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FROM TRIANGULATION to SIMPLEX MESH and VICE-VERSA a Simple and Efficient Conversion

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FROM TRIANGULATION to SIMPLEX MESH and VICE-VERSA a Simple and Efficient Conversion FROM TRIANGULATION TO SIMPLEX MESH AND VICE-VERSA A Simple and Efficient Conversion Francisco J. Galdames1;3 and Fabrice Jaillet2;3 1Department of Electrical Engineering, Universidad de Chile, Av. Tupper 2007, Santiago, Chile 2Universite´ de Lyon, IUT Lyon 1, Computer Science Department, F-01000, Lyon, France 3Universite´ de Lyon, CNRS, Universite´ Lyon 1, LIRIS, SAARA team, UMR5205, F-69622, Lyon, France Keywords: Simplex Mesh, Triangle Mesh, Optimized Surface Interpolation, Surface Mesh Conversion. Abstract: We propose an accurate method to convert from a triangular mesh model to a simplex mesh and vice-versa. For this, we are taking advantage of the fact that they are topologically duals, turning it into a natural swap between these two models. Unfortunately, they are not geometrically equivalents, leading to loss of information and to geometry deterioration when performing the conversion. Therefore, optimal positions of the vertices in the dual mesh have to be found while avoiding shape degradation. An accurate and effective transformation technique is described in this paper, where we present a direct method to perform an appropriate interpolation of a simplex mesh to obtain its dual, and/or vice-versa. Our method is based on the distance minimization between the local tangent planes of the mesh and vertices of each face. 1 INTRODUCTION unfortunately in this case, mesh smoothing is gener- ally high; original shape (curvature) and volume is not Deformable model techniques are widely used in properly respected. In (De Putter et al., 2006), an it- image segmentation tasks. Among these models, erative curvature correction algorithm for the dual tri- simplex meshes are valuable candidates (Delingette, angulation of a two-simplex mesh is proposed. Their 1999), for their favorable characteristics in this type solution provides optimal error distribution between of modeling, as its convenient way to model inter- the two dual surfaces while preserving the geometry nal forces. With this type of meshes, as with trian- of the mesh, but at the price of an iterative global min- gulations, any topology can be described. Triangu- imization over the whole meshes. lations are meshes of triangles in which, if they are In this paper, a new technique is presented, achiev- manifolds, each triangle has three neighboring trian- ing reasonable computation cost and minimal loss of gles, while simplex meshes are meshes of polygons geometric information. From a geometric point of in which each vertex has three neighboring vertices. view, the problem can be reduced to finding an in- Simplex meshes and triangulations are topologically terpolation of the center of each face, and to build duals (Delingette, 1999), and this allows us to natu- the dual mesh accordingly to these points. We pro- rally obtain a simplex mesh by applying a dual oper- pose to use a geometric interpolation, based on the ation to the triangulation, and vice-versa. Moreover, distance to the tangent planes of the vertices of each there are some tasks for which simplex meshes are face. A similar measure has been successfully used not suitable, and thus triangulated meshes are more in (Ronfard and Rossignac, 1996) to perform trian- adapted. Examples of these applications are: genera- gular mesh simplifications. Another equivalent mea- tion of volumetric meshes, rendering and calculation sure has been employed, using this time a summation of area. to obtain a quadratic error in (Garland and Heckbert, Currently, the most common way to perform con- 1997; Heckbert and Garland, 1999). In a more recent versions between triangulations and simplex meshes work, a method for refining triangulations has been is to determine the set of vertices for the final mesh developed (Yang, 2005) using a similar measure. It as the gravity center of each face of the initial is worth pointing out that our global objective is to mesh (e.g. (The Insight Segmentation and Registra- perform a transformation between meshes, and not to tion Toolkit (ITK), 2011)). This technique is fast, but refine them. However, we mainly got inspiration from J. Galdames F. and Jaillet F.. 151 FROM TRIANGULATION TO SIMPLEX MESH AND VICE-VERSA - A Simple and Efficient Conversion. DOI: 10.5220/0003851801510156 In Proceedings of the International Conference on Computer Graphics Theory and Applications (GRAPP-2012), pages 151-156 ISBN: 978-989-8565-02-0 Copyright c 2012 SCITEPRESS (Science and Technology Publications, Lda.) GRAPP 2012 - International Conference on Computer Graphics Theory and Applications this last work. a restricted number of entities is defined, the simplex The paper is organized as follows. In section 2, angle and the metric parameters (Delingette, 1999). we present essential background on simplex meshes, The mesh deformation can be controlled by using their characteristics and relationship with triangula- these entities. tions. The main part concerning the interpolation To perform transformations in any direction be- method used to find the dual mesh is explained in sec- tween these two types of dual meshes, we have to find tion 3. Application of this method to swap between an associated vertex qu of the dual mesh M2 for each meshes is shown in sections 4 and 5, where details face fu of the initial mesh M1. When dealing with can be found for each swap direction. Finally, some triangulations, faces are triangles; and conversely for results are exhibited in section 6, followed by conclu- simplex meshes, faces are polygons whose vertices sions in 7. are generally not coplanar. The resulting mesh M2 should have a regular shape and preserve the geome- try defined by M1, what is far from being straightfor- 2 TRIANGULATION VS. ward. For trying to maintain the geometry, we can im- pose that qu remains close to the tangent planes pi of SIMPLEX MESH each vertex pi defining the face fu. Constraining M2 to have a regular shape, can be achieved by choosing A simplex mesh can be seen as the topological dual qu close to the center of the face fu, i.e. minimize the of a triangulation, each vertex of the simplex mesh distance between qu and all pi. Therefore, we must corresponding to a triangle in the dual triangulation minimize the distance between a point qu and a set of (Fig. 1). However, simplex meshes and triangulations points and planes. A method to achieve the aforemen- are not geometrically duals. