3D wavelet-based multiresolution object representation

*Corresponding author. Tel.: #34-91-664-7442; fax: #34-91-664-7490.E-mail addresses: [email protected] (L. Pastor),

arodri@dtf.".upm.es (A. RodrmHguez), jespa@dtf.".upm.es(J.M. Espadero), [email protected] (L. RincoH n).

Pattern Recognition 34 (2001) 2497}2513

3D wavelet-based multiresolution object representation

Luis Pastor��*, Angel RodrmHguez�, J. Miguel Espadero�, Luis RincoH n�

�Department of Ciencias Experimentales y Tecnolo& gicas, Universidad Rey Juan Carlos, Mo& stoles, Spain�Department of Tecnologn&a Foto& nica, Universidad Polite&cnica de Madrid, Boadilla del Monte, Spain

Received 17 March 2000; accepted 25 September 2000

Abstract

This paper presents a technique for computing multiresolution shape models of 3D objects acquired as clouds of 3Dpoints. The procedure is fully automated and is able to compute approximations for any object, overcoming samplingirregularity if present (sampling irregularity is a common feature of most 3D acquisition techniques; a typical example isstereo vision). The method described here starts by computing an intermediate mesh that meets the subdivisionconnectivity requirement needed to allow the computation of the wavelet transform. The mesh is then adjusted to the 3Dinput data using an iterative deformation process. Finally, a spherical wavelet transform is computed to obtain theobject's 3D multiresolution model. This paper shows a number of real objects acquired with di!erent techniques,including hand-held 3D digitizers. The paper also gives some examples of how multiresolution representations can beused in tasks such as acquisition noise "ltering, mesh simpli"cation and shape labelling. � 2001 Pattern RecognitionSociety. Published by Elsevier Science Ltd. All rights reserved.

Keywords: 3D Modeling; Multiresolution representations; Wavelet-based representations; 3D vision, Automatic shape extraction

1. Introduction

Representing and recognizing objects are two of themain goals of computer vision systems, and are essentialstages for understanding the environment's structure.Object representation or modeling aims at creating thor-ough descriptions of the entities that integrate the per-ceived scene. Sometimes modeling is an objective in itself,as in reverse engineering, where data acquired from realobjects is used as an input to a CAD system. Usually, incomputer vision, computing the representation of anitem is an intermediate stage of the vision system, yield-ing results used by other processes that perform moreabstract operations on the data acquired from the sceneobjects. An illustrative example is object recognition, an

important application itself, and which may also be a keystage in many areas such as robotics, quality control,tracking, etc.

Interest in object representation can be tracked to thebeginnings of computer vision research [1]. Early workconcentrated mainly on representing 2D entities. Never-theless, the development of 3D sensors and sensing tech-niques has stimulated the interest in representing 3Dobjects. The large amount of information involved andthe complexity and speed requirements of the processingtechniques demand the development of e$cient andpowerful object representation methods. Many of the 3Dsensing techniques (particularly in computer vision) pro-vide range data without much structural information,which makes the modeling problem more challenging.

3D geometric models are composed of a large numberof primitive elements, particularly when accuracy is a re-quirement. Model complexity strongly a!ects either thehardware needed for a particular application, the kindsof operations that the system can perform or the max-imum complexity of the objects that can be consideredwhile keeping processing time reasonable. In fact, thisis a situation which is often found; typically, image

0031-3203/01/$20.00 � 2001 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved.PII: S 0 0 3 1 - 3 2 0 3 ( 0 0 ) 0 0 1 7 0 - 9

�Sometimes referred to as tiles or patches.

processing and computer vision applications combinelarge data sets and small processing times. It is wellknown that starting by processing reduced resolutionversions of the input image may result in large computa-tional savings. There are also speci"c tasks, such as edgedetection or texture classi"cation, that bene"t from ana-lyzing input data at di!erent scales. It is therefore notsurprising that hierarchical and multiresolution tech-niques have always raised interest in the computer visioncommunity [2,3].

The development of the wavelet transforms theory hasspurred new interest in multiresolution methods, and hasprovided a more rigorous mathematical framework.Wavelets give the possibility of computing compact rep-resentations of functions or data. Additionally, they allowvariable degrees of detail or resolution to be achieved, andthey are attractive from the computational point of view[4}6]. All these features make them appear as an interest-ing tool to be used for representing 3D objects.

Several 2D computer vision applications usingwavelets have recently been developed [7,8]. Also, some3D wavelet-based object modeling techniques have ap-peared lately in the computer graphics environment[9,10] Nevertheless, similar methods have not been usedin 3D computer vision environments, possibly for twomain reasons:

� Wavelet representations are not translation invariant.This is a signi"cant problem, and di!erent approacheshave been described in the literature to cope with it[11,12].

� The sensors used in 3D vision provide data in a waywhich is di$cult to analyze with standard waveletdecompositions: many 3D sensing techniques providesparse measurements which are irregularly spread overthe object's external surface. This is also important,because sampling irregularity prevents the straightfor-ward extension of 1D or 2D wavelet techniques.

Even though the lack of translation invariance is animportant drawback, we believe that multiresolution ob-ject representations have a bright future in 3D computervision for several reasons, such as:

� Bottom-up scene analysis methods essentially attemptto create hierarchical symbolic representations.Wavelets are excellent for creating hierarchical geo-metric representations, which can be useful in theimage data analysis process.

� Going to 3D implies an important increase in com-plexity. Wavelet decompositions can provide alterna-tive domains in which many operations can beperformed e!ectively.

