Pattern Recognition Letters 30 (2009) 11–17

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Pattern Recognition Letters

journal homepage: www.elsevier .com/locate /patrec

3D shape recursive decomposition by Poisson equation

Xiang Pan a,*, Qi Hua Chen b, Zhi Liu a

a College of Software, Zhejiang University of Technology, 310014 Hangzhou, PR Chinab College of Mechanical, Zhejiang University of Technology, 310014 Hangzhou, PR China

a r t i c l e i n f o

Article history:Received 5 September 2007Received in revised form 5 July 2008Available online 3 September 2008

Communicated by G. Sanniti di Baja

Keywords:3D shapePoisson equationRecursive decompositionGraph cut

0167-8655/$ - see front matter Crown Copyright � 2doi:10.1016/j.patrec.2008.08.014

* Corresponding author. Tel.: +86 571 85290085; faE-mail address: panx@zjut.edu.cn (X. Pan).

a b s t r a c t

This paper proposes a novel algorithm that decomposes the 3D shape into meaningful parts based onPoisson equation. The whole algorithm is divided into three steps. Firstly, shape signature is defined withPoisson equation. Secondly, the binary decomposition based on shape signature is recursively performedto get a coarse decomposition result. Finally, the graph-based minimum cut is used to refine the jaggyboundaries in the initial result. The proposed algorithm not only obtains a set of meaningful parts, butalso is robust in the case of deformation, rotation and other transformations. Furthermore, it can processlarge 3D shapes in an efficient way.

Crown Copyright � 2008 Published by Elsevier B.V. All rights reserved.

1. Introduction

3D shape decomposition is a fundamental problem in part-based recognition (Ullman, 1997). It always plays an importantrole in shape retrieval, deformation, compression, simplification,texture mapping and modeling (Zuckerberger et al., 2002; Li andWoon, 2001). In consequence, 3D shape decomposition has at-tracted more and more research attention

Poisson equation, which is known in mathematical physics,shows a potential application in shape decomposition. Gorelicket al. used Poisson equation for 2D shape decomposition (Gorelicket al., 2004), but they did not analyze the problem for 3D shapesand provide the corresponding solutions. This paper fills the gapand develops 3D shape recursive decomposition by using Poissonequation. In comparison with the Gorelick’s algorithm, our maincontribution is

(1) Robust 3D shape signature. Gorelick et al. only defines 2Dshape signature. This paper extends previous work anddefines Poisson shape signature for 3D shapes. The defined3D shape signature is robust in the case of deformation, rota-tion and crashes.

(2) 3D shape recursive decomposition strategy. Gorelick’s algo-rithm will cause the over-segmentation problem because itdecomposed a 3D shape by using the appropriate percentiles

008 Published by Elsevier B.V. All

x: +86 571 85290668.

of maximum Poisson shape value. The proposed recursivedecomposition strategy can avoid over-segmentation prob-lem effectively.

(3) Refined 3D shape decomposition. The decomposition resultof Gorelick’s algorithm has a noise boundary between twoadjacent parts. The graph cut is used to remove the noiseboundary and thus achieve a fine decomposition result.

The rest of the paper is organized as follows: Section 2 pro-vides a brief review of the related work; Section 3 gives an over-view of the proposed algorithm; Section 4 discusses the definitionof Poisson signature for 3D shapes; Section 5 provides a detaileddescription on how to obtain coarse decomposition results by 3Dshape recursive decomposition. Section 6 discusses how to refinecoarse decomposition results by graph cut; Section 7 provides theexperimental results and analysis for shape decomposition; Final-ly, Section 8 gives a conclusion and recommends for some futurework.

2. Related work

The research in shape decomposition was mainly carried out in2D case. Different rules had been developed for 2D shape decom-position, such as maximal convex parts (Shapiro and Haralick,1979), generalized cylinders (Binford, 1971), short-cut rule (Singhet al., 1999) and minima rule (Hoffman and Richards, 1985). Theserules are used to achieve optimization boundary for decompositionprocess. Following these rules, 2D shape decomposition can be

rights reserved.

12 X. Pan et al. / Pattern Recognition Letters 30 (2009) 11–17

performed by morphological operation (Xu, 2001) or concavity tree(Xu, 1997; Badawy and Kamel, 2005).

