foreshortening in p ersp ective images of rectangles might also create problems
Figure Comparing to the results of Nicholls and Kennedy b we can see that the ratio
of foreshortening can b e quite far from the p erceptually acceptable range It should b e noted that
cub es are not that common and much wider range of foreshortening ratios can b e tolerated for
arbitrary parallelepip eds unless this is something familiarlike a computer terminal or a TV
Spheres Cylinders Humans
All nonorthogonal pro jections of spheres are ellipses The asp ect ratio of these ellipses is a useful
measure of deviation of the image from p erceptually acceptable Figure shows the asp ect ratio
of the image of a small sphere as a function of the pro jection direction Assuming upp er b ound
for the acceptable asp ect ratio we can see that the pro jection direction should not b e less than
pro jection angle
Images of spheres are not that common in painting When they are present they are depicted as
circles regardless of their p osition The most famous example of this typ e is The Scho ol of Athens
by Rafael Figure Pirenne La Gournerie
LinearPersp ective
2
1.9
1.8
1.7
1.6
1.5 aspect ratio 1.4
1.3
1.2
1.1
1 30 40 50 60 70 80 90
projection direction, degrees
Figure Asp ect ratio of the image of a small sphere vs pro jection direction angle
In the b eginning of this chapter we p ointed out the problem with rows of cylinders that is sp eci c
to the linear p ersp ective Another more general problem is the tilt of the axes of the image of a
cross section Figure ma jor axis of the horizontal cross section is always oriented towards the
center of the image The most apparent distortions o ccur in the images close to the lines through
the center of the picture tilted at
Images of humans in exact linear p ersp ectivemay lo ok grossly distorted b ecause of extreme
changes of the asp ect ratios and violations of symmetry Figure a No such asymmetry is
observed in the art human b o dies are typically drawn as if the pro jection p ointwas lo cated directly
in frontofthem
Another problem whichiswell known to photographers is the distortion of the features of a
face resulting from high degrees of p erp endicular foreshortening
Straightness Verticality Texture Density Movement
Linear p ersp ective images of straight lines are straight Linear p ersp ective is the only pro jection
that has this prop erty Klein
Anumb er of problems are asso ciated with the p erception of verticality in linear p ersp ective
images Unless the picture plane is p erp endicular to the ground vertical lines converge or diverge
In some cases Figure this divergence creates desirable artistic e ect in other cases Figure
it is undesirable Even small tilts of the picture plane create considerable distortion in wide angle
images partly due to the violation of the Perkins laws and partly to the divergence of the vertical line and the con ict with the frame of the picture
LinearPersp ective
a
b
Figure The Scho ol of Athens by Rafael a General view b Detail compare the image of the sphere as
painted by Rafael with reconstruction of the central pro jection of the sphere p osition of the center of pro jection were
determined from the images of the columns and walls
Another problem is related to the change in the texture density texture elements of identical
size have more extended pro jections when they are close to the edges of the image This e ect can
b e observed in Figure the tree app ears to b e stretched to the left partly b ecause of the texture
density gradient
For moving linear p ersp ective images computer animations cinema some of the problems as
so ciated with oblique and wide angle pro jections are ampli ed
For example as it is describ ed in Cutting rotating rectangular solids are more often
p erceived as non rigid for oblique pro jections and wide angle p ersp ective Cutting attributes this
fact to the greater p ercentage of frames that don t satisfy Perkins laws for a given viewing p oint
Because of the changes in p erp endicular foreshortening many ob jects app ear to stretch or shrink
when they move across the eld of view
The height of remote ob jects suchasmountains or clouds ab ove the horizon changes when
Other Traditional Persp ective Systems
Figure Two views of New York Cityfrom Feininger it is interesting to note that the author o ers the
second image as an example of rather nonortho dox picture far more interesting than the conventional view
Figure A picture of a building taken with a tilted camera from Feininger we get the impression of
lo oking up as if wewere standing in the street in front of the building
they move from the p eriphery of the picture to the center These changes are quite apparentin
wide angle pictures and in many cases are p erceptually undesirable Figure although correct
geometrically
Other Traditional Persp ective Systems
From the Renaissance until the second half of nineteenth century Western painting and drawing
was based on linear p ersp ective As wehavementioned b efore some deviations from its laws were
common and analysis of p erception provides some explanations for them Artistic p ersp ectiveis
mostly a to ol for depiction of relative p ositions and sizes of ob jects in space its role in depiction of
separate ob jects is much less imp ortant
Other Traditional Persp ective Systems
Figure Beckman Institute courtyard at Caltech photo by the author
In the Middle Ages Europ ean art was quite advanced but didn t makeuseofany pro jection
system in a consistent manner The same can b e said ab out Byzantine and ancient Russian art
Comp ositions created by the artists were often quite complicated and some facts ab out their or
ganization can b e useful to understand Other cultures pro duced evolved forms of pictorial art
indep endently
Perhaps Chinese and Japanese art is the most advanced system di erentfromWestern art Their
approach to the depiction of ob jects might di er in some details but similarities are considerable
the use of color and shading is quite di erent but geometrically there is little or no di erence Most
imp ortantly the approach to the depiction of space adopted in Oriental art is quite consistentand
somewhat di erent from the concept of linear p ersp ective
As it was p ointed out bymany authors for example Hagen Japanese and Chinese artists
typically used oblique parallel pro jection in their drawings As in Western art pro jection was used
mostly for depiction of relative p ositions of ob jects rather than for separate ob jects
Parallel pro jection is just a sp ecial case of p ersp ective pro jection with the pro jection p oint
Other Traditional Persp ective Systems
Figure Tra jectories of p oints in a wide angle motion picture when the camera turns The pro jection angle is
in vertical and horizontal directions
removed to in nity The main problem of this pro jection is that there is no way to combine images
of very large ob jects mountains and small ob jects p eople in the same picture This problem was
often solved in the following way parallel pro jection was used for each part of the picture but some
amount of p ersp ective deminution with distance was intro duced for ob jects lo cated further away As
a smo oth transition b etween these separate areas was imp ossible they were separated by symb olic
clouds
In a sense inside the continuous areas Oriental practice is more consistentthanWestern As
wehave p ointed out the shap e of linear p ersp ective images dep ends up on its p osition within the
image and this fact is typically ignored bythepainters For parallel pro jection this is no longer true
therefore a rigorous construction similar to an aerial photograph is p ossible For oblique pro jections
however it means not the absence of the distortion of shap e but a constant distortion If the angle
between the direction of pro jection and the pro jection plane is greater than all typ es of
Other Traditional Persp ective Systems
Figure Chang Tse tuan c Going Up River at Ch ing ming Festival Time Detail of a handscroll
Palace Museum Peking from Sullivan
distortion are quite small and can b e ignored Of course this is mostly theoretical observation and
no precise geometric constructions were ever utilized
Oriental artists didn t use linear p ersp ective not b ecause they were comp etely unfamiliar with
it A painting by Chang Tse tuan dating as early as XI I century Figure clearly exhibits
p ersp ective deminution Sullivan quotes a Sung dynasty critic Shen Kua who criticized a
tenth century landscapist Li Ch eng for his use of linear p ersp ective Why lo ok at a building said
Shen Kua from only one p oint of view Li Ch eng s angles and corners of buildings and his eaves
seen from b elow are all very well but only continually shifting p ersp ective enables us to grasp the
whole
Another interesting asp ect of Oriental art is o ccasional use of divergent p ersp ective for rectan
gular ob jects The same trend can b e even more consistently observed in the Byzantine and ancient
Other Traditional Persp ective Systems
Figure Eucharist XV century State Russian Museum from Raushenbakh
Russian art Figure It cannot b e attributed to the lack of skill other asp ects of images demon
strate very high skill and artistic talent In Byzantine art in particular it b ecame to large extent
a matter of convention But no religious or symb olic basis is rmly established for this particular
convention and instances of such images in Oriental and Persian art demonstrate that it mighthave
some p erceptual basis
Deregowski and Parker Deregowski et al argue that laterally displaced ob jects i e
those lo cated in the p eriphery of the visual eld are p erceived in this way Anumberofother
explanations were attempted Raushenbakh
This phenomenon is mostly of historical interest b ecause it is likely that most contemp orary
p eople wouldn t nd these images to b e go o d representations of rectangular ob jects It demonstrates
Summary
that p otentially the conventional element in representation is very signi cant it could b e that some
of these images were p erceived as quite acceptable bycontemp oraries
The organization of space in Byzantine and ancient Russian paintings is of greater interest in
some sense the general principle of the representation of space is similar to that of the Japanese
art The scene is sub divided into several parts without connecting elements Each part of the
scene is depicted separately and then they are overlayed The di erence from the Oriental art is in
considerable o cclusion and much less consistent size relationships
Summarizing our observations on di erent artistic cultures that have attempted to depict space
rather than separate ob jects we can observethattheyhave at least one p oint in common all
straight lines are depicted as straight lines This a ma jor restriction on the images of rectangular
ob jects Indeed in all systems parallel or central pro jection is used Divergent p ersp ectivecan
b e considered as central pro jection applied backwards A common trend can b e observed in the
depiction of space the whole comp osition is divided into several parts and each of this parts is
drawn indep endently The pro jection used for each part is close to parallel There are quite a few
exceptions to the ab ove butifany system is ever used consistently it follows this pattern
It should b e noted that other cultures develop ed even more di erent systems of pictorial repre
sentation For example Hagen describ es the system that is used in the art of Northwest Coast
Indians These images app ear even less realistic than Byzantine art to a mo dern Western observer
ant to our discussion and are not particularly relev
Summary
The overview of the p erception of pictures that wehave provided in this chapter will serve as a basis
for the construction of pro jections allowing us to achieve in some cases p erceptually b etter results
than regular linear p ersp ective
In this summary we rep eat the main p oints of our discussion of p erception which are relevantto
the task of constructing pro jections of the three dimensional world into the picture plane
Images of ob jects should b e top ologically similar to a linear p ersp ective image of the ob ject
To b e p erceptually acceptable images of ob jects should satisfy certain criteria suchasPerkins
Summary
laws for rectangular corners have asp ect ratio close to for the images of spheres b ounds on
the p erp endicular foreshortening ratio for parallelepip eds cylinders and humans
Linear p ersp ectiveworks b est for smaller pro jection angles Estimates of the critical angle
vary but for pro jection angles less than we are guaranteed that there will b e practically no
distortion Within these b ounds most of the p erceptual criteria mentioned ab ove are satis ed
This fact is con rmed by the artistic practice in di erent cultures
For wider pro jection angles distortions may o ccur They b ecome quite ob jectionable when the
pro jection angle exceeds Some familiar geometric shap es rectangular corners spheres
cylinders human gures are esp ecially sensitivetosuch distortions Distortions are ampli ed
in moving pictures
Summary
Figure Funaki Screens Rakuchurakugaizabyobu right panel Tokyo National Museum from
Mason
Chapter
Formalization of Perceptual
Requirements
In this chapter we showhow it is p ossible to formalize p erceptual requirements describ ed in Chap
ter use them together with additional technical restrictions to restrict the class of pro jections that
we consider and derive p erceptual metrics that can help to evaluate pro jections from p erceptual
p oint of view
Technical Requirements
To narrowdown the area of the search for p erceptually acceptable pro jections we are going to sp ecify
several additional design considerations They don t haveany p erceptual basis and some of them
are quite restrictive however they make the task of constructing pro jections manageable Resulting
pro jections can b e applied to a wide class of images and the choice of correct tranformation can b e made simple
Structural Requirements
Wewant a parametric family of viewing transformations so that an appropriate one can b e
chosen for eachimage
The numb er of parameters should b e small and they must have a clear intuitive meaning
Construction of the family of pro jections should not dep end on the scene Matching the scene
to a particular pro jection in the family should b e achieved bycho osing an appropriate set of
parameters
Pictures should b e scalable any part of a picture can b e made a separate picture Therefore
no area in a picture can b e considered small enough to b e ignored
Structural Requirements
Structural distortions as describ ed in Section are in general more apparent than nonstructural
distortions It is not di cult to nd exceptions to this rule but given the additional condition of
scalability Section almost any structural distortion would b e highly ob jectionable
This idea allows us to use structural and nonstructural requirements in di erentways we p ostu
late that some structural requirements should b e satis ed exactly considerably reducing the variety
of pro jections that we consider Then nonstructural distortions can b e minimized within the class
of pro jections satisfying structural requirements
Even given the scalability assumption some structural requirements app ear to b e more imp ortant
than others if a requirement is asso ciated with unusual and complicated ob ject geometry it is less
imp ortant than those asso ciated with simpler and more common ob jects
In this section wecho ose several formalizations of certain structural requirements and apply them
to images of al l p ossible ob jects in al l p ossible p ositions wewant our family of pro jections to b e
indep endent of the ob jects that wewant to depict and wewant the resulting images to satisfy a
given set of structural requirements exactly
ts result in the same More sp eci cally we will show that three sets of structural requiremen
structure for the pro jections
the bers of the projection should besubsetsofthe bers of a perspective projection see Sec tion for the de nition of a b er
Structural Requirements
It do esn t mean that any p erceptually acceptable pro jection necessarily has this structure It
rather means that if a pro jection do esn t have this structure there are ob jects that will have unac
ceptable images if they are in certain p ositions in space
For each set of structural requirements wewillprove the statementaboveby assuming the
contrary i e presence of curved or non coplanar b ers and construct an ob ject that will havean
image violating some structural requirement
As the ob jects that we consider in Section are quite simple and common ob ject with planar
faces and straight edges it seems reasonable to assume that most mappings with curved b ers are
likely to pro duce considerable structural distortions for at least some scenes and to concentrate in
the rest of our study on the pro jections with the b er structure of a p ersp ective pro jection which
guarantees absence of structural distortions
De nitions and Preliminary Lemmas
We will use x y for the p oints in the domain of a pro jection a volume in D space and
for the p oints in the range a p oint in the picture plane
A denotes the b oundary of the set A
A denotes the closure of the set A
De nition By a line segment we mean any connected subset of a straight line
We will use the following notation for line segments
a b denotes the closed line segmentbetween the p oints a and b
a b denotes the op en line segmentbetween the p oints a and b
a b denotes the straightlinecontaining the p oints a and b
n m
De nition We wil l cal l a mapping P R R smooth if al l the components of this mapping
areatleast twicecontinuously di erentiable
De nition The set of al l points of the domain of a mapping that map to a xedpoint is cal led
the b er of the mapping at the point
If a b er is a line segment and is not a p oint we will call the straight line containing it the b er
in the domain of a mapping there is b er going through this p oint We will line For each p oint x
denote it F x
Structural Requirements
De nition A pro jection is a smooth mapping P V R R where V is an open
path connected domain in R with the rank of the Jacobian of P equal to in al l points of V
De nition We wil l cal l a continuous smooth curve I R where I is a line segment
an p erturbation of a continuous smooth curve I R if thereisahomeomorphism
I di eomorphism g I I such that the distance jg x xj is less than for any x
De nition A homotopy of a continuous curve I A to another curve I A in
n
A R is a continuous mapping h I A such that h t t and h t t for
al l t I
De nition Two smooth planar curves intersect transversally at a point x if their tangents
at this point arenotparal lel A smooth curve and a smooth surface intersect transversally if the
tangent vector of the curve is not in the tangent plane of the surface
A crossing is an intersection of twocurves that cannot b e eliminated by an arbitrarily small
p erturbation of the curves More precisely
De nition Two planar curves in an open set A R have a crossing inside A if they have
an intersection point in A and thereis such that for any perturbation of any of the intersecting
curves where the perturbed curves stil l intersect inside A
If twocurves intersect transversally in A they have a crossing in A Some non transversal
intersections result in crossing some don t Figure
B A
abc d
Figure Examples of intersections a Transversal curves and have a crossing in any op en set containing the
intersection b Nontransversal curves and still have a crossing in any op en set containing the intersection c
nontransversal and have a crossing in A and don t have a crossing in B d nontransversal and don t have
a crossing in any set
Two closed curves are homotopic in a set if one can b e continuously deformed to the other
without leaving the set
The following de nition is a mathematical expression for having a di erentnumberofholes
We will use without pro of the fact that a smo oth closed curve without self intersections separates the
Structural Requirements
plane into two path disconnected sets one b ounded inside and the other unb ounded outside
The curve can b e retracted to a p ointby a homotopy in the inside area and cannot b e retracted
to a p oint in the outside area
De nition We wil l de ne a hole in the image P O of an object O to bea path connected
set of points H that don t belong to P O if in P O there is a closedsmooth curve without self
intersections such that H lies inside the curve Two holes are di erent if thereisasmooth closed
curve without self intersections such that one hole lies inside the curve and the other outside
De nition The convex hull of a set is the union of al l line segments connecting two points
from this set CH A denotes the convex hul l of the set A
Nowwe proveseveral lemmas that we will use later