This paper presents a fully automated method forcomputing spherical wavelet-based multiresolution

descriptions of objects acquired as clouds of 3D points.Two important features of the method are that it doesnot impose any restriction on the input data, which canbe as sparse and irregularly sampled as desired, and thatit is able to establish structural relationships betweendata points. Within this paper, Section 2 gives an over-view of object representation techniques in computervision. Section 3 describes spherical wavelets. Section4 describes a method to overcome the samples' possiblelack of neighborhood relationships as well as the samp-ling irregularity. Section 5 presents some multiresolutionrepresentations computed from real 3D data. Section6 explores some application areas, such as digitizationnoise removal, mesh simpli"cation and global shape ex-traction. Last, Section 7 summarizes the paper's mainconclusions.

2. Representing 3D objects

Depending on the system's purpose, the objective ofthe modeling stage can either be to collect a set ofdescriptors characterizing the acquired object with re-spect to the system's universe of discourse, or to generatean exhaustive description that allows the object repres-entation to be manipulated afterwards.

It is generally accepted that a good modeling systemshould have the following properties [13,14]:

� Expressive richness.� Stability in presence of errors or noise in the input

data.� Ability to deal with occlusions, which implies some

sort of representation locality.� Capability to re#ect physical measures of the object's

shape.� E$ciency.

Many approaches have been followed to model 2D or 3Dobjects. For planar shapes, moments and Fourier de-scriptors have been used frequently [15]. For 3D sys-tems, the following approaches are classical:Volumetric representations: Spatial occupancy has been

one of the "rst techniques used for representing 3Dobjects [16]. The main drawback of these techniques isthat they are highly ine$cient in the cases where only theobjects' external appearence is relevant, since the ratiobetween `occupieda and `emptya cells is very low. Never-theless, some novel developments using this approachhave appeared recently [17,18].Facet-based meshes: The surface of the objects is de-

scribed by a set of facets,� edges and vertices, where the

2498 L. Pastor et al. / Pattern Recognition 34 (2001) 2497}2513

�Bidirectional re#ectance distribution functions.�A mesh meets the subdivision connectivity condition if it can

be generated from a base mesh through a 1}4 subdivisionprocess.

vertices correspond to the points of the original descrip-tion. In general, the order of the polynomials represent-ing the facets varies in di!erent approaches [19,20]. Forexample, it is possible to "nd planar, cubic, bicubic,quadric or superquadric facets. Generally, methods usinghigher-order facets achieve better mesh compressionrates.Deformable models: This group of techniques use

meshes similar to those used by facet-based methods. Themain di!erence between both is how the mesh is com-puted: deformable models start generating a mesh bycovering a reference surface of variable degree, such asa tessellated sphere [21] or a hyperquadric [22]. Then,an iterative minimization process is applied in order toadjust the mesh's nodes to the 3D input data points,deforming the original reference surface's geometry.Most of the methods propose using similar deformationprocesses, taking the input shape's features into account,while trying to maintain some local or regular propertiesof the initial mesh in order to achieve a stable solution[23,24].Grammatical or structural models: These techniques

assume that it is possible to generate object descriptionsusing a hierarchy of geometric primitives at di!erentabstraction levels, mimicking some theories about objectrecognition processes in the human vision system[14,25,26]. The hierarchy's coarser levels can be used asindexes to accelerate discriminations between very di!er-ent objects, whereas the "ner levels can be used fordiscriminating between similar objects [27}29]. Severalmethods have been suggested following this approach,di!ering in the primitives used and the way to connectthem at the di!erent hierarchy's levels.Aspect graphs: Storing several views (2D projections) of

the 3D shapes connected by a graph to preserve relation-ships among them allows 3D object recognition to beperformed by matching the input 2D image with theavailable views [30,31].

3D Invariants: Extracting 3D invariants allows 3Dshapes to be described by feature vectors in a similar wayto classical 2D approaches [32}34]. Like some of theprevious methods, this approach is only adequate forrecognition purposes.

During the last years, some wavelet-based 3Dmodeling techniques have been applied to the computergraphics arena.

For example, Lounsbery et al. [35,36] present a newclass of wavelets, based on subdivision surfaces, stronglyextending the class of representable functions: whereasprevious two-dimensional methods were restricted tofunctions de"ned in ��, the subdivision schemes present-ed in Refs. [35,36] may be applied to functions de"ned oncompact surfaces of arbitrary topological type. Theypresent several applications of their work, includingsmooth level-of-detail control for graphics rendering,compression of geometric models and animation pre-

viewing. Following Lounsbery et al.'s work, SchroK derand Sweldens [9], have presented a mathematical frame-work for applying wavelet constructions for scalar func-tions de"ned on the sphere. They use the lifting scheme[37,38], to construct biorthogonal wavelets with customproperties, which they use in computer graphics environ-ments for applications such as compressing topographicdata and BDRF� [9], mapping synthetic textures overspherical surfaces [39], interactive mesh editing [40], etc.

Lounsbery et al.'s and SchroK der and Sweldens' contri-butions share a common drawback: they depart from 3Dmeshes with data uniformly distributed over the wholeobject's surface and which meets the criterion of subdivi-sion connectivity.� Eck et al. [41] propose a method forcomputing multiresolution representations of irregularlysampled meshes without subdivision connectivity, suchas those provided by laser 3D digitizers. Eck et al.'smethod is based on the approximation of an arbitraryinitial mesh M to another mesh M� that has subdivisionconnectivity and is guaranteed to be within a speci"edtolerance from M. They start by partitioning the originaltriangular mesh into a small number of regions coveringthe whole object following the concepts of Voronoi dia-grams. Then, a Delaunay-like triangulation is construc-ted by computing the harmonic map h

�of each Voronoi

tile ��

into an appropriate planar polygon P�. The inverse

of h�

provides a parameterization of ��

over P�

which isused to construct a low resolution Delaunay-like tri-angulation K� of the original mesh. Next, they de"nea local parameterization of the triangular regions overtriangular facets of a base mesh K�, again using har-monic maps. The local parameterizations are made to "ttogether continuously, in the sense that they de"ne a glo-bally continuous parameterization �:K�PM. Finally,making a series of J recursive 4-to-1 splits on each of thefaces of K� they get a triangulation K� of K� withsubdivision connectivity. The remesh M� is obtained bymapping the vertices of K� into �� using the para-meterization �, and constructing an interpolating mesh.