With the rapid increase of 3D model and its wide applications,the 3D shape decomposition has received more and more atten-tions. A detailed survey can be found in ( Shamir, 2004; Attene,2006). Among all proposed approaches, watershed decompositionalgorithm is the most popular region-growing approach, which isan extension of watershed segmentation in image processing(Mangan and Whitaker, 1999). The algorithm uses curvature tocontrol the merging operation in 3D shape decomposition. It canuse other measurements, like surface normal and electrical chargedistributions as well (Chen and Georganas, 2006; Wu and Levine,1997). An improved watershed method, called fast marchingwatersheds, is developed by using hill climbing (Page et al.,2003). The watershed algorithm is fast and does not need to specifythe number of parts. However, it is prone to over-segmentation. Toimprove the decomposition results and avoid over-segmentationproblem, distance transform and other more complex transformare applied. For example, Svensson and Baja use distance transformto define object thickness (Svensson and Baja, 2002); Prasad makesuse of constrained Delaunay triangulations to define chordal axistransform (CAT) (Prasad, 2007); Kim et al. takes advantage ofmathematical morphology and convex rule to partition the 3Dshape, namely constrained morphological decomposition (CMD)(Kim et al., 2005). Other 3D shape decomposition algorithms usegeodesic distance to avoid over-segmentation. For instance, basedon geodesic distance and angle between faces, the fuzzy k-meanscluster algorithm easily decomposes the given 3D shape into sev-eral parts (Katz and Tal, 2003); Katz et al. uses geodesic distanceto generate pose invariant scaling for the 3D shape, which is veryuseful to improve the robustness of decomposition results (Katzet al., 2005). Geodesic distance based 3D shape decompositionalgorithms can avoid over-segmentation. However, these algo-rithms need to compute all-pair geodesic distances for the 3Dshape. So they require expensive computation.

3. Overview of the proposed algorithm

This section gives some definitions for 3D shape decompositionand a flowchart of the proposed algorithm. Without losing anygenerality, this paper only focuses on 3D shape representation bytriangle mesh. Other 3D shape representation, like NURBS surfaces,meta-balls (blobby), subdivision surfaces and volumes, can be eas-ily converted into triangle meshes. For a 3D shape S represented bymeshes, it consists of N connected faces fi, i = 1,2,3, . . . , i, . . . ,N, de-noted by set TS. A few terms related to 3D shape decompositionwill be defined as follows:

Fig. 1. Overview of the propose

K-Decomposition: The decomposition result is a set of parts,S1,S2, . . .,Si, . . .,SK, where K is the number of parts. For each partSi # TS, it contains a set of connected faces and satisfies the follow-ing condition:

Si \ Sj ¼ /; i–j

S1 [ � � � [ Si [ � � � [ SK ¼ TS

�ð1Þ

Main part: The main part Smain is one of these parts S1,S2, . . . . . .,SK with the following property: for each part Si – Smain, itis adjacent to Smain.

Fig. 1 illustrates the flowchart of the proposed decompositionalgorithm.

1. The shape signature is computed by using Poisson equation.2. The recursive decomposition strategy is performed to get a

coarse decomposition result.3. The graph cut is used to refine the boundaries between any two

adjacent parts.

4. Poisson signature for 3D shape

In shape decomposition, robust shape signature is very impor-tant. Here, Poisson equation is used to define 3D shape signature.This section will give a detailed description regarding this.

4.1. Poisson equation

Poisson equation arises from gravitation and electrostatics. It isfundamental to mathematical physics. Generally, Poisson equationis a second-order elliptic partial differential equation defined as

r2w ¼ �1 ð2Þ

where function w is called potential function.Though the Poisson equation is most known in electrostatics, it

has a potential advantage in shape analysis. For example, it has afinite set of local maxima related to shape structure. For each pointin boundary, a unique path can be found to each local maximum byfollowing the gradient field.

To use Poisson equation for 3D shape analysis, we first define avolume, whose boundary is a close surface. The binary densityfunction f(x,y,z) for volume can be defined by the followingequation:

f ðx; y; zÞ ¼1; object0; background

�ð3Þ

Secondly, we assume that the potential on the boundary of the 3Dshape is known (Dirichlet boundary condition) and constant. So

d decomposition algorithm.

X. Pan et al. / Pattern Recognition Letters 30 (2009) 11–17 13

the discrete version of Poisson equation for 3D shapes can be de-fined as

DUðx; y; zÞ ¼ �1 ð4Þ

where DU = Uxx + Uyy + Uzz is the Laplacian of U. Poisson equationassigns a value to each internal voxel in volume V. The value de-pends on the relative position of that point within the closed sur-face. In other words, the value measures the average timerequired for particles to arrive at the boundary. The most interest-ing property of Poisson equation for 3D shape decomposition is

1. Robust. Poisson equation is independent of the coordinate sys-tem over the entire domain. Therefore, the shape signaturebased on Poisson equation is robust under rotation. It is alsorobust under noise-distortion, shape crash and rigidtransformation.

