Estimating the Principal Curvatures and the

Darb oux Frame from Real D Range Data

Eyal Hameiri Ilan Shimshoni

Abstract As pro ducts of secondorder computations esti While these features are continuous prop erties of a regu

mations of principal curvatures are highly sensitive to noise

lar surface range data is discrete in nature This is where

Due to the availability of more accurate D range imaging

the challenge of how to extract these continuous features

equipment evaluation of existing algorithms for the extrac

from a discrete approximation of a surface emerges Prin

tion of these invariants and other useful features from dis

crete D data is now relevant The work presented here

cipal curvatures are secondorder derivatives of a surface

makes some subtle but very imp ortant mo dications to two

whichmakes them very sensitivetonoise Thus the main

such algorithms originally suggested by Taubin and

goal of the work presented here is to suggest algorithms for

Chen and Schmitt The algorithms have b een adjusted

to deal with real discrete noisy range data The results of

principal curvatures and Darb oux frame estimations from

this implementation were evaluated in a series of tests on

real noisy data and to test their validity

synthetic and real input yielding reliable estimations Our

The sensitivity of the features to noise imp osed limita

conclusion is that with current scanning technology and the

tions on some early works restricting their use to only the

algorithms presented here reliable estimates of the prin

atures and the Darb oux frame can be extracted cipal curv

sign of the mean and Gaussian curvatures for segmenta

from real data and used in a variety of more comprehensive

In the recognition system tion tasks

tasks

presented in only the normal to the surface which re

quires only a rst derivative of the surface without the

I Introduction and Previous Related work

principal directions is used to recognize D ob jects In

recentyears following the availability of more accurate D

The imp ortance of principal curvatures as invariantfea

sensing equipment as well as the use of p olyhedral meshes

tures of freeform rigid surfaces has b een widely recognized

to approximate freeform surfaces more and more works in

In addition the lo cal co ordinate system consisting of the

volve full principal curvature estimation from discrete data

normal to the surface at that p ointandthetwo principal

in their suggested algorithms

directions the Darboux frame can also b e exploited for a

Two main approaches are b eing taken by all previous

vast variety of tasks The Darb oux frame is not invariant

works The more p opular one until recent years is the

to rigid transformations but it can b e used to establish a

analytical approach In this approach for eachD p oint

lo cal co ordinate frame which is attached to the ob ject at

within the input data a lo cal patch of surface is tted to

each of its sampled p oints

the p oint and its geometric neighbors The tted patchis

These two typ es of features are lo cally asso ciated with

formulated as an explicit function f u v Then the rst

the surface and therefore can be used in a lo cal manner

and second partial derivatives of f u v are analytically ob

For example the principal curvatures can b e used to nd

tained and used within a general and closed analytic form

corresp onding p oints b etween a freeform scene ob ject and

to compute the principal curvatures and directions or the

a library ob ject Then the lo cal Darb oux frame on the two

mean and Gaussian curvatures One analytic metho d dif

ob jects can b e used to register them and test the validity

fers from another by the technique in which the surface

of the matc hed p oints But the features can also b e used

patch is being tted In the patch is a general bi

in a global manner For example in the case of geometric

quadratic patch In an analytic approachwhich uses a

primitives wehave preknowledge of the dierentcharac

cubic bspline tting technique presented in is tested

teristics of the two features when they are collected from

In the patch is also formulated as a biquadratic for

the entire surface of the primitive Then by gathering

mula but the basis functions consist of a dierent triplet of

the feature values from all points of a range scene evi

discrete orthogonal p olynomials and not of the canonical

dence for the presence or absence of geometric primitives

p olynomials

will emerge Furthermore when a primitive is identied in

The second approach extracts the features directly or

the range image its accurate dimensions and orientation

almost directly from the discrete data For example in

can b e recovered from the values of the features In b oth

a Gaussian curvatures op erator was develop ed based

typ es of uses global as well as lo cal it is therefore nec

on a known expression for the Gaussian curvature see

essary to accurately estimate the principal curvatures and

for example as the limit of the ratio of a lo cal surface area

the Darb oux frame from range data

and of its corresp onding area on the Gauss map From a

similar limit of ratio of areas an op erator for the mean

Eyal Hameiri is with the Dept of Computer Science The

T echnion Israel Inst of Technology Haifa Israel Email

curvatures was develop ed directly from discrete data after

eylcstechnionacil

dening sp ecial lo cal areas In the algorithm is based

Ilan Shimshoni is with the Dept of Industrial Engineering and

on an integral formula which denes at each sampled p oint

Management The Technion Israel Inst of Technology Haifa Israel

Email ilansietechnionacil a matrix whose eigenvectors and eigenvalues pro duce the