Lounsbery et al.'s and SchroK der and Sweldens' tech-niques address the modeling problems found in 3D com-puter graphics environments. The application of similarideas to computer vision environments needs to considera number of aspects:

(1) Many 3D vision techniques yield sparse, irregularlysampled measurements where most of the structuralinformation is lost. Typically, stereo vision tech-niques return 3D low-level primitives, such as iso-lated measurements within a point cloud or segments

L. Pastor et al. / Pattern Recognition 34 (2001) 2497}2513 2499

�Higher values of the index j mean higher resolution.�Every element f in <

�can be written uniquely as f"�

�c�f�,

existing positive constants C�

and C�

so that

C��� f ���)�

�

�c���)C

��� f ���.

belonging to the di!erent object's contours ratherthan triangular meshes, either regularly or irregularlysampled.

(2) In computer vision applications, there are other sour-ces of information in addition to geometry alone:color or other spectral information, surface texture,relative motion, etc. All of them give valuable in-formation that can be used to make deductionsabout the perceived scene's structure. Consequently,each sample from the object's surface may contain,besides geometric information, additional data corre-sponding to the surface patch surrounding thesample point.

The method presented here speci"cally addresses theserequirements, overcoming sampling irregularity witha fully automated procedure and representing objectfeatures in an e$cient way. Since the scattered points'surface approximation is carried out by deforming a sur-face with subdivision connectivity, a subdivided icosahed-ron, there is no need to regularize the mesh. With respectto Eck et al.'s, our approach adds the possibility ofprocessing 3D data clouds which do not have any addi-tional structural information.

The following two sections describe spherical waveletsbasis, as introduced by SchroK der and Sweldens [9,39]and the multiresolution representation technique de-scribed in this paper.

3. Spherical wavelets

3.1. Multiresolution analysis

Following Ref. [9], we take the space ¸�"¸

�(S�, dw),

of all square integrable functions de"ned on S�, the sphereof unity radius. A multiresolution analysis is a sequence ofclosed subspaces �<

�� j*0,<

�L¸

�which meets:

(1) <�L<

���� The spaces are nested.

(2) ¸�(S�, dw) is the closure of the union of all

<�

(����<

�is dense in ¸

�(S�, dw)).

(3) For every resolution level j, there are scaling func-tions �

���which are a Riesz basis of <

�,� being

k3K(j) an index set where K(j)LK(j#1).

From the "rst property, we have

����

"��

h�����

������

, (1)

being l3K(j#1), k3K(j). If we have the spaces =�

sothat <

��=

�"<

���and we have a set of functions

�����

� j*0, m3M(j), M(j)LK(j#1)� so that

(1) �����

� j*0, m3M( j )� is a Riesz basis for ¸�(S�),

(2) �����

�m3M( j )� is a Riesz basis of =�,

then the ����

de"ne a spherical wavelet basis. Given that=

�L<

���, we also have

����

"��

g�����

������

(2)

for m3M( j ) and l3K(j#1). In a biorthogonal setting,we have the dual spaces<I

�and=I

�, spanned by the basis

�����

and �I���

, which are orthogonal to ����

and ����

,respectively. A function f3¸

�, can therefore be represent-

ed as

f"����

��I���

, f ����

"����

���

����

. (3)

It is also possible to express ����

, the scaling functions atresolution level j, as a combination of scaling functionsand wavelets at a coarser resolution level j!1:

����

"��

hI�������

������

#��

g��������

������

, (4)

where k3K( j!1), l3K(j) and m3M( j!1).Let �

���be the scaling coe$cients of a function f,

computed at resolution level j. The fast wavelet transformcomputes recursively ��

������ k3K( j!1)�, the coarse

approximations, and ������

�m3M( j!1)�, the detailcoe$cients, both at the following j!1 lower-resolutionlevel:

������

"��

hI�������

����

(5)

and

�����

"��

g��������

����

. (6)

The inverse wavelet transform reconstructs the coe$-cients of the following higher-resolution level j#1 fromthe j lower resolution level's coe$cients:

������

"��

h�����

����

#��

g�����

���

. (7)

3.2. Extension to a triangular facet cover

Without loss of generality, we suppose that the sphericsurface is partitioned by a tiling of triangular facets. Thisis the case when we approximate a sphere S� by theregular subdivision of an icosahedron with tiles¹

���LS�, k3K( j) (see Fig. 1(a)) that satisfy:

(1) S�"�����

¹���

and their union is disjoint.(2) For each j and k, ¹

���is the union of 4 child tri-

angles ¹�����

(see Fig. 2).

2500 L. Pastor et al. / Pattern Recognition 34 (2001) 2497}2513

Fig. 1. Basic model structures. (a) Sphere discretization by sub-dividing an icosahedron j times. (b) Geodesic dome, dual of theicosahedron.

Fig. 2. Parent 8 child relationships between consecutive levelfacets. In the synthesis step, the source is triangle ¹

Hwhich is

divided in order to generate four new triangles. The reverse step,analysis, merges four facets into one.

�Number of neighbors of a geometric primitive, i.e., nodes orfacets.

