A Flexible Similarity Measurefor 3D Shapes Recognition

Antonio Adan, Member, IEEE, and Miguel Adan

Abstract—This paper is devoted to presenting a new strategy for 3D objects recognition using a flexible similarity measure based on

the recent Modeling Wave (MW) topology in spherical models. MW topology allows us to establish an n-connectivity relationship in 3D

objects modeling meshes. Using the complete object model, a study on considering different partial information of the model has been

carried out to recognize an object. For this, we have introduced a new feature called Cone-Curvature (CC), which originates from the

MW concept. CC gives an extended geometrical surroundings knowledge for every node of the mesh model and allows us to define a

robust and adaptable similarity measure between objects for a specific model database. The defined similarity metric has been

successfully tested in our lab using range data of a wide variety of 3D shapes. Finally, we show the applicability of our method

presenting experimentation for recognition on noise and occlusion conditions in complex scenes.

Index Terms—Computer vision, feature measurement, object recognition, similarity measures, pattern recognition.

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1 INTRODUCTION

1.1 3D Similarity Strategies

THE similarity concept is, in a general sense, an

ambiguous and relative concept for humans. This isdue to the fact that men have not got an extrinsic similarity

measure. Intuitively, we say that two objects are similar

through the external appearance or some local/global

features, but we can not measure how much the first object

looks like the second one. Nevertheless, we can always

maintain whether or not they are similar. Even for this case,

different results depending on the perception of each

person would be obtained. In this field, Santini and Jain[1] have carried out a study on similarity measures for

selecting images based on characteristics of human similar-

ity measurement. Anyway, the similarity is usually deter-

mined as a distance between some extracted features or a

vector that combines these [2].Similarity and recognition, in 3D environments, are two

words that frequently appear related in papers. In a general

sense, an object recognition system finds objects in the real

world from an image of the world, using object models

which are known a priori [3]. As we know, there is a wide

variety of approaches that propose different solutions to

recognize 3D objects. Most of them use representation

models based on geometrical features where the solution

depends on the feature matching procedure. Keeping this

area in mind, a short survey of recent works is presented in

the following paragraphs.

For free-form object recognition, several local and globalgeometric characteristics from original data or from thestructured/unstructured polyhedral meshes have beenused [4]. In Shum et al. [5] the correspondence betweentwo original shapes can be obtained by minimizing aspecific distance based on spherical representations. Doraiand Jain [6] use a local curvature feature called shape indexand define a disimilarity measure by comparing theirhistograms for several moments. Recently, Johnson andHebert [7] have been working with polygonal and regularmeshes to compare two objects through spin image concept.This parametric image is obtained by means of linearcorrelation coefficients at surface random points. In [8], [9](Yamany and Farag), a representation that stores surfacecurvature information from certain points produces images,called surface signatures, at these points. As a result of that, astandard Euclidean distance to match objects is presented.Chua and Jarvis code surroundings information at a pointof the surface through a feature called Point Signature [10].An interesting study based on several random distances iscarried out by Osada et al. [11]. They take probabilitydensity functions and cumulative distribution functions todefine disimilarity measures.

Strategies that consider several distances to recognizeobjects are not very usual. Nevertheless, Vandeborre et al.[12] obtain three distances by comparing local and globalfeatures histograms. In order to reduce them to only onedistance, they propose a distance ranking strategy. Lately,Adan et al. [13], propose a recognition process whereseveral distances, based on invariant global features,sequentially reduce the original model database until theobject is identified. Finally, a wide study about similaritymeasures focused on matching can be found in the work ofVeltkamp [14].

View-based object recognition strategies are also foundin the literature. Recently, Serratosa et al. [15] use afunction-described graph (FDG) that gives a compactrepresentation of a set of attributed graphs corresponding

IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 26, NO. 11, NOVEMBER 2004 1507

. A. Adan is with the Departamento de I.E.E. y Automatica, Universidad deCastilla La Mancha, 13071 Ciudad Real, Spain.E-mail: Antonio.Adan@uclm.es.

. M. Adan is with the Departamento de Matematica Aplicada, Universidadde Castilla La Mancha, 13071 Ciudad Real, Spain.E-mail: Miguel.Adan@uclm.es.

Manuscript received 31 July 2003; revised 24 Mar. 2004; accepted 1 Apr. 2004.Recommended for acceptance by R. Basri.For information on obtaining reprints of this article, please send e-mail to:tpami@computer.org, and reference IEEECS Log Number TPAMI-0201-0703.

0162-8828/04/$20.00 ß 2004 IEEE Published by the IEEE Computer Society

to several views of polyhedral objects. Then, the recognitionprocess can be accomplished by comparison between eachmodel and the graph of the unclassified object. In [16], aredundant representation of each object model in terms ofall possible transformation invariant coordinate frames ispresented. In this case, a hash table, which stores such arepresentation, is indexed during the recognition process byusing information invariant features derived from scenedata. Object recognition systems that match an inputsilhouette against a database of stored silhouette viewsare also well known. Lately, a silhouette-based objectrecognition system is evaluated in [17]. Aspect graphtechniques are also used for recognition. Cyr and Kimia[18] measure the similarity between two views of the3D object by a metric that measures the distance betweentheir corresponding 2D projected shapes. This allows themto group similar views into aspects and to represent eachaspect by a prototype. Liu et al. propose the DirectionalHistogram Model for objects that have been completelyreconstructed [19]. For each viewpoint, they obtain thehistogram of thickness distribution that is later formulatedwith spherical harmonics. This kind of representation isinvariant to scaling and rigid transformations but has thedisadvantage of not working with partial views.

One of the keys in this environment is the search forinvariant to scale, translation, and rotation. In previousworks, both local and global strategies can be found. In localapproaches, every point of the surface of the object hasknowledge of the features of its immediate neighbors.Geometrical features such as simplex angle [20], curvatureindex [21], and integral Gaussian curvature [22], [23]correspond to this kind of invariants. For mesh representa-tion, this means that a given node is handled as a single itemisolated from the complete mesh and without any relation-ship to the remaining nodes. On the contrary, globalapproaches consider features that define the whole shapeand are not correlated with the observer viewpoint. Thereare also studies that consider both local-global strategies [11].