principal curvatures and the Darb oux frame The matrix is a nonnegativescalarand n denotes the normal to the

is based on discrete data except for a continuous approxi curveatP The reader can nd more details in

mations of normal sections curves The algorithm in

Now consider S to be a regular surface illustrated in

is based on solving a leastsquares problem at eachvertex

Figure P a p ointlyingonS and C S a regular curve

The leastsquares problem is formulated directly from the

passing through P Let N and n b e the unit length normals

discrete data except for the approximation of cross sections

at P ofS and C resp ectivelyandT the unit length tangent

curves as circles We based our two algorithms on the last

If n is the curvature of C at P and is the of C at P

two nonanalytic metho ds and therefore our suggested al

angle b etween N and n then the quantity T cos

P

gorithms b elong to the nonanalytic approach

is dened as the normal curvature of C at P Note that

for any normal section of S incidentonP the curvature

Most of the curvature estimation algorithms suggested

of this normal section at P equals to the normal curvature

so far had presented their results only on nonenoisy syn

at that p oint

thetic data That is the case in and for example

where satisfactory results have b een achieved for synthetic

ob jects Few others likethework in were also tested

on noisy synthetic data and on real data But as far as

real data is concerned it concluded that curvature esti

mation should b e dealt with caution In full and accu

rate estimation of principal curvatures and Darb oux frame

is presented while the input is a mesh approximating an

articial spatial surface and which is arbitrarily triangu

lated There is no reference to the algorithms p erformance

in case of real data or of synthetic noisy data

Two algorithms presented in for the estimation

of principal curvatures and Darb oux frame from range im

ages have b een mo died in this work We will refer to

these two algorithm as to the T algorithm and C al

gorithm resp ectively The T algorithm which deals cor

rectly with continuous surfaces has b een mo died to deal

Fig The lo cal dierential prop erties of a curve and a freeform

with real discrete surfaces op en or close represented by

surface on whichitlies

a cloud of noisy D p oints In the C algorithm the mo d

ication improved the results for b oth synthetic and real The Meusnier theorem states that all curves in S in

noisy data and also reduced the complexity of the algo

tersecting P and sharing at that p oint the same unit tan

rithm We have tested the p erformance of the mo died

gentvector also share the same normal curvature There

algorithms on syn thetic noisy range data and on real data

fore the quantity T has b een established as the di

P

rectional curvature of S at P along the tangent direction obtained from a Cyb erware D scanner Cases of noiseless

T Note that the sign of the directional curvature is deter synthetic data were also tested esp ecially for comparative

reasons For each surface p oint the features are estimated

mined by the orientation of N In all our discussions from

and a reliability factor is applied to indicate how accurate

nowonN is p ointing out in case of a closed surface or has

the estimations are Wherever the groundtruth is avail

a nonnegative comp onent in the direction of the viewer in

case of an op en one able statistics of the errors resulting from the comparison

were studied

The maximal and minimal values of T along all

P

p ossible tangent directions to S at P are referred to as the

The pap er continues as follows In Section I I we explain

principal curvatures and their two asso ciated tangents as

terms related to curvatures In Section I I I the two original

the principal directions of S at P If the t wo principal cur

algorithms are presented in detail followed by descriptions

vatures share the same value then the principal directions

of the mo dications we have made to them Section IV

are undened It can b e shown that when dened the

is dedicated to our implementation of the suggested algo

principal directions are always orthogonal and thus forming

rithms In Section V representative examples of tests we

together with N an orthogonal triplet known as the Dar

have conducted are presented This section also includes

boux frame While the principal curvatures are invariantto

conclusions ab out the conditions under which the imple

rigid transformations the Darb oux frame is not but it still

mentation should b e used its limitations and the reliability

can b e used to establish a lo cal co ordinate frame whichis

of its results We conclude in Section VI

attached to the ob ject regardless of its orientation



II On Curve and Surface Curvatures

Let and denote the two principal curvatures of S

P P

at P and T T their two asso ciated unit length principal



Let C s denote an arclength parameterized curve pass

directions For every tangent direction T atP where is

ing through a p oint P where C P In this parameter

the angle b etween T and T the Euler formula states



ization the derivative C pro duces the tangenttothe

that

curveat P denoted by T The second degree derivative



sin cos T

P

C gives us the curvature vector denoted by n where P P

b e extracted by solving the linear equations where the III The algorithms and their modifications

co ecients are constant for all p oints on the surface

In this section we will describ e each one of the original

In order to estimate the directional curvature T

P

algorithms in followed by their mo dications to

used in let us consider C s as the arclength param

deal with real noisy range images We shall refer to the

eterized normal section at P along the direction T Ac

algorithm in as the T algorithm and to the one in

cording to the parameterization C P C T and

as the C algorithm

C T N By expanding C s to a Taylor series

P

e obtain around s w

A The T Algorithm

The T algorithm is based up on the construction of a



C sC C s C s O s

matrix expressed as an integral at each D point where

the principal curvatures and Darb oux frame have to be

or

estimated As illustrated in Figure let us arbitrarily



C s P Ts T Ns O s

P

let cho ose T a tangent to S at P For

T denote the tangent to S at P in the direction which

By taking the length of the vectors on b oth sides weget

creates an angle with T The matrix M is then dened

P



as follows

kC s P k s O s

Z



T

Dividing equation multiplied byN by the last equa

T

M T T T d

P P

tion yields



T

N C s P

T O s

P

The integrand in is a matrix which is the pro duct of

kC s P k

a tangentatP and its transp ose Therefore N the normal

to the surface at P isaneigenvector of M asso ciated with

Thus if a point Q lies on S very close to P and T

P

a eigenvalue

denotes the pro jection of PQ on the tangent plane to S at

Let T and T denote the principal directions at P If

P then T can b e approximated as



P

is the angle b etween T and T then



T

N Q P

T

P

T T cos T sin



kQ P k

B T algorithm modications

Thus by using and it can be shown for any se

A surface S is given as a set of no des V where eachone

of them represents a D point lying on S By connect

ing geometrically neighb oring no des into p olygons usually

but not necessarily the p olygons are triangles one can ap

proximate the original continuous surface S as a polyhedral

mesh

Let G b e the set of surface no des containing all the geo

metric neighb ors of a no de v At that no de let M be an

v

approximation of the continuous case equivalent matrix

M We also will refer by T to the tangenttoS at v in the

P i

direction connecting v to v G ie T is the pro jection of

i i

the vector connecting these two no des on the tangent plane

to S at v If is the directional curvature asso ciated with

i

T then

i

X

T

T T M

i i i v

i

v G

i

Fig The T algorithm tangent directions at P

where is prop ortional to the inverse of the geometric dis

i

tance from v to v and the sum of all weights equals

i i

lection of T that the two principal directions at P are the

The use of the weights is not needed in case of a symmetric

other two eigenvectors of M Furthermore if e e

P 

p olyhedral mesh where neighb ors of a no de are scattered

denote the eigenvalues of M asso ciated with T T re

P 

in equal distances more or less around it It is needed

sp ectivelywe can obtain that

in cases of nonsymmetrical meshes esp ecially when G is

dened as the extended neighb orho o d a term that will

 

e e



P P P P b e explained later in this section

Likein the continuous case the normal to S at v is an



are the principal curvature at P There where eigenvector of M As for the principal directions at v

v

P P

fore in the continuous case the principal curvatures can T denote an arbitrary tangent to S at v T and T let



X



 

the principal directions at v and their asso ciated B sin cos

i i i

v v

v G

principal curvatures and let denote the angle b etween T

i

X

and T If is the angle b etween the tangent direction T 

 i i

C sin

i i

and T then by using and the orthogonalityof T



v G

i

and T we obtain that

at all no des In contrast to the continuous case where

X



T

the same set of linear equations in is needed to b e

M T T T T

v

i i i

v v

i

solved at each no de of the discrete case a dierent set of

v G

i

X

equations has to b e established by calculating A B and C

   

sin cos T cos

i i i i

v v

The principal curvatures are extracted by solving and

v G

i

at eachnode v

X

  

T sin cos sin cos



i i i i i

v v

C The C Algorithm

v G

i

The algorithm presented by X Chen and F Schmitt in

The principal direction T is an eigenvector of the matrix

is based up on a construction of a leastsquares problem



M only if sp ecic selections of tangent directions and the

at each vertex to be analyzed for its principal curvatures

v

weights given to these directions are made Practically

and directions

since T is unknown at this stage it is imp ossible to con

 N

trol these selections Furthermore the tangen t directions

are restricted by the sampling in case of real range data or

Q y the triangulation in case of an articially triangulated b ö

T P

mesh Real range images are rather dense and mostly ac

tal vertical or angular sampling in quired in xed horizon T

R n

als In such cases this obstacle do es not emerge thanks

terv è T

to the homogenous scattering of neighb oring no des around

T

each analyzed no de To deal with arbitrarily triangulated

but still not sparse meshes we addressed this discrete

Fig The C Algorithm illustrated on a p olyhedral mesh

case problem by taking homogeneously spread tangentdi

rections while using not only the nearest neighb ors of the

Consider Figure where P Q and R denote no des on a

analyzed no de but further neighb ors as well In our tests

p olyhedral approximation of the surface S Let T denote

this pro cedure has proven to b e eective except for extreme

an arbitrary tangenttoS at P and T denote the tangent

cases This extended neighborhood of the node is also used

which creates an angle with T According to equation

for indirect presmo othing to overcome noise interference

the Euler formula the directional curvature of S at P

as will b e discussed in Section IVA From nowonwe will

along the tangent direction T is given by

refer to no des contained in the extended neighb orho o d of v

as its neighbors By the same arguments and in the same



T cos sin

P

P P

conditions T is also an eigenvector of M

v

Now that wehave the principal directions at v T and



Using trigonometric identities in weget

T wealsohave the angle b etween any tangent direc

i

T A cos B cos sin C sin

P

tion T and the principal direction T As in the con tinuous

i 

case we will use and to obtain

Where

X



T T T

A cos sin

P P

e T M T T T T T

v

i i i

i



v G

i B cos sin

P P

X

   



cos sin cos

C sin cos

i i i i

v v

P P

v G

i

Thus by building or more instances of the linear equa



A B

v v

tion the unknowns A B and C can b e estimated using

X

a direct solution or by solving a leastsquares problem when

T T T

T T e T M T T T

v

  

i i i

 

i

more than equations exist The principal curvatures are

v G

i

given by

X

   

sin cos sin

i i i i

v v

p

A C B A C

v G

i



B C

v v

The principal directions are obtained by rotating T bythe

Where

angle

X



B

A cos

i i

arctan

A C

v G i

In order to build the set of equations the directional cur Our suggestion is to determine the directional curvatures

vatures at P have to be estimated The authors of by using the estimation in

suggested to estimate these curvatures byapproximating a

T

N Q P

curve which lies on the surface S while passing through P

T

P

kQ P k

and two of its neighb ors According to the Meusnier theo

rempresented in Section I I all curves lying on the surface

By this expression we estimate the normal section directly

S and share the same tangent at P also share the same

as a general quadratic curveand do not limit it to a cir

normal curvature at that p oint which is also the surfaces

cle yielding b etter results for all typ es of ob jects except

directional curvature at P along this tangent As shown in

for spheres where all cross sections are indeed circles At

Figure the no de P and two of its geometric neighb ors Q

the same time we are improving the complexity of the al

and R are used to nd a spatial circle passing through all

gorithm to a linear time algorithm

three of them the curvature of this curveat P can b e

P

easily obtained If n denotes the curve normal at P the

IV The Implementation

angle b etween n and the estimated surface normal N and

A Noise Interference

T is the tangent to the curveat P then from the Meusnier

In spite of improvements in D sampling equipment

theorem weknow that

noise in the acquired range image still has a crucial eect

T cos

P P

on the estimation of secondorder features such as principal

curvatures One wayto overcome noise is presmo othing

Thus from every dierentpairof P s neighb ors together

of the surface But an excessive direct smo othing might

with P an instance of can be generated Another

change the lo cal prop erties of the surface resulting in dis

consideration one should takeishowtocho ose the triplets

torted values for the estimated curvatures We therefore

In order to minimize noise eects it is b etter to cho ose P s

apply only minor and very lo cal presmo othing Another

neighb ors in suchaway that the angle between the curve

indirect smo othing is applied by turning not only to the

and surface normals at P will be small as p ossible This

nearest neighbors of a no de during the directional curva

strategy is implemented by picking pairs of no des whichare

ture estimation stage This approach has also has a dis

as close as p ossible to lie on opp osite geometric sides of P

torting eect but it is very minor for limited extensions

This implementation tries to guarantee that equation

of a no de surroundings as was studied in our tests We

is used within an interval where the cosine function slop e is

dene a top ological extended neighborhood of a no de by

mo derate and therefore reduces the eect of p ossible noisy

two parameters For eachnodev see Figure we dene

data on the estimation of the directional curvatures

a ring of initial radius r and width w measured in top o

logical neighb ors that surround v All no des contained in

D C algorithm modications

such an extended neighborhood are used to build M in the

v

Several problems and disadvantages emerge from the

T algorithm In the C algorithm they are used to gener

technique used in to estimate the directional curvatures

ate more instances of equation The values of r and w

First the approximation of a curve which lies on the

surface and passes through the analyzed no de as a cir

cular curve is simply not accurate enough resulting in

unreliable directional curvature estimations whenever the

normal section passing through the analyzed no de diers

from acircle This inaccuracy is even more crucial when

the pair of neighb ors of the analyzed no de are not within

its immediate neighb orho o d Another problem is regard

ing the attempt to approximate curves which are close to

normal sections bycho osing pairs of no des which lie on op

p osite sides geometrically of the analyzed no de It is very

p ossible that a pair of no des which lie on opp osite sides

of the analyzed point do not lie on the normal section

Furthermore since the lo cations of the no des are xed by

the sampling device in many cases even the b est selection

Fig The extended neighborhood of a node v No des within the

of a triplet of no des using the strategy presented by the

gray area are considered as v s neighb ors

authors of is do omed to generate curve whose normal

creates a large angle with the surface normal Thus in w r have to b e adjusted for optimal results according

many cases the attempt to minimize the eects of noise to the level of noise

through the cosine function in equation fails Finally Let us remember that the extended neighborhood also

the need to checkevery two neighb ors of an analyzed no de guarantee the correctness of the T algorithm in the general

in order to nd the b est pairs in the sense of lying on opp o discrete case Therefore regardless of noise in data the

site directions of the no de yields for the algorithm in extension is needed at areas where sampling of the surface