are divided into two subsets S and Lemma If the points of a continuous curve R
S so that S S S S then thereisacommon limit point of S and S
Pro of
Supp ose for eachpointofS and S there is a neighb orho o d which do esn t contain anypoints
of the other set Let x S x S Then there is a continuous curvefromx to x Consider the
inverse images I S and I S It follows from our assumption that for eachpoint
in I there is a neighborhood not containing p oints in I Let the union of all these neighborhoods
be N and the union of neighb orho o ds constructed in the same way for S be N ThenN and N
are op en sets whichcover the whole interval Their intersection is an op en set that do esn t contain
any p oints of I and I therefore it is empty I I Thus wehave a decomp osition of the
interval into two nonempty op en sets which is imp ossible as the interval is connected
Lemma If P V R R is a continuously di erentiable mapping with the rankofthe
Jacobian everywhereequal to then
any ber P of the mapping is a dimensional submanifold of R
the unit tangent vector to this manifold at any point x isacontinuously di erentiable function
of x in a neighborhoodofx for some choice of orientations
the curvature of the ber is a continuous function of x
Structural Requirements
The rst part of the statement immediately follows from Theorem in Warner Let s
prove the second and the third part
Consider a lo cal one to one parametrization of P in a neighb orho o d of a p oint x
x x x x x t The tangentvector at a p oint x isx t P x t Di erentiat
ing this expression with resp ect to t weget
dP x
x t
dx
dP x
As the Jacobian has rank solution of this system exists and is unique up to a constant
dx
Therefore the unit tangentvector is de ned uniquely by the system and the formula
x t
jx t j
dP x
Comp onents of are continous functions of the partial derivatives of the comp onents of
dx
As P is assumed to b e continuously di erentiable the derivatives are continuous functions of x
Therefore x t is a continuous function of x
As the curvature is simply the length of t it can b e proven to b e continuous by di erentiating
the expression for It is imp ortant to note thatx t do esn t dep end explicitly on t
Lemma If al l bers of a projection arelinesegments and any two bers arecoplanar the
fol lowing is true if any two ber lines intersect but do not coincide al l ber lines intersect at the
same point Otherwise al l ber lines areparal lel
Pro of Let f and f be twointersecting but not coinciding b er lines Let x V be a point
outside the plane p de ned by f and f and f be the ber line of x Asf should b e coplanar to
f it can b e either parallel to it or intersect it But if it is parallel to f it is parallel to the plane
p and therefore do esn t intersect f But it isn t parallel to f either therefore f and f are not
coplanar Thus f should intersect f and f But it can intersect the plane p only in one p oint so
it should b e the common p ointoff and f
Consider a p oint x in the plane p and the b er line f corresp onding to it The same reasoning
applies to f and the pair f f
Structural Requirements
Supp ose f and f do not intersect Supp ose some b er f intersects f Ifitisnotinp thenit
cannot b e parallel to f and it do esn t intersect it which is imp ossible If it is in p then a b er line
f which is not in the plane should intersect it in twopoints f f and f f which is imp ossible
Holes in Images
Given our de nition of holes description of example a on Figure obtains exact meaning
Condition H An image of an ob ject should not haveanumb er of holes di erent from
the numb er of holes for any p ersp ective pro jection of the ob ject
Lemma Given Condition H al l bers of the projection should be line segments
Idea of the Pro of Supp ose there is a b er which is not a linear segment We construct a
tub e around this b er in suchaway that no straight line can get through the tub e Therefore no
p ersp ective pro jection of the tub e can have holes in it The b er go es through the tub e without
intersecting the walls therefore there is at least one hole in the image of the tub e under P which
contradicts the Condition H
A2 x2 N
f y
T
A1
x1
Figure Construction of Lemma Pro of
Structural Requirements
Supp ose a b er f is not a line segment Cho ose a p oint x on the b er where the curvature is not
zero There is a neighb orho o d of the p oint x where the curvature is not zero b ecause the curvature
is continuous Lemma
As the b er f is a one dimensional submanifold of R with induced top ology we can nd an
op en neigb orhho o d N of x in f which is di eomorphic to an op en interval and N N f where
f f
NisanopensetinR Wecancho ose N and N so that N is a ball
f
Consider the endp oints x and x of N IfN is chosen in the interior of f they should b e
f f
outside N As they are limit p oints of N they should b e on N These are the only intersections
f
of f with N
ver of f with op en balls centered at the p oints of f Let f b e the closure of N Consider a co
c c c f
Cho ose the balls so that for the p oints of N their closures are inside N Asf is b ounded and
f c
closed it is compact Therefore there is a nite sub cover Cover and there is suchthatanyball
f
c
of the radius less than centered at a p ointof f is inside Cover Let T Cover N
c f f
c c
T has the following prop erties T f N f T fx x g Ifapoint y is inside T there is
f
apoint y of f such that jy y j AsN is a ball the convex hull CH T will b e contained in
f f
T will b e contained in N Therefore CH T f N N and CH
f
Consider the line segment x x As the curvature of f in N is not zero ther is a p oint y on
this line segmentwhichisnotonf there are neigb orho o ds M of f and M of y whichdon t
c f c y
intersect Cho ose small enough for T to b e inside M and for the ball of radius to b e inside M
f y
De ne the ob ject O to b e CH T Consider T N By construction the only part of T that
can intersect N should b elong to the b oundary of the balls B x and B x centered at x and
x Let A B x N A B x N For su ciently small A and A are disjoint i e
T N is a disjoint union of A and A AsA and A are intersections of a ball and a sphere A
and A in N are circles De ne the ob ject O as O n T n A A A A Wethrow
T with N away T and the interior of the intersection of
Consider a p ersp ective pro jection Q with center outside O Letl b e a straight line going through
the center of pro jection a b er line of the p ersp ective pro jection and let b e the corresp onding
p oint in the picture plane Wewanttoshow that there are no holes in the image of O AsO is
convex its p ersp ective pro jection image is convextoo anditiseasytoshow that it do esn t have
holes As Q O Q O no p oint that do esn t b elong to Q O can b e a p oint of a hole in Q O
Structural Requirements
Wehave to consider only the p oints of Q O n Q O If Q O n Q O the line l intersects O
but do esn t intersect O i e intersects O inside T
No p oint outside T can b e connected byacontinuous curvetoapointinT without crossing T
All the p oints of T except those that are in N are in O If the straight line l intersects O but
do esn t intersect T it should go through the p oints of A or A
Supp ose l intersects only one of these sets Consider a plane through l whichintersects N only
are on one side of this plane the only p oints of O that are on inside A Then all the p oints of O
the other side b elong to T Therefore for anycurveinO there is a homotopyin R that contracts
it to a p oint without intersecting l The image of O has the same prop erty with resp ect to the
image of l whichisapoint Therefore there is no closed curveinQ O thathas inside and
cannot b elong to a hole in Q O
Supp ose l intersects b oth sets Any line going through the balls of radius centered at x and
x go es through a ball of radius centered at y But this ball do esn t intersect T therefore there
isapointonl whichisoutside T and inside N Then the line should intersect T n N whichis
part of O Thus is in the image of O
ever has a hole Fib er f do esn t intersect We conclude that Q O has no holes in it P O how
O by construction It intersects the set A at the p oint x Consider the image of A A A
is a circle therefore its image can b e considered as a smo oth closed curve Again if is su ciently
small we can assume that the directions of the b ers going through the p oints of A are within a
small solid angle and are transversal to the sphere N The image of A A is one to one and
the curve S P A do esn t have self intersections
Clearly x P A is in the interior of this curve and therefore there is a hole in the image of
O
Threfore P O and Q O have di erentnumb er of holes for anychoice of Q and Condition H is
violated
Lemma Given Condition H al l bers should becoplanar
Idea of the Pro of The idea is similar to the pro of of the previous lemma If there are two
non coplanar b ers we can construct an ob ject with tub es through the ob ject along the b ers in
Structural Requirements
suchaway that one cannot see through b oth tub es at the same time There will b e at least two
holes in the image of the ob ject under P and no more than one under P
a 2 F(x 2) C ε
a’
b’
1 C ε b
F(x 1)
Figure Construction from Lemma
Pro of
Supp ose wehavetwo non coplanar b ers f and f
As wehave proven all the b ers are line segments Cho ose a curve connecting two p oints on
the b ers f and f We can separate all the p oints on the path into two sets S thesetofall
p oints for which the b er F x is coplanar with f and S the rest of the path Using Lemma
we can nd a p oint x suchthatinany neighborhood of this point there are p oints from S and S
Let B x b e a ball contained in V Cho osing x tobeapointfromS and x to b e a p oint from
that are not coplanar and b oth intersect B x We can pick S we obtain b ers F x and F x
p oints a a F x andb b F x inside B x
Consider two cylinders C and C of radius and with axes a a and b b Let l and l be
resp ectively We can cho ose and C line segments connecting p oints on the opp osite faces of C
in suchaway that for anychoice of l l they are not coplanar and C and C are inside B x
Appropriate choice of a a band b can also ensure that the faces of the cylinders are inside the
b oundary of the convex hull CH C C
Structural Requirements
Consider an ob ject O de ned as CH C C n Int C Int C Any p ersp ective pro jection
of this ob ject has at most one hole a straightlinewhichintersects the convex hull but do esn t
intersect O should enter at one of the cylinders faces and exit at the opp osite face of the same
cylinder If there are two disconnected regions in the image in which b ers of the p ersp ective
pro jection b ehave in this way wehavetwo coplanar straight lines b ers of a p ersp ective pro jection
are coplanar that intersect CH C C only inside C and C resp ectively But by construction
this is imp ossible It can b e shown that each cylinder can pro duce no more than one hole in a
p ersp ective pro jection
On the other hand as a a and b b b elong to the b ers and the b oundaries of the faces of
P a andP b the image P O has at least two holes cylinders map to curves enclosing the p oints
By Lemma all b er lines are either parallel or intersect at one p oint Therefore b ers of P
are subsets of b ers of a p ersp ective pro jection
The main problem with the argumentabove is that the ob jects that we construct are rather
unusual and the resulting structural distortion is not very signi cant it is not obvious when weare
supp osed to b e able to see through a curved pip e and when we are not In the following subsections
we will to consider simpler ob jects
Lo ops Folds and Twists
In this section we consider the following conditions which formalize Figure d e but are more
restricitve We describ e this set of conditions mostly for historical reasons this is the set describ ed
in Zorin and Barr The conditions used in the next section formalize essentially the same
requirements in a b etter way
The main advantage of this formalization is that we don t have to deal with smo othness prop erties
and the pro ofs b ecome simpler
Condition L The mapping of any line segment l to its image P l is one to one every
where or it is a p oint
This condition prevents lo ops in the images of lines It is more restrictive not only it do esn t
allow lo ops but also folds that is situations when the image is not one to one but is still simply connected
Structural Requirements
Condition T The mapping of a subset of a plane m to the image P m is one to one
everywhere or nowhere
This condition prevents twists in the images of planes
Lemma Given condition L al l bers of the mapping P are line segments
x 3
x2 -1 ξ) P ( y
x
1
Figure Construction from Lemma
Pro of
Consider a b er P Supp ose it is not a line segment Let x and x be two p oints of the
b er Then P x x
Let S P x x S P n x x It is p ossible to show that there is a p oint
x P such thatinany neighb orho o d of this p oint there are p oints of S and S Let B x be
a ball contained in V the domain of the mapping We can cho ose x x and x P tobe
three non collinear p oints in B x
Consider the image of the triangle x x x As the whole triangle cannot map into this
would imply that dimP contradicting Condition L there is a p oint y in the triangle such
that P y Consider the segment x y y where y x x If P y then the
image of x y is not one to one but contains at least twopoints If P y then the same
Thus the assumption that a b er can contain non collinear p oints contradicts is true for x x
Condition L
Note that this lemma do esn t rely on the Condition T
Lemma Any two bers arecoplanar
Structural Requirements
x 1 y 1 x2 y 2 -1 P ( ξ ) 1 -1 ξ
P ( 2)
Figure Construction from Lemma
Pro of
Supp ose two b ers are not coplanar Cho ose two p oints on these b ers Consider a continuous
path connecting these twopoints that is contained in V Such a path exists b ecause V is assumed
to b e path connected As in the previous lemma we can separate all the p oints on the path into
two sets S the set of all p oints x for which the b er P P x is coplanar with P P x and
S the rest of the path In the same manner we can nd a p oint x suchthatinany neighb orho o d
of this p oint there are p oints from S and S Again let B x b e a ball contained in V Let P
x and P betwo non coplanar b ers corresp onding to p oints in B x Consider two p oints
and x on P and y and y on P The image of x x is not one to one it follows
from Condition T that the image of x x y is not one to one anywhere Let be a pointin
the image As the intersection of the b er P with the plane of the triangle consists of more
than one p oint and the b er is a line segment the whole b er b elongs to the plane of the triangle
But the same thing will b e true for y y x Cho ose P x y As two triangles share
P x x y and the b er P shouldbeinthe the edge x y P y y x and
planes of b oth triangles i e in x y As it follows from continuity should contain x y This
is imp ossible b ecause x and y map to di erent p oints Therefore there can b e no two b ers that
are not coplanar
As b efore from the two lemmas that wehaveproven it follows that the b ers ofP are subsets
of b ers of a p ersep ective pro jection
Structural Requirements
p ’ 2 y2 x2 y2 l y2 y2’
M 2 c P P(c) ’ y1 P(a) x1 y1 l1 P(M 1) -1 P ( ξ ) a ψ P(M 2) M 1 ψ yy 11’
ξ ψ’
ψ
Figure Constructions from Lemma the image of the triangle y ay is twisted
Lo ops and Twists
The following statement is the precise expression of the idea that the image of a line segment should
b e a curve with no lo ops although not necessarily one to one
Condition L The image of a line segment should b e a simply connnected curve
Twists in the image of a plane can b e describ ed more precisely in the following way supp ose
an ob ject has a at face with several straight edges intersecting only at the endp oints In this case
no p ersp ective pro jection pro duces images of edges crossing each other at some p oint in the middle
unless they coincide This leads us to the following formalization
Condition T The image of a pair of coplanar line segments should have no crossing
points that are not images of an intersection p ointofthesegments
Again we prove the same twotyp es of lemmas as in previous sections
Lemma Given Conditions L and T every ber is a line segment
Pro of
Supp ose a b er P has non zero curvature in a neighb orho o d of a p oint
Cho ose twopoints x x on the b er P inside this neighb orhho o d Consider two parallel
going through the p oints x and x resp ectively l l V Cho ose these line segments l and l
intervals so that x and x are their interior p oints and the plane through l and l is transversal to
all the b ers going through l and l
Structural Requirements
The images of the line segments l and l intersect at the p oint Suppose is an intersection
p oint such that there is no p ossibly one sided neighborhood of where the curves P l andP l
coincide
There should not b e a crossing Condition T so the intersection at shouldn t b e transversal
Consider a plane p through the line segment l which is crossed by P transversally The
b ers of the mapping P going through the line segment l map it into a smo oth curve in the plane
p which is a one dimensional submanifold Condition L Therefore the tangentvector to the
image of l is unique at all p oints As the plane p is transversal to the b er P wecan
in suchaway that for each b er going through cho ose a one dimensional neighborhood of x in l
apoint in this neighb orho o d there is only one intersection with the plane p this is p ossible due to
continuity of the tangentvector Lemma Restriction P j of the mapping P R R to the
p
plane p is di erentiable and the p oint x is nonsingular b ecause of the transversality assumption
Therefore in a neighborhood of x it can b e inverted and the inverse mapping will have the same
smo othness inverse function theorem Consider T P j It maps the neighb orhho o d N of to a
p
neighboorhood N of x the curve P l to a smo oth curve in the plane p As is an intersection
l andP l p oint such that there is no p ossibly one sided neighborhood of where the curves P
coincide the same is true for x T l andl in p Then there is a p oint x in N on T P l where
the tangentvector is not parallel to l Consider l which lies in the plane p and go es through the
p oint x The curve T P l intersects l at that p oint and is transversal to it Therefore the same
is true for the images P l and P l contradicting Condition T We conclude that anyintersection
p ointof P l andP l m ust have a neighb orho o d p ossibly one sided where the curves coincide
Picktwopoints and in this neighborhood Let y P l y P l
y P l y P l
Consider the planar quadrangle y y y y The images of the edges y y and y y coincide
by construction
ery p ointofl there is a planar neighb orho o d in the plane p of the quadrangle y y y y For ev
q
where the mapping P j is one to one b ecause at the p oints of l b ers are transversal to the plane
q
by construction Due to the compactness of l there is an such that there are no suchpoints that
are closer than to y y l Pickapoint a on y y which is closer to l than Denote
p oints on y y and y y which are at the distance from y and y as b and b Denote the
Structural Requirements
quadrangle y y b b as M In the same waywe can construct the quadrangle M near l We
can cho ose M M and a in suchaway that P M P M and P y a P M
Consider the line segments y y and y a
The image of y y is a curve connecting P y andP y P y P y y crosses P b b
at a unique internal p oint P c b ecause the mapping of M to the picture plane is one to one and
y y intersects b b at one interior p oint c P c y lies in the interior of P M and connects
two p oints on the b oundary P y a lies in the interior of P M and connects two p oints on the
b oundary
WecaneasilyshowthatM n y a is separated by y a into twosets S and S with the
y there is a path connecting twopoints of M and lying inside M that do esn t following prop ert
intersect y a if and only if the p oints b elong to the same set De ne these sets to b e the sets of
p oints on the di erent sides of the line y a in the plane of the quadrangle Consider twopoints
s and s from di erent sets and a connecting path t Cho ose a unit normal vector n to the line
y a Let d be the vector from y to t Consider the scalar pro duct d n If it is negativefor
s it should b e p ositivefors Therefore there is a p ointonthecurve where it is zero continuity
and so the curveintersects y a
This prop erty transfers to P M and to P M due to the inclusion P M P M The
p oint P c b elongs to P b b As P b b is part of the b oundary and the only b oundary p oints
of P y a are P y and P a they don t intersect as P j is one to one Therefore P b b
M
lies entirely in one of the sets P S orP S It also lies in the same set as P b because P b and
P b can b e connected by a path that do esn t cross P y a The p oint P y lies in the other set
y as y and b are in di erent sets and P P y We conclude that P y a should intersect