2. Geometry structure related. Poisson equation can describe theshape structure in a mathematical meaning well.

4.2. Definition of poisson 3D shape signature

The discrete Poisson equation, however, cannot be applied tothe 3D shape S based on triangle mesh directly. So we first con-struct the corresponding volume V for the 3D shape S. For each ver-tex in S, it can be mapped to a voxel in volume V. As shown in thefollowing Fig. 2.

We use a voxelization algorithm based on Z-buffer proposed in(Karabassi et al., 1999). The heart of this voxelization algorithm isto place the 3D mesh into a boundary box. It gets projection depthinformation from each face of the boundary box. Therefore, for apair of slides along the x axis, we can get min and max value(X1,X2) of depth value. Similar for (Y1,Y2) and (Z1,Z2). Then voxel-ization for each point (x,y,z) in volume V can be defined as the fol-lowing equation:

f ðx; y; zÞ ¼1; X1 6 x 6 X2;Y1 6 y 6 Y2; Z1 6 x 6 Z2

0; otherwise

�ð5Þ

Fig. 2. The mapping from mesh vertex (red circle) to voxel (blue) (during the voxelizationinterpretation of the references to colour in this figure legend, the reader is referred to

Fig. 3. The distribution of Poisson shape signature for some 3D shapes (The green color mThe difference among the Poisson shape signature for parts is very obvious. Poisson sh(right)). (For interpretation of the references to colour in this figure legend, the reader i

Through the mapping from vertices in S to voxels in volume V,we can get shape signature in the volume instead of in the meshsurface. Based on the definition of Poisson equation, it will assigna value for each internal voxel in Volume V. The value, namely Pois-son shape measure, is denoted by ci for each voxel {vijf(x,y,z) = 1}.Finally, for each face fi 2 S, its Poisson 3D shape signature wi forface fi can be obtained by the following equation:

wi ¼P3

j¼1ðmaxðciÞ; i 2 NeighborkðviÞÞ3

ð6Þ

here each voxel vi has k-ring adjacent voxel set Neighbork(vi). Thevalue k is assigned to be 2 in our implementation. As shown inFig. 3, two adjacent parts of the 3D shape have significantly differ-ent Poisson 3D shape signatures. In addition, the shape signature re-main robust under rigid transformation and part crashes.

4.3. Poisson solver

As for the above Poisson shape signature, how to solve the Pois-son equation is a remained problem. Discrete Poisson equationwith known boundary condition can be defined by linear system.It can be represented as the following form:

Af ¼ b ð7Þ

Furthermore, the coefficients matrix A is a symmetric positive def-inite matrix. So the solution of Poisson equation is reduced to solvethe sparse linear system. To speed up the computation, we useCholesky decomposition:

A ¼ LT � L ð8Þ

here L and LT are the Lower Triangular Matrix and its transposerespectively. The Cholesky decomposition is a special kind of LUdecomposition. The computation cost, however, is almost half ofother LU decomposition algorithms. After Cholesky decompositionof coefficient matrix A, the linear system can be solved by back sub-stitution with the following equation:

process, each vertex in mesh surface S has a corresponding voxel in volume V). (Forthe web version of this article.)

eans lower value of Poisson 3D shape signature, while the red color means higher.ape signature remains robust under part crash (middle) and rigid transformation

s referred to the web version of this article.)

The histogram of poisson shape signature

40

60

nt

14 X. Pan et al. / Pattern Recognition Letters 30 (2009) 11–17

LTy ¼ b

Lx ¼ yð9Þ

For the implementation of the above solution, we recommend usingTAUCS open-source package (Toledo, 2003). TAUCS is a C library ofsparse linear solvers developed by Tel-Aviv University and has ahigh efficiency.

5. Recursive coarse decomposition

After obtaining the shape signature, each face fi has Poisson 3Dshape signature wi. We then focus on the decomposition of 3Dshapes composed of a main part (e.g., the torso of an animal) andseveral connected exterior parts (e.g., the head, limbs, and tail).

5.1. Recursive binary decomposition

Before performing recursive binary decomposition, a core facefcore for the 3D shape should be defined. The core face can be ex-tracted by the following equation:

fcore ¼ ffijwi ¼maxðwjÞ; fj 2 TSg ð10Þ

After obtaining the core face, the binary decomposition is per-formed recursively to achieve coarse result. In recursive process,binary decomposition will generate a new shape part Si from faceset FR. The process is recursively performed until the exit conditionis satisfied. Then the remained faces in set FR are the main part ofthe 3D shape. The whole process can be shown in the following ta-ble and the symbol Geod denotes geodesic distance between anytwo faces.