Let �(¹���

) be the spherical area of a triangular patch onS� at resolution level j, and de"ne the functions

����

" ¹���

and �����

"�( ¹���

)�� ¹���

, (8)

where ¹���

is equal to 1 inside the patch ¹���

, and zerooutside ¹

���. Then, the spaces <

�"span��

����k3K(j)�

de"ne a multiresolution analysis of ¸�(S�,dw), with

����

and �����

as scaling and dual scaling functions. Froma patch ¹

��H, and its children ¹

����� �������, obtained

through subdivision as in Fig. 2, we can construct thebio-Haar [39] wavelets as

����

"2(������

!(� (¹�����

)/�(¹�����

)) ������

), (9)

in order to have their integral vanishing. The set of dualfunctions is de"ned as

�I���

"1/2(�������

!����H

). (10)

SchroK der and Sweldens present in Ref. [9] a number ofvertex-based wavelet bases which are more e$cient forperforming certain geometric operations. In conse-quence, vertex basis seem better "t for computer graphicsenvironments than facet basis such as the bio-Haar de-scribed above. In this paper, facet rather than vertex basis

have been used, since they are better adapted to com-puter vision environments: in addition to geometricaldata, each node should be able to gather other sources ofinformation such as motion, texture, color, etc. whichcould be computed from the area associated to eachfacet. If vertex basis are used, the vertex' support regionsat di!erent resolution levels are not nested. On the otherhand, the price to be paid while using facet basis isa slight increase in complexity for performing operationsbased on the object's geometry.

4. Overcoming irregularly sampled data

Generality is the "rst aspect that has been taken intoaccount in this paper. The main objective was to developa fully automated procedure for achieving 3D multi-resolution representations that could be used in anycomputer vision environment, and which could take ad-vantage of any 3D data measuring technique. Previousmethods for 3D modeling in computer graphics environ-ments departed from hypothesis that hold only for ob-jects which have either been synthesized or acquired withdense samplings. In these cases, therefore, it is reasonableto assume sampling regularity, or at least, it is easier tocompute additional structural information such as edgesand facets lists.

The technique described here, presented "rst in Ref.[42], does not need any additional assumption. In conse-quence, it is able to compute and e$ciently handle multi-resolution super"cial representations of objects acquiredas irregularly sampled 3D clouds of points without anyoperator assistance.

In this method, we must "rst de"ne a representationdomain in which we can map the original input data* the 3D coordinates of each acquired point * andwhere we can have the possibility of handling the repres-entation at di!erent resolution levels. We achieve this bycreating an icosahedron and recursively subdividing eachof its facets in order to get a regular polyhedron with20�4 triangular faces. This polyhedron can then beapproximated to a sphere of unity radius (see Fig. 1(a)).Using the multiresolution analysis described in the pre-vious section, we can propagate the geometry of theobject at di!erent resolution levels. Higher values ofn correspond to "ner levels of detail on the object. On theother hand, lower values of n represent a coarser descrip-tion of the object.

The dual of the icosahedron is a geodesic dome,a wireframe mesh of hexagonal cells (see Fig. 1(b)), whereeach node of the mesh has valence� three (like the facets

L. Pastor et al. / Pattern Recognition 34 (2001) 2497}2513 2501

Fig. 3. Representation of the deformation orientation process inorder to compute the objective function d

.

in the icosahedron). An advantage of the dual is the lowerdimensionality of the primitives that compose the mesh(3D points and segments), which allows higher degrees offreedom in the process of adjusting it over the scattereddata.

Once the representation domain has been de"ned, it isnecessary to adapt this geodesic dome to the input data'sshape. The deformation process can be broken down intothe following steps, following the solution proposed byIkeuchi et al. [43]:

(1) Determine for each node ( face) of the geodesic dome(tessellated sphere), the closest 3D point in the data"le corresponding to the object being modeled.

(2) Perform an iterative deformation of the wireframemesh while the average sum of local errors exceedsa "xed threshold. Every local error is de"ned by thedistance between a node of the geodesic dome and itscorresponding closest point, as determined in theprevious step. Each node is deformed to match theobject, according to an approximation force F

�and

a curvature force F�

controlling the local regularitycurvature, both de"ned for the actual position P

of

each node at time t. The new position of the nodeat time P

��is given by: P

��"P

#F

�#F

�#

d(P !P

��), where d represents a damping coe$c-

ient a!ecting the rate of convergence.

In the work of Ikeuchi et al., the "rst stage of the deformation process is performed manually, establishinga connection between the main features of the object'sshape and the nodes of the lower resolution mesh. Suc-cessive mesh subdivisions only take into account anEuclidean metric.

In this paper, the process has been fully automated,proceeding in the following steps:

(1) Determine an initial number of mesh nodes depend-ing on the number of points acquired from the ob-jects' surface. Experimental tests have proven that aninitial number around 1/4 to 1/16 of the input objectpoints yield good approximations.

(2) The relationship between nodes and input datapoints is established considering not only Euclideandistance, as suggested by Ikeuchi et al., but includingalso a term that takes into consideration the defor-mation direction. This directional distance can beexpressed as follows:

If �l��(�N���

then d"

n ) l

�n�#0�2 ) �l�,

else d"MAXFLOAT,

(11)

where d

is the directional distance between the inputdata point P

�and the mesh node N

�, d is the Euclid-

ean distance, N�

is the vector passing through the

center of mass of the cloud of points and the nodeN

�, l is the vector linking the point P

�and the node

N�, and n is the normal to the plane de"ned by the

node's neighbours <�, j"1, 2, 3 (see Fig. 3).

With these modi"cations, it is possible to compute theshapes shown in this work via a fully automated process.Using the directional distance produces more accurateapproximations, as shown in Fig. 4.

When the deformation process is "nished, we map thepositions of each node of the deformed geodesic dome tothe corresponding facet of the icosahedron, which mayalso gather other information in addition to geometry. Atthat moment, we can apply the analysis process to theicosahedron in order to reduce the representation's levelof detail. The geometry of the object is recovered atresolution level j if we compute the dual of the icosahed-ron, which corresponds to a wireframe mesh topologi-cally equivalent to the geodesic dome.