P c y at an interior p oint This intersection is a crossing takeinterior p oints s and s of P S
and P S lying on the curve P y a There are neighb orho o ds of these p oints that are contained
in P S andP S resp ectively Thus if the p erturbation do esn t take s and s of the curveoutof
these neighb orho o ds and do esn t take the part of the curvebetween s and s out of P M then the
intersection is not eliminated This is the case for su ciently small p erturbations b ecause the part
Therefore of the curvebetween s and s lies entirely inside P M The same is true for P c y
P y a and P c y violate Condition T and wegetacontradiction Therefore all b ers should
have zero curvature i e all b ers should b e line segments
Structural Requirements
Lemma Given Conditions L and T all bers arecoplanar
Pro of
Assume the that the opp osite is true In the same way as in Lemma we construct two
continuous parts of non coplanar b ers f and f inside a ball which lies completely in V Cho ose
two p oints x and x on b ers f and f resp ectively
Cho ose a p oint x on x x sothatifwe draw a plane p through the b er going through x and
x x the b ers f and f won t lie in the plane
The tangentvector to the b er is a di erentiable function of the p ointin R Therefore the
angle b etween a xed direction and the b er F x t where x t is a parametrization of x x is
a di eren tiable function We can cho ose the direction so that it is not a constant Then there is a
line segment inside x x where the derivative of the angle with resp ect to t has a constant sign
Then on this line segment all b ers have di erent angles with the xed direction
d
A
n’ C n
B
Figure Unit sphere construction from Lemma
In addition we can cho ose x and x close enough to ensure that no b ers along x x lie in
the plane p which is p erp endicular to p and go es through x x The following construction on
the unit sphere where p oints corresp ond to the directions of the b ers helps to explain whywecan
makesuchchoice of x x p and p
The directions of b ers ll a closed area A on the unit sphere This area do esn t include the
p oint corresp onding to the direction of x x x and x b elong to di erent b ers Let d be a unit
vector corresp onding to the direction of x x with an arbitrary choice of orientation
Structural Requirements
p’ f2 pp’ x2 f 1 y2 y y1 T T(y 1) 2 x
x2
x1 p x1 x2 x1 y
1 T(y 2)
Figure Construction from Lemma the image of the quadrangle x x y y is twisted
Possible directions of the normals n and n of the planes p and p form a circle C in the plane
p erp endicular to d with arbitrary choice of orientation
Directions p erp endicular to d form a band B a union of the large circles of the sphere This
band do esn t include the circle of p ossible directions of p p b ecause A do esn t include d Asa
result the intersection of B and C can b e made arbitrarily small This intersection consists of two
connected parts coresp onding to sets of opp osite directions By making them su ciently small we
can guarantee that the directions p erp endicular to those in C B are not in B Thenwe can cho ose
n in C B and n in C p erp endicular to n This will satisfy the conditions stated ab ove
Cho ose a half space with resp ect to the plane p and pro ject the parts of the b er contained in this
half space into p We can always cho ose p in suchaway that these pro jections are on di erentsides
x the p oint corresp onding to n in the circle C should b e b etween the p oints of intersection of x
of large circles corresp onding to the p erp endiculars to the directions of f and f with C
The plane p is transversal to the b ers in some neighborhood N of x x Therefore the
mapping of this neighb orho o d to the picture plane will b e one to one and smo oth On the plane p
wecancho ose a neighborhood N of x x which maps into the image of the neighb orho o d N
Cho ose the p oint y N on the pro jection of the b er f into the plane p and y N on the
pro jection of the b er f so that y and y are on one side of x x Cho ose the pairs of p oints
in suchaway that the intervals y x and y x don t intersect
Let s show that there is a neighborhood of x x such that all b ers that lie in the plane p and
intersect this neighb orho o d also intersect x x Supp ose for any there is a b er f that lies in
p intersects x x at a p oint outside x x and go es through a p ointwhich is closer than to
x x By construction there is a b er f intersecting but not containing x x that lies in the
plane p Bycho osing small we can make the lines containing f and f intersect arbitrarily close to
Structural Requirements
x x i e inside V This is imp ossible b ers cannot intersect Therefore we can cho ose y and
y in suchaway that the only b ers in p intersecting the quadrangle y x x y intersect x x
The mapping T P j P maps this quadrangle to the plane p Consider the b er going
N
through y Ify is su ciently close to x the angle b etween the b er going through y and the
plane p is close to the angle b etween f and p
The plane p sub divides the space into two half spaces H and H LetH b e the half space that
p lies in the same half plane with resp ect contains part of the b er f whose pro jection into the plane
to x x asy and y By our choice of p the pro jection of f H lies in the other half plane
Therefore the pro jection into p of the part of F y which lies in H will b e close to the pro jection
of f Bycho osing y close enough to x x we can guarantee that it lies entirely in the same
half plane of p as the pro jection of f and do esn t intersect x x As p p F y p T y
pro jects to a p ointon x x We conclude that T y isnotinH and is therefore in H Inthe
same wa ywe can show that T y isinH Thus T y and T y are in di erent half planes of p
with resp ect to x x T y y intersects x x atthepoint x asthe ber F x lies in p and
do esn t b elong to x x If there is a path inside the image which go es from a p oint of the image
on one side of the line x x toapoint on the other side there should b e a p oint z x x on
the path b ecause no p ointinx x y y is mapp ed to a p ointof x x outside x x Therefore
x y y is sub divided into two disconnected parts by x x the set x
Let D b e an op en neighb orho o d of T x x y y with x x n x x excluded It has the
same prop erty with resp ect to x x and T y y iscontained in the interior of D AsF x p
and F x p there are neighb orho o ds of the endp oints of x x in x x that don t contain
any p oints of of b ers that lie entirely in p Therefore we can picktwopoints x and x on x x
that don t coincide with x or x and such that for all x x x n x x F x p
De ne D D n x x n x x D has the same prop ertyasD Consider p erturbations of
x x and T y y Su ciently small p erturbation do esn t move x x out of a band around
ve x and x into D andtheendpoints of T y y out of their neigh x x It also do esn t mo
b orho o ds that have emptyintersection with the band T y y Then p erturb ed x x retains the
dividing prop erty for D and the endp oints of T y y are still in di erent sets Therefore the
intersection is preserved under small p erturbations and is a crossing This contradicts Condition T
We conclude that the b ers should b e coplanar
Structural Requirements
This is su cient Lemma to prove that the b ers of the pro jection P coincide with the
b ers of a p ersp ective pro jection
Decomp osition
Wehaveproved in previous sections that various sets of structural requirements result in the following
prop erty for pro jections
Theorem Given Condition H from Section or Conditions L and T from Section
or Conditions L and T from Section a projection P satisfying these conditions should have
bers that are subsets of the bers of a perspective projection
From this theorem it immediately follows that we can decomp ose the pro jection P in twoways
describ ed in the following
Corollary Assume that the conditions of Theorem hold Let betheperspective projection
coresponding to P
is a central projection with center C Assume in addition that the region V lies entirely in
ewithrespect to a plane going through the center of ThenP can bedecomposed one half spac
in two ways
asacomposition of a central projection into a plane fol lowedbya smooth one to one
plane
transformation T of the plane or
plane
asacentral projection into a sphere with center C fol lowed by one to one smooth
spher e
mapping T of the sphere into the picture plane
spher e
is a paral lel projection Then P can bedecomposedasacomposition of an orthogonal central
projection into a plane fol lowed by a smooth one to one transformation T of the
plane plane
plane
It is imp ortanttopoint out that the theorems were proven for a path connected domain V
Therefore if all the ob jects in a picture that are sub ject to the structural conditions describ ed ab ove
can b e separated into sets with each set lying in a separate path connected domain for example
foreground middleground and background a di erent p ersp ective pro jection can b e chosen in the
Nonstructural Requirements
decomp osition for each set It is interesting to see actual realization of similar principles in some
Western and Japanese paintings Section Section
Nonstructural Requirements
As wehavementioned in the b eginning of this chapter we will use nonstructural requirements to
cho ose pro jections in the class de ned by the structural requirements Due to the more quantitative
nature of these requirements the appropriate way to formalize them is di erent Rather than
de ning which images satisfy the requirement and which images don t wede ne error functions
that give us a measure of a nonstructural distortion Formally sp eaking these are functions on the
set of all p ossible images Functions of this typ e are di cult to deal with Therefore it will b e more
convenient to de ne error functions on the p oint of the picture plane in suchaway that the error
function at each p ointgives an estimate of the maximal distortion of a small image at this p oint
In this section we consider only pro jections on a path connected op en domain V satisfying
Theorem Therefore for each pro jection we can de ne the center of pro jection unless all b ers
are parallel and decomp ositions as describ ed in Section We will assume that the b er structure
of the pro jection is xed i e pro jections and in the decomp ositons are xed
spher e plane
De nitions
Let s establish some notation for the pro jections that satisfy the conditions of the theorem
plane
W R p
V R plane
H
H
P
H
T
spher e plane
H
H
Hj
T
spher e
W S
spher e R
We will consider pro jections P V R R from an op en connected domain V into
the picture plane which are comp ositions of the pro jection from the center O into the
plane
intermediate plane p and a mapping T W R R W V Wecan
plane plane plane plane
cho ose the plane p so that that the distance from O to the plane is L
P can b e also represented as a comp osition of central pro jection from O into the sphere of
and some mapping radius L with center at O intermediate sphere V R S spher e
Nonstructural Requirements
T W S R W V We will assume that the image of V in the sphere
spher e spher e spher e
b elongs to a hemisphere
If the b er structure of the pro jection P is that of a parallel pro jection we will use only the rst decomp osition
picture plane π
η ψ ρ intermediate sphere (η,ξ) ξ O
ζ (θ,ζ) L θ p x y φ r (x,y) intermediate plane
(x,y,z)
Figure Co ordinate systems
Let s intro duce rectangular co ordinates x y and p olar co ordinates r on the plane p rect
angular co ordinates and p olar co ordinates in the plane On the sphere we will cho ose
angular co ordinate system and lo cal co ordinates in the neighborhoodofa xedpoint
u L v L sin Figure
The corresp ondence b etween T and T is given by the stereographic pro jection S R
plane
r tan
Zero Curvature Condition
The requirement from Section is p erhaps the easiest to formalize
Zero curvature condition Images of line segments should have zero curvature at each
point
It would b e desirable to use a measure of deviation from this condition similar to Watt s measure
Section Watt s measure however is de ned for a particular viewing p osition while our
Nonstructural Requirements
measure has to b e indep endent of the viewing conditions A natural candidate for such measure is
the curvature of the image of a straight line at a p oint
As the curvature has the dimension of inverse length we need a scaling factor to make the measure
indep endent of the size of the picture As a scaling factor wecancho ose some characteristic length
The most appropriate choice app ears to b e either a typical size of a line segment that can b e found
in a picture or the size of the picture The later choice gives a conservative measure assuming the
worst case there are images of straight lines crossing the whole eld of view
Let s qualitatively compare this measure with Watt s measure Watt s measure dep ends on the
viewing conditions and therefore wehavetomake assumptions ab out these conditions Due to the
spatial limits of acute vision describ ed in Section we can assume that the picture o ccupies no
more than of the eld of view Most pictures are viewed from a distance when their angular
size is not less then Given these estimates wehave the viewing distance l equal roughly to
to times the size of the picture Let l k R where R is the size of the picture The area
between a chord of length d and a segment of a curve with radius of curvature r is to the rst order
of approximation d r Assuming d to b e of the order of magnitude of the size of the picture
d k R where k and close to one by the order of magnitude The Watt s functional can b e
written as C k k R r where C is less than unity b ecause we usedachord instead of a least
squares lines Therefore it is roughly prop ortional to the curvature times the size of the picture
with the co e cient of the order
If we consider the decomp osition P T we can observethat satis es the
plane plane plane
zero curvature condition Therefore wehave to consider only the mapping T We will denote
plane
comp onents of T x y whichisapoint in the picture plane by x y x y
plane
The curvature dep ends not only on the p oint but also on the direction of the line whose image
we are considering As an error function for the zero curvature condition at a p oint x we will use
an estimate of the maximum curvature of the image of a line going through x
Let s estimate the maximum curvature of the image of a line de ned by x v t in the intermediate
pro jection plane in the p oint T x
plane
d
where d is the change in the direction of the tangentvector of the curve By de nition
ds
along the segment ds
d
Let w b e the tangent of the curve t w Then dt
Nonstructural Requirements
dw dw dw
jw j w
dt dt dt
j j
jw j jw j jw j
In our case the curve is de ned by the equation t T x v t To estimate the curvature
plane
dw
j j
jw j
dt
and the minimum of for a xed p oint x from ab ove we will estimate the maximum of
jv j jv j
d
Using F x v t rF v for a scalar function F we can write
dt
dw dv dw t
rw v rw v
dt dt dt
d d
jrw jrw j jv j jr j jr j jv j
dt dt
jr r v j jr r v j jv j j v r r j j v r r j jv j
j jv j j e r r j j e r r j e r r j e r r
x y x y
Expanding j e r r j j j j j and other terms in similar way we get
x xx xy
q
dw t
j j j j j j j j j j j j jv j
xx yy xy xx yy xy
dt
Now let s estimate jw j from b elow
jw j r v r v
v v v v
x y x y x y
y y y x x x
Av Bv v Cv
y x
x y
We can nd suchthat
w w
u v
p p
cos sin
w w w w
u v u v
Thus
jw j A cos B cos sin C sin
Nonstructural Requirements
Minimizing this expression w r t weget
p
A C B jv j jw j A C
Finally we get the estimate for the curvature
q
j j j j j j j j j j j j
xx yy xy xx yy xy
p
j j
A C A C B
To simplify calculations we will use the square of the curvature
j j j j j j j j j j j j
xx yy xy xx yy xy
p
K T x y
plane
A C A C B
If wesetK T x y we see that all of the second derivatives of and should b e equal
plane
to zero and so T should b e a linear transformation This coincides with the main theorem of
plane
a ne geometry whichsays that the only transformations of the plane which map lines into lines
are linear transformations
Direct View Condition
Perceptual distortions of the shap e of rectangular corners spheres b o dies cylinders etc Sec
tion don t o ccur when we use narrow angle p ersp ective pro jection i e when the principal ray
go es through the ob ject and the angular subtense of the ob ject when viewed from the pro jection
p oint is small Therefore the following condition can b e considered a generalization of a number of
nonstructural p erceptual requirements
Direct view condition It is desirable for the pro jection to b e close in a neighborhood
of each p oint to the p ersp ective pro jection into a plane p erp endicular to the b er line
going through this p oint
First we consider the case when the b er structure of the mapping coincides with the b er
structure of a central pro jection
We can observe that the pro jection into the sphere is lo cally close to the pro jection into
spher e
the tangent plane of the sphere which is p erp endicular to the b er lines of P
Nonstructural Requirements
Therefore if we use the decomp osition P T wehave to construct the mapping T
spher e spher e
which is lo cally as close to a similarity mapping as p ossible Formally it means that the di erential
of the mapping T which maps the tangent plane of the sphere at each p oint x to the plane
spher e
T R R coinciding with the picture plane at the p oint f x should b e close to a similarity
f x
mapping The di erential DT can b e represented by the Jacobian matrix J of the mapping
f x
T at the p oint x
spher e
In co ordinate form this can b e written as
J I
f x
for some
For lo cal co ordinates u v on the sphere the matrix J can b e written as
f x
u v
C B
J
A
f x
v u
Then for to b e satis ed it is necessary and su cientthat
jJ j
f x u v v u
Note that these equations formally coincide with Cauchy Riemann conditions If we consider the
sphere and the plane to b e complex manifolds then the equations ab ove are necessary and su cient
conditions for T to b e a conformal mapping
spher e
For global spherical co ordinates wehave
u v
sin
u v
sin
Wewillsay that mappings T and P satisfy direct view condition if they satisfy the condi
spher e tions
Nonstructural Requirements
We also need of measure of deviation from the direct view condition A nondegenerate linear
transformation J is a similarity transformation if and only if jJwj jw j do esn t dep end on w If this
ratio dep ends on w thenwe de ne the direct view error function to b e
jJwj jJwj
min max D T
spher e
jw j jw j
D T can b e used as the measure of non directness of the transformation at the
spher e
p oint
This measure has a simple geometric meaning Supp ose wehave a small sphere at some p oint
in space The image of the sphere on the intermediate sphere is a small disk Then this disk is
mapp ed by T into the picture plane Ideally this image should b e a circle and as it was noted
spher e
in Section the asp ect ratio of the image is a reasonable measure of distortion in this case
The functional that wehave de ned is the deviation from of the asp ect ratio of the image of a
small sphere This measure can b e also related to the maximal o ending angle in the Perkins laws
Section
jJwj jJwj
and max We will use the following notation Let s nd the explicit expression for min
jw j jw j
E
u u
F
u v u v
G
v v
Then jJwj Ew Fw Gw
v
u v
This expression is identical to the expression in Section so we immediately get
p
jJwj
E G E G F min
jw j
p
jJwj
max E G F E G
jw j
Finally for the direct pro jection error function D weget
Nonstructural Requirements
p
E G E G F
p
D T
spher e
E G E G F
Using the corresp ondence b etween the intermediate sphere and the plane we can write D as a
function of T x and y
plane
The only transformations that satisfy the direct view condition exactly are the conformal map
pings from the sphere to the plane
As it can b e easily shown if T is linear corresp onding T is not conformal
plane spher e
Nowwe consider the case of pro jections P with parallel b ers In this case the pro jection to
the intermediate plane has the direct view prop erty All conformal transformations of the plane
will preserve the direct view prop erty Then the conditions b ecome standard Cauchy Riemann
equations
The same measure of deviation from the direct view condition can b e used but in this case
wetake the lo cal co ordinates u v to b e the same as
In this case any comp osition of orthogonal and similarity transformation of the plane will b e con
formal and linear and will satisfy the direct view condition But any pro jection T where
plane plane
T is a comp osition of orthogonal and similarity transformation is equivalent to an orthogonal
plane
parallel pro jection up to a scaling factor Therefore we arrive at an imp ortant conclusion
The direct view condition can b e satis ed simultaneously with the zero curvature con
dition within the class of pro jections describ ed by Theorem only if the pro jection
coincides with an orthogonal parallel pro jection to the plane up to a similarity transfor
mation of the plane
ErrorFunctionals
In view of the statement in the previous subsection we can see that only orthogonal parallel pro jec
ultaneously It is quite obvious tion satis es b oth the direct view and zero curvature conditions sim
that parallel pro jection is not a satisfactory choice in many cases In Section we examine some
reasons for that Therefore we are also interested in pro jections that have the b er structure of a
central pro jection
Since for central pro jections the direct view and zero curvature conditions cannot b e satis ed
Nonstructural Requirements
exactly we should nd a way to de ne the optimal tradeo b etween these twotyp es of distortions
In order to do this it is convenient to use global error functionals which indicate the quality of the
pro jection with resp ect to a given condition
In order to estimate the total error for a pro jection wehavetocho ose a measure and integrate
the error function over the domain of the T we will assume that the direct pro jection error
plane
function is converted to the intermediate pro jection plane co ordinates D D T x y The
plane
most conservativeway to de ne the curvature and direct pro jection functionals is to use L measure
which results for continuous D and K in
K T sup K T x y D T sup D T x y
plane x y plane plane x y plane
p
Other p ossibilities are for example L measures
ZZ ZZ
p p
p p
K T x y dxdy D T D T x y dxdy K T
plane plane plane plane
x y x y
Optimization Problem
Initially we don t makeany assumptions ab out the form of the functional that wecho ose except
that it should b e some functional norm in the space of smo oth functions Assuming T to b e
plane
smo oth K T if and only if K T x y almost everywhere The same is true for
plane plane
D Then the only functions satisfying K T are linear functions and the only functions
plane
satisfying D T are those derived from conformal mappings of the sphere onto the plane
plane
Therefore we cannot make b oth functionals equal to zero for anychoice of the norm So wewant
foragiven value of one functional to nd a function T such that the other functional has the
plane
smallest p ossible value
minimize K T sub ject to D T
plane plane
or
minimize D T sub ject to K T
plane plane
Nonstructural Requirements
Wewant to nd a family of functions T orT solving the constrained optimization
plane plane
problems stated ab ove In each case we get a function or
We can reduce the constrained problem stated ab ove to an unconstrained one in the following
way
Consider a linear combination of the functionals F T K T D T If
plane plane plane
T minimizes F T for a xed then K T isminimal for the xed value of D T
plane plane plane plane
D T As wechange thevalue of D T will increase up to the maximal value whichis
plane plane
attained when Therefore instead of minimizing K T for xed values of D T we
plane plane
can minimize the functional F T for varying from to The same is true for minimizing
plane
D T for xed values of K T The problem can b e restated as follows
plane
minimize F D K for
Wealsohave to sp ecify the b oundary conditions in order to make the problem well de ned This
can b e done by xing the frame of the picture that is the values of T on the b oundary of W
plane
To ensure that the mapping is one to one and the inverse is smo oth the Jacobian of the mapping
should not b e zero anywhere
It would also b e desirable not to increase one distortion anywhere in the image b eyond the
level that is p ossible when the other distortion is It is a somewhat restrictive requirementand
p otentially can b e relaxed if it is p ossible to obtain considerable reductions in maximal distortion
at the exp ense of a small increase of of small values of distortion It is unlikely for the distortion
of curvature if the curvature is in some part of the image small increases in the curvature are
likely to b e observable b ecause of the threshold nature of the p erception of straightness Watt et al
It is p otentially p ossible to gain some improvementby relaxing the same requirement for the
direct view condition small distortions of shap e are undetectable for most ob jects
Mathematically the restrictions ab ovecanbeformulated as
K x y K x y D x y D x y
conf lin
where K x y is the value of the error function at the p oint x y for a conformal mapping
conf
D x y and D x y is the value of the error function for the identity mapping the only lin
Nonstructural Requirements
one which has K x y K x y dep ends on the choice of the conformal mapping this
conf
choice can b e arbitrary b ecause all conformal mappings result in D x y and any is su cient
if condition holds
In Section we will explore a particular case of this optimization problem for T with
plane
central symmetry
It is also interesting to consider weighted error functions that is w x y K T x y and
K plane
w x y D T x y where w x y and w x y are nonnegativeweights
D plane K D
The weights of the error functions can b e interpreted as the relative imp ortance of a particular
typ e of p erceptual error close to the given p oint in the image It can b e user sp eci ed or calculated
using information ab out the ob ject space
In this case the total error can be de ned as the norm kw x y K T x y
K plane
w x y D T x y k and the optimal transformation
D plane
K D dep endencies As wehaveshown the original optimization problem can b e reduced to
minimization of one functional F This functional do esn t have a go o d p erceptual interpretation A
good waytocharacterize the optimal family of transformation is bythe K D pairs corresp onding
to each memb er of the family
If we solve the optimization problem we get a one parametric family parametrized by
which de nes a parametric curve K D This curveallows to see to what extent distortions
can b e eliminated simultaneously
It is necessary to emphasize that the optimization problem formulated in this section is not equiv
alent to the task of minimizing distortions It is a sp eci c formalization of this task with relatively
low dep endence on the scene contents Pro jections can b e improved p erceptually byintro ducing
more dep endence on scene contents The trado is b etween the di culty of the construction of an
ween appropriate pro jection and the quality of the resulting image We tried to cho ose a path in b et
in which the choice of pro jections is limited but some improvement can still b e achieved
Chapter
Construction of Pro jections
Previous Work
Anumb er of reasons lead to the constructions of picture pro jections that were di erent from the
linear p ersp ective There were three main motivations p erceptual Raushenbakh Reggini
Stark artistic Ernst Inakage Flo con and Taton and technical
Goncharenko et al considerations
Perceptually based systems
Reggini s p ersp ective Reggini s system of p ersp ective Reggini is based on Thouless for
mula Thouless a Thouless b Thouless rewritten as
i
t p
r d
where t is the distance from the pro jection p oint to the ob ject d is the distance from the
pro jection p oint to the plane and i is a constantbetween and The pro jection of a p ointis
computed in the following way let r b e the distance from the p oint to the principal ray the ray
Perceptually based systems
through the pro jection p oint p erp endicular to the pro jection plane Let t b e the distance from
the p oint to the pro jection plane A p oint X in three dimensional space can b e sp eci ed by three
numbers r t in the cylindrical co ordinate system with the origin coinciding with the pro jection
p oint and the axis coinciding with the principal ray
In this system the rays sets of p oints mapping to a single p oint in the pro jection plane are
curved In the previous chapter wehave discussed the problems that are asso ciated with this
curvature
The main goal of Reggini s system is to provide for changing parallel foreshortening as a result
close ob jects have smaller degree of parallel foreshortening The e ective viewp oint determined by
the intersection p oint of the tangent lines to the rays at a given distance from the picture plane
moves away from the pro jection plane as the distance to the ob ject increases
Raushenbakh s system Raushenbakh s system Raushenbakh is based on an idea similar to
Reggini s Instead of using Thouless formula he derives similar curvilinear systems under more gen
eral assumptions ab out the relationship of the apparent size on the pro jected size More sp eci cally
Raushenbakh assumes that the apparent size is related to the pro jected size by the formula
s sF D
where s is the apparentsize s is the size of pro jection onto a xed plane D is the distance
from the plane to the ob ject He suggest a technique for measuring F D
The result of his derivations is a family of pro jections di erent systems corresp ond to di erent
tradeo s b etween vertical and horizontal deviation of the depicted size from the apparentsize
Both systems that were describ ed ab ove are based on the assumption that the apparentsize
should b e depicted We should note however that the size constancy works also for pictures
although less than for the real scenes Still there is a di erence in the p erception of size in pictures
and in the real world and some intermediate values b etween apparent and pro jected size might create
the most appropriate impression These systems are also useful for understanding some deviations
from linear p ersp ective in paintings and naive art some of the representations of ob jects that are
consistently present in the pictures could result from unconscious attempts to depict the apparent size
Perceptually based systems
We b elieve that main problem of these systems is that the distortion asso ciated with relativesize
can b e found mostly in wide angle images and a numb er of other distortions such as the distortion
of shap e in these image tend to b e more prominent The systems describ ed ab ove might result
in reduction of these distortions but nothing in their construction guarantees that The imp ortant
problem of the curvature of images of straight lines is not addressed either
We don t describ e two more systems Flo con s and Stark s b ecause we didn t have the original
references available and we could not nd satisfactory accounts of their work
Artistic systems
In a sense any pro jection system falls into this category b ecause any system can b e used for artistic
purp oses We describ e several systems that were created sp eci cally for artistic purp oses
Escher s p ersp ective M C Escher is famous for geometric construction in his paintings He
used unuasual p ersp ective constructions in manyworks and one of his exp eriments Figure is
particularly relevant to our discussion From Escher s notes that he had made when he was working
on this woodcut weknow that he was trying to create a smo oth transition b etween di erent views
of a building with the viewing direction changing from horizontal to almost vertical Resulting
construction is similar to the panoramic pro jection describ ed b elow and allows to avoid distortion
of rectangular corners in all parts of this complicated construction
Systems describ ed byM Inakage Anumb er of pro jections were describ ed in Inakage
Inakage de nes curvilinear perspective as a linear pro jection of the three dimensional space in to
a curved surface This de nition implies that the surface is pro jected into the plane to get the nal
image Therefore this de nition is equivalent to the decomp osition given by Theorem
D warping extends the curvilinear p ersp ective allowing to change the metrics of space pro duc
ing curved rays Any continuous system can b e considered an instance of curvilinear p ersp ective
combined with D warping
Sp eci c case of inverse perspective is describ ed by the formulas
z z
y y x x
d d
Perceptually based systems
Figure M C Escher High and Low lithograph
Perceptually based systems
where x y z isapoint in space x y is it s pro jection and d is the distance from the pro jection
p oint to the pro jection plane
Another example of p ersp ective constructions are the photographic collages byDavid Ho ckney
Ho ckney
As wehave mentioned in Chapter deviations from linear p ersp ective in the art are probably
more common than rigorous p ersp ective constructions Rejection of linear p ersp ectivewas one of
the theoretical foundations of cubism The systems that wehave describ ed ab ove are just several
examples of attempts to use alternative metho ds of pro jection
Technical Projections
Extreme distortions of linear p ersp ective for angles close to and imp ossibility of using it for
pro jection angles greater than prompted several practical solutions primarily in lens design
For extremely wide angles greater than degrees linear pro jection is never used b ecause of
the distortions and di culties in creating appropriate rectilinear lenses Fish eye lenses are based
on pro jections that are di erent from linear p ersp ective All of these pro jections have straightrays
unless the light propagates in a medium with variable refraction co e cient
Several pro jections used in lens design are describ ed in Goncharenko et al The b er
structure of all these pro jections is identical to the central pro jection all of the b ers are straight
lines intersecting at the center of the lens which can b e called the pro jection p oint Moreover these
pro jections have axial symmetry Let p b e the axis of symmetry which corresp onds to the raygoing
ts Let b e the angle b etween the direction from through the center of the lens and its fo cal p oin
the pro jection p oint to a p oint in space X We will use a p olar co ordinate system r on the
pro jection plane with the origin at the intersection p oint of the axis and the plane Then the image
of the p oint X is determined by a function r F
Three pro jections describ ed in Goncharenko et al are shown in the table b elow f is the fo cal length of the lens
Axially Symmetric Projection
name formula examples of lenses
orthographic r f sin Fisheye Nikkor OP
Kowa Fisheye Olympus Zuiko Fisheye
equidistant r f
Nikkor Fisheye
Asahi Sup er Takumar Fisheye Minolta MC Fish
equalized r f sin
eye
Another example of a pro jection that is widely used in photography is the panoramic pro jection
If the lm in a panoramic camera is placed on a cylindrical surface the resulting pro jection resem
bles equidistant pro jection in one direction and regular linear p ersp ective in the other Panoramic
pro jection was advo cated by Feininger as the only correct p ersp ective on the grounds that
parallel lines should always b e seen as convergenttoapointatin nity
The pro jections that wehave describ ed in this section were mostly ad hoc solutions to the
problems of wide angle images No consistent e ort was made to minimize geometric distortions
Axially Symmetric Projection
In general the optimization problems formulated in Section are very di cult In this section
we will consider the case of axially symmetric pro jections The transformation T for these
plane
pro jections can b e written in p olar co ordinates as
r
In this case the problem can b e reduced to a one dimensional problem Unfortunately even in
this form direct solution of the optimization problem is quite di cult We shall make some estimates
of the optimal values of the functional and show that some simple families of pro jections are close
enough to optimal We shall consider some applications of one of these families in Chapter
Reformulation of the Problem
Using Equations for the transformation T we can simplify the expressions for error functions
plane
After substitutions we obtain
Axially Symmetric Projection
r r r r
R K r R
min
r r r
where R is the scaling constant
The expression ab ove is for the transformation T R R The direct pro jection
plane
functional was derived for T S R The formulas relating T and T are
spher e plane spher e
r tan
The direct view functional has the form
d
max
d sin
d
min
d sin
Using r tan weget
max r
r
D r
min r
r
We can notice that in b oth cases the dep endence on the angular co ordinate completely disap
p eared so now the problem is one dimensional
Weshouldalsonotethatif do esn t dep end on the only consistentway to de ne the b oundary
conditions is to sp ecify the value of on a circle i e for r R where R is the size of the picture
we will assume that we use it as the scaling factor in the curvature error function
As the transformation T is required to b e one to one and continuous everywhere
plane
and r everywhere In order for the error functions to b e nite at In general it
r for all r b ecause otherwise we get an in nite compression factor at is desirable to have
the p oints where
Moreover if wewant the transformation to b e smo oth at zero
The optimization problem for the sup norm now takes the form
r D r minimize F max K
r
sub ject to R R r
Axially Symmetric Projection
Limiting Cases
Consider the cases and In the rst case the optimal solution is r r In the second
case we obtain a di erential equation
p
r r
which can b e easily solved The solution satisfying R R is
p
r R
p
r
r R
The expression for the comp osition of the stereographic pro jection and which can b e obtained
by substituting r tan into the expression for is more intuitive
tan C tan
where C is a normalization co e cient and results in the following expression for
R tan arctan r
r
tan arctan R
This mapping has a simple geometric interpretation Figure rst the plane is pro jected to
the intermediate unit sphere and then repro jected back from the p ole
1
r ρ( r )
Figure Construction of DP r
Axially Symmetric Projection
This mapping has an imp ortant role in the rest of the thesis We will call it direct view trans
formation and denote it DV r R
In Figure we compare this transformation with transformations implemented in the sheye lenses
θ 1
θ _θ sinθ 2sin_ θ 2tan tanθ 2 2
abcde
Figure Pro jections orthographic a equalized b equidistant c DP r d linear e
Lower Bound of the Optimal Values of F
In this section we will nd a lower b ound of the optimal values of the functional
First we observe that
min F min max F r
R
min F R R R R min F R R R R
min F R x x R
x x
Therefore we can estimate the minimal value of the functional from b elowby minimizing the
function
x x
R
x R x max R F x
min x R x
If x should b e zero at the p oint where the minimum is achieved We can consider the
Axially Symmetric Projection
function F of only one variable x Assume we shall consider the case case later
Then we can divide the functional by and use
p p
Consider three ranges of x x R R x and x
p
x R In this case the function can b e rewritten as
x
R
F x x R
x R x
The derivative of the function is
x x
R
F x x R
x R x
In this range of x the derivative is clearly negative Therefore if the minimum is attained in
p
R this range it must b e attained at the rightendpoint
x In this case the function takes the form
x R x F x x R
R
The derivative of the function is
Axially Symmetric Projection
F x x R x x R
R
The derivative is p ositive in this range therefore the function is increasing and the minimum
can b e attained only at the left endp oint x
p
R x In this case the function and the derivative are
x
R
F x x R R x
x
x x
R
x R F x x R
x
At the right endp oint of the interval the derivativeisalways nonnegative at the left endp oint
it might b e p ositive or negative
The second derivative of the function in this range is
Axially Symmetric Projection
x x x x
R R R
F x x R R
x x x
It can b e easily seen that each term is p ositive inside the interval Therefore the rst derivative
increases inside the interval and has no more than one ro ot If it is p ositive at the left endp oint
there are no ro ots otherwise there is one ro ot
Using the continuity of the function we conclude that the minimum is attained at a p oint in the
p
R x foranyvalue of interval
If it is easy to see that in this case the minmal value of F is and is attained at
p
x R
If the value of D at R is known we can nd r
The K D curve for the b ound is shown in Figure for R eld of view
Second Bound
The b ound that wehavederived in the previous section is not very tight which it can b e easily
seen for In this case the minimum of the functional is and the optimal solution is unique
Section The values of K in this case are much higher then the b ound found in the previous
section In this section we shall make some additional assumptions ab out and nd a b etter b ound
for the optimal pairs K D in a narrower class of functions We b elieve that this b ound is still
far from the optimal it is however the b est we could obtain Although the optimization problem
that we consider can b e easily reduced to a problem of optimal control in the form of Mayer with
inequality constraints Pontriagin the necessary conditions that we could nd don t yield a
go o d estimate of the optimal values of the functional F or of the optimal solution in particular
b ecause the second derivative of the Hamiltonian with resp ect to the control vanishes identically
