SHAPE SEGMENTATION FOR SHAPE DESCRIPTION

Transcription

1 SHAPE SEGMENTATION FOR SHAPE DESCRIPTION Olga Symonova GraphiTech Salita dei Molini 2, Villazzano (TN), Italy Raffaele De Amicis GraphiTech Salita dei Molini 2, Villazzano (TN), Italy ABSTRACT In this paper we present a new approach for segmentation of 3D models. The goal of the proposed approach is the decomposition of a model into large simple and small complex components. The top-down decomposition is achieved through the iterative bisection of the model. The bisection process is represented in the hierarchical tree structure. Further the obtained segments are checked on the possibility to be merged into larger simple segments. The results of such segmentation can be used later in the fields of shape analysis and shape description. KEYWORDS Model Segmentation, Hierarchical Subdivision. 1. INTRODUCTION AND RELATED WORK The problem of shape retrieval becomes more challenging due to the increasing number of 3D models available on the Internet. As a consequence the need of methods for shape description arises. Shape descriptors should fully represent a shape but at the same time should be compact enough to allow efficient retrieval. A good solution could be a descriptor obtained as the result of decomposition of a model into components corresponding to meaningful parts of the shape, extraction of the most distinctive features and their union into one compact structure. This structure can describe the connectivity relations between the features represented by geometrical or morphological characteristics. However, it can be difficult to describe the shape of all components in a unique way. Mortara et al. (2004) categorized the extracted segments of a model into three groups having one, two and more boundaries. For each group the authors provided several different shape characteristics. The other shape segmentation technique which object is further application in retrieval process was proposed in (Zhou and Huang, 2004). The efficiency of any segmentation technique depends on its future application and thus could be evaluated only according to the earlier posed target. In (Shamir 2006) and (Attene 2006) the authors give a broad overview of segmentation techniques depending on a type of extracted segments and a target of a technique. In our previous work (Symonova et al 2006) we use a Reeb graph as a shape descriptor for models with an arbitrary genus. A Reeb graph is defined as the quotient space of the Morse function defined on the manifold representing a model. The process of Reeb graph construction includes insertion of a number of contours inside the mesh of the model and tracing their evolution. Insertion of the contours is analogous to segmentation of the model, where each segment is represented by a node in the graph. Moreover, due to the fact that the Reeb graph describes the topology of the model, its nodes should reveal topological changes of the manifold representing the model and the edges of the graph should represent connections between them. Consequently the preliminary segmentation should decompose the shape into homotopical components. The current work is the extension of (Mortara 2004) and (Patanè 2004) where the authors segmented a model into components with different number of boundaries. Segments with one boundary correspond to the

2 minimal or maximal critical regions. A segment containing two boundaries corresponds to a regular region. For these two types of segments the further shape analysis is provided in (Mortara 2004), so that they are classified as a planar, sharp or blend segment with one boundary, or cylindrical or conical segment with two boundaries. Segments with three and more boundaries represent branching parts. They correspond to saddle areas. No further shape analysis is provided for this kind of segments. Keeping in mind these observations we conclude that regular regions as well as maximum and minimum regions are easier to be characterized geometrically than saddle regions. On the other hand, in many areas, where shape analysis is applied, it is necessary to reach the compromise between exhausting shape description and better shape compression. For this reason we consider that it is important to maximize the regions which can be easier described geometrically (from now on called simple) and to minimize the regions difficult for the further analysis and representation (called complex). In the current work we follow the idea of Mortara (2004) to divide a model into segments with different number of boundaries. Moreover, we take into account the fact that the shape of segments with one and two boundaries is easier to analyze. As a consequence, we propose a new iterative method for segmentation of a model which aims to maximize segments with one and two boundaries by minimizing segments with more boundaries. 2. ITERATIVE MODEL SEGMENTATION In this section we give the theoretical background and practical details of the proposed approach for model segmentation. 2.1 Theoretical Background Consider a smooth function f defined on a compact manifold M and suppose that an interval = { p p M, f ( a) f ( p) f ( b)} does not contain a critical point of f. Then two level sets f 1 ( a) and M ab f 1 ( b) are diffeomorphic. Moreover, if the level sets f 1 ( a) are regular values of f (they are diffeomorphic to a circle), then the segment Mab is diffeomorphic to a finite cylinder with two boundary components. Consider now what happens when there is a critical point between two level sets f 1 ( a) and f 1 ( b). Theorem 1 (Fomenko and Kunii 1997) Let the function f be Morse function on a compact smooth n-manifold M (without a boundary), and lower set = { x f ( x) a}. M a 1. Suppose that 0 is a critical value of f, and that there is only one critical point P 0 in Then M ξ can be obtained from M ξ by attaching an n-dimensional λ -cell. M ξ, ξ, with index λ. 2. The manifold M can be built by starting with the empty set, and attaching in succession a finite number of n dimensional λ -cells, one λ -cell for each critical point of f of index λ. That is, f determines a cell decomposition of M. From the theorem it follows that two level sets f 1 ( a) of the interval Mab have different homotopy types if the interval contains a critical point. If the boundary of the interval Mab (a < b), denoted as M, is the union of two level sets f 1 ( a), then if f 1 ( a) or f 1 ( b) is empty then there is a minimum, or respectively maximum in M M, (Hirsch 1976). If we trace the changes in the sequence of the level sets f 1 ( p) of the interval Mab of a 2D manifold M ( a p b) which are diffeomorphic to a circle, then each minimum corresponds to the introduction of a circle, i.e. attaching a 2-cell (Wallace 1968). Similarly, according to the Theorem 1, extinguishing of a circle between two level sets f 1 ( a) correspond to the maximum occurrence in the interval Mab, or, in other words, to attaching of 0- cell. Finally, the union of two circles in the interval Mab, with consequent formation of a new circle, corresponds to a saddle point on it, i.e. attaching 1-cell. Similarly, there is a saddle in the interval Mab when there occurs a disconnection of one circle, which forms two new circles (the situation opposite to the described before). Due to the fact that a Morse function can be defined on every compact smooth manifold, Theorem 1 implies that

