日本シミュレーション学会論文誌，Vol.3, No.1, pp.1-10, 2011 1 論 文 Mesh Segmentation Using the Platonic Solids † Maria Savchenko*, Luis A. Diago*, Vladimir Savchenko**, Olga Egorova* and Ichiro Hagiwara* Abstract Mesh segmentation has become an important step in model understanding and can be used as a useful tool for different applications, for instance, modeling, computer aided design (CAD), and reverse engineering. In this paper, we present a novel application of the platonic solids to find direction vectors for grouping the surface mesh elements. Normal vectors of the faces of the selected platonic solid are defined as the direction vectors. Our algorithm divides a polygonal mesh into the color regions (segments) with polygonal elements with normals that correspond to the direction vectors. Results of experiments on real 3D models demonstrate the performance and efficiency of the proposed algorithm. The original contribution in this paper is using normals of the faces of the platonic solids as the direction vectors for grouping mesh elements of the 3D surface meshes. Key words Mesh segmentation, The platonic solids, Combinatorial optimization, Feature extraction latest algorithms and evaluation results. The authors conclude 1. Introduction that each algorithm has benefits and drawbacks and future Mesh segmentation has become an important step in research works on mesh segmentation will be useful. In the model understanding and can be used as a useful tool for paper [4], a new curvature based algorithm which segments different applications, for instance, modeling, CAD, and the mesh into several regions is described. The authors extend reverse engineering. Part decomposition gains attraction since the region growing algorithm to unstructured three- it simplifies the problem with multi-part, complex objects dimensional surface meshes. After seed vertex selection and into several subproblems each dealing with their constituent sorting the vertices by their filtered absolute curvature, single, much simpler parts. Mesh segmentation is included in regions are growing from each vertex in order of ascending many mesh processing algorithms: morphing, improvement, curvatures. The algorithm makes removing small holes in compression and more. In [1] the survey and definition of the regions: all vertices not assigned to a region, but generic algorithms for the major segmentation techniques are surrounded by region vertices, are assigned to their introduced. Simultaneously, existing methods handling surrounding region. Estimation of surface features is a main triangular meshes segment models into surfaces instead of part of the segmentation process and object recognition. into meaningful parts. A number of feature detection and mesh Features are intrinsic properties of the 3D shape, which segmentation techniques have been proposed recently. In [2] include local geometry and topology [5]. The problem of an exhaustive overview of 3D mesh segmentation feature decomposition is very important and many solutions methodologies examining their suitability for CAD models were proposed for feature detection. The paper [5] discusses is presented. In the paper, a categorization of the existing 3D the extension of a scale-space decomposition approach for mesh segmentation methods is proposed and the basic feature extraction. In addition, the authors discuss the conclusions about different methods are drawn. The authors performance of the technique used to extract features from also present criteria and features used for each segmentation CAD data in polyhedral representation. They demonstrate method. The paper [3] provides a comparative study of the results of the feature extraction on noisy data. In the paper [6], the authors present a simple, automatic method to recover * Tokyo Institute of Technology the sharp features that are lost by reverse engineering or by ** Hosei University † 2010 年 9 月 29 日受付 2010 年 12 月 16 日再受付 remeshing process. Identification of chamfer triangles is 1 2 日本シミュレーション学会論文誌 第 3 巻 第 1 号 2011 年 based on the initial identification of the smooth edges and six filters that color the edges, vertices, or triangles, based on the colors of their adjacent or incident elements are used. The work [7] describes a hybrid algorithm which while Fig. 1 The platonic solids. denoising regularizes triangle meshes on flat regions for further mesh processing preserves crease sharpness for image denoising problem [15]. faithful reconstruction. A clustering technique, which combines K-means and geometric a priori information, is 2. The platonic solids suggested. When K-means does not give a very satisfactory The platonic solids, described by Plato in his Timaeus, vertex partition, the authors propose hierarchical K-means. are some of the simplest, if not the simplest, of polyhedra. A Clustering repeats on the unresolved cluster until all sub- platonic solid is a convex polyhedron composed of convex, clusters consist of vertices with only one type of feature - congruent, regular polygons (faces). There are five such solids corner, edge, or