As shown in Table 1, the main problem of decomposition processis to find the peak face fp in binary decomposition (Section 5.2 willgive a detail description on how to find the peak face). Once thepeak face has been found, the binary decomposition can be per-formed easily with the following property: if the face fj belongs tothe part Si, its distance Geod(ffar, fj) is shorter than Geod(ffar, fp).Fig. 4 further shows the process of binary decomposition.

Table 1Recursive binary decomposition

Step 1: Initialize the face set FR = TS and part count i = 0. Get core face fcore by Eq.(10)

Step 2: Generate a new part Si = /Step 3: Get the geodesic distance farthest face ffar 2 FR to fcore.Step 4: Find a peak face fp in the geodesic path between fcore and ffar.Step 5: Insert faces fj into set Si if they have Geod(fj, fp) < Geod(ffar,fp)Step 6: Set FR = FR � Si and i = i + 1. If the exit condition is satisfied, go to Step 7,

else goto Step 2Step 7: Set main part Smain = FR.

Fig. 4. The process of binary decomposition. (For interpretation of the references to co

5.2. Finding the peak face fp

Two steps are required to find peak face fp. Firstly, an adaptivethreshold wT is found for Poisson 3D shape signature wi in face setSF, where SF denotes a set of faces along geodesic path from facefcore to ffar. Because the property of Poisson equation is related withshape structure, the histogram of Poisson 3D shape signature alongthe geodesic path shows double peaks. Fig. 5 shows an example ofhistogram. Therefore, the threshold can be found by maximum var-iance between clusters. Maximum variance among clusters hasbeen widely used in image binarization (Otsu, 1979). It is veryeffective to get threshold when histogram distributions have dou-ble peaks.

Secondly, the face set SF is classified into two classes by thresh-old wT, denoted by SF1 and SF2. The numbers of faces in the two setsare N1 and N2 respectively. The peak face is found by the followingprocedure: Suppose fcore 2 SF1 and ffar 2 SF2, faces along the geode-sic path from fcore to ffar, are scanned one by one. When the count ofscanned faces reaches to N1, the current face will be marked aspeak face fp.

5.3. Exit condition of recursive process

For the face sets fi 2 SFj, j = 1,2, their average Poisson 3D shapesignatures are computed by the following equation:

wj ¼X

wi=Nj; f i 2 SFj; j ¼ 1;2 ð11Þ

So the difference between two parts can be estimated by the follow-ing equation:

lour in this figure legend, the reader is referred to the web version of this article.)

0

20

1 51 101 151 201 251

Poisson shape signature

Face

cou

Fig. 5. Histogram based on Poisson shape signature. (We normalize the Poissonshape signature into [0,255] for visualization).

Table 2The computation time for all-pair geodesic distance and the proposed algorithm

Model (#faces) All-pair geodesic distance (s) The proposed algorithm (s)

Hand (3094) 20.3 1.47Lion (9996) 242.42 2.17Horse (16843) 718.75 2.36Human (32386) 4371.43 2.42Ant (48048) 7854.67 2.48

X. Pan et al. / Pattern Recognition Letters 30 (2009) 11–17 15

�wdiff ¼ j �w1 �w2j=maxðw1;w2Þ ð12Þ

The bigger value wdiff means less similarity between two parts, sothe termination condition for recursive process is wdiff < wT andthe value wT is set to be 0.4 in our implementation. When the abovecondition is satisfied, it means the algorithm has decomposed the3D shape into meaningful parts successfully. Otherwise, the algo-rithm continues recursive process in order to get a new part.

6. Fine decomposition

The recursive process has decomposed the 3D shape into differ-ent parts. The decomposition result, however, will be very coarseand has jaggy boundary. The main reason is that the binary decom-position can not obtain precise boundary by adaptive threshold.Therefore, a post-processing is required to refine decompositionresult. Based on the minima rule from cognitive studies, a gooddecomposition for the 3D shape is along concave edge. So the dualgraph of the 3D mesh is constructed firstly, then a minimum cut fordual graph is performed to get fine decomposition result.