3 any compact smooth manifold has cell decomposition (Fomenko and Kunii 1997). Further we present the algorithm for cell decomposition of the manifold representing a 3D model. 2.2 Hierarchical Mesh Segmentation The idea of hierarchical segmentation consists in associating a segment of the mesh to a node of the tree, where the tree represents the segmentation process. Initially the whole model represents a unique segment, and it is associated to the root of the tree. After subdivision of a segment into several new components they will be stored as its child nodes. We choose left child-right sibling tree structure to represent iterative segmentation process. Our choice is determined by the fact that we do not know in advance how many new segments we obtain at each subdivision step or, in other words, we cannot forecast the number of child nodes created at every iteration. We perform mesh segmentation depending on the value of shape measuring function defined on a model. We experimented with height function, distance from barycenter and integral geodesic distance functions (Hilaga 2001). The process of mesh segmentation starts from inserting an initial number of contours N into the mesh representing the root node. The inserted N contours subdivide the mesh into N+1 intervals M = ( M 0 = M min, M1) ( M1, M 2)... ( M N, M N + 1 = M max ). The process of contour insertion is analogous to the one described in (Attene 2003). We enumerate all contours in order to use contour indices in the further merging process. All inserted contours are isotopic to a circle. If we insert a contour exactly at a level where the measuring shape function possesses a critical value, then the contour is degenerate and is represented by a point, a disk or by tangent circles. In this case we slightly shift the current isolevel and insert a new contour. After insertion of contours we subdivide the mesh into separate segments. We require that two segments do not have any common vertex, edge or triangle. To satisfy the requirement we duplicate the vertices and edges comprising the contours and update their membership for the adjacent triangles. At this step the original mesh consists from N+1 or more connected components. We extract each connected component from the mesh and associate it as a child node. From here the iterative segmentation starts. The system calculates the genus g = (E-V-T-B+2)/2 (Mäntylä 1988) for each leaf node of the hierarchical tree, where E, V, T, and B are the number of edges, vertices, triangles, and boundaries correspondingly. If a segment has non-zero genus or it has more than two boundaries then it is classified as a complex region. For such segments a new contour is inserted in the middle. The inserted contour subdivides the component into two or more new segments. They are stored as child nodes of the processed segment. In order to make more efficient the use of memory, we do not keep the triangle mesh representing the bisected component, because now it can be represented as the union of the triangulations of its children. The iterative bisection continues until all components represented as leaf nodes of the tree are classified as simple or have the size less then the predefined threshold. Together with the triangulation which represents a segment we store the list of boundary indices. The list will be used in the further process of merging (see Figure 1(a)). Figure 1. Hierarchical segmentation. (a) tree left child - right sibling for representation of iterative segmentation; (b) segmentation and merging processes When the iterative bisection terminates, simplification of the obtained segmentation is performed. Simplification merges two adjacent simple components which could have been created on the previous step of bisection. Two segments can be merged if they have a common boundary, or in other words, if the