1.2 Our Strategy

Adaptable strategies are rarely considered in literature. Inthis case, every node of the model has an extendedknowledge of its surroundings and all or part of it can beused for further tasks. The main difficulty in using thatidea, in a solid modeling environment, is to define a modelwith an appropriate topology and connectivity. Lately, wehave defined the Modeling Wave Set (MWS) on sphericalmesh models. MWS structure [24] has been designed tomaintain new and wider relationships among subsets ofmesh nodes.

Therefore, the difference between most strategies andours is that we are able to compare two objects taking anypart of information of the model. So, local, half, global,contiguous, or discontiguous spatial information could bechosen to recognize an object. In order to express thisconcept of adaptability to different specifications, we havelabeled our method as “flexible.” As it will be seenthroughout the paper, the best strategies correspond tothe cases where not local or isolated information butdistributed information around every point of the surfaceof the object is chosen. Consequently, an adaptable (or

flexible) similarity measure contrary to previous fixedsimilarity measures is defined in our case.

A few methods referenced in previous section are close

to ours. To make a deeper comparison with them it isnecessary to know our approach in detail. That is why we

have included this comparison in Section 5.We have structured the paper as follows: In Section 2,

MW model is briefly explained whereas the Cone-Curva-ture feature is widely discussed throughout Section 3.

Section 4 is devoted to defining and analysing the similaritymeasure and presenting the recognition algorithm. Section 5

compares our method to similar approaches. Similarityexperimentation is shown throughout Section 6. Applica-

tions to recognition, including performance on noise andocclusion conditions are shown in Section 7. Finally,

conclusions are dealt with in Section 8.

2 MWS REPRESENTATION MODEL

Since the present paper focuses on similarity and recogni-

tion, it is not possible to explain in detail our representationmodel. Complete information about this can be found in

[24]. Nevertheless, it is necessary to give a brief explanationto understand further concepts.

2.1 MWS Concept

Our solid representation model is defined on a mesh of

h nodes with 3-connectivity relationship from the tessella-tion of the unit sphere. Let TI be this initial spherical mesh

and TM the mesh fitted to the object. For building TM, TI isdeformed until it fits the object as follows: First, original

range image is normalized by applying a scaling factor tomake the object be bounded by the unit sphere. Then, the

nodes of TI are approximated to the normalized objectsurface. Finally, the deformed mesh is regularized (Fig. 1a).

Several deformation processes can be carried out through

an iterative strategy where each node of the mesh isexposed to a set of forces. Details of such methods can be

found in [20] and [25].Finally, several geometric features are extracted from TM

and mapped into TI. Therefore, TI is implemented as a data

structure for representing 3D objects in a simplified andnormalized manner. So, all the models have the same

number of nodes and equal mesh topology.After taking an initial node N of TI, a topological

structure called Modeling Wave (MW) organizes the rest of

the nodes of the mesh in disjointed subsets following a newrelationship. In this sense, a three-neighbor relationship is

just a kind of local topology. Each subset contains a groupof nodes spatially disposed over the sphere as a closed

quasi-circle, resulting in subsets that look like concentricrings on the sphere. Since this organization resembles the

shape of a wave, this has been called Modeling Wave (MW).Consequently each of the disjointed subsets is known as

Wave Front (WF) and the initial node N is called Focus. Of

course, MW structure remains in TM (Figs. 1b and 1c).Formally, MW is defined as follows:

Definition 1. Let us call Modeling Wave with Initial Focus N

a partition fF1;F2; . . . ;Fqg of TI verifying

1508 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 26, NO. 11, NOVEMBER 2004

IÞ F1¼fNg IIÞ Fjþ1¼VðFjÞÿ[

jÿ1

m¼1

Fm; 8 j : 1::qÿ 1; ð1Þ

where V is the neighborhood relationship of three connectivityimposed for TI, F

j is the jth WF generated in the process and itis contiguous with Fjþ1.

From the previous definition it can be deduced that anynode of TI may be Focus and, therefore, it can generate itsown MW. Therefore, h different MWs can be generated in amodel. Let us call all the possible MWs that can begenerated over TI Modeling Wave Set (MWS).

Consequently, several types of node relationships can bedefined in the model. They have been classified accordingto three different complexity levels: 1) Neighborhoodconnectedness: each node is 3-connected to its three standardneighbors. 2) Wave Front connectedness: each node ist-connected to t nodes (where t varies for each WF) by meansof the Wave Front structure. 3) Modeling Wave connected-ness: each node is h-connected to all the h nodes by means ofthe Modeling Wave structure.

2.2 Mapping on MWS

The last phase of model building consists of computing

discrete values of some three-dimensional object features

and storing them conveniently in the MWS structure. So, it

is possible to map local features of TM over TI considering

3-connectivity, like discrete curvature measurements ([26]),

color features, distance to the origin of coordinates, etc. But,

it is also possible to map global features, like the principal

directions of the solid or other features, based on the MW.Due to the multidimensionality of MWS, this representa-

tion gives us information about features from as many

viewpoints as the number of existing MWs. In fact, each

MW defines a feature exploration viewpoint or reciprocally,

from each viewpoint a features space, organized in Wave

Fronts, can be obtained. So, it is important to note that it is

possible to have a complete description of the 3D object

from many viewpoints in an integrated model. Of course, it

is not necessary to store the h subspaces of features because

they can be obtained just by changing the focus and going

through its MW (see Fig. 2).

AD�AAN AND AD�AAN: A FLEXIBLE SIMILARITY MEASURE FOR 3D SHAPES RECOGNITION 1509

Fig. 1. Example of modeling. (a) Mesh to solid fitting process. (b) MW drawn over TI. (c) TM with several WFs plotted on it and a detail of jthWF of N.

Fig. 2. (a) Generating of feature spaces on MWS. Each one can be associated with a specific Focus. (b) WFs printed over the mesh modelcorresponding to several MWs.

3 CONE-CURVATURE FEATURE

3.1 Definition

Although several kinds of features have been mapped intoTI in previous works [13], [26] in this case Cone-Curvature(CC) is defined as a new and intuitive feature based on theMW structure taking into account the location of the WFsinside the model TM.

Definition 2. Let N be Initial Focus on TM. We call jth ConeCurvature of N, the angle of the cone with vertex N whosesurface is fitted to the jthWave Front of theModeling Waveassociated to N. Formally:

�j¼signðF jÞ��

2ÿ

1

tj

X

tj

i¼1

ji

�

�

�

�

�

�

�

�

�

�

with ji ¼ffCjNNi; Ni 2 F j; ð2Þ

where tj is the number of nodes of F j and Cj is the barycentreof F j. The range of CC values is ½ÿ�=2; �=2�, being the signassigned taking into account the relative location of O, Cj, andN , where O is the origin of the coordinate system fixed to TM.Negative values are for concave zones, values next to zerocorrespond to flat areas, and positive values correspond toconvex zones. Fig. 3a illustrates this definition.

Note that a set of values f�1; �2; �3; . . . ; �qg gives anextended curvature information around N until the qth WF,where the word “curvature” has a nonlocal meaning. So, foreach node N, a set of q values could be used for exploring itssurroundings (see Fig. 3b).

On the other hand, coming back to the wholerepresentation in the sphere TI, it can be said that acorrespondences vector C¼fc1; c2; c3 . . . cqg where cj :TI!ÿ�=2; �=2½ � j¼1; . . . ; q, q being the number of Wave Frontconsidered, is established for all the nodes of TI. Fig. 3cillustrates the last concept. In this, every correspondencecj has been represented as a mapping of the jth CC overthe tessellated sphere TI where the size of the spotssymbolizes the values of CC.

Therefore, for two poses of an object a, vectors Cða0Þ andCða00Þ are related through a rotation of TI. So, anequivalence class CðaÞ is associated to the object a whereCða0Þ 2 CðaÞ and Cða00Þ 2 CðaÞ.

The whole Cone Curvature information is stored in aCC-matrix �, of hxq dimension; h being the number ofnodes of TI and q the number of CCs considered. The choiceof a particular row gives the vector f�1; �2; . . . ; �qg for aspecific Focus. Note that � is invariant, unless row

permutations, to changes in the pose of the object. So,models for the same object in different poses provide thesame CC-matrix with different h indexing.

3.2 Analyzing the Cone Curvature Concept

In order to complete and clarify in more detail the previoussection, several reflections about the CC concept areincluded as follows:

3.2.1 Robustness of CC to Noise

Fig. 4 plots a set of CCs (second to 18th CCs) for differentnodes of a mesh model. In Fig. 4a, the location of the nodescorrespond to different areas in the mesh. Note that theirCC distributions are very distant. Nevertheless, in Fig. 4b,where the nodes are in the same zone, the correspondingCC distributions are nearer, except in the second and thirdorder. Obviously, this is due to the fact that for the first WFsthe measure of CC is more sensitive to errors.

In an ideal case, CC is invariant to transformations (scale,translation, and rotation). In other words, there is atheoretical perfect CC correlation for the same point of theobject in two different poses. Unfortunately, there areseveral uncontrolled noise sources that reduce such acorrelation. Errors are mainly due to the 3D sensorresolution, the modeling process and the effects of meshdiscretization/resolution.

In order to quantify the effect of the noise on CC, weperformed an experiment to measure the correlationbetween the CC for the original and corrupted versions ofan object. For it, Gaussian noise was introduced to the

1510 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 26, NO. 11, NOVEMBER 2004

Fig. 3. (a) Definition of CCs. (b) Visualization of the CCs vector for a node N. (c) Visualization of the CCs vector mapping over the sphere TI. The size

of circle/rhombus represents the value of CC.

Fig. 4. CCs for scattered (a) and near (b) nodes. The X-axis and Y-axis

correspond to CC-order and CC value, respectively.

original position of the points of the model mesh. Severalvalues from low to high Signal to Noise Ratios (SNR) weretested and the corresponding correlation values werecomputed. The correlation coefficient for each SNR is theaverage of the correlation coefficients on all corruptedpoints. This experiment was repeated for 70 objects in ourlab on real conditions. Fig. 5a shows the correlationobtained for each object and for different SNR values. Astable performance of the correlation for all objects takencan be seen, having a mean standard deviation of 0,0141. Itis also evident that for usual SNR ratios (> 20 db) very goodresults are obtained. So, above 20db the correlation is higherthan 0.9, which demonstrates that CC representation isrobust enough when noise is added.

Fig. 5b shows the results of a study of the CCperformance to noise but from the CC-order viewpoint. Inthis case, the X-axis is the CC-order and the Y-axis is thecorresponding correlation coefficient. Different curves,corresponding to different SNR also appear on it. Severalinteresting comments can be extracted from this graphic,especially for curves corresponding to 20db, 26db, and34db. Note that all curves are increasing which means thatthe correlation increases as the order increases. On the otherhand, the correlation is always low for low orders whereasit is high for high orders. This result implies that the higherthe CC-order the less sensitive the CC is to noise.

The last conclusion agrees with the experiment in Fig. 4.Therefore, small variations in the location of the nodes inthe mesh involve high variations in first orders of CCwhereas for upper orders such variations do not mean-ingfully affect it. It can be said that an error filtering effectappears as the order increases. Keeping this in mind, localfeatures are not suitable enough for characterizing an objectunless errors are minimized. This idea will be consideredagain in the next experimentation sections.

3.2.2 The Spatial Continuity of CC

Spatial continuity of CC means that CC along any path ofnodes is a continuous function for any CC-order consid-ered. A visualization of this property, in the discrete sense,can be appreciated in Fig. 6a where the CC value is plottedover the mesh as a specific gray level. Since the CC is a

AD�AAN AND AD�AAN: A FLEXIBLE SIMILARITY MEASURE FOR 3D SHAPES RECOGNITION 1511

Fig. 5. Performance of CC to noise. (a) Correlation coefficient for several

SNR ratios in a database of 70 objects. (b) Correlation coefficient versus

CC-order.