The elementry estimate that we derive in this section supp orts the idea that the interp olations
derived in the next section are close to optimal but of course do esn t prove it
Axially Symmetric Projection
We make the following assumptions ab out
D K are nondecreasing functions of r
p
r D r is less than for r r for all r i e D r
p
r r
The function is concave
The rst assumption is motivated by the desire to avoid larger distortions closer to the center
of the picture i e to preserve the same typ e of distribution of distortions as for the limiting cases
r r and r DP r
The second assumption follows from inequalities
The third assumption means that there is no overcompression in the radial direction Consider
an image of a small sphere For p ersp ective pro jection it is an ellipse with its ma jor axis oriented
along the direction to the center of the picture For DP r the image is a circle Wewould exp ect
intermediate pro jections to pro duce images of intermediate shap es i e ellipses with ma jor axis still
oriented along the direction to the center but with smaller asp ect ratio
The last assumption is motivated by the fact that in a regular p ersp ective image the amountof
the shap e distortion increases towards the edges and the compression should increase
K r K where
r r
K
Due to Assumption K is p ositive Note that As Assumption
j j
Let u Then
u u
K K u K
r r
Wehave assumed that K increases Supp ose u decreases at some p oint than K should
dK
increase Therefore at that p oint Di erentiating we get the following expression for the
dr
derivative
Axially Symmetric Projection
dK
u u u u
dr r r r r
It is non negativeif
u
u
r
u
u r
r
Supp ose that u increases on r Then
Z
u u r
u maxu u du
r r r
r
If wecho ose r to b e the maximalinterval on which u do esn t decrease it will decrease on
an interval r r As u r increases on r the following estimate can b e made
r
u r
u r u u r
r
u u r max
r
r r
u r
r
p
It remains to estimate u r r First we estimate Denote d D Then
p
r r d
p
r and integrating from r to we get Dividing by r
Z Z
dr dr
p p
ln d
r r
r r
The integral in is equal to the function DP r de ned in Section Finally weget
d
DP r DP r
From and we obtain
p p
r r r r
u
d
DP d DP These b ounds can b e directly used in b ecause the rst b ound is an increasing function of
Axially Symmetric Projection
u r and the second b ound is a decreasing function of u r Due to the Assumption the value
p
of u isknown as D r is nondecreasing D d therefore d
We can further improvethelower b ound for D close to in the following way the estimate
still holds for maximum of u on r Supp ose the estimate is maximal for some r Then the
value of K will b e greater than the lower estimate for K on the interval r plus the maximum
of the estimate of u
The plot of the resulting b ound is shown in Figure
K 2
1.5
1
0.5
0 0 0.2 0.4 0.6 0.8 1
D
Figure Bounds from Section upp er curves and Section lower curve
We can further narrowdown the range of the functions that we are considering and assume that
u increases In this case the estimate can b e further improved as the value of u will b e greater
than the maximum of Figure
Interp olations
We couldn t nd a direct solution of the optimization problem but given the estimates of
the previous sections we can evaluate interp olations b etween the solutions for the cases and
describ ed in Section
We suggest two p ossible interp olations
linear interpolation describ ed bytheformula
Axially Symmetric Projection
p
R r
p
r r R
r R
geometric interpolation describ ed by
arctan r
tan
r R
arctan R
tan
The second case corresp onds to the following geometric construction DP r is the pro jection of
the plane to the unit sphere from the center of the sphere followed by repro jection from the p ole If
wemove the p ointfromwhichwe repro ject from the sphere to the plane from the p ole to the center
we get a smo oth interp olation from r to DP r describ ed by
If we compare the K D curves for these interp olations with the estimates derived in the previous
sections b oth families of functions satisfy the assumptions for all estimates wecanseethatnot
much improvement can b e gained by nding b etter approximations to the solution of the optimization problem Figure
K
2
1.5
1
0.5
0 0 0.2 0.4 0.6 0.8 1
D
Figure The K D curves from top to b ottom are linear interp olation geometric interp olation estimate from
Section estimate from Section lower curve
It is imp ortant to rememb er that p erception of curvature and of some shap es such as spheres and
rectangular corners has a threshold nature Therefore once a p erceptual requirement is violated
by a su ciently large margin p erceptual saturation is likely to o ccur and even large di erences
Axially Symmetric Projection
do not makemuch di erence Our rst estimate was made without any assumptions ab out and
it still indicates that for lowvalues of D K should b e quite large
Nevertheless the di erence b etween the interp olations that we prop ose and the estimates is still
considerable and it is at least of theoretical interest to obtain b etter estimates of the optimal K D
pairs and of the optimal solution
Extentions
Both one parameter families that wehave describ ed can b e easily extended to larger families
parametrized by the function By allowing dep ending on the angle we obtain greater exi
bility in the choice of transformation appropriate for a particular picture
The equations for the pro jections take the form
p
r R
p
r r R
r R
arctan r
tan
r R
arctan R
tan
In this case is similar to the user sp eci ed weights describ ed in Section If we assume
that changes slowly the expressions for direct view and curvature error functions can b e applied
and for each direction the problem can b e solved indep endently Therefore optimal values for
each direction are close to those obtained for the axially symmetric case
For fast changes in however this do es not apply andtwo dimensional optimization is necces sary to compare the family to the optimal solution as de ned in Section
Chapter
Implementation and Applications
Implementation
Wehave implemented the transformations describ ed in Section and Section The im
plementation of our pro jections is quite straightforward The pro jection of Theorem
plane
practically coincides with the standard p ersp ective parallel pro jection There is however an im
p ortant implementation detail that is absent in some rendering systems As wementioned b efore
our center of pro jection need not coincide with the p osition of the camera or the eye It is chosen
according to p erceptual requirements For instance it can happ en that the most appropriate center
of pro jection for an o ce scene is outside the ro om Figure c In these cases it is necessary
to haveamechanism for making parts of the mo del invisible These parts of the mo del should
participate in lighting calculations but should b e ignored by the viewing transformation This can
b e done using clipping planes
The D part of the viewing transformation T can b e implemented as a separate p ostpro
plane
cessing stage In this way the transformation can b e applied to any p ersp ective image computer
generated or photographic The only additional information required is the p osition of the center of
pro jection relative to the image The basic structure of the implementation is very simple
Implementation
for all output pixels i j do
p
r i j
setpixel i j
interpolated color r i r r j r
end
For the geometric interp olation the inverse function can b e calculated directly using
the formula
arctan R r
r tan arctan tan
R
Equation can b e used in the cases of constant and variable
For linear interp olation the inverse function canbecomputednumerically with any
of the standard ro ot nding metho ds such as those found in Numerical Recipes Press et al
As wehave mentioned in Section for mo derately wide angles the di erence b etween any
transformation from the family that we are considering and the identity transformation is quite small
That means that the initial approximation of identity for anynumerical ro ot nding pro cedure is
good and even the dichotomymethodworks quite well There is no particular advantage in using
the linear interp olation at least in our implementation but it was the rst one that wehave
implemented and some of our images were generated using this transformation
color x y function computes the color for any p oint x y with real co ordinates The interp olated
in the original image byinterp olating the colors of the integer pixels This interp olation can b e done
using convolution with any reasonable lter Using a wider lter slows down the computation but
virtually guarantees the absence of aliasing
Implementations of and are practically identical to the implementation describ ed
ab ove There is a signi cant di erence in the case of the values of cannot b e precomputed
b ecause they dep end on Using is preferable b ecause the inverse function can b e computed
explicitly
We also need a mechanism that allows user to sp ecify desired values of for various directions
in the picture One way to implement this is to allow user to sp ecify the values of atanumber of
p oints compute the direction from the center of the picture to each p oint and use a closed spline
curvetointerp olate the values of for all directions see for example Hill
Deep and Shallow Scenes When Do We Need a Wide Angle Projection
The algorithm is linear in n and should b e ab out as fast as any other image transformation
change of the size ltering or other geometric transformations Our non optimized program in
which no attempt was made to eliminate extra function calls in inner lo ops etc to ok ab out sec
on an HP to transform a by image with a lter of width
Deep and Shallow Scenes When Do We Need a Wide Angle
Projection
As wehave discussed in detail in Chapter narrow angle p ersp ective pro jections are likely to pro
duce p erceptually acceptable images while wide angle pro jections are likely to pro duce distortions
and are practically unusable for extremely wide angles Figure a As wehaveshown in Chap
ter and Chapter it is in general imp ossible to nd a pro jection whichwould pro duce undistorted
images for any scene given a xed angle of pro jection
Before we try to use compromises suggested in Chapter wehavetoanswer the following
question when is it p ossible for a given scene to use a narrow angle pro jection instead of a wide
angle one
A general direction from the scene to the pro jection p oint is determined byourchoice of the
side of the ob jects that wewant to see in the picture we assume that our preferences are consistent
with the b er structure of the p ersp ective pro jection i e we don t want to see around the corners
or oppp osite sides of one ob ject at the same time We are only free to cho ose the distance b etween
the pro jection p oint and the scene
Supp ose wewantanobjectofsize s is lo cated at the distance l from the center of pro jection
to have approximately the same size as an ob ject of size S lo cated at the distance l L S s
s the distance l As the ratio of the sizes of their images for a p ersp ective pro jection is Sl l L
determining the pro jection angle can b e estimated from Sl l L s The pro jection angle
can b e found from the expression tan S l L The resulting estimate fo r l is S s L Of
course S need not b e a size of an ob ject it can b e a distance b etween ob jects and in general any
characteristic length in the direction p erp endicular to the principal ray We can consider S to b e the
width of the scene in the background and s to b e the width of the scene in the foreground L
is just the distance b etween the closest and most remote ob jects in the scene depth of the scene
Deep and Shallow Scenes When Do We Need a Wide Angle Projection
Therefore the pro jection angle has to b e large if the di erence b etween the foreground width of the
scene and the background width of the scene is large compared to the depth of the scene
We shall call a scene shal low if the ratio S s L is small if this ratio is large then we
shall call the scene deep
Deep scenes require wide angle pro jections while narrow angle pro jections can b e used for shal
low scenes
Almost any indo or scene is inherently shallow b ecause the ob jects in such scenes typically have
comparable sizes Therefore in most cases for closed spaces of relatively small size we can get away
with using the p ersp ective pro jection It is imp ortanttonotethatthechoice of the pro jection p oint
for a scene do esn t have to re ect an actual physical p osition of the observer in the scene For
example any real life photo showing a whole wall of an o ce would require a very wide pro jection
angle However in computer graphics nothing prevents us from cho osing a pro jection p ointinan
unreachable p osition in or outside the ro om Our choice can b e based purely on the p erceptual
quality of the picture Minimization of shap e distortion requires narrow angles of pro jection At
the same time extremely small angles of pro jection result in a lack of parallel foreshortening and
excessive p erp endicular foreshortening for some ob jects Therefore a mo derate pro jection angle is
most preferable In Figure we can see pictures of the same part of the o ce from three pro jection
p oints corresp onding to the pro jection angles and Most p eople prefer the pro jection in
the middle In fact the number was determined by showing to p eople a set of pictures with the
pro jection angle varying from to and asking them to c ho ose the nicest picture The variance
of the results was small nob o dy chose any image outside the limits
The e ect of the change of the pro jection angle on the image of a deep scene is radically di erent
For a deep scene the general layout and relative sizes of the ob jects are unchanged only for a small
range of angles Consider Figure The image has the desired comp osition but distortions
are clearly visible the men close to the edges of the picture app ear to b e bulkier than the men in
the center of the picture although their shap e was identical The shap e of the head app ears to b e
distorted in some cases and the men close to the edge of the picture app ear to b e asymmetrical
The two images are radically di erent in the rst case when all the ob jects are still present
in the picture the men in the picture almost disapp ear In the second case the pyramids in the
background are absent and relative sizes of the images of the men are changed
Deep and Shallow Scenes When Do We Need a Wide Angle Projection
a
b
c
Figure Example of a shallow scene a pro jection angle b pro jection angle c pro jection angle
Position of the camera is shown on the right for each picture Note that for image the camera is outside the ro om
Applications of Correcting Transformations
For deep scenes the pro jection angle is practically determined by the scene and comp osition of
the picture and cannot b e varied In this case transformations of the typ e describ ed in Section
can b e applied to decrease the distortions in the p ersp ective image The results of applying the
transformation to the picture is shown in Figure b Note the reduction in the distortion
of shap e of the men As a tradeo the barrel distortion is intro duced but due to the absence of
straight lines its only manifestation is a slant of some gures
Striking distortions o ccur in wide angle motion pictures esp ecially when the camera rotates
around a xed vertical axis pan They are primarily asso ciated with two factors
whenanobjectmoves to from the edge to the center of the picture the shap e of its image
changes from distorted to non distorted This change is noticeable and pro duces an impression of
non rigidity
the vertical co ordinate of ob jects that are lo cated at a large angle from the horizon changes
considerably cf Figure This e ect app ears unnatural
These e ects are reduced if transformations like or are applied to the frames
Summarizing the discussion ab ove we suggest the following recip e for the choice of an appropriate
pro jection for an image
Determine if the scene is shallowordeep
If the scene is shallow cho ose a p erceptually optimal pro jection angle which is likely to b e in
the range
If the scene is deep determine the minimal pro jection angle that is acceptable for the scene
Then cho ose a transformation from or which pro duces the b est results
Applications of Correcting Transformations
In the previous section we describ ed the conditions which require a wide angle pro jection under the
assumption that we are restricted in our choice of pro jection only by the comp osition of the scene
In the case of photography there are several additional restrictions there should b e no obstacles
between the camera and the ob jects that must app ear in the image and the size of the camera
cannot b e made arbitrarily large For example it is imp ossible to make a picture of a ro om from
Applications of Correcting Transformations
a
b
c
d
Figure Example of a deep scene a pro jection angle b pro jection angle correcting transformationwith
applied c pro jection angle keeping the gures of men approximately the right size d pro jection
angle keeping the pyramids in the eld of view
Applications of Correcting Transformations
a pro jection p oint b ehind a wall As a result the pro jection angle can b e chosen only in a limited
range and some pictures must use a wide angle pro jection
In many cases the b est camera p ositions are physically inaccessible or there is no time to make
agoodchoice of the p osition
All these reasons explain why wide angle images are inevitable in many cases even for shallow
scenes Transformations can b e used to correct for distortions of shap e in such
images
The p osition of the center of pro jection is usually known for computer generated images but is
more di cult to obtain for photographs For photographs it can b e calculated if weknow the size
of the lm and the fo cal distance of the lens used in the camera Alternatively it can b e computed
directly from the image if there is a rectangular ob ject of known asp ect ratio present in the picture
App endix B
If our task is correcting a distorted photographic image it is imp ortant to understand what typ e
of distortion is present
For example distortions related to verticality Section can b e corrected by a linear trans
formation as it was known to photographers for a long time Feininger Figure
ab
Figure Photos from Feininger a The original photo with convergentverticals b Same photo printed
with slanted easel the slantischosen so that the verticals b ecome parallel
Distortion of shap e can b e corrected using our transformations In some cases these trans
formations also decrease distortions of relative size due to the excessive p erp endicular or parallel foreshortening
Other Applications
Let s consider in greater detail an example of application of transformations and
The head of the author in Figure a is considerably distorted and his right shoulder app ears
to b e longer than the left He also app ears to havemuch more muscle than in real life The shap e of
small ob jects on the table is distorted The rectangular corner indicated in the left part of Figure a
also lo oks distorted cf Section
After application of the transformation for the author lo oks much more like himself
and the rectangular corner lo oks b etter The straight edges in the left part of the picture are curved
If we use the transformation with gradually increasing from on the left to on the
right we can eliminate the curvature of edges while leaving the corrections in the left part o the
picture intact Unfortunately the rectangular corner do esn t lo ok right again we cannot haveboth
zero curvature and correct shap e at the same place
Again wegive a short recip e for correction of wide angle photos
If vertical ob jects in the picture app ear to b e falling apply a linear transformation which
makes convergentverticals parallel
If distortions of shap e are present apply or Transformation is su cient
when all long straight lines in the picture pass close to the center of the picture otherwise
cho ose close to in for the directions where there are straight lines and no ob jects
of simple or familiar shap e and close to in the directions where there are such ob jects and
no long straight edges
Other Applications
There are two more applications of our transformations that might b e useful
Zo oming If it is necessary to extract a part of a wide angle image the ob jects in the sub image
are likely to lo ok quite distorted In the context of the whole image this distortion app ears to b e more
acceptable but when a part of the picture is clipp ed the distortions b ecome more ob jectionable
If the clipp ed part is small no long straight edges will app ear in the picture and transformation
with close to can b e applied b efore clipping The results are demonstrated in Figure d and e