The positions of the wireframe mesh's nodes at levelj have been computed from the points adjusted to theinput data through the recursive application of equations12 or 13, which compute new values after one analysisstep, moving from level j#1 to level j:

���H

"

��� �

������

) � (¹�����

)

� (¹��H

), (12)

���H

"���� �

�������/4, (13)

���

"������

!���H

for l"1, 2, 3. (14)

Eq. (13), which avoids computing the weights associ-ated with irregularly sampled distributions, can be used

2502 L. Pastor et al. / Pattern Recognition 34 (2001) 2497}2513

Fig. 4. In#uence of the directional distance d

in the deformation process. (a) Clouds of points from the tip of an arrow. (b) Meshcomputed using the Euclidean distance. (c) Mesh computed using the directional distance d

.

whenever geometric accuracy is not a requirement, orwhen there is an appreciable amount of sampling regu-larity.

In the synthesis process, we must recover the originalvalues going from coarser level j to "ner level j#1:

������

"���

#���H

for l"1, 2, 3, (15)

������

"

���H

) � (¹��H

)!��� �

������

) � (¹�����

)

�(¹�����

), (16)

������

"4���H

!

��� �

������

. (17)

Eq. (17) must be used instead of Eq. (16) during synthesiswhenever Eq. (13) is used during the analysis stages.

The same multiresolution analysis and synthesis pro-cesses can be applied to any other kind of informationassociated to each of the input sample points, such ascolor, surface texture, etc.

The method described above can generate multiresolu-tion representations of objects acquired with any 3Dsensing technique. The method's only restriction con-cerns the object's shape: only objects with a genus equiva-lent to that of a sphere can be modeled exactly. Highergenus objects can also be approximated by the repres-entation technique described here, but it has to be notedthat holes will disappear from the representation when-ever the resolution falls below certain threshold. In anycase, this kind of behavior seems reasonable in a multi-resolution representation.

5. Multiresolution representation examples

This section presents "ve examples acquired with dif-ferent sensors that can be used to assess the possibilitieso!ered by multiresolution representations. In some ofthem (Figs. 5(a) and 7(a)), the input data is a cloud of 3Dpoints acquired from the object's surface with the help ofa hand-held 3D digitizer. This device is a tactile low-pricesensor manipulated by a human operator, which allows

the simulation of any other kind of sensor that providesscarce and irregularly sampled measurements. The ac-quired data in all of the examples provides exclusivelygeometric information (only the points' coordinates arekept).

Fig. 5 shows di!erent resolution versions of a closedsurface approximation of a human skull. The data setsdisplayed in Figs. 5(b)}(h) are a set of triangular facetmodels, derived from the dual polyhedron of the geodesicdome (topologically equivalent to the recursively dividedicosahedron), by computing the geometric center of eachhexagonal cell of the wireframe mesh at level j. Thereduced resolution "gures show that the overall objectshape is well-kept up to resolution level 1 (5th super"cialapproximation), although the number of object pointshas been greatly reduced.

Fig. 6 shows di!erent resolution versions of a vase,acquired with an inaccurate sensor. The data sets dis-played in Figs. 6(b)}(f) are a set of wireframe models,using the adjusted 3D points as nodes. Figs. 6(b) and (c)show a rough surface, due to acquisition errors. Thoseerrors fade away with other details when the resolutionlevel is decreased; this smoothing e!ect should not beconfused with that obtained through low-pass "lteringoperations such as those performed in Figs. 10 and 11,where the original resolution level is kept.

Fig. 7 presents again closed surface representationscomputed from the dual polyhedra of the geodesic dome.In this case the input data has been acquired from a com-position of geometric "gures with a hand-held 3Ddigitizer. The input cloud has been preprocessed in orderto considerably reduce the number of acquired points inthe object's right-half. Figs. 7(b)}(f) show that the methodcopes well with this strong sampling irregularity, produ-cing only a less-accurate approximation in the object'ssub-sampled right-half.

Fig. 8 shows di!erent views of a human vertebra. Theinput cloud of points has been computed from volumet-ric data obtained through 3D tomography, which allowsthe selection of sample points at the desired spatial loca-tions. The result is a dense cloud of uniformly distributed

L. Pastor et al. / Pattern Recognition 34 (2001) 2497}2513 2503

Fig. 5. Representation of a human skull. (a) Original cloud of points from a human skull. (b) Closed surface obtained from the adjustedmesh over the input data (20�4� facets). (c) 1�� Super"cial approximation (20�4� facets). (d) 2�� Super"cial approximation (20�4�

facets). (e) 3�� Super"cial approximation (20�4� facets). (f) 4� Super"cial approximation (20�4� facets). (g) 5� Super"cial approxima-tion (20�4� facets). (h) 6� Super"cial approximation (20�4� facets).

points. Figs. 8(a) and (b) show a rendered view and thecomputed input cloud. Figs. 8(c)}(e) present di!erentresolution wireframe approximations.

Last, Fig. 9 shows di!erent resolutions wireframe andsurface approximations of an amphora. In this case, alaser-based sensor coupled to a rotating table have beenused for the 3D digitization, resulting in a more denseand regular sampling.

From a computational point of view, the most com-plex operation is the computation of the intermediaterepresentation (the wireframe model), since selecting the

closest input point to each mesh node has a complexityO(m log n). Computing the direct and inverse wavelettransforms is quite fast (O(n)), although visualizingobjects which have resolution levels beyond 5 is slowwithout specialized graphics hardware. Most images pre-sented in this paper have been obtained with the Geom-view [44] and Xwave [45] programs; Geomview hasbeen also used to remove occluded edges from the wire-frame model (note that Geomview displays hexagonalcells with uniform grey levels, even though the cells arenot planar).