Fig. 6. (a) Illustration of second, eighth, 14th, and 18th CCs. Note the discontinuity in second CC and the continuity for the rest. (b) CC models for

several orders. The continuity of CC for each patch of the mesh when the order grows can be appreciated. (c) Visual representation of the

f�1; �2; . . . ; �18g for the set of marked nodes. (d) Examples where the CC values increase or decrease in the sequence f�1; �2; . . . ; �18g.

value that corresponds to vertices instead of patches, thecolor is assigned to the patch through the CC’s average oftheir vertices. This is just made for visualization reasons.Several colored meshes corresponding to several CC ordersare plotted. At this point, it is necessary to remark on how aconsistent representation of the object implies low fre-quency components or color continuity in a mathematicalsense. In other words, high frequency implies that somekind of error is present in the 3D image. As it can be seen forlow orders (case c2), the sensitivity to noise is appreciatedagain, the discontinuity being evident whereas for upperorders (remaining cases) no high frequency components areappreciated. Of course, this could also happen if the surfaceof the object was highly irregular or sharp, but such kinds ofsurfaces have not been used in our database.

3.2.3 The Continuity on the CC-Order

Finally, there is another aspect that requires attention. It canbe said that there is also continuity in the CC vectorf�1; �2; . . . ; �qg. In Fig. 6b, a sequence of models for severalorders is presented. It can be appreciated that every patch ofthe mesh smoothly changes its grey level when j grows.This property can be shown better in Fig. 6c where a visualrepresentation of the CCs for a set of close nodes has beenplotted (remember that the color is assigned to the patchthrough the CC’s average of their vertices). At this point, itis important to note that it can be thought that CC vector isalways an increasing sequence but the CC value canincrease or decrease in the sequence depending on thekind of surface taken. So, the notion that the CC becomesmore regular as the depth increases in not true at all. Fig. 6dgives an example of that for another sets of nodes.

Finally, note that such a representation (i.e., sphericalrepresentation) is applicable only for genus 0 objects.Consequently, any cross-sectional curve for each depth(i.e., WF) in the mesh model will always be a simple closedcurve because of the Wave Front it is.

4 FROM SIMILARITY MEASURE TO RECOGNITION

4.1 Similarity Measure

The meaning of h and q dimensions of the CC matrix �

could be explained as follows: The choice of a particularrow is equivalent to selecting the Focus of the MW; in otherwords, where it begins to explore the object. The choice of qprovides a CC value corresponding to a specific depth fromthe Focus; the higher order the more depth (see Fig. 3).

Keeping this in mind, distance dj between two modelsTI

m and TIn is defined as:

djðTIm; TI

nÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

X

h

k¼1

ð�mðk;jÞ ÿ�n

ðk;jÞÞ2

v

u

u

t ð3Þ

where sorted distributions of jth CC are taken for bothmodels. Note that for each couple of models (m,n) adistance-vector fd1; d2; . . . ; dqg is obtained. So, it is possibleto analyze the suitability of considering near (low depth),half (medium depth) or far (high depth) informationaround the Focus.

The problem with summarizing such a distance-vector toa single value is a multiobjective optimization problem thatinvolves finding out a Pareto optimal solution. If thesolution is Pareto optimal there is not another solution thatgives better solutions to the objectives of the problem (see[27]). For that, we have taken two classical strategies.

First, we have considered a solution dA, where allvariables are equally meaningful. Such a solution is givenby Zeleny and Yu in Compromise Programming ([28], [29]).According to this, we obtain a distance (pseudometricdistance) where the norm of fd1; d2; . . . ; dqg is:

dAðTIm; TI

nÞ ¼X

q

j¼1

dj TIm; TI

nð Þ�

�

�

�

p

!1=p

: ð4Þ

If p ¼ 1, this expression is reduced to a Tchebycheffmetric:

dAðTIm; TI

nÞ ¼ max djðTIm; TI

n�

�

�

�; j ¼ 1; . . . ; q: ð5Þ

On the other hand, we have taken a second strategy thatweighs up the objectives through coefficients and mini-mizes the balanced sum (Zadeh [30]). If the solution isunique, then it is a Pareto optimal solution (see [27]). In ourcase, a distance dB is defined as:

dB TIm; TI

nð Þ ¼X

q

j¼1

wj � dj TIm; TI

nð Þ; ð6Þ

where wj represents the weighting of the jth distance dj.In practice, the weighting in dB has a clear meaning. For

example, for a vector fw1; w2; . . .wqg with low values forlow orders the high sensitivity to noise in the closedneighborhood of the Focus will be filtered. On the contrary,for low values of high orders, information far from the Focus(and, therefore, not meaningful) will be rejected. Moreover,in order to measure a distance based on depth, a weightingassignment a priori is always possible. The next section willbe devoted to analyzing the results after taking severalweighting strategies.

Once the distance d has been defined and considering amodel database where D is the maximum distance, a binaryrelationship through a membership function s : X �X !0; 1½ � is established as follows:

sðTIm; TI

nÞ ¼ 1ÿ dðTIm; TI

nÞ=D; ð7Þ

where d may be dA or dB without distinction (sA or sB,respectively). Obviously, sðTI

m; TInÞ 2 0; 1½ �.

As we know, a similarity relationship must satisfy thereflexive, symmetric, and triangle inequality properties.These properties can be formally defined as:

1Þ sðTIm; TI

mÞ ¼ 1

2Þ sðTIm; TI

nÞ ¼ sðTIn; TI

mÞ

3Þ sðTIm; TI

nÞ � maxk

max 0; sðTIm; TI

kÞþsðTIn; TI

kÞÿ1� �

:

ð8Þ

Reflexive and symmetrical properties are obviouslyverified in our case. The third property is known as Maxÿ� transitivity and it is also verified since it has been defined

1512 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 26, NO. 11, NOVEMBER 2004

through a pseudometric distance (Bezdek [31]). Such arelationship is called likeness relation. Unlike the crisp case,we assign each couple of objects a real value where valuesnear to zero imply low similarity and values near to onemean high similarity.

Finally, we have defined a Similarity Matrix S for adatabase as:

S ¼ fsmn =smn ¼ sðTIm; TI

nÞg; 8ðTIm; TI

nÞ 2 X �X: ð9Þ

Note that unlike others approaches [7] where a previousselection of points is carried out, in our method all nodes ofthe mesh are considered for defining the similarity measure.Moreover, there are not restrictions, like high or lowcurvature [8], [9], on the choice of the focus nodes.