Other Applications
d
a
e b
c
Figure An example of application of to a wide angle photo a Original picture note the
distortion of the shap e of the head and of the marked rectangular corner in the right part of the picture b Transfor
mation with applied the straight edges in the right part of the picture are curved c Transformation
applied changes from to d Zo om of the head in the original photo e Zo om of the head in the transformed
photo b
Summary
Extremely wide angle images As wehave seen Figure a linear p ersp ective is unsuitable
for extremely wide angles Rather than using traditional sheye views which overcomp ensate
for the distortions of the p ersp ective pro jection we can use one of our transformations Visible
distortions are practically inevitable but for our transformations they are likely to b e less than for
the p ersp ective pro jection and sheye views
Summary
Here is a summary of p ossible applications of the transformations that we prop ose
Creation of wide angle computer generated pictures with minimal distortions
Reduction of distortions in photographic images
Creation of wide angle motion pictures with reduced distortion
A b etter alternative to sheye views
Zo oming of parts of a wide angle picture
Summary
a b
c d
Figure Extremely wide angle images a Linear p ersp ective b Angular sheye equidistant pro jection
c Hemispherical sheye orthographic pro jection d Our direct pro jection Images b and d are quite similar
but in c we can observe excessive compression of the ob jects in the margins the tables app ear to b e narrower and
the ball do esn t lo ok spherical
Chapter
Conclusions and Future Work
Conclusions
Let s reiterate the main p oints of the thesis b elow
Wehaveprovided a review of p erception of geometry in pictures and of the usage of p ersp ective
systems in the art Chapter
Wehave suggested a classi cation of p erceptually imp ortant geometric prop erties of images
structural and nonstructural whichisconvenient for the study of general pro jections of the
space into the plane from a p erceptual p oint of view Section
Wehave shown that a numb er of simple structural requirements such as the absence of lo ops
in the images of straight lines the absence of twists in the images of planes when applied to
all p ossible ob jects that can o ccur in a picture lead to the conclusion that the b ers of the
pro jection should b e subsets of the b ers of a p ersp ective pro jection Section
Wehaveintro duced twotyp es of error functions which can b e used as quantitative measures
of distortions of curvature and shap e Section
Future Work
Wehave shown that the error functions cannot b e simultaneously identically zero for any
pro jection except orthogonal parallel For each error function however there are pro jections
for which the error function vanishes identically Section
Wehave describ ed error functionals and formulated optimization problems that can b e used
to construct pro jections with minimized distortions Section
Wehave found estimates of the optimal values of error functionals for the axially symmetric
case and demonstrated that under certain assumptions simple interp olations havevalues of
error functionals close to the optimal Section
Wehave describ ed situations when distortions are likely o ccur and suggested metho ds to
decrease them Chapter
Future Work
There are a numb er of directions in whichthiswork can b e completed and extended
The most immediate goals are
to nd b etter estimates of the optimal values of functionals and approximations to the solution
of the optimization problem in Section
to consider other cases of simple symmetries for example pro jections similar to cylindrical
A promising direction of work whichwe hop e to pursue is suggested by the fact that conformal
transformations preserve the direct view error function That means that we can try to decrease cur
vature or achieveimprovement in some other p erceptual factor without causing additional distortion
of shap e if we use conformal transformations of the picture plane
The decomp osition theorem from Chapter applies to the case of connected regions of space It
is interesting to consider the ways of splitting the space into several disconnected parts and using a
di erent pro jection for each of these parts
ts of the scene It is also p ossible to consider We tried to avoid dep endence on the conten
transformations that dep end strongly on our knowledge of the geometry of the ob jects in the scene
In this case error functions havetobeevaluated only for the actual images of ob jects rather than
for all p ossible images and b etter results can b e achieved at the exp ense of much greater complexity
and lesser generality of the algorithms
App endix A
Review of the Exp erimental Studies of
Perception of Pictures
This is a brief exp osition of exp eriments describ ed byvarious authors It is not in anyway complete
Our main purp ose here is to give the reader a general idea ab out stimuli viewing conditions tasks
and results of the exp eriments
We tried to describ e all the exp eriments using consistent terminology from Section Thus
on numerous o ccasions we had to change the original terms to their equivalents
In most cases wesay nothing ab out theories of p erception that the exp eriments were supp osed
to test On the contrary we tried to separate the exp erimental data and the theory and to get rid
of the bias intro duced by the theory into the interpretation of the results
We sub divided the pap ers into several sections according to what we b elievewas the main
exp erimental sub ject of the pap er
A Robustness and Perception of the Picture Surface and Frame
Figure A Stimulus from Hagen b
A Robustness and Perception of the Picture Surface and Frame
Hagen b Hagen M A In uence of picture surface and station p oint on the ability
to comp ensate for oblique view in pictorial p erception Developmental Psychology
Exp eriment
Sub jects Children years old years old and adults in each group
Stimuli Photographs of triangles and squares against a textured background horizontal plane
Figure A Three relative sizes of the images in the pictures bigger was and of
the smaller wotyp es of stimuli were used color prints and slides
Viewing conditions Peephole mono cular viewing two viewing directions and viewing
distance equal to the pro jection distance
Task The sub jects were instructed to p oint to the bigger ob ject
Results Correct station p oint increased the p ercent of correct choices for adults and year old
children for slides and decreased for prints For year old children the p ercent of correct answers
was very small indicating insu cient understanding of p ersp ective pictures
Lumsden Lumsden E A Problems of magni cation and mini cation An ex
A Robustness and Perception of the Picture Surface and Frame
planation of the distortions of distance slant shap e and velo city In Hagen
pages
Rosinski and Fab er Rosinski R R and Fab er J Comp ensation for viewing p oint
in the p erception of pictured space In Hagen pages
E A Lumsden R R Rosinski and J Fab er describ e some unpublished exp eriments by W C Purdy
Sub jects Unknown
Stimuli Texture gradients pro jected onto a screen Visible p ortion of the pro jection was circular
and lled with texture oblique central pro jection of a piece of transparent lm with a grid of lines
approximately inch apart Two p ositions of the light source and the lm source to lm distance
remained constant were chosen so that they resulted in magni cation and of the tilted lm
Four pro jection directions of the lm were used and The pro jection
directions were chosen so that each image coincides with the image one step b elow if pro jected from
the viewing p oint
Viewing conditions Fixed viewing p osition on the other side of the screen bino cular viewing un
certain viewing angle limited to by an ap erture
Task Sub jects were asked to adjust a rectangular plate to a p osition parallel with the slantofthe
textured surface in the picture
Results For magni cation judged slant corresp onds to the depicted slant one step b elow that
is the slantisperceived as if the pro jection was made from the viewing p oint For magni cation
viewing p oint coincides with the pro jection p oint judged slantwas in corresp ondence with depicted
slant with the same constant error
alence of magni cation to magni cation one step b elow is not surprising as Note The equiv
the distance from the viewing p oint to the screen remained constant and the resulting stimulus was
practically identical
Rosinski and Fab er Rosinski R R and Fab er J Comp ensation for viewing p oint
in the p erception of pictured space In Hagen pages
Exp eriment
Sub jects Unknown
A Robustness and Perception of the Picture Surface and Frame
Stimuli Photos and line drawings of gray square parallelepip eds cm in height and varying in
width Three faces of each parallelepip ed were visible Pro jection p ointwas cm away from the
picture plane Orthogonal central pro jection was used
Viewing conditions The viewing p ointwas lo cated cm cm cm away from the picture on
the principal ray This corresp onds to te viewing angles of approximately and and
magni cations
Task Sub jects were asked to estimate the size more sp eci cally width of the ob jects in the pictures
Estimate were mo dulus free in abstract units
Note it is entirely unclear from the text what were the sp eci c instructions to the sub jects and
what abstract units were to b e used
Results There was no signi cantchange in apparent relative size of the ob jects when the
viewing p ointwas changed
Exp eriment
Sub jects Unknown
Stimuli Computer generated images of slanted lattices displayed on a CRT Pro jection p oint for the
pictures was cm away from the screen Central orthogonal pro jection
was used Depicted orientation varied from to in increments
y this Viewing conditions Viewing p ointwas on the principal ray to the picture cm awa
corresp onds to magni cations in the range Viewing was bino cular
Task Observers were asked to make direct judgement of the apparent slant in degrees
Results Judged dep ended on magni cation mini cation this dep endence was more signi cant
for slants closer to or and less signi cant for the medium range For magni ed pictures
dep endence was more apparent than for mini ed
Note The task of judging the slant in degrees is quite complicated it app ears that the data were
distorted by the absence of exp erience of matching numerical measure with a particular slant The
main p oint however is that the apparent slant dep ends to some extent on magni cation Purdy s
exp eriment app ears to b e more reliable
Exp eriment
Sub jects Unknown
A Robustness and Perception of the Picture Surface and Frame
Stimuli Computer generated images of slanted lattices in this exp eriment pro jection p ointwas
always at cm from the screen The range was in increments same as in Exp eri
ment
Viewing conditions Viewing p ointwas lo cated on the principal ray cm
away from the picture which corresp onded to magni cation in the range Viewing was
bino cular
Task Same as in Exp eriment
Results Practically no dep endence of the judged slant on the magni cation was observed
Rosinski et al Rosinski R R Mulholland T Degelman D and Farb er J Pic
ture p erception An analysis of visual comp ensation Perception Psychophysics
Same exp eriments are describ ed in Rosinski and Fab er in less detail
Exp eriment
Sub jects college students
Stimuli Photographs of a slanted strip ed rectangle slantvaries in the range orthogonal
central pro jection Surface was rotated around a vertical axis exact lo cation of the axis was not
sp eci ed Kub ovy in his account of the exp eriment sp eci es the vertical axis going through the
center of the rectangle Photographs were cut into trap ezoidal shap es coinciding with the outlines
of the images of the rectangle
Viewing conditions Viewing was mono cular through an ap erture and the photograph was placed
in a viewing b ox where no frame was visible Viewing direction was either or the distance
from the picture plane was equal to the pro jection distance
Task Sub jects were asked to set the inclination of a vertically pivoted palm b oard to match the
apparent slant
Results
Judged slant di ered considerably for and viewing direction
De ne Projected slant as equal to the slant of the rectangle that would pro duce the same image
if pro jected from the viewing p oint When the viewing p oint coincides with the pro jection p oint
A Robustness and Perception of the Picture Surface and Frame
pro jected slant is equal to the depicted slant It was found that the dep endence of the judged slant
on the pro jected slant for viewing direction was very similar to the dep endence of the judged
slant on the depicted slantfor viewing direction The authors conclude that under restricted
viewing conditions apparentslant is determined by the pro jected slant
Exp eriment
Sub jects Same
Stimuli Same
Viewing conditions Viewing was bino cular the photos were in a frame without a viewing b ox
no ap erture was present Viewing direction was or viewing distance was equal to the
pro jection distance
Task Same
Results There were practically no di erence b etween judged slants As pro jected slant in this
exp eriment di ered considerably the authors conclude that the apparent slant is determined by the
depicted slant for unrestricted viewing conditions
Hagen et al b Hagen M A Jones R K and Reed E S On a neglected
variable in theories of pictorial p erception Truncation of the visual eld Perception
Psychophysics
Exp eriment
Sub jects
Stimuli a Iso celes triangles whose height and base were in The triangles were placed
vertically on the table Distance was measured from a round marker placed on the table in away
from the sub ject The distance b etween the markers and the triangles was and
in b Slides of the scene cm
Viewing conditions Viewing was mono cular Three viewing conditions were used for the real scene
xed p osition with untruncated eld of view p eephole mm rectangular truncation cm
slot cm away Slides were viewed from the pro jection p oint conditions for the slides coincided with the rectangular truncation
A Rigidity in Motion Pictures
Task Sub jects were asked to repro duce with a tap e the size of the triangles and the distance from
the marker to the triangles
Results
The distance was consistently underestimated for the truncated eld of view there was a consid
erable compression of the scene i e the relative distances decreased The e ect was similar for the
three truncated conditions slides rectangular truncation p eephole
Intercept extrap olated apparent zero distance was the greatest for slides and signi cantly
non zero for all conditions except the untruncated viewing
Size estimates for the truncated eld of view were lower than the estimates for the untruncateed
eld of view for the real scene for slides they were higher at smaller distances but lower at longer
distances
A Rigidity in Motion Pictures
Sub jects
Stimuli Orthogonal and oblique direction of pro jection parallel and central pro jections pro
jection angle of nearly rectangular rotating rigid and nonrigid solids on a computer display
The images were wireframe with hidden lines removed Twotyp es of deformations were used for
non rigid solids a ne angles b etween edges are unchanged and non a ne Figure A Defor
mations were p erio dical with the p erio d equal to the p erio d of rotation Extent of deformation was
of the smallest dimension of the solid For the oblique pro jection the asp ect ratio
of the ob jects was changed to make them app ear less narrow
Viewing conditions Unrestricted viewing conditions the viewing angle not sp eci ed but constant
Duration of each trial was sec p erio ds of rotation
Task Sub jects were asked to rate the rigidity on the scale of to indicating high con dence
that the ob ject was rigid
Results
Parallel pro jection makes it more di cult to detect non rigidity
in viewing direction do esn t makeany di erence The change of
A ne deformations are more di cult to p erceive
A Rigidity in Motion Pictures
Figure A Left Nonrigid stimuli from Cutting
Figure A Right Stimuli from Exp eriment Cutting
Small deformations are not recognized very well
Exp eriment
Sub jects Same
Stimuli Stimuli of the same typ e Central pro jections were used Pro jection angles were and
pro jection directions andavariable direction in the range withthe
p erio d equal to one half of the p erio d of rotation Figure A
Viewing conditions Two viewing angles were used and
Task Same
Results
For parallel pro jection wehave less non rigidity p erceived for all slants of the screen
For p ersp ective pro jection there is considerable non rigidity for slant
Simulated and physical distance pro jection p oint p osition and viewing p oint p osition did not
a ect judgements
Among non rigid stimuli non a ne transformations were much easier to p erceive
Relation b etween wider angle p ersp ective and viewing direction radii pro jection just
lo oked bad from viewing direction
Exp eriment
Sub jects
Stimuli Rigid nonrigid stimuli pro jection direction was comp ensated by the viewing
A Size and Distance in Pictures
direction of the same magnitude radii pro jection distances corresp onding viewing angles
are
Viewing conditions Viewing distances were radii
Task Same
Results Rigiditywas preserved no comp ensation for slant o ccurred
A Size and Distance in Pictures
Smith b Smith O W Judgements of size and distance in photographs The
American Journal of Psychology
Exp eriment
Sub jects p er viewing condition
Stimuli Photographs of a plowed eld extending from the foreground to distant hills stakes
in each photographs were lo cated on a circular arc yd away from the pro jection p oint the height
of the stakes varied in the range in in in increments An ft gap in the middle of the arc
p ermitted sub jects to see the th stake at a distance stakes were identical in all pictures th
stakewas or in high and it was or yd All p ossible combinations
were used Two more photographs were obtained by air brushing out all the detail except for the
image of one stake in high at yd from the pro jection p oint On one of them shadows were
painted All photos were by in
Viewing conditions Mono cular viewing from a xed p osition through an ap erture Two viewing
boxes were used one resulted in magni cation and the other in magni cation the viewing
distance was equal to and of the pro jection distance The b oxes were black inside back lit
slides were used Data for similar pictures from two di erent studies were used restricted viewing
from the pro jection p oint bino cular viewing and real scene viewing
Task Sub jects were asked to judge the size of the distant stakebymatching it with a near stake
estimate the distance to the distant stake and stakes in the foreground in yards and the heightof
the distant stake in feet Imp overished photographs were judged either b efore or b efore and after the regular photographs
A Size and Distance in Pictures
Results
Matches of height and judgements of height in feet didn t vary muchwiththechange of viewing
conditions Height judgements were overestimates
The judgements of distance increased for magni cation and decreased for magni cation
Standard deviation increased with the distance and was quite high so only qualitative conclusions
are valid
Qualitative trend for the imp overished photographs is the same distance judgements dep end on
the viewing p oint but distances were considerably underestimated if the imp overished photographs
were viewed rst Size judgements in imp overished photographs were not signi cantly di erent
Smith Smith Patricia C S O W Ball throwing resp onses to photographically
p ortrayed targets Journal of Experimental Psychology
Exp eriment
Sub jects for Exp eriment for Exp eriment
Stimuli In a ro om by ft there were marked target lo cations at ve distances
mfromthe viewer Figure A The viewer was used with ap ertures which resulted
in approximately elds of view Below the viewer a curtain could b e
op ened and through this op ening the sub jects could toss a ball at a target Next to the viewer
another curtain could b e op ened and through this op ening the sub jects could observe the targets
with unrestricted head motion
Figure A Diagram of the ro om of the exp eriments Smith
A Size and Distance in Pictures
Viewing conditions viewing conditions were used mono cular viewing through the viewer with
one of the p ossible ap ertures bino cular and mono cular unrestricted viewing
Task Sub jects were asked to throw the ball at one of the targets for di erent viewing conditions re
stricted conditions rst in Exp eriment unrestricted conditions rst in Exp eriment Immediately
after the toss the viewer or the curtain was closed
Results Restricted eld of view resulted in larger distance to the p oint of impact of the ball There
was no signi cant dep endence on the size of the ap erture and b etween mono cular and bino cular
