A Meaningful Mesh Segmentation Based on Local Self-similarity Analysis Zhi-Quan Cheng, Gang Dang and Shi-Yao Jin PDL Laboratory, National University of Defense Technology, China [email protected]

Abstract On the basis of minima rule from the cognitive theory, this paper presents an algorithm decomposing a mesh into smaller parts by feature contours, gotten from the minima negative curvature vertices. The algorithm is carried out in two steps. Firstly, to avoid over-segmentation, our method excludes unimportant local adjacent similar contours. Secondly, the remnant salient contours are automatically completed to form short loops around mesh’s parts, constrained by two near parallel cutting planes that are determined by principal component analysis of all vertices. The algorithm has been demonstrated on many meshes, and the results show that it not only can perceptual group the adjacent self-similarity regions, but also can achieve reasonable segmentations.

1. Introduction In recent years, mesh segmentation [1][2] (or partition) has become a key ingredient in many mesh manipulation methods. One type is called part segmentation, which focuses on efficient procedure to produce meaningful segmentation results (or called shape decomposition), which are in close agreement with human shape perception. The minima rule [3], from the cognitive theory, has already been used for meaningful segmentation of meshes by [4][5][6]. The minima rule states that human perception usually divides an object into parts along the concave discontinuity of the tangent plane. However, these

a)minimum curvatures show

studies may be more reasonable if they had considered meshes with complicated surface. In fact, the adjacent similar parts of a mesh should be treated as a whole [7]. Especially, the model of classical sculpture (xyzrgbdragon), ornamented architecture (ornamental column), or animal fossil (dinosaur), always has local similar parts surrounded by the valleys. For these models, previous algorithms would possibly divide them into over-segmentations (dinosaur model in Fig. 7.a), even wrong components (column 1 model in Fig. 7.b). In general, we make the following contributions: z Partial similarity. We address the phenomena of adjacent part-in-a-whole in the human perception theory area, and discuss the problem that local similarities result [8] in the limitation of previous algorithms. z Meaningful segmentation method. We propose a mesh decomposition algorithm, which partitions a given mesh into multiple meaningful parts. Based on the minima rule, our approach makes use of salient contours to form short loops around mesh’s parts. Our method is carried out in two steps (Fig. 1). Section 3 identifies a mesh’s feature fields (Fig. 1.a), and defines the local structure descriptors to detect the local similarities (Fig. 1.b). Section 4 (Fig. 1.c) priors the salient feature contours, and completes them to form closed loops to partition the mesh, and reject the meaningless loops. In addition, we review related works in section 2, and demonstrate our effectiveness of our method with a number of meshes in section 5. At last, the conclusion is made in section 6.

b)local self-similarity detection

Figure 1. Process of our algorithm

978 -1-4244-1579-3/07/$25.00 © 2007 IEEE

c) mesh segmentation

2. Related work Unlike traditional segmentation methods, which can be classified into growing regions, merging regions, hierarchical clustering and spectral analysis types, reviewed by [1], the algorithm based on the minima rule has used concave area to segment models. Page et al. first employed the minima rule and used the region growing watershed algorithm [4] to partition, which couldn’t cut a part if the part boundary contains nonnegative minimum curvatures. To overcome the shortcoming, Page et al. then used the factors proposed in [5] to compute the salience of parts by indirect super-quadric model. Until now, only Lee et al. [6] concentrated on feature boundaries on the mesh surface, and used scissor operator to form closed loop around parts. Similar to [6], we consider the salient feature contour and present a simple and heuristic method to close them. More important, the selfsimilarity limitation of previous algorithms, mentioned in section 1, is repaired by local similarities detection. 3D shape matching utilized geometric properties to compare shape, and have always been applied in shape retrieval field. Until now, most of the researches have paid attention to global similarity of the 3D models, but few studies have attempted to partial match [9]. Spin image [10],[11] was generated by range scanners to recognize and register 3D objects. Shape Contexts [12] have also been used for describing the shapes of local regions. In contrast to whole shape descriptors for various properties of shapes, selecting a subset of local descriptors is a well known technique for speeding up retrieval. A center-surround filter of curvature across multiple scales on a shape was used to select salient regions for shape matching [13]. Shilane and Funkhouser [14] defined distinction as the retrieval performance of a local shape descriptor. Gal et al. [8] augmented part-in-whole matching by considering salient features based on curvature properties.

3. Partial similarity detection Psychophysical experiments [7] have showed that the human visual system decomposes complex shapes

into parts based on valleys and treats adjacent selfsimilar features as a whole. To finish the segmentation, we define a feature value on each vertex by the minimum curvature value (Fig. 2.a), which is calculated by the finite-difference local neighbor computation presented by Ruisinkiewicz [15]. And then, to overcome the irregular distribution of the curvature value, a thresholding process is adapted. the values are normalized as Lee’s [6] rule: “If k(v) is the minimum curvature value at a vertex v , the normalized value is Cf(v)= (k(v) - u)/s , where u is the mean and V is standard deviation of k(v) over all vertices of the mesh”. We assign the normalized value Cf(v) to each vertex v of the mesh as the feature value. And then, we use hysteresis thresholding [16] on the Cf(v) value to define feature field (Fig. 2a). The upper bound for the hysteresis is set as -1.3 and lower bound is -0.9. By connecting the vertices that pass the thresholding, we construct regions (Fig. 2.b) on the mesh surface defined by triangles that contain these vertices. And then, local adjacent self-similarity properties of a mesh (Fig. 2.c), are extracted. The feature regions, masked by the adjacent local selfsimilarity areas (Fig. 2.d), could not be used to define feature contours partitioning the mesh. To generate a rather small and yet effective set of local surface descriptors, we define a local surface patch and associate it with a descriptor that represents the patch. There is an important concave constraint condition that must be met at the patch quadric fitting scheme. That is the valley boundary mechanism, which requires the concave areas to be the partial shapes border. Firstly, we select the mesh vertices, whose neighbor vertices include at least two vertices in the detected field region, and mark them as the willing conquering candidates. Our method is efficient in the sense that we deal with a subset of all vertices and ignore the areas far from the feature regions. Secondly, for each candidate vertex, we use the region growing watershed algorithm [4] to conquer the largest possible quadric patch, which is adjacent to defined feature region. And we ruled out the vertices, included in the local quadric, from the sorted list. After that, the iterative process continues to detect the next quadric

low

high

a)feature field

b) feature regions

c)local self-similarity

Figure 2. Process of the partial similarity

d)feature regions masked by similarities

until all marked vertices have been processed. Our method is greedy in the sense that in each step we define the largest possible area around the selected vertex that meets the prescribed error threshold. We use 10í4 of the model bounding box diagonal length for all the examples shown in this paper. We measure the quality of a quadric fitting using the sum of squared algebraic distances of the fitted points from the fitting surface. In the following, for each local quadric patch of the model, a representative point at the center of its mass is selected, and it is associated with the highest curvature across the patch. The salient geometric features, which are non-trivial shape regions that have a high curvature value relative to their surrounding, are constructed by growing a cluster of descriptors. The same saliency grade S expression of a patch F consisting of d descriptors is computed as [8]. S = ¦ W1 Area ( d )Curv(d )3  W2 N ( F )Var ( F ) (1) d F

where, Area(d) is the area of the patch associated with d relative to the sphere size, Curv(d) is the minimal curvature value associated with d, N(f) is the number of local minimums or maximums curvatures in the cluster, Var(F) is the curvature variance in the cluster, the two weighted factors W1 and W2 are both set as 0.5. The expression Area(d)Curv(d)3 marks the saliency of the region, by considering the relative size and minimal principle curvature. The function W2N(F)Var(F) defines the interesting degree of the cluster, and adds up to the saliency of the cluster. After the representative salient features of the model have been detected, each of them is recorded by the saliency grade values, as expressed by equation 1, and inserted into a hash table. By the hashing, we can easily find neighboring partial matches between the current testing part and those stored in the hash table. The neighboring distance threshold value is 0.1 of the model bounding box diagonal length for all the models. The Fig. 3 shows the results of the local adjacent selfsimilarity extraction in other models.

4. Meaningful segmentation After the local adjacent self-similarity detection, we extract feature contours from the left feature regions. We accomplish this goal using the Hubeli’s skeletonizing method [[16]]. Then, after sorting these contours (section 4.1). we compute each feature contour’ orientation of two restrained parallel planes in section 4.2, its loop completion method is presented in section 4.3, and the acceptance condition is finally explained in section 4.4.

4.1. Priority of candidate feature contours To cut the mesh, a single specific feature contour should be chosen from the candidates, obtained from before feature contour extraction stage. It’s important to determine the order of contours, because our approach prevents the partitioning loops from crossing each other. Therefore, in our implementation, the priority of a feature contour Ȗ, define by equation 2, would be calculated by three factors: length, centricity, and perpendicular quality. Priotiy (J ) Length(J ) * Centricity (J ) *K (J ) (2)

z

Length( J ) = ¦ length(e) , where length(e) is eJ

z

z

the length of an edge e. Centricity(Ȗ) = (1-Center(Ȗ).x/halfX)* (1Center(Ȗ).x/halfY)* (1-Center(Ȗ).x/halfZ), where function Center(Ȗ) is the barycenter of Ȗ, halfX, halfY, and halfZ respectively stands for the three half axes length of the model’s oriented bounding box. = 1-Min(Angle(xAxis,mainDir), K( J ) Angle(yAxis,mainDir), Angle(zAxis,mainDir))/(Pi/2), where perpendicular quality K ( J ) is computed from the minimum angle between cutting plane’s normal and the most parallel medial axis.

4.2. The orientation of two restrained parallel planes

a)buddha

Figure 3. detection

b)column1 c)colume2

Instances

of

d)dinosaur

partial

similarity

By thorough analysis, we develop a simple “Sandwich” cutting approach that has two parallel restricting planes. And the key of this method is how to define the orientation of the two planes. Since every feature contours has an associated set of vertices, a good idea is that we can find the best fit plane whose distances summation to this set of points is the least. This kind of least distance planarity computation is of course quite common. The standard technique,

2

a)

b)

c)

d)

e)

1

a)b) The examined neck part and its local magnified area c)d) The local frame computed from matrix Z in different view e) Closed loops Figure 4. The way to find the orientation of sandwich’s two parallel planes based on principle component analysis, is to construct the sample covariance matrix: 1 k (3) Z ¦ (Vi  V )(Vi  V )T k 1 i 1

where V is the mean of the vertices V

(¦ i Vi ) / k .

The three eigenvectors of the matrix Z determine a local frame with V as the origin. The covariance matrix Z is computed using quadric metrics. Among the three eigenvectors, the one corresponding to the smallest eigenvalue is like to contour’s general direction (see red vector in Fig. 4.c and 4.d), and then the orientation of sandwich would be one of the other two eigenvectors. The alternative determination is done by using feature contour completion tactic explained in the 4.3 section. We compare the two closed loops formed from the two eigenvectors, and select the one corresponding to the loop whose zone isn’t crossing any existed segmentation loop and its length is the shortest (see the loop flaged by ĸ in the Fig. 4.e). While the other (corresponding to the loop flagged by ķ in the Fig. 4.e) would be excluded beyond all doubt. For the instance showed in Fig. 4, the appropriate is the green eigenvector.

4.3. Sandwich: close feature contours to form cutting loops on mesh surface For a selected feature contour Ȗ which is always open, we need complete it to form a closed loop around the mesh. The most crucial issue in closing feature contour Ȗ is how to restrict the shortest path between its endpoints, which is from one endpoint to the other by going over the other side of the mesh instead of the natural shortest route. Lee et. al. [6] finished this work by using four complex functions: distance, normal, centricity, and feature, and they adopted four experiential parameters determined by tedious experiments. On the contrary, we solve the problem in a simple and intuitive way, which restricts

the contour completion by two parallel cutting planes and an isolating zone, and the structural diagram of our approach can be seen in Fig. 5. The parallel cutting planes generate a restricted region, and have a great impact on the feature contour closing path by constraining the path only in the area. However, if the shortest path between the contour’s endpoints, which can be archived by the common Dijkstra's algorithm, is not limited, it would be apt to stop at the original feature contour rather than go through the other side of the mesh. So, to avoid this problem, an isolating zone, which is an approximate oriented bounding box (OBB), is added to prevent the shortest path from going wrong way. The OBB (see Fig. 5.c) is created as the following rules: the center is located at the position of distance d from the middle vertex of the feature contour, the three normalized vector are initialized as the before three eigenvectors accordingly, and the three half axis’s length are set as 3d in the oriented direction and 2d in the others. Especially, distance d is defined as follow. ­°3* GAPpoint , if GAPpoint ! LNGedge (4) d ® °¯ 3* LNGedge , else where GAPpoint is the average distance between the adjacent points of the contour, and LENedge is the average edge length of the mesh. Two restricted plane The closed loop

2d

d

3d 2d 2d

a) b) c) The isolating zone a) One contour; b) The components of sandwich (front view); c) The diagram of isolating zone (side view).

Figure 5. The structural diagram of feature contour completion

During the process of feature contour completion, the feature function also considers other contours located in the same restricted area. Therefore, we enable two feature contours far from each other to be connected in the looping. In order to accept the accurate segmenting feature loops, we use the identical part salience criterion [6] to check whether they are significant enough or not. The criterion is the combination of three factors of a part, which include area, protrusion, and feature. Since our approach is also heuristic, it is possible that there is rare manual rejection for some model.

partition it in wrong way. All the before deficiencies have been overcome by our approach, since the shortcoming is resulted by the local adjacent selfsimilarity properties of a mesh. Besides these, the results of bunny and horse model are also different. [6] has decomposed the bunny model (Fig. 7.c into five parts: two ears, the head, the body, and the tail, while our proposed approach automatically partitions it into more parts: two added back legs, one front leg, two eyes, and one mouth. For the horse model (Fig. 7.d), their algorithm cuts the horse’s head at the unsuitable position, while our approach can get more logical horse head.

5. Results and discussion To demonstrate our algorithm, we present some results for a series of triangle meshes in Fig. 6. It is obvious that these models in Fig. 6 have been accurately segmented into parts beyond the local selfsimilarity parts-in-a-whole treatment, since the parts are fitted with human visual perception results. a) disonar

a) xyzrgb-dragon

b) rabbit

b) column 1

c) bunny c)buddha d)column 2 Figure 6. Instances of segmentation results We now investigate experimental results using the algorithm we have presented. We focus on a visual comparison in Fig. 7, which demonstrates a simple comparison of our approach to Lee et al. [6]. The [6] has been reproduced with the help of its authors. As shown in Fig. 7.a, the method in [6] automatically segments the disonar mesh into more excessive parts, since there are so many feature curves would be treated as candidate feature contours, and for the column 1 model as shown in the Fig. 7.b, [6] would

d) horse Figure 7. A visual comparison of [6] algorithm (left) with our approach (right)

6. Conclusion In this paper, we attempt to use the idea of local self-similarity to guide the process of mesh segmentation. This is very appealing and leads to excellent results, which has been proven very well. The methodology we used is based on an analysis of the surface to define a set of local structure descriptors that captures the essence of the geometry of the original mesh. Based on the local adjacent partial matching method, we also presented a new automatic mesh segmentation approach for those meshes with complicated surface, which have local similar protrusion parts. Based on minima rule, our partitioning approach targets the cutting positions along the concave discontinuity of the tangent plane. The feature contour loop completion process is limited by two parallel restricted planes and one isolating zone, which guide the feature contour loop to go over the mesh in direct and straight way. We have briefly compared our approach to Lee et al. [6] and presented some results for a series of meshes. These results demonstrate the power of our method to decompose an object into meaningful visual parts. We believe that much room remains for further meaningful mesh segmentation. First, no algorithm can provides a topological guarantee in that a surface of any genus can be eventually divided into a collection of topological disks. Second, user would like to see the segmenting algorithm being applied to more complex models such as the Stanford’s David and others. Finally, the various levels of a multi-resolution mesh especially pose more challenging difficulties due to noise of various resolutions in the meshes.

Acknowledgments We would like to thank Yunjin Lee for her earnest explanation at their paper and the help in the reproducing process. The bunny, horse, and other models are provided by Stanford Graphics Laboratory, the Caltech Multi-Res Modeling Group, 3D Meshes Research Database by INRIA GAMMA Group, and Aim@SHAPE Shape Repository.

References [1] A. Shamir, “Formulation of Boundary Mesh Segmentation”, In: Proc. of 3DPVT, 2004, pp. 82-89. [2] M. Attene, S. Katz, M. Mortara, G. Patane, M. Spagnuolo, and A. Tal, “Mesh Segmentation - a Comparative Study”, In: Proc. of SMI, Japan, 2006, pp. 14-25. [3] D.D. Hoffman, and W.A. Richards, “Parts of recognition”, Cognition, 1996, 18, pp. 65-96.

[4] D.L. Page, A.F. Koschan, and M.A. Abidi, “Perceptionbased 3D triangle mesh segmentation using fast marching watersheds”, In: Proc. of IEEE Conf. on Computer Vision and Pattern Recognition, 2003, vol. II, pp. 27-32. [5] D.L. Page, M.A. Abidi, A.F. Koschan, and Y. Zhang. “Object representation using the minima rule and superquadrics for under vehicle inspection”, In: Proc. of 1st IEEE Latin American Conference on Robotics and Automation, 2003, pp. 91-97. [6] Y. Lee, S. Lee, A. Shamir, D. Cohen-Or, and H.-P. Seidel, “Mesh Scissoring with Minima Rule and Part Salience”, Computer Aided Geometric Design, 2005, 22, pp. 444-465. [7] Wright R.D., Visual Attention. Contributors. Oxford University Press. New York, 1998. [8] R. Gal, and D. Cohen-Or, “Salient Geometric Features for Partial Shape Matching and Similarity”, ACM Trans. Graph, 2006, 25(1), pp. 130-150. [9] J. W. Tangelder, and R.C. Veltkamp, “A survey of content based 3d shape retrieval methods”, In: Proc. of International Conference on Shape Modeling and Applications, 2004, pp. 145-156. [10] A. Johnson, and M. Hebert, “Using spin-images for efficient multiple model recognition in cluttered 3-D scenes”, IEEE PAMI, 1999, 21(5), pp. 433-449. [11] A. Frome, D. Huber, R. Kolluri, T. Bulow and J. Malik, “Recognizing Objects in Range Data Using Regional Point Descriptors,” In: Proc. of Eighth European Conf. Computer Vision, 2004, vol. 3, pp. 224-237. [12] D. Huber, A. Kapuria, R.R. Donamukkala, M. Hebert, “Parts-based 3-d object classification”, In: Proc. of the IEEE Conference on Computer Vision and Pattern Recognition, 2004, vol.2. pp. 82-89. [13] T. Gatzke, C. Grimm, M. Garland, and S. Zelinka, “Curvature Maps for Local Shape Comparison”, In: Proc. of the International Conference on Shape Modeling and Applications, 2005, pp. 246-255. [14] P. Shilane, and T. Funkhouser. “Selecting Distinctive 3d Shape Descriptors for Similarity Retrieval”, In: Proc. of the IEEE International Conference on Shape Modeling and Applications, 2006, pp. 18-28. [15] S. Rusinkiewicz, “Estimating Curvatures and Their Derivatives on Triangle Meshes”, In: Proc. of 3DPVT, 2004, pp. 486-495. [16] A. Hubeli, and M. Gross, “Multiresolution feature extraction for unstructured meshes”, In: Proc. of IEEE Visualization, 2001, pp. 287-294. [17] D. Hoffman, and M. Signh, “Salience of visual parts” , Cognition, 1997, 63, pp. 29-78.

A Meaningful Mesh Segmentation Based on Local ...

[11] A. Frome, D. Huber, R. Kolluri, T. Bulow and J. Malik,. “Recognizing Objects in Range Data Using Regional Point. Descriptors,” In: Proc. of Eighth European Conf. Computer. Vision, 2004, vol. 3, pp. 224-237. [12] D. Huber, A. Kapuria, R.R. Donamukkala, M. Hebert,. “Parts-based 3-d object classification”, In: Proc. of the ...

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