4.2 Recognition

At this point, there are severalways to continue the presentedwork. Among them, we are interested in accomplishingrecognition tasks and carrying out clustering algorithms aswell. In this paper, we will deal with the first one.

The task of object recognition is easy when the similaritymeasure is efficient enough. This comes down to a mini-mization task. Since our method is based on 3D models, themodels of the same object registered in different poses arenot identical. This means that similarity measures for thesame object will be near to 1, but not necessarily 1.

For a sample a0 with model TIa0 , we define the

recognition of a0 in the data base of n objects as

Cða0Þ 2 CðaÞ , sðTIa0 ; TI

aÞ ¼ MinfsðTIa0 ; TI

iÞ; i ¼ 1; . . . ; ng

ð10Þ

As it has been said in Section 3.1, we recognize an objectwhen we assign Cða0Þ to the class CðaÞ.

5 COMPARISON WITH SIMILAR APPROACHES

The solution and strategy for 3D similarity/recognitionpresented here is close to some of the works cited inSection 1.1. Nowadays, three known approaches exist thatcould be considered closer to our work than others. Thesecorrespond to Chua and Jarvis with point signature [10],Jhonson and Hebert with spin image [7], and Yamany andFaray with surface signatures [8], [9]. After presenting ourmethod, it is a good time for making comparisons.

Point signature encodes information on a 3D contour of apoint P of the surface. The contour is obtained byintersecting the surface of the object with a sphere centeredon P. The information extracted consists of distances of thecontour to a reference plane fixed to it. So, a parametriccurve is computed for every P and it is called point signature.An index table, where each bin contains the list of indexeswhose minimum and maximum point signature values arecommon, is used for making correspondences.

The idea of obtaining information on a contour is alsoapplied in our case through the WF concept. Nevertheless,WFs are contours that are inserted in the mesh modelbecause the MW topological structure remains after thetessellated sphere is deformed and fitted to the surface ofthe object. Therefore, it is not necessary to compute thelocation of the contours on the model. Obviously, the kind

of feature handled by Chua and Jarvis is different to ours,but it is remarkable that only one contour, depending on theradius r of the sphere, is used for constructing a pointsignature. This shows that point signature as an isolated andlocal concept, depending on P and r. Since CC concept islinked to the MW one, CC offers wider information aboutthe surroundings of a point P from the first to last orderand, more important, this information is always available.Then, the user can select the CC order or the set of orders (inothers words, the contours) depending on the specificationsof its experiment and the available partial data. So, if wechoose jth CC, we know what it is exactly because it isinside the whole structure MW. Therefore, CC is not anisolated concept. This makes our representation morecompact.

Spin image representation encodes information not for acontour but for a region around a point P. Two geometricalvalues (�, �) are defined for the points of a region and a2D histogram, with � and � as coordinates, is finallyconstructed.

The first aspect to keep in mind is the sensitivity to errorsor noise. In this case the normal on P is used to compute �

and �, which implies that spin image is very sensitive tosources of error like variation in scanned points coordinatesor variation of location of the nodes in the mesh model.Theses variations produce variation on the normal surfaceand finally on the spin image. In our case, as it has beenproven in Sections 3.2 and will be shown in Sections 6 and7, low orders (second and third) are also sensitive to errorand give the worst results for characterizing objects. In fact,we have excluded them in recognition tasks. Fortunately,we can use higher CC orders where sensitivity to errors/noise is highly damped, to explore and recognize objects.

Although spin images are generated from models using apolygonal approximation to the surface, this kind ofrepresentation deals with more dense information thanours. In order to not have an excessive quantity ofinformation, the spin image’s dimensions are reduced. Thismeans that a local or short exploration around every pointis considered. In spite of that, the correspondence betweenscene and models is carried out for a random selection ofpoints. Therefore, another obvious difference with ourmethod is the extension of the region taken around eachpoint. Certainly, we have less local information but moreknowledge on an extensive region. Comparing the size ofspin image’s and CC, maintaining equal resolution ofmeshes. The ratio of stored information by point of themesh is m:n

q , (m;n) being the spin image dimensions and q thenumber of WFs used. For the whole model the storedinformation ratio by object is h:m:n

q , h being the number ofnodes of the mesh. Since CC is computed for each WF, wecan estimate that at least m:n > 10q (for example, a conecurvature vector for q ¼ 2; . . . ; 18 correspond to 640 nodes)and, finally, the ratio can be evaluated > 10h. This provesthat our method uses a lot less information to solvesimilarity/recognition problems.

Surface signature representation has several commonpoints with spin image. In this case, surface curvatureinformation is extracted to produce 2D images called surface

AD�AAN AND AD�AAN: A FLEXIBLE SIMILARITY MEASURE FOR 3D SHAPES RECOGNITION 1513

signature where 2D coordinates correspond to othergeometrical parameters related to local curvature.

One of the aspects that may be added in this analysis isthat this kind of representation has been focused onmatchingin 3D scenes more than on accomplishing 3D shapessimilarity studies whereas our method is more focused onthe latter. In fact, in this method, the points with lowcurvature are eliminated and surface signature is onlygenerated for the rest of the points. Consequently, surfacesignature models can be considered as incomplete. This ideais not applied to our models because the curvature vector iscomputed for all points of the mesh. In this sense, theCC model is complete.

The last point implies that this representation hasimportant 3D shape restrictions. So, it can not be used forpolyhedral objects and objects with flat or smooth surfaces.Our representation works well for polyhedral and 3D free-shapes but, of course, it also has other shape restrictions.For example, we can not model objects with genus differentto zero (with holes) and objects with sharp surfaces.

6 SIMILARITY EXPERIMENTATION

We have evaluated both similarity measures sA and sBcorresponding to the euclidean dA and weighted dBdistances, respectively. As it has been said in Section 4.1,distance dB weighs up the CCs values depending on thedepth level over the MW structure. So, different combina-tions with respect to different criteria could be chosen forexploring the object (model) and testing how well ourapproach works. For that, we have considered a wide rangeof criteria which are shown in Table 1. Explorations for onlyone WF correspond to B1 and B6 cases (for low levels), B2

(for medium level), and B7, B8 for (high levels). Explora-tions through several WFs correspond to the rest of thecases: contiguous wave fronts and near to the focus in B3 andnot contiguous Wave Fronts for B4 and B5. Next to Table 1, avisual representation of the CC orders considered for everycase has been included.

The experimentation has been accomplished with a realdatabase consisting of 70 3D objects of genus zero. Sharpsurfaces or surfaces with holes have been excluded in ourdatabase. Fig. 7a shows the database used in this work.3D information is given by a Gray Range Finder sensor thatprovides 640 x 480 points in the 3D space ((x,y,z)

coordinates). We have worked with low-medium resolutionmeshes of 1,280 nodes and around 40 WF’s for the wholemesh. Nevertheless, for computing the similarity measure,only the first 18 Cone-Curvatures, corresponding torespective WFs, have been considered. This means that,when the first node (Focus) is fixed, around half of themodel is used to compare objects, the other half being usedwhen the mesh is explored from its antipode node. In fact,the last values of CC (near the antipode focus) may varywidely and such information is not meaningful. That is whythis information has not been used for recognition tasks.

In Fig. 7b, we present the Similarity Matrix S for A2 (0 =black, 1 = white) and Fig. 7c shows sorted values for allrows of S and several representative objects in our databasehave been marked on it. Note that these curves giveinformation on whether the object is more or less equal/different to the rest of the objects in the database. Forexample, objects No. 35, 66, 50, 10 are relatively morecommon than objects No. 8, 28, 40, 2.

A similarity distribution function gives clearer informa-tion about the properties of the Similarity Matrices. That iswhy we have computed the similarity distribution functionsfor all the cases ðA1;A2; . . . ;B8Þ. Fig. 8a shows them. Notethat, all cases can be considered as similar gaussiandistributions, where only the distribution for the B1 caseappears different from the rest. Table 2 presents the meansimilarity and the standard deviation for each case. Uni-formity of these values is evident, 0.6905 being the finalaverage similarity with a standard deviation of 0.0239(3.45 percent). So it can be said that, for the cases consideredin Table 1, globally, the database has a stable similaritymean. Theoretically, the B1 case, which takes only the secondorder, gives a little more discrimination, but the dispersionaround themean is also bigger than the rest. The influence ofthe high variation of second CC to noise is the cause of thatperformance. In this sense, Fig. 8b shows the mean similarityin B1 (second CC) and B4 (third, sixth, ninth, 12th, 15th, 18thCCs) for 70 objects of the database. Comparing both graphicswe can appreciate that the results given in the first case aremore random than in the second one.

Thinking about the recognition problem, we are inter-ested in obtaining conclusions after exploring the similaritymatrix for values close to 1. Consequently, we havecomputed the percentages (with respect to the total numberof objects of the database) for high values of similarity.

1514 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 26, NO. 11, NOVEMBER 2004

TABLE 1Distances and Cases Tested

These quantities are obtained as follows. After setting s, wecompute for each object the percentage of the number ofobjects that have similarity higher than s. The final value isthe average values for all the objects. Fig. 8c plots theexpected percentages for similarities higher than 0.8 (�),0.85(o), 0.90(+), and 0.95(*). It can be seen that very lowpercentages (< 5 percent) are expected for similaritieshigher than 0.9. This means that, in our metric anddatabase, theoretically very few objects are expected to bevery similar. Taking into account these results we concludethat CC allows us to discriminate enough objects in thedatabase.

7 APPLICATIONS FOR 3D RECOGNITION

In order to prove the efficiency of our approach inrecognition tasks, we have performed a complete studyon its performance for isolated objects, in noisy environ-ments and complex scenes.

7.1 Recognition Test for Isolated Objects

First, we pretend to analyse the performance of the similaritymeasures for isolated objects. In this experiment, we haveobtained four models for each object corresponding to four

different poses sensed (see Fig. 9). So, we have worked withfour sets of 70 models which will be called G1, G2, G3, andG4. Six pairs of groups have been chosen for experimenta-tion and, for each pair, every object of the first component issearched in the database of the second component. There-fore, we have tested our method for 420 cases.

Table 3 summarizes the results of such experimentation

where three success averages for the first, second, and third

candidates are shown for each couple of groups. In general,

themethodworkswell, but the results arebetter in somecases

than in others. Themost satisfactory results correspond toA1,

A2, B3, B4, and B5 cases where no local but CC distributed

information has been considered (see Table 1). So, averages

for the first candidate are A1-97:8 percent, A2-97:5 percent,

B3-96:2 percent, B4-97:8 percent, and B5-96 percent.For cases with only local information (information of only

oneWF), smaller percentages are obtained for B2-94 percent,B6-92:3 percent, and B7-93:3 percent. Finally, when thelocal information taken corresponds to the WFs extreme(j ¼ 2 and j ¼ 18), the success average as first candidate ismeaningfully worse (B1-55:8 percent and B8-87:5 percent).

Consequently, we can conclude that using first orders of

CC exclusively, worse results are expected because their

AD�AAN AND AD�AAN: A FLEXIBLE SIMILARITY MEASURE FOR 3D SHAPES RECOGNITION 1515

Fig. 7. (a) Polyhedral and free-form objects belonging to the database. (b) Similarity Matrix. (c) Sorted similarity functions for all objects. Numbers 35,

66, 50, 10 (up) and 8, 28, 40, 2 (down) are marked.

1516 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 26, NO. 11, NOVEMBER 2004

Fig. 8. (a) Similarity distribution functions. (b) Mean similarity for all objects corresponding to B1 and B4. (c) Percentages of objects that have

similarity higher than 0.8 (�), 0.85(o), .0.90(+), and 0.95(*).

TABLE 2Mean Similarity and Standard Deviation for Each Case

Fig. 9. Examples of objects in different poses.