non-feature. In the paper [8] characterizations (see in Fig.1(from left to right)): a tetrahedron - a four-sided and techniques for detecting geometric features in surface shape composed of four equilateral triangles, a cube - a six- mesh are presented. A method using line extraction for the sided shape made of squares, an octahedron - an eight-sided purpose of hexahedral mesh generation is described in [9]. shape made of triangles, a dodecahedron - a twelve-sided In the paper, surface mesh groups are defined by grouping shape composed of pentagons, an icosahedron - a twenty- the mesh faces on their degree of adaptation. In [10] authors sided shape made of twenty triangles. present algorithm for splitting the input mesh into reliefs. It has been well proven geometrically and topologically They do this by using the set of defining vectors: the simplest that the five known platonic solids are the only which can case - the set of 6 vectors pointing along the coordinate axes possibly exist in three-dimensional space. They have been of a cube. However, the choice of the basic vector is an analysed in great topological detail and have been completely interesting problem, which was not considered yet. In [11] defined mathematically. the iterative merging of adjacent triangles of the 3D mesh is The platonic solids have served mathematicians and presented. The surface is segmented into regions with a physical scientists in numerous applications, and their new similar normal property. A feature-shape model is generated applications could be found to solve engineering problems. by extracting the feature lines, which are defined as a set of mesh-lines shared by two different surface-mesh groups. In 3. Segmentation algorithm based on the feature line extraction algorithm [12], accurate method to platonic solids estimate the discrete curvature is used. The approach is based We suggest a new and simple method for mesh on utilizing discrete differential operators on piecewise linear segmentation based on the platonic solids. Our method meshes which allows avoiding costly preprocessing steps provides the decomposition of the mesh into groups of the such as surface fitting technique or constructing an implicit mesh elements according to the information, which is coming surface [13]. The paper [14] presents an algorithm to extract from the normals of the faces (the direction vectors) of the features lines from a point cloud without curvature estimation. platonic solid. The angle differences between the normals of The algorithm uses a first order segmentation for receiving each mesh element and the normals of the faces of the platonic initial information about the location of the feature lines. solid are used as a criterion for grouping elements. Those The problem of denoising (or smoothing) 3D meshes has direction vectors of the platonic solid are selected to form a received a lot of attention in recent years due to the increasing group of elements (a segment), for which the angles θ with use of 3D scanning technology. Meshes supplied by laser the element normals nx, ny, nz are minimal. Input model is scanning devices often carry high-frequency noise in the represented as a polygonal mesh. Output model consists of a position of the vertices, so a mesh smoothing algorithm is set of disjoint, constituent segments, whose union is identical required to rapidly remove noise while preserving real details to the input model. Direction vectors of the platonic solid in the acquired data. Most techniques for 3D mesh smoothing and embedding the platonic solid into X, Y, Z- coordinates have predecessors in the literature on the significantly simpler of a given model are shown in Fig. 2. 2 M. Savchenko, L. Diago, V. Savchenko, O. Egorova and I. Hagiwara：Mesh Segmentation Using the Platonic Solids 3 4. Extensions of the algorithm 4.1 Local mesh denoising technique Mesh denoising (correction of the “alien” colors) is produced in each separated segment. By the separated (a) (b) segment we mean a set of mesh elements that have the same Fig. 2 Direction vectors (the cube) (a); Embedding the platonic color and are topologically conjugated (one color may define solid (the tetrahedron) into the X, Y, Z-coordinates of the model (b). several distinct segments). This segment can possess the “alien” elements. So, the correction technique is based on improving of these “alien” elements. Our denoising technique includes three steps: (1) correcting all elements from the list of “alien” elements, (2) updating the list of “alien” elements, (3) repeating this process iteratively until the list of “alien” elements will be empty. The color correction is done by the rotation of the “alien” planes according to a predefined normal, which is calculated as follows. The neighbors of each “alien” element can be elements with the color of the given segment and elements of the “alien” colors. We precede averaging only normals of the elements with the color of the given segment. Rotation of the “alien” normal N to the averaged normal of their Fig. 3 Scheme of the algorithm.