a quadratic time and space complexity is inhomogeneous

D Final implementation B Surface Normal Estimations

There are several known techniques to estimate the sur

Both algorithms have b een implemented and found to

face normals at discrete surface p oints All metho ds per

pro duce almost identical p erformance when synthetic sur

form some kind of averaging or compromising pro cess b e

faces with or without noise were analyzed The T al

tween dierent evaluations of the surface normal at the

gorithm yields b etter results in cases of real noisy range

analyzed no de Wehave found that for our purp oses real

images The reason is that the C algorithm is based on

range images where the data is dense and homogeneously

solving a leastsquares problem whichmakes it quite sensi

sampled there isnt much dierence b etween the metho ds

tive to noise When the noise can not b e neutralized eec

We preferred to take the normal as the weighted average of

tively enough by presmo othing likein the case of a real

all the normals of the planes whichcontain the no de and

noisy data where statistics of the noise are unavailable it

two of its adjacent neighb ors We use the angle created

leads to less accurate estimations On the other hand the

at the no de when connecting it with two of its adjacent

C algorithm can deal with any nonsymmetric sampling or

neighbors as weight

with any arbitrary triangulation in case of articial data

C The Reliability Estimator

V Experimental Results

Since noise in the sampled p oints is not homogenous the

Of all previous related works only the one in included

accuracy of the feature estimates is not homogenous either

evaluation of the algorithms p erformance with real noisy

Therefore it will b e very helpful for future applications to

range data In ve dierent metho ds to estimate princi

assign to each analyzed p oint an estimator to the reliability

pal curvatures were tested and none of them proved to yield

of its extracted features We use the estimated principal

sucient reliabilitywith real data Other previous works

curvatures and Darb oux frame at an analyzed p ointtoap

that wehave examined had only b een tested on nonnoisy

proximate the surface lo cally and then compare it to the

simulated data probably due to the sensitivity of the fea

lo cations of the p oints in the lo cal neighborhood captured

tures to noise Our suggested improvements yielded b etter

by the sampling device Smaller dierences between the

results in the general case with simulated data as well As

two suggest greater reliability of the extracted features

for the principal directions probably b ecause of diculty

Wewishtoevaluate the reliability of the features k k



in obtaining the ground truth non of the previous works

T T and N as were estimated at p oint P Let Q be a



had presented results in regard to principal directions

neighbor of P and T the pro jection of PQ on the tangent

Where it is p ossible to obtain the ground truth we stud

plane to the surface at P An approximation of the normal

ied the statistics of the error b etween estimated values and

section passing through P and in the direction of Q can b e

real values In other cases we examine the results qualita

obtained from equation

tively from illustrations of the features histograms and by

lo cally examining the feature values at key p oints on the

C sP Ts T Ns

P

surface An example for some of these illustrations can b e

found in Figures and One of our main to ols is the D

T is computed by using the extracted principal curva

P

histogram in whichthe values of the principal curvatures

ture in equation and s is the curve parameter Now we

are used as the histogram entries We term this histogram

examine how close is Q to the approximate normal section

as the principal curvatures histogram

Let C s denote the closest p ointonC to Q Tond s

 

we minimize the Euclidean distance b etween Q and C s

A Examining the Eects of Sampling Density and

The square of the error is given by

Smo othing

Q C s Q P Ts T Ns

P Using synthetic data we studied the eects of the sam

pling density the distorting eects of the presmo othing of

The optimal s is found by equating the derivativeofthis

data and of the extended neighborhood analysis in order

expression to zero Using the orthogonality of the Dar

to establish an optimal platform for real data analysis At

boux frame we obtain the condition for an extremum in

the same time we ob jectively tested the algorithms on noisy

the squared error expression

synthetic geometric primitives where the ground truth can

be obtained We added Gaussian noise in all co ordinates



T s Q P N T s Q P T

P P

with sd which is to p ercents of the sampling inter

val in our real data Wechecked the statistics of the error

For eachrealrootofthe last p olynomial we can calcu

between the estimated values and the values which were

late the distance of C s from Q and takethe one which

extracted analytically In principal curvatures the units

minimizes it to be s We rep eat this pro cedure for each



are cm In the vectors of the Darb oux frame the error

Q in G the immediate neighb orho o d of P The reliabil

i

is the angle between two vectors and therefore units are

ity is prop ortional to the inverse of the reliability factor as

real and radians denotes the mean dierence between

dened in

s

estimated values and denotes the standard deviation of

X

Q C s

i i 

this dierence

i

jGj

Q G

i

In Figure typical results of a test which examine the

a b

a

c d

Fig Synthetic torus a Maximal principal directions b Min

imal principal directions c Maximal principal curvatures brighter

means greater absolute value d Principal curvatures histogram

eect of sampling density is presented We can see that the

b

Fig Examining to ols for Darb oux frames demonstrated on a de formed cub e a Maximal principal directions b Minimal principal ó-ê ó-T