TABLE 3Recognition Results for Isolated Objects

corresponding CC values are highly sensitive to noise. So,

these CC values could change with small variations in the

locations of the nodes from one model to another. On the

other hand, high orders of CC do not provide meaningful

information of the object which is also sensitive when the

pose of the object changes. On the contrary, the best results

correspond to the cases where not low-depth or isolated

information but medium-depth or distributed information

around every node of the mesh is chosen. This means that,

using the object model and following different criteria, a

wide variety of parts of the object (or depth levels) can be

chosen in order to recognize an object. This property can

obviously be useful for performing further recognition

approaches in partial and occluded scenes as will be

discussed in Section 7.3.To make a deeper evaluation of these recognition values,

it is essential to take into account the kind of database

handled. Looking at Fig. 7a, it is clear that groups of objects

exist that look quite similar according to the measure s and

also according to a human or geometric sense. For example,

cube shapes (35, 36, 37, 38, etc.), prism shapes (10, 18, 19),

cylinder shapes (62, 63, 64, etc.), and round shapes (41, 42,

43, etc.) appear on it. Table 4 presents, for a set of

representative objects, their three most similar objects. This

gives visual information of how difficult the recognition of

each object is theoretically. Therefore the recognition for

objects belonging to such groups is expected to be worse

(objects 35, 62, 51, 14, 47) than for free shape ones (objects

29, 21, 25, 8, 12, 4, 6, 2). Obviously, this question has an

influence on our experimentation by decreasing the average

recognition ratios. Consequently, we think that averages

higher than 90 percent are good enough in our database.

Finally, it is important to point out that not all the shapes

that apparently look similar are also similar according to

our similarity measure s and on the contrary. For example,

objects 62 and 66 that could be labeled as cylinders are very

distant for us.

7.2 Recognition with Noise Added

Similarity measure robustness to noisy on unorganized datahas been tested in a new experiment. In this, gaussian noiseat different SNR is injected in the x, y, and z components ofthe points given by the sensor. Then, the previous recogni-tion experiment (Section 7.1) is repeated again. So, ourrecognition approach is evaluated in noisy environments.

The results of the experiment are summarized in two

graphics in Fig. 10. In both, the X-axis is the Signal to Noise

Ratio and the Y-axis is the success percentage for the first

candidate (Fig. 10a) or for first, second and third candidates

(Fig. 10b). In these graphics, the curves corresponding to the

cases defined in Table 1 appear (different combinations of

CC-orders). So, we can evaluate and compare the perfor-

mance of the recognition method in noisy environments for

each case. From now on, we will make several comments

taking into account Fig. 10b.

Obviously, the results for recognition improve when

SNR increase. Note that, at around 20 db, there are

percentages above 90 percent in several cases (A1, B4, B5,

B7, and B8) whereas for more than 25 db only cases B1, B6,

and A1, give poor results (under 40 percent). Keeping in

mind the election chosen for B1 (only second order) and B6

(only fourth order), it is evident that worst results occur

again when low and isolated orders are chosen. In the case

of A1, where all orders are considered, the influence of

lower orders is also evident. On the contrary, in B4 (third,

AD�AAN AND AD�AAN: A FLEXIBLE SIMILARITY MEASURE FOR 3D SHAPES RECOGNITION 1517

TABLE 4The Three Most Similar for a Set of Objects

Fig. 10. Recognition versus SNR. Success percentage for (a) the first candidate and (b) first to third candidates.

sixth, ninth, 15th, 18th), B5 (fifth, 10th, 15th orders), B7 (10th

order), and B8 (18th order) where sparse or isolated-high

orders are taken, very good results are obtained.In summary, we can conclude that the recognition

method is robust enough to noise when the CC vector

chosen does not include low orders. As more low-orders

were included the worse were the results obtained.

7.3 Extension to Cope with Partial Objects

Although the main objective of this paper is to present and

analyze the representation based on CC as a suitable 3D

similarity measure, this section presents an extension of it

showing its effectiveness in practical scenes including

partial objects with occlusion and cluttering.Nowadays, it is widely accepted that to deal with a

single view in real complex scenes is a hard and difficult

task that frequently implies that previous processes or steps

must be solved before accomplishing recognition. This is

our case, where 3D segmentation and partial modeling

steps must be carried out before solving higher-level tasks.

Therefore, we will present just a minimum version of this

point focusing it on similarity when part of the information

is available. It is certain that this point could be the topic of

a second paper. Anyway, information on the segmentation

procedure and partial model definition can be found in [32]

and [33].When part of the surface of an object is available, we just

adapt our method taking the set of complete Wave Fronts

that appear in the partial model corresponding to the partial

view. Consequently the metric dB given in (6) is used to deal

with partial information. Note that cases B1 to B8 were

defined in Section 5 taking into account different criteria

about contiguous/not-contiguous, isolated/disperse, close/

far information around the focus. Now, it is time to define

what kind of strategy can be used depending on the partial

data given by the sensor. For example if, after setting a

focus, 2, 3, 4, 5, 6, and 7 WFs were complete, we could

choose B1, B2, and B3 or any other combination of WFs to

carry out the matching with the complete model.In fact, we have already analyzed the performance of our

method in such cases in previous sections. Nevertheless, in

this case, where the object can be partially occluded, we

have less information than in a just partial model extracted

from the complete model. As a consequence of that, the

recognition is expected to be more ambiguous, which isobvious in the field of recognition with partial objects.

Fig. 11a presents a complex scene with occlusion andclutter where the cloud of points of the surface of the sceneis given by the sensor. After exploring the 3D data, adistributed segmentation technique separates their consti-tuent parts [33]. Figs. 11a and 11b show the 3D pointscorresponding to each segment and identify them in theintensity image.