2504 L. Pastor et al. / Pattern Recognition 34 (2001) 2497}2513

Fig. 6. Multiresolution representation of a vase. (a) Input cloud of points. (b) Adjusted mesh over the input date (20�4� nodes). (c)1�� Wireframe approximation (20�4� analysis coe$cients). (d) 2�� Wireframe approximation (20�4� analysis coe!.). (e) 3�� Wireframeapproximation (20�4� coe!.). (f) 4� Wireframe approximation (20�4� coe!.).

6. Processing multiresolution representations

3D computer vision, the process of analyzing 2D im-ages or 3D data to make inferences about the nature ofthe perceived scene, is a challenging task. On one hand,sensors provide limited information about the object'sfeatures (at least, because of self-occlusions). On the otherhand, real objects show a very high variability; evenalterations in the environmental conditions may createstrong changes in the way the scene is perceived.

Multiresolution representations which keep both thecoarse approximation and detail coe$cients at each in-termediate level, such as those computed here, are re-dundant: every approximation can be computed fromany of the higher-resolution coarse approximations(direct transform), or from the lower-resolution's detailand approximation coe$cients (inverse transform).

These data redundancy is very useful, because it allowsthe resolution level which is most appropriate for ful"ll-ing a particular task to be selected [46,47]. Also, di!erent

L. Pastor et al. / Pattern Recognition 34 (2001) 2497}2513 2505

Fig. 7. Multiresolution representation of a composition of geo-metric blocks. (a) Input cloud of points. (b) Closed surfaceobtained from the adjusted mesh over the input data (20�4�

facets). (c) 1�� Surface approximation (20�4� facets). (d) 2�� Sur-face approximation (20�4� facets). (e) 3�� Surface approxima-tion (20�4� facets). (f) 4� Surface approximation (20�4� facets).

levels can be processed in parallel, and their output canbe combined for increased robustness [7,48]. Addition-ally, detail coe$cients from di!erent resolution levels cangive shape information at di!erent scales [49]. Froma computational viewpoint, multiresolution representa-tions trade memory for complexity, since processing "rstlow-resolution versions of the perceived data "rst canproduce large computational savings [50,51].

As stated before, the main objective of this paper is todescribe a technique for computing 3D multiresolutionrepresentations without any operator intervention in thebelief that these kinds of representations can help sim-plify the analysis process of 3D objects. For instance, thissection presents some examples of operations performede!ectively and e$ciently using the multiresolution rep-resentations described in the previous sections. In par-ticular, the following processes are shown:

� Preprocessing: acquisition noise removal.� Data reduction: simplifying the input 3D point cloud

without loosing signi"cant detail.� Object shape analysis: extracting the acquired object's

global shape, to be fed to a shape-tagging process.

6.1. Acquisition noise removal

In 3D environments, the sensors' miscalibrations orlack of accuracy result in noisy depth estimates. Figs.10(a) and 12(a) show two examples of objects' surfacespresenting spikes due to acquisition errors. These errorsfade away gradually as the model resolution is decreased,as shown in Fig. 6.

Wavelet transforms' detail coe$cients can be seen asthe di!erences between `predicteda and `actuala values,(the predictions are made from lower resolution coarseapproximations) [37,52], or as the output of di!erentbandpass "lters. Detail coe$cients can be used for "lter-ing, since details at speci"c resolution levels might havelow signal-to-noise ratios. In the representations used inthis paper, detail coe$cients represent local shape vari-ations, and are therefore strongly a!ected by high-fre-quency digitization noise such as that present in Figs.10(a) and 12(a).

Figs. 10(b)}11(d) and Fig. 12(b) present di!erent resultsachieved by "ltering the objects in Figs. 10(a) and 12(a),respectively. The vase's case is very interesting, because itshows the e!ects of "ltering at di!erent resolution levels.

Figs. 10(a) and (b) show the two highest resolutionapproximations to the original data. Figs. 10(c) and (d)show the same two highest-resolution approximationsafter a "ltering stage performed using the highest-resolu-tion detail coe$cients. This "lter has removed the high-resolution digitization noise quite e!ectively.

Figs. 10(b)}(d) exhibit some lower-resolution noisewhich has not been removed by this "rst "lter. Figs. 11(a)and (b) show the two highest-resolution approximationsagain, but this time after performing a "ltering stageusing only the second highest resolution coe$cients. Thelower frequency noise appearing in Figs. 10(b)}(d) hasbeen removed, although the high-resolution noise is pres-ent again.

Last, Figs. 11(c) and (d) show the results of "lteringusing the two highest-resolution detail coe$cients. Thevase's surface now presents a smooth surface.

2506 L. Pastor et al. / Pattern Recognition 34 (2001) 2497}2513

Fig. 8. Multiresolution representation of a human vertebra (8(a)). (a) Rendered view of a human vertebra obtained through 3D tomorphology. (b) Input cloud of points. (c) Adjusted mesh over the input data (20�4� nodes). (d) 1�� Wireframe approximation (20�4�

analysis coe$cients). (e) 2�� Wireframe approximation (20�4� analysis coe!.).

Fig. 12(b) shows the results of "ltering Fig. 12(a) usingthe highest-resolution level detail coe$cients.

6.2. Data reduction

3D meshes, such as those displayed in this paper, showa large redundancy degree, just like 2D images do. Forexample, if we want to represent an object preservingsome areas of "ne detail, a high-resolution scanning ismandatory. However, other areas of the object might notneed such a high-resolution representation. The prob-lems created by this redundancy are possibly more acutein the 3D case, since the data sets are more complex andthe processing algorithms more computationally de-manding.

It is evident that large reductions in mesh complexitycan be achieved by eliminating nodes in smooth or #atareas, where the nodes' positions can be accurately pre-dicted from neighboring points. Meshes can be simpli"edthis way without loosing signi"cant geometrical informa-tion.