4 intersection of their lists of boundary indices B i is not empty B i B j. The merging process is inverse to the process of mesh division. Figure 1(b) illustrates the iterative segmentation and further merging processes. In this way, the bisection step increases the number of segments but at the same time minimizes the complex areas. The simplification step reduces the number of the created segments by enlarging the size of simple areas. As the result of this process the model is segmented into large simple and small complex components. 3. RESULTS AND DISCUSSION In this section we show some results of the described segmentation. We used the height function, distance from the barycenter and integral geodesic distance as measuring shape functions for segmentation. We tested the proposed approach on models taken from different domains. Some of them were downloaded from Aim@Shape Shape Repository (Aim@Shape 2006). The results of our experiments showed that for the CAD models the better segmentation is obtained using the height measuring function. Here, under the better results we mean the segmentation which gives maximally large simple segments. Such results can be explained by the fact that CAD models are usually correctly oriented during the design process. However such segmentation is not invariant to the position of a model in space and, thus, requires a preliminary alignment. The Figure 2(a) shows the results of the segmentation of the CAD model using height measuring function and the function of the distance from barycenter. On the contrary, for free form models, like animals and toys, better results are obtained when the measuring function is the distance from the barycenter or integral geodesic distance. Figure 2(b) illustrates segmentation of the free form model using all three measuring functions. Figure 2. Model segmentation. (a) segmentation of the CAD model using height and barycenter distance functions; (b) segmentation of the free form model using height, distance from the barycenter and integral geodesic distance functions. The measuring function of the distance from the barycenter as well as integral geodesic distance function gives the advantage of being invariant to rotation. Figure 3(a) shows the segmentation of Homer model randomly rotated in space. Integral geodesic distance function has the advantage to provide mesh segmentation invariant to the posture of a model. Figure 3(b) shows the result of segmentation of the free form model in different poses using integral geodesic distance function. Figure 3. Model segmentation a) the model is located differently in space (the distance from the barycenter function); b) the model has different poses (integral geodesic function). 3.1 Conclusions and Future Work In this work we presented the new part-type segmentation technique. The target of our method is to decompose a model into wherever possible large components, which can be easily characterized by

5 geometrical parameters in the future applications, and smaller more complex components. We consider a simple component to be topologically equivalent to a disk or a cylinder, whereas the complex components are those having branching shape. The proposed top-down approach takes a whole model as a unique component in the beginning and continues to bisect it extracting new connected components. The process terminates when the complex segments have the size less then the predefined threshold. We implemented the proposed approach using the height, distance from the barycenter and integral geodesic distance measuring functions. The future work will focus on the investigation of a possible application of the proposed technique in the field of the shape retrieval. For the shape description in the retrieval process we use Reeb graph, where each node represents a component obtained after running the segmentation process described in this work. Figure 4 shows our first results. Figure 4. Reeb graphs for free form and CAD models. The number of nodes of the resulting Reeb graph is minimized due to merging simple segments. In future we will also focus on the shape analysis of each segment separately. ACKNOWLEDGEMENTS This work has been supported by the CFP6 IST NoE AIM@SHAPE. REFERENCES Aim@shape. Attene, M. et al, 2003, Shape understanding by contour-driven retiling. The Visual Computer, 19(2-3): Attene, M. et al, 2006, Segmentation and shape extraction of 3d boundary meshes. Computer Graphics Forum, State-ofthe-Art report. Proceedings Eurographics Fomenko, A. T. and Kunii, T. L, Topological Modeling for Visualization. Springer-Verlag Tokyo, Inc., Hong Kong, China. Hilaga, M. et al, 2001, Topology matching for fully automatic similarity estimation of 3d shapes. In SIGGRAPH 01: 28th Annual Conference on Computer graphics and interactive techniques, New York, USA. Hirsch, M. W, Differential Topology. Springer-Verlag New York Inc., New York, USA. Mäntylä, M, Introduction to Solid Modeling. Computer Science Press, Rockville, Maryland, Mortara M. et al, 2004, Blowing bubbles for the multi-scale analysis and decomposition of triangle meshes. Algorithmica, Special Issues on Shape Algorithms, 38(2): Patanè, G. et al, 2004, Para-Graph: Graph-Based Parameterization of Triangle Meshes with Arbitrary Genus. In: Computer Graphics Forum, vol. 23 (4) pp Shamir A, 2006, Mesh segmentation - a comparative study. In Proceedings Shape Modeling International (SMI06).Washington, DC, USA. Symonova O. et al, 2006, Topological Descriptor for CAD Model with Inner Cavities. In Proceedings of the 4th Eurographics Italian Chapter Conference. Catania,Italy. Wallace, A., Differential Topology: First Steps. W.A. Benjamin Inc., New York, USA. Zhou, Y and Huang, Z, 2004, Decomposing Polygon Meshes by Means of Critical Points.In Proceedings of the 10th International Multimedia Modelling Conference. Washington, DC, USA.