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A Computational Model of View Degeneracy and its Application to Active Focal Length Control
Wilkes, David; Dickinson, Sven J.; Tsotsos, John K. https://scholarship.libraries.rutgers.edu/discovery/delivery/01RUT_INST:ResearchRepository/12643458520004646?l#13643535820004646
Wilkes, D., Dickinson, S. J., & Tsotsos, J. K. (1997). A Computational Model of View Degeneracy and its Application to Active Focal Length Control. Rutgers University. https://doi.org/10.7282/T3R214W9
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A Computational Mo del of View Degeneracy and its
Application to Active Focal Length Control
David Wilkes
Department of Computer Science
University of Toronto
Sven J Dickinson
Rutgers University Center for Cognitive Science RuCCS and
Department of Computer Science
Rutgers University
John K Tsotsos
Department of Computer Science
University of Toronto
Abstract
We quantify the observation by Kender and Freudenstein that degenerate views o ccupy
a signi cant fraction of the viewing sphere surrounding an ob ject This demonstrates that
systems for recognition must explicitly account for the p ossibility of view degeneracy We show
that view degeneracy cannot b e detected from a single camera viewp oint As a result systems
designed to recognize ob jects from a single arbitrary viewp oint must b e able to function in
spite of p ossible undetected degeneracies or else op erate with imaging parameters that cause
acceptably low probabilities of degeneracy To address this need we give a prescription for
active control of fo cal length that allows a principled tradeo b etween the camera eld of view
and probability of view degeneracy
Introduction
Image segmentation is the task of transforming a signal level image description into a symbolic
description consisting of sets of segments or features of some sort Examples of features in common
use are homogeneous regions and also discontinuities describ ed by line segments or curves Such
features may b e di cult to b oth identify and interpret due to the accidental alignment of spatially
distinct scene parts b elonging to a single ob ject or multiple ob jects as seen from the camera
viewp oint This alignment is often referred to as view degeneracy This sup erp osition of features
can lead to erroneous part counts or bad parameterizations of a mo del For example if two ob ject
edges are abutted and collinear one longer edge is seen in their place
Given arbitrarily high resolution such alignments would not b e a problem b ecause they would
o ccur only for a vanishingly small fraction of the p ossible viewp oints Unfortunately cameras
and feature extracting op erators have nite resolution We shall see in this pap er that this enlarges
the set of viewp oints that are problematic
We b egin by de ning view degeneracy precisely and showing how our de nition covers many
interesting classes of degeneracy We develop a mo del of degeneracy under p ersp ective pro jection
It is parameterized by the distance to the ob ject to b e recognized the separation of features on
the ob ject the minimum separation of the corresp onding image features in order for them to b e
detectible as distinct features and camera fo cal length The mo del gives the probability of b eing in
a degenerate view as a function of the mo del parameters The evaluation of the mo del at realistic
parameterizations indicates that degenerate views can have signi cant probabiliti es
We will show that there is a signi cant additional problem for recognition systems op erating from
a single arbitrary viewp oint Namely view degeneracy is not detectible from a single viewp oint
As a result single viewp oint systems must b e able to p erform unambiguous ob ject identi cation
in spite of p ossible degeneracies The alternative is to minimize the probability of degeneracy We
will show that the probability of degeneracy is very sensitive to camera fo cal length This will lead
nally to a prescription for fo cal length control that allows a tradeo to b e achieved b etween the
width of the eld of view and probability of view degeneracy
De nitions of degeneracy
Kender and Freudenstein presented an analysis of degenerate views that p ointed out prob
lems with existing de nitions of the concept The de nitions that they prop ose in place of earlier
problematic ones incorp orate the notion that what constitutes a degenerate view is dep endent on
what heuristics the system uses to invert the pro jection from three dimensions to two One of the
de nitions incorp orating this system dep endence is the negation of the de nition of a general view
point A general viewp oint is such that there is some p ositive epsilon for which a camera movement
of epsilon in any direction can b e taken without e ect on the resulting semantic analysis of the
image This de nition was noted by Kender and Freudenstein to b e incomplete in the sense that
there exist general viewp oints on certain ob jects that are unlikely to allow a system to instantiate a
prop er mo del for the ob ject relative to other views of the same ob ject e g the view of the base of
a pyramid They argue for somehow quantifying degeneracy according to the amount of additional
work exp ected to b e required to unambiguously instantiate a mo del corresp onding to the ob ject
b eing viewed
We are interested in a de nition that is indep endent of the particular recognition system in use
and are willing to trade away the completeness of the de nition to achieve this Figure illustrates
a de nition of degeneracy that covers a large subset of the cases app earing in the literature The
de nition has to do with collinearity of the front no dal p oint of the lens and a pair of p oints on or
de ned by the ob ject We shall consider a view to b e degenerate if at least one of the following
two conditions holds
a zero dimensional p oint like ob ject feature is collinear
with another zero dimensional ob ject feature and
the front no dal p oint of the lens
a zero dimensional ob ject feature is collinear with object point feature
object point feature
front nodal point object point features
object point feature
front nodal point
Figure Our de nition of view degeneracy The two cases are the collinearity of the front no dal
p oint of the lens with two ob ject p oint features top or with one ob ject p oint feature and a p oint
on one line b ottom or line segment
some p oint on either
a an in nite line or
b a nite line segment
de ned by two zero dimensional ob ject
features and the front no dal p oint
Each of the sp eci c imaging degeneracies enumerated by Kender and Freudenstein for p olyhedral
ob jects b elongs to one of these types Below we discuss each of their example degeneracies in turn
Vertices imaged in the plane of scene edges This is encompassed by case b ab ove and is
depicted in gure The lower right scene edge on the upp er ob ject is the line segment of case b
The vertex on the lower ob ject is the zero dimensional ob ject feature
Parallel scene lines imaged in their own plane We depict this situation in gure This is
case a with an in nite line The p ortion of the viewing sphere in which the degeneracy o ccurs is
that for which the front no dal p oint of the lens is in the plane de ned by an end p oint of the edge
on the front ob ject and the other scene line extended to have in nite length
Coincident scene lines imaged in their own plane This generalizes to coplanar lines imaged
in their own plane We depict this situation in gure This is equivalent to case b using an
in nite line An endp oint of the edge on the right hand ob ject may b e used as the zero dimensional
feature The edge on the left hand ob ject may b e extended to de ne the in nite line
Figure An example of view degeneracy in which a vertex is imaged in the plane of a scene edge
Figure An example of view degeneracy in which parallel scene lines are imaged in their own plane
Figure An example of view degeneracy in which coplanar scene lines are imaged in their own
plane
Figure An example of view degeneracy in which p erfect symmetry gives preference to a at D
interpretation of the lines
Perfect symmetry Kender and Freudenstein include p erfect symmetry in the enumeration of
types of degeneracy b ecause they say that it leads to a tendency to interpret the symmetric structure
as at An example is shown in gure In the case of a radial symmetry this is equivalent to
case with the two zero dimensional p oints chosen to b e any two distinct p oints on the axis of
symmetry In the case of symmetry ab out a plane this is equivalent to case a with b oth features
chosen arbitrarily to lie in the plane of symmetry Note that in the case of symmetry the features
de ning the degeneracy are more likely to b e abstract p oints on the ob ject e g the centroid of
an ob ject face rather than p oints extractible directly by low level vision The use of abstract
ob ject p oints rather than obvious feature p oints has no impact on the estimation of probabilities
of degeneracy that follows What matters is that there is some means to enumerate the types of
degeneracy that are imp ortant to the approach This enumeration is discussed later in the pap er
Dep endence on e ective resolution
If a computer vision system had in nite ability to resolve ob ject features that were close to one
another then all of the degeneracies discussed ab ove would have in nitesimal probability of o ccur
rence Figure shows typical lo ci of p oints on the viewing sphere at which instances of each type
of degeneracy o ccur assuming in nite resolution Each case degeneracy o ccurs at a pair of p oints
on the sphere surface Each case a degeneracy o ccurs at a circle of p oints on the sphere surface
Each case b degeneracy o ccurs on a sector of a circle of p oints on the sphere surface Since these
lo ci of p oints are of lower dimensionality than the viewing sphere itself the probability that the
viewp oint is on such lo ci is vanishingly small
In nitesimal probabilities of view degeneracy would p otentially allow strong inferences to b e
made ab out scene structure Consider a certain coincidence C of simple ob ject features A and
B If C o ccurs in the image much more frequently than exp ected due to view degeneracy then many
of the o ccurrences must b e due to a regularity in the domain of interest i e the frequent physical
coincidence of A and B This regularity is useful for recognition if C reduces the uncertainty in
ob ject identity The utility of C for recognition may b e expressed more formally as the average Viewing sphere Position of point-point degeneracy
Object point
Object line Object point
Circle of point-line degeneracy
Position of point-point
degeneracy
Figure Lo ci of degeneracy with in nite resolution This illustrates degeneracies of b oth case
type involving two D ob ject features and case type involving a D ob ject feature and an
in nite line de ned by ob ject features
number of bits of information gained by observing C An expression for this is
X
P r obj ect ijC
P r obj ect i log
P r obj ect i
i
or
X
P r obj ect i
P r obj ect i log
P r obj ect ijC
i
This is also called the asymmetric divergence or mean information gain p er observation of C
Kender and Freudenstein p oint out that the nite resolution of real systems has the p otential
to make the probability of the degenerate views signi cant or even a certainty This is due to the
thickening of all of the lo ci of p oints on the viewing sphere corresp onding to degeneracies to include
viewp oints from which the key features are almost collinear with the front no dal p oint of the lens
Figure illustrates this thickening This would have a negative impact on systems that assume no
degeneracy If the coincidence C discussed ab ove o ccurs more frequently due to degeneracy than
exp ected then expression ab ove will b e smaller than exp ected since the apparent coincidence of
comp onent features A and B will now o ccur more often for ob jects in which A and B are physically
separated than with in nite resolution
We will use the phrase e ective resolution to capture all system dep endencies in our mo del As
a result the de nition is a general one to b e sp ecialized to each system as needed
De nition The e ective resolution in the image is the minimum separation of a pair of p oints
corresp onding to distinct ob ject features needed in order to detect b oth features Viewing sphere
Circle of point-point degeneracy
Object point
Object line Object point
Band of point-line degeneracy Circle of point-point
degeneracy
Figure The lo ci of degeneracy of the previous gure with nite resolution The regions of the
viewing sphere a ected by degeneracy are large The circle of the previous gure is actually a thick
band on the sphere The p oints from the case degeneracy are actually disks
E ective resolution is a function of b oth actual camera resolution and the metho d used for fea
ture detection For example one may estimate the e ective resolution to b e the width of the
convolution kernel used to extract ob ject edges or some function of the kernel width and edge
length chosen to mo del actual system p erformance
Overview
The reason for using the ab ove de nitions of degeneracy and e ective resolution is that they have
allowed us to construct a system indep endent quantitative mo del of view degeneracy We can
predict the probability of b eing in a degenerate view as a function of the geometric parameters
of the key ob ject features and the e ective resolution This is useful b ecause it allows one to
decide the imp ortance of degenerate views in various imaging situations In particular we note
that degenerate views have signi cant probability in many realistic situations
In the next section we present our mo del and its analysis Section presents an example
parameterization of the mo del corresp onding to a real application The result shows a signi cant
probability of view degeneracy The implications of this for recognition systems is discussed in
Section along with various means of coping with degeneracy The sensitivity of the probability
of view degeneracy to fo cal length leads to a prescription for fo cal length control discussed in
Section Finally we conclude with a recapitulation of the results B
r φ f A' α θ α A s R
B'
Figure Perspective pro jection of a pair of p oints We will determine the fraction of all orientations
of line AB in relation to the camera for which the image separation s of A and B is small enough
to interfere with their resolution into distinct features
Mo del
Degeneracy based on a pair of p oints Figure shows the p ersp ective pro jection of a pair
of feature p oints A and B For simplicity our analysis assumes that A is centred in the image
We are interested in the separation s of the images of the two p oints as a function of the other
parameters shown in the gure The parameters are
f distance from image plane to rear no dal p oint
R distance of A from the lens front no dal p oint
r separation of the two ob ject p oints
orientation of B with resp ect to
the optical axis
apparent angular separation of the p oints
We have that
s
tan
f
and that
q
2 2
2 2 2
sin r cos sin r
tan
R r cos cos
Let s b e the e ective resolution as de ned earlier Equating the two right hand sides we will solve
c
for given s s and The choice to solve for is arbitrary We obtain two solutions and
c 0
at which s s
1 c
q
2
2 2 2 2 2 2
r f s R s r Rs f
c c c
1
A
cos
0
2 2
cos r f s
c
q
2
2 2 2 2 2 2
Rs f r R s r f s
c c c
1
A
or cos
1
2 2
r f s cos
c
1
and exist provided that the arguments for their resp ective cos functions are in the interval
0 1
Where they exist there are intervals of for which s s The intervals are and
c 0
Where they do not exist it is b ecause the entire range of values from to gives s s
1 c
at the given value Figure shows the sections of the sphere swept out by p oint B in Figure
as ranges over a small interval and ranges over the interval For each small
0 0
interval in the ratio of the area for which s s to the total area of the sphere swept by p oint B
c
is g where
shaded ar ea g
total ar ea
Z Z Z Z
+ +
0 0 0
2 2
r cos d d r cos d d
2
r
0
0 1 0
Z Z
0
2
cos d r cos d
2
r
0
1
sin sin
0 1
sin sin
0 1
Thus we may estimate the probability that s is smaller than some critical value s by doing a
c
numerical integration of g over all values to determine the fraction of the sphere swept by
and that gives s s
c
Degeneracy based on a p oint and a line Figure shows the p ersp ective pro jection of a
line segment of length l with endp oints B and C and a p oint A We simplify our calculations by
centering B in the image Without loss of generality we choose an image co ordinate system that
0 0 0
causes the image of the line to lie along one of the co ordinate axes Let A B and C b e the images
of A B and C resp ectively In addition to the parameters used in the case of two p oints we have
the additional parameters
w angle b etween the line segment and
the optical axis
0 0
a length of the line segment B C
p ossibly in nite
0 0
b separation of A and C
0 0
c separation of A and B
0
t distance b etween B and the
0
nearest p oint to A on the in nite
0 0
line through B and C
0
The parameter t is used to determine whether a p erp endicular dropp ed to the line through B and
0 0 0
C falls on the segment b etween B and C φ=π/2
φ=φ 1 φ=φ 0 φ=π
φ=0 θ +∆θ θ 0 0
surface element
rdφ
rcos φ dθ
Figure The regions of the sphere swept by p oint B for and The
0 0
shaded parts are the areas for which s s c
C A
l r φ ω B' f θ R B c t
s A' a
b
C'
Figure Perspective pro jection of a line segment and a p oint We will determine the fraction of
all p ossible orientations of A with resp ect to line BC for which the image separation of A and BC
is small enough to interfere with the detection of A as a separate feature
We have
f
s r cos sin
R r cos cos
f
t r sin
R r cos cos
f
a l sin
R l cos
q
2 2
s t a b
p
2 2
s c t
We are interested in the distance of the image of the p oint from the line segment This distance d
is
s if t and t a
b if t a d
c if t
We considered two p ossibiliti es for the integration for the probability that d is less than some critical
value d The approach that is analogous to the approach used in the rst case leads to numerical
c
evaluation of a three dimensional integral An easier metho d would b e to p erform a Monte Carlo
integration by randomly choosing triples uniformly on the surface of the viewing sphere
and tabulating the fraction of the trials for which d d
c
Combining probabilities of individual degeneracies Finally if we have estimates of the
probabilities of degenerate views based on individual pairs of p oints and p oint line segment
pairs we need to combine the probabilities of individual degeneracies to get the probability that at
least one degeneracy exists This nal step is somewhat application dep endent b ecause di erent
classes of ob jects have di erent degrees of interaction among their features If we were to assume
indep endence among all of the p oints and line segments in the ob ject set and if the ob jects were
wire frame ob jects so that all features were visible all of the time then the task would b e easy All
features would b e able to interact with each other to cause degeneracies This mo del is appropriate
with sensors or ob jects for which transparency or translucency is common e g X ray images
For computer vision there typically exist pairs of features that do not interact to form a de
generacy simply b ecause they are at opp osite p ositions on an opaque ob ject We will examine the
case of a p olyhedral world in order to provide an example of the analysis necessary to determine
overall probabilities of o ccurrence of degeneracy In a p olyhedral world features are clustered to
b e in coplanar groups faces reducing the probability of degeneracy from the value that would b e
estimated assuming indep endence of the features
In the case of single isolated convex p olyhedral ob jects we can take a face by face approach
to estimating the probability of degeneracy Sp eci cally we will only examine interactions among
p oints and lines b elonging to the same face since features not sharing a face are relatively less likely
to interact An edge on view of a face is equivalent to our case a degeneracy where the p oint used
is an arbitrary vertex and the line used is an edge on the opp osite side of the face extended to
have in nite length
To arrive at a global estimate of degeneracy we could simply sum the probabilities of degeneracy
for each face However since parallel faces cause strongly overlapping regions of the viewing sphere
to b e degenerate faces should b e divided into equivalence classes with one class p er face orientation
For example a cub e consists of three sets of faces with three distinct orientations We may
estimate the probability of degeneracy for the cub e by summing the probabilities of three faces one
representing each equivalence class Finally we can take into account the degeneracy due to radial
symmetry by summing the p oint pair degeneracy estimates for each axis of radial symmetry in the
ob ject
If we de ne terms as follows
p the probability of an individual p oint pair degeneracy
00
p the probability of an individual line p oint degeneracy
01
n the number of axes of radial symmetry
r
n the number of face equivalence classes
f
where n do es not include axes of symmetry lying in planes of symmetry then the overall probability
r
of a degenerate view p is estimated by
d
no f ace deg ener acies
z
n n
r
f
p p p
d 00 01
z
no sy mmetr y deg ener acies
The ab ove expression corresp onds to an assumption of indep endence of the axes of radial sym
metry and face groups We conjecture that this is more realistic than assuming no overlap b etween
the regions of degeneracy as one would assume by simply summing the individual probabilities of
degeneracy In a world with multiple ob jects in each image there is a greater degree of indep en
dence among the features since the p ositions of the ob jects with resp ect to one another are less
likely to b e as ordered as the p ositions of individual features within an ob ject The ab ove enumera
tion would provide an optimistic lower b ound on the probability of degeneracy Summation over all
feature pairs as in the wireframe case would provide a p essimistic upp er b ound on the probability
of degeneracy
Parameterization of the mo del
We have parameterized the mo del to match the situation in our own exp erimental setup as describ ed
in detail in and presented in overview in We have used the actual camera pixel size to
compute the critical feature separation s The ob ject related parameters are those of an ob ject
c
set consisting of origami gures that is used in our exp eriments Figure tabulates the results
with the basic parameterization and examines the sensitivity of the probability of a degeneracy to
each of the mo del parameters by p erturbing each parameter in isolation
We see that for example for a mo del system with a parameterization to recognize small tabletop
ob jects from a range of half a metre there are one in ve o dds that the view encountered will b e
degenerate The sensitivity analysis demonstrates that there are realistic parameterizations of the
mo del for which probabiliti es of degeneracy are very signi cant We also tried a parameterization
with f and s set to match human foveal acuity of twenty seconds of arc The probabilities of
c
degeneracy were negligably small This may explain why the imp ortance of degenerate views to
computer vision has traditionally b een underestimated
Basic parameterization
parameter value
f metres
R metres
r metres
l metres
s metres pixels
c
n
f
n
r
Results at basic parameterization
probability value
p
00
p
01
p
d
Perturbations of basic parameterization
p erturbation p p p
00 01 d
f m
f m
R m
R m
r m
r m
l m
l m
l m
5
l m
s pixel
c
s pixels
c
n
f
n
f
n
r
7 4 4
human fovea
Figure Results for a tabletop vision system The basic parameterization corresp onds to our
exp erimental setup One fth of the views on a single ob ject are degenerate The probability of
degeneracy varies signi cantly as mo del parameters are p erturb ed Object
Picture from camera 1
Camera 1 Camera 2 identical images
Degenerate viewpoint Non-degenerate viewpoint
Figure The camera in the left scene is at a p osition of degeneracy The camera in the right scene
is not since no alignments are present yet the two camera images are identical This demonstrates
that degeneracy is not detectable from a single viewp oint
Implications for recognition
Our mo del and realistic parameterizations of it provide quantitative arguments that so called de
generate views should b e taken into account by designers of recognition algorithms If one excludes
degenerate views from the ob ject representations or viewp oints used by a system then one must
have a mechanism for detecting degeneracy when it o ccurs and moving out of the degenerate view
p osition It is imp ortant to note that even detection of degeneracy requires camera motion Using
our de nition of view degeneracy we have
Theorem It is imp ossible to detect view degeneracy from a single camera viewp oint
Pro of Assume to the contrary that we have a degeneracy detector D that takes a single image
as input and determines whether it contains instances of view degeneracy We may cause D to fail
as follows We construct a pair of scenes such that each scene results in the same camera image
but one is degenerate and the other is not Then whether D rep orts that the image is degenerate
or not D will b e in error on one of the two scenes Figure diagrams such a pair of scenes The
idea is to acquire a camera image at a p osition of degeneracy then print the image acquired If
one then acquires an image of the print from an appropriate face on viewp oint the result can b e
made indistinguis habl e from the image of the original scene The rst image a face on view of a
cub e is degenerate a small motion of the camera will change the number of faces of the cub e that
are visible This second image however is from a non degenerate viewp oint small motions of the
camera result only in minor distortion of the camera image In fact changes in viewp oint with
resp ect to the print of the scene of up to degrees result in smo oth changes in the image features
2
For recognition to b e successful from a single viewp oint therefore degenerate views must b e
usable even if the degeneracy is not explicitly detectible
D ob ject recognition systems based on simple indexing features such as p oints or lines can
avoid problems asso ciated with various degeneracies if they can recover a su cient number of
nondegenerate features from the image e g With a few features accurate p ose estimation
is p ossible and the pro jected aligned mo del can b e used to verify other mo del features in the image
Voting techniques can also accumulate evidence for a degenerate view or mo del orientation with a
small number of nondegenerate features
D ob ject recognition systems that attempt to recover more complex qualitative features such
as geons do not rely on accurate p ose estimation and geometric feature veri cation
Instead they attempt to recover complex volumetric primitives which serve as their indexing
primitives for recognition Each of the ab ove systems assumes a nondegenerate view of a primitive
so that it can b e uniquely recovered and used as an index
In the D recognition work of Dickinson Pentland and Rosenfeld complex volumetric
primitives are also recovered from a single D image A distinctive feature of this approach is the use
of part based aspect matching techniques to recover the primitives from the image This approach
requires the estimation of the probabilities of di erent asp ects including the so called degenerate
ones The probabilities were estimated by determining what set of regions was visible from each
p oint on a discretized viewing sphere Given an edge based segmentation technique the so called
degenerate views or asp ects were found to have nontrivial probabilities More recently Shimshoni
and Ponce note that accidental views will app ear over a nite region of the viewing sphere
when considering nite resolution Assuming orthographic pro jection they present an algorithm
for constructing the nite resolution asp ect graph of a p olyhedron
Single view approaches to recognition can b e successful provided that the camera view even if
degenerate is not ambiguous in the sense of b elonging to more than one stored ob ject mo del If
the mapping from a recovered view to an ob ject mo del or part is ambiguous then the various
interpretations should b e rank ordered to provide a more e cient search A more e ective
solution to dealing with ambiguity is to take an active approach Wilkes has explored the
idea of moving the camera to a viewp oint advantageous to recognition This approach also meets
with success but at the cost of signi cant viewp oint changes Dickinson Christensen Tsotsos and
Olofsson use a probabilistic asp ect graph to guide the camera from a p osition where the view is
ambiguous to a p osition where the view is unambiguous
A nal alternative is somehow to reduce the probability of view degeneracy We see from
the sensitivity analysis in Section that the probability of degeneracy p is sensitive to b oth fo cal
d
length and e ective resolution Thus we may consider controlling the aiming of a variable resolution
foveated sensor using an attentional mechanism of some sort or we may control fo cal length This
latter approach is the sub ject of the next section
Prescription for fo cal length control
There is a clear tradeo b etween wide angle of view and probability of view degeneracy Wide angle
is desirable for initial ob ject detection in the presence of uncertainty in camera or ob ject p osition
b ecause the ob ject of interest is more likely to fall in the eld of view if fo cal length is small On the
other hand small fo cal length increases the probability of view degeneracy A similar tradeo has
b een noted in the work of Brunnstr om who uses a wide angle of view to detect ob ject junctions
and then zo oms in to increase image resolution for junction clasi cation The existence of such a
tradeo suggests that there can b e a principled choice of fo cal length in many circumstances
Supp ose that we have known constants p the maximum acceptable probability of view de
0
generacy and s the critical separation of image features Then the fo cal length should b e set
c
to the smallest value that gives a probability of degeneracy p no larger than p This may b e
d 0
done automatically by a vision system provided that the necessary parameters of our mo del can
b e determined
We may rewrite all of the expressions involving parameters l R and r equations
r l
in terms of the ratios and
R R
r
1 2
r r
2
2 2 2
s f s f s
c c c C B
R R
1
C B
cos
0
A
2 2
f s cos
c
r
1 2
r r
2
2 2 2
f s f s s
B c c C c
R R
1
B C
or cos
1
A
2 2
f s cos
c
f r
cos sin s
r
cos cos R
R
r f
t sin
r
cos cos R
R
f l
sin a
l
R
cos
R
r l
Thus if we can estimate and we can completely parameterize the mo del computing p via
d
R R
the ab ove equations plus equations and
r l
and as If the current fo cal length f is known then we may make a rough estimate of
R R
follows We may sample the distribution for the apparent separation s of neighbouring features in
the image getting an average apparent separation s to minimize the e ects of view degeneracy
in this sampling we should use a large fo cal length If all pairs of adjacent features had exactly
the separation r in D space and all relative orientations of such feature pairs in D space were
equally likely then the exp ected value for the image separation s of feature pairs would b e given by
pr obability density f or tilt
Z
z
f r
2
E s cos d cos
R
0
z
imag e leng th if pair tilted by
f r
R
r
Thus s E s gives us an estimate of
R
r s
R f
r l
if the predominant degeneracies in the domain of interest are of the type We may set
R R
l
of case b or set if the predominant degeneracies are of the type of case a Which
R Probability of degeneracy vs. focal length probability pdPlot.data 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 focal length in mm
0.00 20.00 40.00 60.00 80.00 100.00
Figure Probability p as a function of fo cal length f at the basic parameterization
d
parameterization is chosen typically do es not a ect the order of magnitude of the resulting p
d
values Figure shows how p varies with fo cal length for our basic parameterization of Section
d
In general the fo cal length f giving p p is found by applying a zero nding metho d to the
d 0
function g f p p where p is found by evaluating our mo del as describ ed ab ove Since the
d 0 d
g (f )
n
function is smo oth and has only one zero we recommend a Newton iteration f f
n+1 n
0
g (f )
n
0
where f is any p ossible fo cal length value for the system and g f is the derivative of g with resp ect
0 n
g (f + ) g (f )
and the iteration is terminated when successive iterations change to f approximated by
2
f by less than To summarize given a sampling of image feature separation at a known large
fo cal length we can solve for an up dated fo cal length that gives a probability of view degeneracy
not to exceed a given target value
Conclusions
We have demonstrated that view degeneracy is a frequent enough o ccurrence to warrant considera
tion in the design of computer vision systems The fact that view degeneracy is not detectible from
a single viewp oint means that systems that attempt recognition from a single arbitrary viewp oint
must b e able to do so from degenerate as well as non degenerate views The alternative is to take
steps to minimize the probability of degeneracy
The probability of view degeneracy may b e reduced by several means One is to move the camera
to a viewp oint advantageous for recognition Another is to use an attentional mechanism to
aim a high resolution fovea like sensor for feature detection A nal economical p ossibili ty is the
control of fo cal length discussed ab ove
Acknowledgements The authors wish to thank the Natural Sciences and Engineering Research
Council of Canada the Information Technology Research Centre and the Canadian Institute for
Advanced Research for nancial supp ort The authors are members of the Institute for Rob otics
and Intelligent Systems IRIS and wish to acknowledge the supp ort of the Networks of Centres of
Excellence Program of the Government of Canada and the participation of PRECARN Asso ciates
Inc The second author is the CP Unitel Fellow of the Canadian Institute of Advanced Research
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