1 1 directions ó-N ì-ê 1 ì-T 1 ì-ê 2 ì-N

ó-T2 ó-ê ì-T

2 2

a b

Fig Synthetic cone in dierent densities mean and sd of

dierences a In principal curvatures b In Darb oux frame

denser the sampling is the smaller the standard deviations

are although b oth mean and sd of b oth curvatures are

a b

stabilizing very rapidly for very lowvalues that represent

even sparser sampling rates then those of atypical range

image We conclude at least for geometric primitives that

the algorithms yield very accurate results

In Figure we present statistics of typical test which

examined the distorting eect of the extended neighbor

hood It was conducted at a sampling density equal to the

density in our real tests which is ab out to sam

ples p er square cm We can see that errors in the principal

directions are low and stable for all the checked neighbor

hoods The principal curvatures b ecome quite distorted

c d

when we use neighb orho o ds with a radius of more than

Fig Examining to ols demonstrated on a deformed cub e a

to neighbors

Maximal principal curvatures b Minimal principal curvatures In

In another series of tests we checked what results

a and b brighter gray indicates greater absolute value c Relia

bility factor brighter areas indicate higher reliability d Principal

are obtained with noisy primitives when combining pre

curvatures histogram

smo othing with an extended neighborhood Typical results

are demonstrated on a synthetic noisy torus in Figure

By examining the statistics of the errors in all features we

p erformance we also tested them on dierent ó-T algorithms ó-ê 2

2

freeform surfaces are Laser Scanner Mo del mo del

ó-T The Cyb erw 1

ì-ê2 ed us to acquire all the D images used to p erform the ó-ê serv 1

ì-T2

real data tests All quantitative tests yielded similar re ì-ê

1 ì-T

Here we present results of only two such tests A

1 sults

sphere with a measured radius of cm was scanned

The surface was analyzed after three iterations of smo oth

a b

ing by a Gaussian lter with windowsizeof and with

Fig Synthetic torus in dierent neighb orho o d extensions mean

ring parameters of r and w The real value of

and SD of dierences a In principal curvatures b In principal

b oth curvatures should have b een With the C al

directions

gorithm we got a mean of and standard deviation of

0.06 0.015

With the T algorithm we got a mean of and

0.05

standard deviation of We also compared the statis

0.04 0.01

y factors to those obtained in synthetic

ì-N tics of the reliabilit

data These values were a little higher than those

0.03 noisy

1/cm. ó-ê Radians

2

thetic noisy data indicating lower reliabilitybut 0.02 0.005 ó-N of the syn ó-ê

1

alues were very low The range image and some 0.01 still the v ì-ê

1 ì-ê2

qualitative results are shown in Figure 0 0 0 1 2 3 4 5 0 1 2 3 4 5

Number of smoothing iterations Number of smoothing iterations

a b

x10 -3 0.015 3

2.5 ì ì-T2 0.01 2 ì-T 1 1.5

factor

Radians

0.005 1

Reliability ó ó-T2

ó-T 0.5 1

0 0 0 1 2 3 4 5 0 1 2 3 4 5

Number of smoothing iterations Number of smoothing iterations

c d

a b

Fig Synthetic noisy torus analyzed with a neighb orho o d radius

of in dierentnumb er of presmo othing iterations mean and sd

of dierences a In principal curvatures b In normals c In

principal directions d Reliability factor mean sd

can see the tradeo b etween the eect of the reduction of

noise and the surface deforming eect caused by smo oth

ing

From tests such as ones describ ed in this section we con

clude that bycombining extended neighb orho o ds with mi

c d

nor presmo othing of data the algorithms extract reliable

Fig Real sphere a Range image b Principal curvatures

estimations of the features under typical noise conditions

histogram c Principal curvatures brighter level indicates greater

Another b enet from these tests is that nowwe can estab

absolute value d Reliability factor brighter gray indicates b etter

lish an optimal platform to analyze real data whose sam

reliability

pling density and noise levels were simulated in these tests

As a conclusion in our tests on real data wecombine pre

A cylinder slightly tilted in the X Y plane with a mea

smo othing of no more than iterations of Gaussian lter

sured radius of cm was sampled The cylinder was also

of size and an extended neighb orho o d of radius that

analyzed using the same parameters as the sphere With

is no more than to neighbors

the T algorithm we got a mean of should havebeen

and sd of for the maximal principal curvature

B Real data

For the minimal one we got mean of which is exactly

the real value and sd of With the C algorithm Even though most of the ground truth in real data can

we obtained the following means and standard deviations not be obtained in order to have some quantitative es

and for the maximal principal curvature timates of the results we tested the algorithms on sim

and for the minimal one The results obtained for ple geometric primitives spheres and cylinders In these

real cylinders were a little b etter than those obtained for primitives part of the ground truth can b e obtained in a

real spheres This is due to the fact that the spheres have limited level of accuracyby measuring the physical dimen

more b oundaries which are almost parallel to the sampling sions of the primitives For a qualitative examination of the