Wavelet coe$cients can be used for assessing potentialcandidates to be eliminated in the mesh-simpli"cationprocess. For example, Fig. 13 presents the results of

performing one and two steps in a procedure for fusingnodes at resolution level 4. Figs. 13(a) and (b) show levels4 and 3 from the vertebra's original multiresolution rep-resentation, respectively. Data at level 4 (Fig. 13(a)) isanalyzed for a selective fusion procedure. Level 3 (Fig.13(b)) is an upper limit for the fusion, since each lowerresolution approximation is achieved by fusing everynode at the immediately higher resolution level.

The selective fusion proceeds by analyzing a surfacerugosity measure based on detail coe$cients: only thosenodes with a rugosity below the speci"ed threshold willbe fused. The results achieved after one fusing step arepresented in Fig. 13(c); it can be seen that nodes in #atareas follow a pattern identical to that at resolution level3, while nodes in higher curvature areas such as thevertebra's protuberances or central plateau's boundaryfollow a pattern identical to resolution level 4. Fig. 13(d)shows further node elimination after a second fusing step.

6.3. Object shape analysis

Shape variations due to sensor noise or intra-classpattern variability produce changes in object appearancewhich hamper shape analysis or recognition processes.

L. Pastor et al. / Pattern Recognition 34 (2001) 2497}2513 2507

Fig. 9. Multiresolution representation of an amphora. (a) Inputcloud of points (30.731 points). (b) Adjusted mesh over the inputdata (20�4�nodes). (c) Closed surface obtained from the ad-justed mesh over the input data (20�4� facets). (d) 1�� Wireframeapproximation (20�4� analysis coe$cients). (e) 1�� Surface ap-proximation (5.120 facets). (f) 2�� Wireframe approximation(20�4� analysis coe$cients). (g) 2�� Surface approximation(1.280 facets). (h) 3�� Wireframe approximation (20�4� analysiscoe$cients). (i) 3�� Surface approximation (20�4� facets).

Fig. 10. Filtering results for a vase acquired with an inaccuratesensor. (a) Adjusted mesh over the input data. Initial resolutionlevel: 4 (20�4� nodes). (b) 1�� mesh approximation for the meshin "ure 10(a). Resolution level: 3 (20�4� analysis coe$cients). (c)Adjusted mesh from Fig. 10(a) "ltered at resolution level 3, afterone synthesis step to compute the original resolution level (4). (d)Adjusted mesh from Fig. 10(a) "ltered at resolution level 3.

Coarse shape approximations extracted from multi-resolution representations can be used to make the pro-cess more robust, since small or intermediate shapevariations are lost whenever resolution falls below a cer-tain threshold.

Additionally, the analysis procedures speed up, sincethe input data sets are signi"cantly reduced. As an illus-trative example, data points from intermediate repres-

entation levels for 3D objects, such as those presented inthis paper, have been fed to a shape labelling processpresented in Ref. [53]. This process computes qualitativedescriptions of the global shape of 3D objects, departingfrom clouds of points in ��, such as those produced byany of the representation or processing techniques de-scribed in the previous sections. As output, the labellingprocess associates the input data with a volumetric labelthat qualitatively describes the cloud's global shape. Thesystem is also able to decompose the input object intodi!erent volumes with a homogeneous shape. First, thesystem computes a hierarchy of 2D and 3D primitivesthat are subsequently grouped from the lower levels up inorder to create a 3D shape description. The method canbe decomposed in the following steps [53]:

(a) Realignment. Once the cloud of points has been ac-quired, a realignment stage is performed in order toreorient the cloud along the maximum elongationaxis (MEA).

(b) Cloud decomposition. The cloud of points is decom-posed into a continuous sequence of bins or slices

2508 L. Pastor et al. / Pattern Recognition 34 (2001) 2497}2513

Fig. 12. Filtering results for a golf club's head acquired with an inaccurate sensor. (a) Adjusted mesh over noisy input data. Initialresolution level: 4 (20�4� nodes). (b) Filtering the mesh in Fig. 12(a) at resolution level 3 and reconstructing the original resolutionlevel (4).

Fig. 13. Data reduction of a human vertebra starting from Fig.13(a). (a) Adjusted mesh over the input data (20�4� nodes). (b)1�� mesh approximation. (c) Mesh in 13(a) after one selectivefusion step. (d) Mesh in 13(a) after two selective fusion steps.

Fig. 11. Filtering results for the vase in Fig. 10(a). (a) Adjustedmesh from Fig. 10(a) "ltered at resolution level 2 and reconstruc-ted until level 4. (b) Adjusted mesh from Fig. 10(a) "ltered atresolution level 2 and reconstructed until level 3. (c) Adjustedmesh form Fig. 10(a) "ltered at resolution levels 2 and 3 andreconstructed until level 4. (d) Adjusted mesh from Fig. 10(a)"ltered at resolution levels 2 and 3 and reconstructed until level 3.

taken along the MEA. A plane which is perpendicu-lar to the MEA is assigned to each slice, and all of thepoints within each bin are orthographically projectedonto its corresponding bin's plane.

(c) 2D Qualitative primitive extraction. The original cloudof points is discretized over the MEA, and the 3Danalysis problem can be decomposed into a series of2D shape extraction cases. This process producesa geometrical shape tag which is assigned to each bin.

(d) 2D Shape tag association and global shape computa-tion. Grouping successive 2D shape tags permits thelocal recovery of the third dimension. In conse-quence, the output of the grouping process is a collec-tion of consecutive local 3D descriptions that can be

L. Pastor et al. / Pattern Recognition 34 (2001) 2497}2513 2509

Fig. 14. Overall qualitative shape extraction process for the amphora in Fig. 9. (a) Planes: 1..216 Shape: Increasing circumference.(b) Planes: 217. .432 Shape: Decreasing circumference. (c) Planes: 433..628 Shape: Convex circumference. (d) Planes: 1..103 Shape:Decreasing circumference.

recursively associated, reducing the number of localdescriptions and the level of description detail alongthe MEA. Finally the label that best describes theobject's global shape is returned.