unrestricted viewing The order of the unrestricted and restricted viewing conditions didn t a ect
the qualitative relationship b etween the results for restricted and unrestricted viewing conditions
Exp eriment
Sub jects
Stimuli Same except that the sub jects observed photographs of the scene in the viewing apparatus
instead of the scene itself For unrestricted viewing conditions the real scene was used Photographs
were made from the correct viewing p oint no attempt was made to match the colors or intensities
photographs were black and white with yellow target markers paintedonthem
Task Same Sub jects were not told that photos are used after the exp erimenttheywere asked if
they sawanything unusual in the scene visible through the viewer
Results
The data were adjusted in the following way the di erence b etween means in unrestricted viewing
conditions identical for Exp eriments and was added to the means for restricted conditions
to correct for the di erence b etween groups of sub jects Adjusted results were very close for pho
tographs and real scene restricted viewing
There was practically no di erence b etween bino cular and mono cular unrestricted viewing
No sub ject rep orted that he had seen photos in the viewer or noted anyunusual qualityofwhat
he had seen
Exp eriment We omit this exp eriment b ecause we don t nd its results relevant to our purp ose
Smith a Smith O W Comparison of apparent depth in a photograph viewed
from two distances Perceptual and Motor Skil ls
A Orientation and Layout
Exp eriment
Sub jects
Stimuli in back lit photograph of a corridor ft long
Viewing conditions Mono cular viewing through a p eephole from two viewing distances and
of the pro jection distance from the viewing p oints on the principal ray to the picture Viewing
angle was and magni cation and A viewing b oxwas used to restrict sub ject s
eld of view to the image area
Task Sub jects were asked to estimate the distance to the front edge of the corridor and the length
of the corridor in paces
Results The means for the greater viewing distance were more than two times greater than the
means for the smaller viewing distance
A Orientation and Layout
Halloran Halloran T O Picture p erception is array sp eci c Viewing angle
versus apparent orientation Perception Psychophysics
Exp eriment
Sub jects
Stimuli Pictures similar to those used in Rosinski et al Black and white strip ed rectangles
depicted at slant around a vertical axis the axis was lo cated
at the right edge of the rectangle
Viewing conditions Unrestricted viewing from the viewing distance equal to the pro jection distance
and the viewing direction of
Task Sub jects were asked to repro duce the slant with a p ointer pivoted around a vertical axis
Results For large slants and large displacements viewing direction of or exceeding those
of Rosinski et al there was a considerable dep endence of the estimated slant on the viewing p osition
A Orientation and Layout
Figure A Left Winslow Homer The Fog Warning from Halloran
Figure A Right Scene reconstruction from Halloran
Exp eriment
Sub jects Same
Stimuli Pictures from Exp eriment which depicted and slant transformed in the manner
that corresp onds to the viewing p ositions of Exp eriment twoframeswere present the trans
formation of the frame of the original and pictures and a rectangular frame of the nal
picture
Viewing conditions Viewing p ointwas xed viewing direction was
Task Same
Results No di erence from Exp eriment except for a small nearly constant digression towards pic
ture plane Compare to the apparent distortions of the pictures of slanted pictures as demonstrated
in Pirenne
Exp eriment
Sub jects
Stimuli Multicolored repro duction of Winslow Homer s painting The Fog Warning by cm
Figure A
Viewing conditions Unrestricted viewing from p ositions the following are pairs viewing distance
viewing direction m m m m m m
m
ere instructed a to orient the p ointer in the direction that the dory in the Task The sub jects w
picture was headed with resp ect to the picture plane b to orientthepointer in the direction of the mother ship on the horizon from the man s p osition
A Orientation and Layout
Results
There is a consistent shift in b oth judged orientations with the change in the viewing p oint As
the viewing direction angle increases the judged angle of orientation of the b oat and of the direction
to the ship increases
The rotation of the judged orientation of the b oat approximately corresp onds to the mean rotation
of the ob jects reconstructed using sets of cues Figure A
Set the xed height of the man and the length of the b oat
Set the rectangle from the stern width to the corresp onding width in the front of the b oat
Set the xed length and width of the b oat except and viewing directions
Set oars are of equal length
The b est t is given byaveraging set which results in no rotation with any of the sets
The change of di erence of the angle of orientation of the b oat and the direction to the ship is
small over interval the judged angle of orientation of the b oat
Exp eriment
Sub jects
Stimuli Mono chrome repro duction of the picture from Exp eriment
Viewing conditions Unrestricted viewing from viewing p oints from Exp eriment with the viewing
direction the viewing distance was scaled as the size of the picture was di erent
Task
a Sub jects were asked to make separates judgementof
the plan view orientation of the dory that is indicated by its long axis disregarding all other cues
the orientation with resp ect to the picture plane of a straight line connecting the blades of the
oars
the angles b etween the line connecting the blades of the oars and the b oat axis
To accomplish this a pair of p ointers pivoted on a common axis were used to representtwosides
of the angle to b e estimated
b Sub jects were asked to identify the correct b oat among plan view outline drawings with
length to width ratios of and Results
A Orientation and Layout
Mean length axis orientation was approx For a similar viewing p osition in Exp eriment
the mean orientation was indicating some in uence of other cues
The oar crossing angle implied by separate judgements of the oar direction and the b oat axis
direction was while the mean oar crossing angle judged directly was
There were strong individual di erences
The results of this exp eriment suggest that cues can in uence judgments in di erentways
dep ending on the task and that the ability to separate cues di ers among sub jects
Exp eriment
Sub jects
Stimuli stimuli similar to Goldstein Figure A except they were p ersp ective line drawings
each depicted a cm ball with one or two cm square dots on it One dot was lo cated on the
intersection of the central p erp endicular to the picture plane and the sphere the image of this dot
was in the center of the image of the ball the other one was on the big circle of the ball which lies
in the plane p erp endicular to the picture plane rotated from the p osition of the rst dot
Viewing conditions Unrestricted viewing the viewing distance equals the pro jection distance
cm viewing direction was
Task For the two dot picture sub jects were asked to indicate the angle b etween dots with reference
to the center of the ball for the single dot pictures sub jects were asked the dot s direction from the
picture plane Pointer arrangement from Exp eriment was used
or single dot pictures the average judgements were for the center dot and for Results F
the side dot resulting in the relative angle of For the two dot pictures the mean judgement
was
Exp eriment
Sub jects
Stimuli Same as in Exp eriment
Viewing conditions Same as in Exp eriment and also from the viewing p oint on the principal ray
cm away from the picture
A Orientation and Layout
Figure A Left Stimuli from Exp eriment Goldstein
Figure A Right The Village of Bequigny by Theo dore Rousseau Goldstein
Task Sub jects were asked to indicate spatial orientation of the dory using a p ointer with two degrees
of freedom
Results
Mean judged ab out angles were for viewing direction and for viewing
direction Compared to the Exp eriment the rotation b etween p ositions was twice as much
Mean judged pitch angle b etween the axis of the b oat and the horizontal plane was and
observers indicated no or little increase others indicated substantial increase
Goldstein Goldstein E B Rotation of ob jects in pictures viewed at an an
gle Evidence for di erent prop erties of twotyp es of pictorial space Journal of
Experimental Psychology Human Perception Performance
Exp eriment
Sub jects
Stimuli Line drawings of ro ds Figure A with di erent swings around a vertical axis Depicted
swings were The ro d was mm across
Viewing conditions Viewing was mono cular p osition of the righteyewas xed The pictures
ere rotated so that the viewing direction was and The viewing w
distance was cm corresp onding to the viewing angle of approximately for viewing direction
Task The task was to set the direction of a p ointer parallel to the direction of the horizontal ro d in the picture
A Orientation and Layout
Results All the ro ds app ear to b e rotating as the viewing direction changes The amountof
rotation increases with depicted angle from for the ro d up to for the ro d
Exp eriment
Sub jects Same
Stimuli Black and white photograph of the painting The Village of Bequigny by Theo dore
Rousseau Figure A
Viewing conditions Same as in Exp eriment The viewing angle was for the viewing direction
Task Observers were asked to judge the orientation of the road A the rut in the road B the
house C and the line de ned bytwo trees D
Results Road rotates the house rotates the trees rotate the rut rotates Mostly
con rms the results of the Exp eriment although the house rotates more then the direction b etween
trees while app earing to b e more p erp endicular to the road with viewing direction
Exp eriment
Sub jects Same
Stimuli Similar to those in the Exp eriment but the length of the horizontal ro ds was only mm
only disks are left
Viewing conditions Same as in Exp eriment
Task Same as in Exp eriment
Results The amount of rotation signi cantly increased for small angles up to for ro d
but remained unchanged for larger angles The di erential rotation decreased
Exp eriment
Sub jects Same
Stimuli Same as in Exp eriment
Viewing conditions Same as in Exp eriment
Task Observers were given a sheet of pap er with two parallel lines on it indicating the direction of
the road The task was to indicate the p osition of the rut of the house and of the trees relativeto the road
A Orientation and Layout
Figure A Left Stimuli from Exp eriment Goldstein
Figure A Right Results for Exp eriments and Goldstein
Results There was little dep endence on the viewing direction There was some change in the
relative distances but almost no change in the orientation
Goldstein Spatial layout orientation relative to the observer and p erceived
pro jection in pictures viewed at an angle Journal of Experimental Psychology
Human Perception Performance
Exp eriment
Sub jects
Stimuli Black and white photographs of three vertical dowels with horizontal strip es on a homege
neous white surface against a homogeneous white background Figure A The size of the picture
was by cm
Viewing conditions Mono cular viewing from a xed p osition with viewing distance equal to the
pro jection distance and viewing direction Viewing direction
was changed by rotating the picture
Task Observers judged the spatial layout of the dowels by arranging three discs on a piece of pap er
A Orientation and Layout
Figure A Left Drawings of the face stimuli from Exp eriment Goldstein
Figure A Right A line drawing of the sphere from Exp eriment Goldstein
Results Averaged results are shown in Figure A for viewing directions Size BC
was scaled to the same size for all observers Judged layout changed only slightly with the viewing
direction
Exp eriment
Sub jects Same
Stimuli Same
Viewing conditions Same
Task Observers were asked to set a p ointer parallel to the directions BA BC CA
Results Total changes of the directions were BA CA BC Thelayouts
reconstructed from the directions are shown in Figure A These layout di er greatly from each
other
Exp eriment
Sub jects
Stimuli Seven photographs of a frontally oriented human face with di erent gaze directions resulting
from lo oking at seven di erent targets p ositioned at intervals in the range of directions from
to Figure A
Viewing conditions The viewing was mono cular p osition of the righteyewas xed The pictures
as were rotated so that the viewing direction w
Task The task was to set the direction of a p ointer parallel to the direction of the gaze in the picture
Results Total rotation of the judged direction of the gaze was for the depicted gaze of
A Orientation and Layout
and only for the gaze and ab out for gaze Total amount of rotation increased
towards gaze
Exp eriment
Sub jects
Stimuli Similar to the photographs from Exp eriment in addition to the frontal p osition of the
face three additional head orientations were used and
Viewing conditions Same
Task Same
Results
The results were qualitatively similar to the results of Exp eriment The maximum of total
rotation is shifted to smaller angles for and head orientations
Judged gaze direction is relatively indep endent from the head orientation
Exp eriment
Sub jects
Stimuli Line drawings of twovertical cylinders similar to those used in Exp eriments and Black
on white pictures were used for viewing in the light and negatives were used for viewing in the
dark
Viewing conditions Similar to Exp eriments and Two mo des of viewing were used in the light
regular line drawings and in the dark back lit negatives
Task Sub jects were asked to orientapointer in the direction parallel to the direction from cylinder
A to cylinder B
Results Viewing the picture in the dark when the picture frame and surface are invisible increases
total amount of rotation of the judged direction for oblique orientation
Exp eriment notnumb ered in the pap er
Sub jects
Stimuli Photograph of a sphere with a marked p oint either at horizontal direction from the center
of the sphere or at counted from the direction parallel to the picture plane Figure A
Viewing conditions Exact conditions unknown Viewing direction was
A RectangularCorners
Task Sub jects were asked to judge the directions from the center of the sphere to the dots presum
ably using a p ointer
Results The average judgements of the orientations of the p oints A and B were and
which results in calculated angular separation of
A RectangularCorners
Perkins Perkins D N Visual discrimination b etween rectangular and nonrect
angular parallelepip eds Perception Psychophysics
Exp eriment
Sub jects
Stimuli drawings of b oxes some of which could have b een and some of which could not havebeen
orthogonal parallel pro jections of rectangular b oxes according to the rst Perkins law Section
There were di erent drawings the rest were pro duced by rotating the original drawings by
The original drawings all had angle C Figure A and angle A varied from
to Three edges radiating from the central vertex were cm size of the pictures was in
Viewing conditions Unrestricted viewing from a p osition feet away from the pictures Three
metho ds of presentation were used straight when pictures were presen ted one by one
samples when two pictures with A equal to nonrectangular case and rectangular
case were p ermanently displayed next to the judged picture
pairs when pictures were sorted into two piles and the rectangularity status rectangular
nonrectangular b orderline was di erent for corresp onding cards in two piles
Task In the rst two cases the task was to sayifthebox in the picture app eared to b e rectangular
In the last case the task was to saywhichbox app ears to b e more rectangular
Results For each pro cedure the p ercentage of correct judgements for rectangular pictures was close
to for nonrectangular it varied from for the straight pro cedure to for pairs Any
boxwas shown in orientations Distribution of the numb er of rectangular judgements maximum
is shown in Figure A
A RectangularCorners
B
A C
Figure A Left One of the stimuli for Perkins
Figure A Right Results of Perkins
Perkins Perkins D N Comp ensating for distortion in viewing pictures
obliquely Perception Psychophysics
Exp eriment
Sub jects
Stimuli Same as in Perkins
Viewing conditions First sub ject judged rectangularity of the pictures from viewing direction
the viewing distance ft cards only Next two viewing directions were used and
A susp ended wire frame ensured correct viewing direction otherwise viewing conditions were
unrestricted The choice of the viewing directions was motivated by the following observation
classi cation of the repro jection of the drawings into an oblique plane according to the Perkins laws
change for a large numb er of pictures around In the neighb orho o d of and classi cation
of these pro jections were stable cf Section
Task Sub jects were asked to judge each picture as rectangular or nonrectangular b ox
Results All b oxes for each of the two viewing directions could b e classi ed into one of four cate
gories using four p ossible combinations of two criteria whether the picture satis es the rst Perkins
law and whether the pro jection of the picture onto the plane p erp endicular to the viewing direction
satis es this law The b orderline case was excluded
A RectangularCorners
Figure A Left Stimuli from Shepard
Figure A Right Three dimensional ob jects from Shepard
Judgements were made mostly according to the classi cation of the picture not of the pro jection In
the cases of a con ict the p ercent of judgments coinciding with the picture classi cation was lower
but still high ab ove in all cases except orthogonally nonrectangular pro jectively rectangular
There was very little pro jective in uence at
Shepard Shepard R N Psychophysical complimentarity InPerceptual Or
ganization Kub ovy M and Pomerantz J R editors pages Lawrence
Erlbaum Asso ciates Publishers Hillsdale New Jersey
Exp eriment
Sub jects Unknown
Stimuli line drawings consisting of three radial lines meeting at di erent angles at one p oint
Figure A
Viewing conditions Circular ap erture in frontofthedrawing otherwise unrestricted viewing
Task Sub jects were asked if each pattern could b e a pro jection of a a corner of a cub e all three
angles of a tetrahedron all three angle or of a planar Mercedes Benz sign all three
A RectangularCorners
angles Figure A Sub ject could rotate eachdrawing in its plane around the center with
a remote control
Results Dashed lines in Figure A indicate the b oundaries of the areas of orthogonal parallel
pro jections of corresp onding spatial corners For the corners there is no di erence b etween orthogonal
parallel and central pro jections assuming that the principal ray of the central pro jection go es through
the p oint where the lines meet The size of the circles is prop ortional to the fraction of p ositive
answers
Figure A Results of Shepard
For the cubic corner there is close agreement of threshold of p ositiveanswers with orthogonal
pro jection b oundaries This con rms Perkins rst law
For the Mersedes Benz gure the agreementwas less than for the cubic corner and for corner
even less However most of the change was in the direction of decreasing the ranges of acceptable con gurations
A Foreshortening
A Foreshortening
Nicholls and Kennedy a Nicholls A L and Kennedy J M Angular subtense ef
fects on p erception of p olar and parallel pro jections of cub es Perception Psy
chophysics
Exp eriment
Sub jects
Stimuli Line drawings of a cub e orthogonal parallel pro jection and central pro jection with and
pro jection angles
Viewing conditions Pro jections were viewed mono cularly viewing angles were
Task The sub jects were asked to compare the pictures with a standard the picture with the
pro jection angle viewed at a distance corresp onding to the viewing angle
Results
Mo derate parallel foreshortening was always the b est parallel pro jection was second b est
and strong parallel foreshortening was the worst
Parallel pro jection is p erceived as distorted only at close distances only for it was worse than
strongly convergent
Exp eriment
Sub jects
Stimuli Same as but the image with mo derate degree of parallel foreshortening pro jection
angle p ositioned at the distance corresp onding to the viewing angle was used as standard
Viewing conditions Same
Results Order of preference was preserved except for the closest distance when the pictures
with the strongest parallel foreshortening was preferred over the parallel pro jection no parallel
foreshortening but they were still judged to b e slightly worse than the pictures with mo derate
parallel foreshortening
Exp eriment
Sub jects
A Foreshortening
Figure A Left Stimuli from Nicholls and Kennedy b Exp eriment
Figure A Right Stimuli from Nicholls and Kennedy b Exp eriment
Stimuli Same
Viewing conditions The viewing angles were and Standard image
