3D-PIC: POWER ITERATION CLUSTERING FOR SEGMENTING THREE-DIMENSIONAL MODELS Zahra Toony ⋆ , Denis Laurendeau ⋆ , Philippe Giguère ⋆⋆ , Christian Gagné ⋆ ⋆ Computer Vision and Systems Laboratory, Department of Electrical and Computer Engineering, Université Laval, Québec, QC, Canada ⋆⋆ Department of Computer Science and Software Engineering, Université Laval, Québec, QC, Canada ABSTRACT Segmenting a 3D model is an important challenge since this oper- ation is relevant for many applications. Making the segmentation algorithm able to find relevant and meaningful geometric primi- tives automatically is a very important step in 3D image process- ing. In this paper, we adapted a 2D spectral segmentation method, Power Iteration Clustering (PIC), to the case of 3D models. This method is fast and easy to implement. A similarity matrix based on normals to vertices is defined and a modified version of PIC is implemented in order to segment a 3D model. The proposed method is validated on both free-form and CAD (Computer Aided Design) models, on real data captured by handheld 3D scanners, and in the presence of noise. Results demonstrate the efficiency and robustness of the method in all cases. Index Terms — 3D mesh segmentation, 3D similarity matrix, meaningful clustering 1. INTRODUCTION Segmenting a 3D model is an important step in different applica- tions, such as object recognition and matching, mesh manipula- tion, texture mapping, reverse engineering and other applications which require the structure of the object shape. Segmenting a 3D model can be applied on point clouds [1, 2] or on meshes (with triangulation) [3, 4, 5, 6]. Two surveys presented in [7] and [8] on 3D mesh segmentation approaches classify the segmentation methods in different groups. One of the categories that attracted more attention is the spectral analysis group. The spectral graph theory was first introduced by [9] and then adapted by [10] as Nor- malized Cuts for image segmentation. These types of approaches define a similarity matrix of the model, calculate the Laplacian function based on similarities and determine eigenvectors in order to segment the models. 3D mesh segmentation using a spectral clustering method was done by [4] for the first time. It defines the similarity matrix based on likelihood of faces belonging to the same segment. This algo- rithm conducts the segmentation along concave regions and thus occasionally identifies unexpected segments. The authors then de- fined another 3D segmentation method in [5] based on partial sim- ilarity matrix. Afterwards, they combined the advantage of using spectral clustering to 2D contour analysis in [6]. This method needs some parameters to be set and is sensitive to noise. Re- cently, a method was introduced to segment 3D CAD mesh mod- els [3]. This approach first classifies the CAD model into sparse Emails: zahra.toony.1@ulaval.ca, denis.laurendeau@gel.ulaval.ca, philippe.giguere@ift.ulaval.ca, christian.gagne@gel.ulaval.ca and dense regions, and then uses the Hough transform and mean- shift segmentation to segment each region. However, it tends to fail in the presence of noise. Power Iteration Clustering (PIC) was introduced in 2010 as a simple and scalable clustering algorithm [11]. The authors com- pared their method with spectral clustering approaches which dif- fers in how a low-dimensional vector is found from the similarity matrix. In spectral clustering approaches, the bottom eigenvectors are segmented but in PIC, a weighted combination of eigenvectors is utilized and thus enabling the generation of results better or at least similar to those obtained using conventional spectral meth- ods [11]. Here, we present a three-dimensional PIC (3D-PIC) al- gorithm, adapted from the 2D version. The rest of the paper is organized as follows. The details of two-dimensional PIC are represented in Section 2. Section 3, presents our novel three-dimensional PIC. The experimental re- sults and comparison with other methods is presented in Section 4 and finally, Section 5 concludes this paper. 2. 2D-PIC The Power Iteration Clustering (PIC) method is a spectral analysis approach which first defines a similarity matrix and then embeds the data in a low-dimensional sub-space [11]. Spectral clustering algorithms, such as Normalized Cuts [10], usually find the Lapla- cian of the similarity matrix and perform the low-dimensional em- bedding on this matrix. However PIC finds a normalized sim- ilarity matrix instead of the Laplacian. It then performs the low- dimensional embedding with an approximation of the “eigenvalue- weighted linear combination of all the eigenvectors of the normal- ized similarity matrix” [11]. This method is efficient in time and space, in comparison with other spectral analysis approaches and since it considers a weighted combination of all vertices, its re- sults are also better or at least similar to those of other spectral clustering methods. The details of this method are presented in Algorithm 1. 3. OUR PROPOSED METHOD: 3D-PIC As we desire to segment a 3D model in a meaningful way such as a human would do, we need to separate different meaningful ge- ometric features from each other. In spectral analysis approaches, the features of the models are described through the similarity ma- trix. Therefore, a key component is a similarity measure that can maintain the primitives with the same properties in a segment. To be able to discriminate different primitives like planes and cylinders in 3D space, and to find the edges between primitives we are proposing that the direction of the normals to the vertices 978-1-4799-1369-5/13/$31.00 c  2013 IEEE