Fig. 14 shows the results achieved after applying thislabelling scheme to an input cloud of points extractedfrom one of the amphora's intermediate resolution levels(Fig. 9). After selecting the MEA axis, the amphora isdecomposed into a set of regions, according to theirshape. Finally the global tag decreasing circumference isselected for the overall object's shape. Note that approx-imately two-thirds of the amphora's pro"le correspondsto that label.

7. Conclusions and future work

The method described here bridges the gap betweenwavelet-based multiresolution representations developedfor computer graphics environments and the speci"cproblems found in computer vision environments, wherethere is a stronger need for fully automated procedures.

To the "eld of 3D multiresolution object representa-tion, wavelets o!er a sound theoretical framework. Mostimportant, using a wavelet-based approach for comput-ing 3D multiresolution representations allows the trans-ference of research work performed in other applicationareas to computer vision environments. This is a strongadvantage, because there is a very large research e!ortdevoted nowadays to wavelets in a wide diversity ofapplication areas. The present paper gives two examplesof typical wavelet applications ported to 3D object rep-resentations: compression and "ltering.

Working with multiresolution representations o!ersinteresting possibilities. Processing wavelet and coarseapproximation coe$cients can facilitate certain tasksduring the preprocessing or analysis stages. Some exam-ples, such as noise removal and mesh simpli"cation, havealready been mentioned in this paper. Other examplescan be found in the bibliography: FOR instance, Reissell[50] suggests using wavelet coe$cients for identifyingsmooth surface sections to plan mobile robot pathsthrough natural terrains. Also, within CAD or graphicsenvironments, Stollnitz et al. [5] point out the bene"tso!ered by multiresolution editing. Last, an advantage

2510 L. Pastor et al. / Pattern Recognition 34 (2001) 2497}2513

that is evident from the examples given here is thatmultiresolution models o!er a dramatic reduction inmodel complexity and allow the user to choose the "del-ity with which the original data is modeled, letting himtrade "delity for complexity.

From the 3D object recognition point of view, multi-resolution representations have the advantage of provid-ing a set of simpli"ed versions of the object's shape, whichcan be used for extracting di!erent information to beused in the recognition process. This paper gives anexample, using coarse approximations for computing theoverall global shape. Further work is needed in this area.

Perhaps the method's main drawback is that, to beprecisely modeled, the input object must have a genusequivalent to that of a sphere, since holes within theobject's surface are lost whenever resolution decreasesbelow a certain threshold. Somehow this behavior isreasonable, given that details such as super"cial texture,protuberances or holes should disappear as the repres-entation becomes coarser. In any case, when analyzing3D shapes (for example, in recognition environments),holes can be dealt with as independent entities, therebypreserving the method's usefulness.

Future research lines include:

� Using other basis functions that might achieve betterapproximations with fewer coe$cients than the bio-Haar basis used in this paper.

� Performing shape analysis and other processingstages, such as those presented in Section 6, in thewavelet transform domain.

� Dealing with objects with higher genus.

Acknowledgements

This work has been partially funded by the SpanishCommission for Science and Technology (grants CICYTTAP94-0305-C03-02 and CICYT TIC98-0272-C02-01)and Madrid's Autonomous Community (grant06T/020/96). The authors gratefully acknowledge thehelp provided by BeleH n Moreno (3D skull data) andJaime GoH mez (amphora).

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About the Author*LUIS PASTOR received the BSEE degree from the Universidad Polit'eecnica de Madrid in 1981, the MSEE degreefrom Drexel University in 1983, and the Ph.D. degree from the Universidad PoliteH cnica de Madrid in 1985. Currently he is Professor inthe University Rey Juan Carlos. His research interests include computer vision (3D modeling and recognition) and parallel computing.

About the Author*ANGEL RODRIDGUEZ received his degree in Computer Science and Engineering and the Ph.D. degree from theUniversidad PoliteH cnica de Madrid in 1991 and 1999 respectively. His Ph.D. is centered on the tasks of modeling and recognizing 3Dobjects in parallel architectures. He is an Associate Professor in the Photonics Technology Department, Universidad PoliteH cnica deMadrid, Spain and has published works in the "elds of parallel computer systems and computer vision. He is an IEEE and an ACMmember.

About the Author*J. MIGUEL ESPADERO received his degree in Computer Science and Engineering from the UniversidadPoliteH cnica of Madrid in 1998. Since then he has been a Ph.D. student at the Photonics Technology Department in the same university,

2512 L. Pastor et al. / Pattern Recognition 34 (2001) 2497}2513

with the research theme `Data Modeling and Automatic Recognition using Waveletsa. Nowdays he is a Teaching Assistant in theUniversidad Rey Juan Carlos of Madrid. His research interest include 3D objects representation and recognition, computer vision,image processing and wavelets.

About the Author*LUIS RINCOD N received his degree in Computer Science and Engineering from the Universidad PoliteH cnica deMadrid in 1992. From 1993 to 1997 he has been an Assistant Professor in the Electrical, Electronic and Control EngineeringDepartment in the Universidad Nacional de EducacioH n a Distancia (UNED). Since 1997, he is an Assistant Professor in theExperimental Sciences and Technology Department in the Universidad Rey Juan Carlos of Madrid and teaches Computer Structureand Technology and Computer Architecture. He has published works in the "eld of computer vision and quality control. His researchinterests include computer vision, pattern recognition, quality control and parallel architectures.

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