pro jection mo derate degree of parallel foreshortening set at a p osition corresp onding to the viewing
angle
Results Mo derate parallel foreshortening was still rated the b est at all distances but now parallel
pro jection had lower ratings than strongly divergent
Nicholls and Kennedy b Nicholls A L and Kennedy J M Foreshortening and
the p erception of parallel pro jections Perception Psychophysics
Exp eriment
Sub jects
Stimuli Line drawings of a cub e in oblique parallel pro jection One face was always depicted as a
square oblique lines representing the lateral edges of the cub e met the horizontal at
the degree of p erp endicular foreshortening the ratio of the length of the edges of the fronttothe
length of the oblique lines was and Figure A
Viewing conditions Unrestricted viewing
Task Sub jects were asked to cho ose b est drawings among the set with a xed angle
A Foreshortening
Results ratio was consistently preferred
Exp eriment
Sub jects
Stimuli Three wo o den blo cks with square cross sections with length to width ratios of and
line drawings with receding lines equal to of the frontedges
Figure A
Viewing conditions Unrestricted viewing
Task The sub jects examined one out of three wooden blocks After that they were asked to cho ose
the b est drawing
Results
For blo ck picture was preferred for blo ck was preferred and for blo ck
was preferred In rst two cases the preferred p erp endicular foreshortening was in the
third case Ab out of the sub jects made the dominating choice
Exp eriment
Sub jects
Stimuli Same blo cks as b efore but the pictures were either frontal orthogonal parallel pro jections
or frontal orthogonal central pro jections Figure A
Viewing conditions Unrestricted viewing
Task Same
Results For parallel pro jection and blo cks the preferred p erp endicular foreshortening
was for blo ck For central pro jections it was and resp ectively Ab out of
the sub jects made the dominating choice
Hagen and Elliot Hagen M A and Elliot H B An investigation of the re
lationship b etween viewing condition and preference for true and mo di ed linear
p ersp ective with adults Journal of Experimental Psychology Human Perception
e Performanc
A Foreshortening
Figure A Stimuli from Nicholls and Kennedy b Exp eriment
Figure A Stimulus from Hagen and Elliot
Exp eriment
Sub jects college students in Exp eriment college students in Exp eriment
Stimuli sets of images of p olyhedra Figure A generated by the metho d describ ed in Reggini
see also Section convergence degree from to Angular size from to
Viewing conditions a mono cular p eephole from xed distance from and viewing directions
b free viewing D solids were shown prior to the exp eriment
Task a assign rank in the range to b Sort the pictures from the most to the least natural
lo oking
Results Parallel pro jection is consistently rated the b est from b oth viewing p oints For the free
A Foreshortening
viewing the result is same the only exception are the pictures of cub es is the most frequently
chosen
Exp eriment
Sub jects college students
Stimuli Same
Viewing conditions Viewing p oint coincided with the pro jection p oint for the degree of convergence
regular p ersp ective Mono cular viewing p eephole
Task Sub jects were asked to rank the pictures from most to least natural lo oking or from most to
least accurate drawing
Results Parallel pro jection has the highest rating No dep endence on the form of instructions was
observed
Exp eriment
Sub jects college students
Stimuli Sets D F from Exp eriments
Viewing conditions Viewing p oint coincided with the pro jection p oint for the degree of convergence
mono cular viewing p eephole Stimuli were presented three at a time each picture was oriented
p erp endicular to the viewing direction
Task Sub jects were asked to cho ose the most and the least natural drawing from each triplet
Results Parallel pro jection was chosen as the most realistic more frequently than any other degree
of convergence
Note This study was critisized by other authors The images in sets C E F di ered not only in the
degree of convergence but also in visibility of faces of the p olyhedra The use of Reggini s p ersp ective
seems to b e unnecessary b ecause the degree p ersp ectiveconvergence parallel foreshortening could
be varied simply bychanging the viewing p oint
Hagen M A Elliott H B and Jones R K A distinctive Hagen et al a
characteristic of pictorial p erception The zo om e ect Perception
A Visual eld
Exp eriment
Sub jects college students
Stimuli Picture sets D F from Hagen and Elliot typ es of stimuli a black and white line
drawings normal light b Dayglo uorescent pap er under normal light c Dayglo stimuli under
ultraviolet light
Viewing conditions Mono cular viewing p eephole the viewing p oint coincides with the pro jection
p oint for the degree of convergence Three pictures from the same set were presented simultane
ously The viewing direction was D ob jects were shown to the sub jects b efore the exp eriment
Task Sub jects were instructed to cho ose the most realistic and natural lo oking drawing out of three
Results
Under the normal light parallel pro jection was preferred for all typ es of stimuli
For Dayglo stimuli under ultraviolet light there was no particular preference
Exp eriment
Sub jects
Stimuli Dayglo stimuli from Exp eriment under ultraviolet light
Viewing conditions Mono cular viewing with head motion allowed
Task Same as in Exp eriment
Results Results are similar to those obtained under normal light parallel pro jection is preferred
Exp eriment
Sub jects
Stimuli Same
Viewing conditions Bino cular viewing with xed p osition of the head
ask Same T
Results Slightchange in the direction of normal light preferences but not nearly as radical
A Visual eld
Finke and Kurtzman Finke R A and Kurtzman H S Mapping the visual eld
A Visual eld
in mental imagery Journal of Experimental Psychology General
Exp eriment
Sub jects
Stimuli A circle cm in diameter with eight radial lines A disk cm in diameter was placed in
the middle each half of the disk contained gratings or cycle p er degree cp d Gratings had
the same numb er of cp d and were p erp endicular to each other Figure A The disk in the middle
was oriented so that the gratings were parallel or p erp endicular to one of the drawn diameters of
the circle
Figure A Stimuli from Finke and Kurtzman
Viewing conditions The viewing p ointwas xed cm away from the disk resulting in the angular
size of the small disk approximately and the size of the circle approximately Viewing was
bino cular
Task
a Sub jects were asked to move their eyes along each diameter until they could not see that the
disk in the middle consists of two distinct parts A small red dot on the end of a ro d was used as a
xation p oint
b The sub jects were trained to imagine the pattern in the middle the actual pattern was removed
and the same exp erimentwas p erformed
Note The details of instructions are imp ortant for part b see the pap er for details Results
A Straightness and Curvature
The average eld size b oth for real and imagined stimuli decreased with increase in the spatial
frequency of the gratings for cp d for cp d for cp d
The measured visual eld was longer horizontally than vertically asp ect ratio for
cp d resp ectively
Imagery elds were in close corresp ondence with the p erceptual elds
Exp eriment
Sub jects
This exp erimentwas designed to check the in uence of knowledge or exp ectations on the results of
part b of the previous exp eriment Sub jects were asked to guess the outcome of the exp eriment
that was describ ed to them actual stimuli were demonstrated
Results Resulting eld sizes for and cp d gratings were considerably lower than in Exp eriment
the eld size estimate for cp d was closer to the cp d estimate contrary to the results of the
Exp eriment Average predicted visual elds were circular no eccentricity sub jects however
did predict considerable eccentricity in the correct direction
Note Exp eriment was mostly irrelevant for our purp oses and is omitted
A Straightness and Curvature
Watt et al Watt R J Ward R M and Casco C The detection of deviation
from straightness in lines Vision Research
Exp eriment
Sub jects authors
Stimuli Various p erturbations of a straight line segment presented on the screen of an oscilloscop e
The segments were arc min in length the size of the p erturbations along the line was in the
e S like Eachtyp e range arc min There were typ es of p erturbations vernier like bump lik
of p erturbation had various typ es of smo othing none discontinuous constant bumps Gaussian
linear Figure A Figure A Figure A
The maximal amplitude of the p erturbation in the direction p erp endicular to the bump varied The
A Straightness and Curvature
Figure A Left Vernier likestimulifrom Watt et al
Figure A Right Bump likestimuli from Watt et al
Figure A S like stimuli from Watt et al
resolution of the displaywas arc min The direction sign of the p erturbation varied randomly
The orientation of the line varied randomly in a range around the vertical axis
Viewing conditions Unrestricted viewing from the distance m
Task The sub jects indicated the sign of the presented stimulus if they could not decide they
guessed
Results On the basis of the data threshold amplitudes for various stimulus typ es and size were
computed
The threshold amplitude was never b elow arcsec
For vernier p erturbations the threshold decreased with size of the p erturbations for the other two
typ es it decreased for smaller sizes of the bump but increased for larger sizes
The b est t to the data was provided bythehyp othesis that the threshold is determined by
the area of the maximal bump with resp ect to the least square ro ot straight line through the data
threshold value arc min if the amplitude is greater than arc min
Exp eriment a
Sub jects Unknown
Stimuli Pairs of lines on the screen of an oscilloscop e One of the lines was straight the dots of the
A Straightness and Curvature
other line were p erturb ed according to a Gaussian distribution with varying means The length of
the lines varied
Viewing conditions Same
Task Sub jects were asked to cho ose the p erturb ed line
Results According to a computer simulation the largest bump rule would pro duce threshold vs
length dep endence with exp onent Probability summation bumps are detected with some
probability rule gives exp onents b etween and Complete summation all bumps are
used corresp onds to the exp onent The exp onents found in the exp erimentwere signi cantly
less than and greater than weak probability summation
Exp eriment b
Sub jects Unknown
Stimuli Bump like p erturbation with no smo othing from the Exp eriment was used Line length
was arc min A wide range of p erturbation lengths was considered
Viewing conditions Same
Task Same
Results It app ears that for larger p erturbation sizes more than haveoftheline thebumpson
the ends of the line are detected rather than the large bump in the middle This results in increase
of the threshold for larger p erturbation sizes cf Exp eriment
Exp eriment a
Sub jects Unknown
Stimuli Lines with two Gauss like bumps on one side with length arc min displayed on the
screen of a oscilloscop e A random Gaussian p erturbation with variable standard deviation and zero
mean was added
Viewing conditions Same
Task Same as in Exp eriment
Results Amplitude threshold was plotted against the standard deviation of the added noise As
suming the existence of internal noise in determination of p osition of the p oints it was estimated to
b e approximately arc sec
A Straightness and Curvature
Exp eriment b
Sub jects Unknown
Stimuli Lines with three typ es of p erturbations the rst second and third derivatives of a Gaussian
with variable spatial extent
Viewing conditions Same
Task Same as in Exp eriment
Results Threshold decreases up to certain size of the p erturbation than slightly increases This
critical size value is relatively indep endentofthetyp e of the p erturbation approximately arc min
This might b e an indication that there is a limit for spatial integration
Ogilvie and Daicar Ogilvie J and Daicar E The p erception of curvature Cana
dian Journal of Psychology
Exp eriment
Sub jects
Stimuli Black curved lines mm wide on a in photographic plate The radii of curvature
were and m Twochord length were used and m De ne
normalized curvature as the chord length divided by the radius of curvature Normalized curvature
approximates the displacement of the end p oint of the line with resp ect to the tangent at the starting
p oint expressed as a fraction of the chord length For mm chord length it was equal to
Viewing conditions The viewing p ointwas m away from the display resulting in angular subtens
of the lines of and arc min A cm ap erture was used to restrict the eld of view to
Viewing was mono cular
torientations vertical horizontal oblique Task Lines were presented to the sub jects in di eren
Sub jects were asked to indicate if the line is curved up or down left or right Each line was
presented times lines were presented in random order
Results Calculated average thresholds were b est correlated with the segment angular area which
was calculated using an approximate formula chord length sagitta where sagitta is the
A Verticality
distance from the middle of the arc to the chord Threshold was in the range log sq sec
sq min
A Verticality
DiLorenzo and Ro ck DiLorenzo J R and Ro ck I The ro d and frame e ect as a
function of the righting of the frame Journal of Experimental Psychology Human
Perception Performance
Exp eriment
Sub jects
Stimuli A luminous vertical ro d inside a luminous frame The tilt of the frame could b e adjusted
Viewing conditions Unrestricted viewing from a xed p osition viewing angle in a dark ro om
only the frame and the ro d were visible
Task The sub jects were shown the frame tilted at After min the sub jects were asked to
adjust the frame to the vertical p osition After that the ro d tilted at and the frame tilted at
in the same direction were shown The sub jects were asked to adjust the ro d to the vertical p osition
the frame remained tilted Same exp erimentwas done with the head of the observer tilted at
Results The average righting e ect the tilt of the single frame after adjustment was the
average ro d and frame RFE e ect was and for counterclo ckwise and clo ckwise directions
When the head was tilted the e ects increased
Exp eriment
Sub jects
Stimuli Similar to the Exp eriment with an internal frame added angular subtense
Viewing conditions Same
Task The outer frame was presented in an upright p osition for min Than adjusted if the observer
p erceived it as tilted After that it was returned to the ob jectivevertical Then the inner frame was
added at a tilt of and Exp eriment was p erformed with the inner frame as the frame Same
exp erimentwas p erformed with tilted head
A Verticality
Results No righting e ect or RFE were observed in b oth cases
Exp eriment
Sub jects
Stimuli Same as Exp eriment
Viewing conditions Same
Task Similar to Exp eriment but with outer frame tilted and inner frame vertical
Results Approximately righting e ect for b oth frames and RFE were observed When
the head was tilted the righting e ect increased for b oth frames RFE e ect increased mostly
insigni cantly
App endix B
Reconstruction of the center of
pro jection from a pro jection of a
rectangle
Supp ose we know that a picture contains a p ersp ective pro jection of a rectangle which is a nonde
generate quadrangle Let l b e the line going through the center of pro jection O p erp endicular to
the pro jection plane p Figure B
Let s assume in addition that we know the p osition of the intersection p oint O of the line l and
the pro jection plane p If the picture was made with a regular camera then O is the center of the
picture
Intro duce a co ordinate system with the origin at p oint O z axis directed along the normal to the
plane and x and y axes parallel to the pro jection plane
Let x x x x b e the known pro jections of the vertices of a rectangle with unknown vertices v
v v v Let d OO d Let e b e the unit vector in the direction of d Then for all j
x d j
App endix B Reconstruction of the center of projection from a projection of a rectangle
vj
x y j
O O’ z d l
x p
Figure B
We will use parentheses for the scalar pro duct of a pair of vectors and for the triple pro duct of
a triple of vectors We will use brackets for the vector pro duct
The fact that x is the pro jection of v can b e expressed as
j j
x v d
j j
v B
j
d
The sides of the rectangle should b e equal
v v v v B
The lengths of the diagonals should b e equal
jv v j jv v j
which is equivalentto
v v v v B
Subtracting v v v v which follows from B from B and simplifying we get
v v v v B
Substituting the expression for v from B into B and B results in j
App endix B Reconstruction of the center of projection from a projection of a rectangle
v e x v e x v e x v e x B
d
x x v e v e x x v e v e B
d d
Multiplying the vector equation B by x x and x we get the following system of equations in
variables v e v e v e we leave v e as a parameter
x x v e x x x x x x
C C B C B B
C C B C B B
C C B C B B
v e B
x x v e x x x x x x
C C B C B B
A A A
x x v e x x x x x x
The matrix of the left hand side of this system is the Gram s matrix of the vectors x x x By
assumption the pro jection of the rectangle is not a line segment Therefore the rank of this matrix
must b e and the system has a unique solution
We can also note that we can express the determinant of the matrix in the system B as
D x x x where x x x is the triple pro duct x x x This is a consequence of
the following general identity
ae af ag
ab c efg B
be bf bg
ce cf cg
The solutions of the system are
x x x x x x
v e
v e
x x x x x x
D
x x x x x x
x x x x x x
v e
v e
x x x x x x
D
x x x x x x
App endix B Reconstruction of the center of projection from a projection of a rectangle
x x x x x x
v e
v e
x x x x x x
D
x x x x x x
Using identity B the solutions can b e simpli ed
x x x x x x x x x
v e v e v e
x x x
x x x
x x x x x x
v e v e v e v e
x x x x x x
Substituting these expressions for v e v e v e into B and dividing by
v e x x x we get the following equation
x x x x x x x x x x x x x x x x B
The z comp onentofthevectors x x and x is equal to d Manipulating the determinants we get
x x x x x x
i j k i j k
d x x x
x x x i j k x x x
i j i j
k k
d d d
x x x x x x
k j j i k i
C B
d
A
x x x x x x
k j k i j i
x x x x x x x x x x
i j k j j k i j j k
B C
d d
A
x x x x x x x x x x
j i j j j j i
k k k
We denote this determinant D Note that
ij k
jD j j x x x x j B
ij k i j j k
aretwo dimensional vectors in the pro jection plane x x x
i
i i
Further x x x x d x x x x d Equation B takes the form
App endix B Reconstruction of the center of projection from a projection of a rectangle
D D x x d D D x x d
which is equivalentto
D D x x D D x x d D D D D
This equation has a unique solution if
D D D D B
Consider the geometric interpretation of the quantities x x x It is the volume of the paral
i j k
x x taken with the p ositivesignifthevectors form a right hand triple lelepip ed spanned by x
j k i
and with the negative sign if the vectors form a left hand triple
Assume counterclo ckwise ordering of the vertices of the quadrangle x x x x if welookdown from
O Figure B This assumption can b e made without loss of generality b ecause if wechange the
ordering the signs of all the volumes that we consider will change simultaneously which do esn t a ect the form of B
O
3 p 4 2
1
Figure B
From Figure B we see that
x x x V ol ume O x x x Volume O
App endix B Reconstruction of the center of projection from a projection of a rectangle
x x x V ol ume O x x x Volume O
where Oijk denotes the pyramid with ap ex O and base ij k The heights of all pyramids are equal
to d AsD x x x d jD j S where S is the area of the triangle ij k Thiscanbe
ij k i j k ij k ij k ij k
also seen from B
We can use the information ab out the signs of x x x to write B in the following form
i j k
d S S S S S S x x S S x x
Condition B is
S S S S B
4
c d C 3
1 a b
2
Figure B
The diagonals of the quadrangle divide it into four parts Denote their areas by a b c d
Figure B Then condition B means that
c b d a d c a b
After rearrangementwe get
c a b d
which means that nonadjacent pieces must have unequal areas This is a necessary condition for d
to b e determined from B
Denote y x x Then the formula for d is
ij i j
App endix B Reconstruction of the center of projection from a projection of a rectangle
x x j y y jj y y j x x j y y jj y y j
B d
j y y jj y y j j y y jj y y j
The distancefrom the center of projection to the projection plane is given by B whenever the
right hand side is de ned and positive For the right hand side of B to be de neditisneccessary
and su cient that the area of the triangle C is not equal to the area of the triangle C and the
area of the triangle C is not equal to the area of the triangle C FigureB
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