After that, we can independently work with eachsegmented part. For each 3D segment we build thespherical partial model, extract the foci and the completeWFs for each one of them. Among them, we establish a setof CC orders, �, and a set of foci, ÿ, of the partial model.Finally, ÿ is matched in the database of complete models. Inessence, this matching process on a spherical model isequivalent to matching spheres under rotation R. Conse-quently, (3) and (6) are updated for ÿ and �, as follows:

djðTIm; TI

nðRÞÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

X

k2ÿ

ð�mðk;jÞ ÿ�n

Rðk;jÞÞ2

s

; ð11Þ

dB TIm; TI

nðRÞð Þ ¼X

j2�

dj TIm; TI

nðRÞð Þ; ð12Þ

where TIm is the partial spherical model and TI

nðRÞ is acomplete spherical model rotated. R can be computedthrough correspondences node to node in both spheres butin practice, most cases are rejected after imposing athreshold on dj. So, among possible matchings, dB isevaluated only for very few cases. Table 5 shows thematching success percentage versus the correspondencesreduction percentage in this phase. As it can be seen,15 percent of the nodes of the complete models TI

n can beconsidered as candidates to be matched with the foci of thepartial model TI

m. Finally, (12) is applied in order to findthe best theoretical candidates.

1518 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 26, NO. 11, NOVEMBER 2004

Fig. 11. Example of segmentation of a complex scene.

TABLE 5Matching Success versus Reduction

Fig. 12 shows the procedure for one segment marked inthe scene of Fig. 11. Fig. 12a illustrates the partial rangedata, the part of the sphere taken for modeling and thepartial model obtained. In Fig. 12b, we present two of thepossible sets of foci that could be chosen for matchingdepending on the WFs considered. So, ÿ1 are plotted on themodel for �1 ¼ f5; 6; 7; 8; 9; 10; 11; 12g and ÿ2 for �2 ¼ f4g.For the first one, the WFs corresponding to each focus of ÿare printed over the mesh model in Fig. 12c. Obviously, asmore elements � had, the smaller ÿ is. In our case andtaking into account the conclusions given in Section 6,Figs. 12b, and 12c, lowest orders (second and thrid) arerejected in practice, but no more restrictions have beenimposed for choosing ÿ. This allows us to evaluate what thebest election is depending on the partial data. Finally, thematching of the partial data with the best candidates of thedatabase is presented in part Fig. 12d. Obviously, severalcorrect matchings can occur due to the scarce informationavailable in the partial view.

8 SUMMARY AND CONCLUSIONS

This paper is devoted to presenting a new strategy forobjects recognition based on a flexible similarity measurewhich allows us to compare two 3D shapes taking differentparts of the object.

The method is implemented building mesh models thathave got a Modeling Wave topological organization. MWestablishes a n-connectivity relationship in 3D objectmodeling meshes. Through the new feature called Cone-Curvature, which originates from the MW concept, extendedgeometrical information around every node can be handled

in a structured manner. So, for each depth level, CC feature

is defined and, therefore, each node has got an associated

CC vector.Keeping this concept in mind, both Euclidean and

weighted distances have been considered to define a

similarity measure between models for a specific database.

Reflexive, symmetrical, and triangle inequality properties

are verified in this relationship and, therefore, can be

considered formally as a similarity measure. Finally, the

recognition problem is easily solved minimizing such a

measure.A wide experimental test of our similarity measure has

been accomplished in our lab using real range images. In

order to prove the efficiency of our approach in recognition

tasks, a study on its performance for isolated objects, noisy

environments, and complex scenes has also been presented.

Therefore, major problems like partial matching and

recognition with presence of occlusion can be solved

through our method. After that, we have concluded that

the best results correspond to the cases where not low-

depth or isolated information but medium-depth or dis-

tributed information around every node of the mesh are

chosen. In fact, excellent success percentages have been

obtained in those cases.Finally, a comparison of our method to closer approaches

completes the manuscript. Taking into consideration the

results presented, we think that this method could be used

for future research on 3D computer vision. For example,

adaptation to multiresolution models and clustering are

two topics that we are currently exploring.

AD�AAN AND AD�AAN: A FLEXIBLE SIMILARITY MEASURE FOR 3D SHAPES RECOGNITION 1519

Fig. 12. (a) 3D partial data extracted of the scene (I), partial sphere chosen for modeling (II), its corresponding partial model (III). (b) Set of foci ÿ1 for

�1 ¼ f5; 6; 7; 8; 9; 10; 11; 12g and ÿ2 for �2 ¼ f4g. (c) WFs for each focus of ÿ1. (d) Several matching possibilities.

ACKNOWLEDGMENTS

This research has been supported by the CICYT Spanish

project PDI2002-03999-C02. Thanks to the Associate Editor

and the anonymous reviewers for their valuable comments

and suggestions.

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Antonio Adan received the MSc degree inphysics from both the Universidad Complutenseof Madrid and Universidad Nacional de Educa-cion a Distancia (UNED), Spain, in 1983 and1990, respectively. He received the PhD degreewith honors in industrial engineering from theUNED. Since 1990, he has been an associateprofessor of Castilla La Mancha University(UCLM) where is leader of computer visionprojects in the Ingenierıa de Sistemas y Auto-

matica Group (ISA-UCLM). His research interests are in patternrecognition, 3D object representation, 3D segmentation, 3D sensors,and robot interaction on complex scenes. During this time he has mademore than 40 international technical contributions on prestigious journals(CVIU, IVC, PR) and conferences/workshops (LNCS, ICIP, ICRA, ICPR,etc.). He was awarded the 28th Annual Pattern Recognition SocietyAward corresponding to the best paper in 2001. He is currently on theeditorial board of Pattern Recognition Journal and he is member ofIEEE, IEEE Computer Society, IEEE Signal Processing Society, andIEEE Robotics and Automation Society.

Miguel Adan received the MSc degree inmathematics from the Universidad Complutenseof Madrid, Spain, in 1979 and the PhD degree inindustrial engineering from the Applied Mathe-matics Department in the U.N.E.D. (Madrid,Spain), in 2001. He has been working asgrammar-school teacher from 1980. Since1995, he has been an assistant professor ofCastilla La Mancha University (UCLM). He hasworked on nonconvex multiobjective optimiza-

tion from 1995 to 2002 and, currently, his research interests are incomputer vision topics: shape similarity, clustering, and patternrecognition. He has published several works in optimization journals(JOTA, AML, EJOR, etc.) and lately in computer vision conferences. Heis researcher in computer vision projects of the Ingenierıa de Sistemas yAutomatica Group (ISA-UCLM).

. For more information on this or any other computing topic,please visit our Digital Library at www.computer.org/publications/dlib.

1520 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 26, NO. 11, NOVEMBER 2004