direction and therefore more noise in their range images

Illustrations of some qualitative results are shown in Figure

a b

a b

c d

Fig Atoy mo del of a sheep a Range image b Maximal

principal curvatures c Minimal principal curvatures In b and c

brighter gray indicates greater absolute value d Reliability factor

brighter gray indicates b etter reliability

c d

Fig Real cylinder a Range image b Reliability factor

brighter gray indicates b etter reliability c Principal curvatures his

togram d Principal directions within an enlargement of p ortion of

the real cylinder mesh

From the quantitative results we can see that the accu

racy of the features estimation is not as go o d as in the syn

thetic noisy data case but still the means of the principal

Fig Atoy mo del of a sheep Principal curvatures histogram

curvatures are quite accurate The problem is the large

standard deviations Similar scattering around means of

the principal curvature values were obtained in all similar

of totally dierent depth These less accurate areas can

tests with primitives The cause is that noise in real data

be clearly seen in the illustration of the reliability factor

do es not have a Gaussian distribution and therefore is not

Other problematic areas can b e seen wherever a higher level

b eing smo othed eectively enough In a general freeform

of noise is noticed in the range image illustrations

ob ject where values of principal curvatures can not b e es

The only quantitative estimators that we still have in

timated by their mean value wecanavoid b eing misled by

the case of general freeform surfaces are the means and

the less accurate estimations by using the reliability fac

standard deviations of the reliability factors These statis

tor which is assigned to each estimated set of features If

tic values were higher than in the case of noisy synthetic

we lo ok at the reliability illustrations we can notice that

primitives but still low enough to indicate go o d accuracy

by the the problematic areas were identied and lo cated

in the over all estimations of the extracted features

reliability factor the darker areas

One additional asp ect that emerged here is the dier

Other series of tests were conducted with general free

ence between the curvatures histograms of two or more

form surfaces Qualitative results of a representative ex

similar in general shap e and size ob jects The issue of al

ample of such tests are illustrated in Figures and

most uniqueness of a principal curvatures histogram can

be exploited in a recognition pro cess of general freeform

By examining the illustration and the features quantita ob jects which is based on global analysis of the features

tivevalues at key p oints we can estimate the results We This is due to the fact that as long as the same areas of

found the results to b e reliable except at p oints whichlies an ob jects are scanned its curvatures histogram remains

along b oundaries that separate between two scene areas the same regardless of its orientation or p osition Full

or patrial comparison between a curvatures histogram of frame from real D range data and its application to the recovery

of primitives Masters thesis The Technion

a library ob ject and the histogram of asampled scene or

E Hameiri and I Shimshoni Using principal curvatures and

ob ject can b e the basis of such a recognition pro cess We

darb oux frame to recover D geometric primitives from range

images In Symp on D Data Processing Visualization Trans

did not however explore this asp ect in this work

mission

BKP Horn Computer Vision MIT Press Cambridge Mass

VI Discussion

AE Johnson and M Heb ert Using spin images for ecient

In this work wehave presented two previously published

ob ject recognition in cluttered D scenes IEEE Trans Patt

algorithms aimed to extract the Darb oux frame and prin

Anal Mach Intel l May

estimation P T Sander and S W Zucker Stable surface

cipal curvatures at eachpoint on a freeform surface We

In Proceedings Eighth International Conference on Pattern

have suggested subtle mo dications for these metho ds im

Recognition Paris France October IEEE Publ

CH pages

proving their results considerably when dealing with real

G Taubin Estimating the tensor of curvature of a surface from

range images and making them more generally applicable

a p olyhedral approximation In Proc Int Conf Comp Vision

for practical use

pages

B C Vemuri A Mitiche and J K Aggarwal CurvatureBased

Dierent asp ects of the mo died algorithms their limi

Representation of Ob jects from Range Data Image and Vision

tations and ways to overcome noise in the data were pre

Computing May

sented and discussed An estimator for a lo cal reliability

N Yokoya and M D Levine A hybrid approach to range image

segmentation In Ninth International Conference on Pattern

of the extracted lo cal features was develop ed

Recognition Rome Italy November pages

We have tested the suggested algorithms on synthetic

Washington DC Computer So ciety Press

noiseless and noisy data to estimate its correctness and

accuracy and to establish an optimal analyzing platform

for the analysis of real noisy data We then made a series of

tests to estimate the accuracy of the estimations with real

range images Our tests show that b oth algorithms pro duce

very accurate estimations when synthetic noiseless or noisy

range images were analyzed The results with real data

were less accurate when compared to the synthetic noisy

cases but they still were very reliable and accurate enough

for practical usage This was demonstrated in another work

of ours in which we used the estimated features to

recover D geometric primitives from real range images

We conclude that it is now p ossible to use the princi

pal curvatures and the Darb oux frame as a stage within

a vast variety of more complex applications But when

using these estimated values it is extremely imp ortantto

takeinto account that the accuracy of the results ma yvary

considerably Therefore they should b e incorp orated with

some lo cal estimation for their reliabilityaswas done with

our reliability factor

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