To generate dual graph, searching region needs to be defined.The searching region RS contains the faces, whose geodesic dis-tance to the boundary is smaller than a pre-defined threshold TH(Each face fi 2 RS denotes a node in dual graph). Two additionalnodes T1 2 Si and T2 2 Smain is also needed. With all defined nodes,a dual graph can be constructed for fine decomposition. The nodeT1 has a connection with face fi 2 Si \ RS, and T2 has a connectionwith face fi 2 Smain \ RS, as shown in Fig. 6:

After generating the dual graph, the fine decomposition is per-formed by maximum flow (minimum cut) algorithm from sourcenode T1 to target node T2 (Goldberg and Tarjan, 1988). For twoadjacent faces fi and fj, the capacity function for dual graph canbe defined by the following equation.

capði; jÞ ¼1

1þ cos tði;jÞavgðcos tÞ

; if ði; jÞ–T1; T2

1 otherwise

(ð13Þ

where the value avg(cost) is the normalization factor and cost(i,j) isdihedral angle, defined by the following equation:

cos tði; jÞ ¼ 1eta � ð1� cosðfi; fjÞÞ

ð14Þ

where factor eta is equal to 1.0 if it is a concave angle, otherwise 0.3.The above algorithm makes the cut along the concave edges andproduces a good result.

7. Experiments and discussion

Experiments are carried out to test the performance of the pro-posed algorithm. The algorithm is implemented in a commodity PCwith Pentium 1.8 GHz CPU and 512 MB Ram, The used 3D shapeshave the number of faces varying from 3000 to 40,000. Four indi-

Fig. 6. The dual graph for fine decomposition.

vidual experiments are conducted. Firstly, the computation timeis analyzed and results show efficiency of the proposed algorithm.Secondly, the proposed algorithm is verified to better than Gore-lick’s algorithm. Thirdly, the proposed algorithm is proved to be ro-bust under different transformations. Finally, decompositionresults of other 3D shapes are provided.

Firstly, we compare the computation time of all-pair geodesicdistance with the proposed algorithm. All-pair geodesic distanceneeds to compute the geodesic distance for an arbitrary pair offaces, so the time complexity is O(N̂3) for N faces. The proposedalgorithm only needs to compute the geodesic distance from coreface to another face. The time complexity is (K + 1) � O(N) for Kparts. Therefore, the computation time for the proposed algorithmis much less than all-pair geodesic distances. Table 2 shows thecomputation time for different 3D shapes.

Secondly, experiments show the proposed algorithm is morestable than Gorelick’s algorithm. The latter sets a threshold forextracting the main part. Here the threshold is set to be 60% ofhighest Poisson shape value. As shown in Fig. 7, Gorelick’s algo-rithm gets a correct decomposition result for the ‘‘horse” but failsfor the ‘‘cat”. As a comparison, the proposed algorithm is stablefor both 3D shapes.

Thirdly, experiments show that the proposed algorithm is ro-bust under different transformations. Fig. 8 shows the decomposi-tion results of transformed shapes, including rigid, rotation, scaleand shear transformation. All decomposition results of trans-formed shapes are consistent with that of the original 3D shape.

Finally, Fig. 9 presents Poisson 3D shape signatures and decom-position results for other 3D shapes. For each 3D shape, the distri-bution of Poisson 3D shape signatures represents the intrinsicgeometry structure and the decomposition results shows themeaningful parts of the 3D shapes.

Fig. 7. The proposed algorithm is more stable than Gorelick’s algorithm.

Fig. 8. Decomposition results for the 3D shapes in different transformations.

Fig. 9. Shape part signature and decomposition results of some models. (The left is the distribution of Poisson shape signatures, and the right is the decomposition result.)

16 X. Pan et al. / Pattern Recognition Letters 30 (2009) 11–17

X. Pan et al. / Pattern Recognition Letters 30 (2009) 11–17 17

8. Conclusion and future work

This paper proposes 3D shape recursive decomposition basedon Poisson equation. The proposed algorithm can not only decom-pose the 3D shape into meaningful parts, but also avoid jaggedboundaries and over-segmentation problems. Furthermore, thealgorithm is robust in the case of different transformations andefficient for large 3D shapes.

Our current work is focusing on improving decomposition re-sults by using other part salient decomposition measurements, likeconvex hull. Our future work is to apply Poisson equation in otherapplications, mainly 3D shape matching and skeleton extraction.

Acknowledgements

This research work was supported by China Natural ScienceFoundation (Grant No: 60703001). Part of work is supported byNatural Science Foundation of Zhejiang Province, China (GrantNo: Y106203, Y106329). Author would like to thank You Qian inIndiana University for her proof-writing and useful comments.Thanks are also given to the anonymous reviewers for their kindcomments.

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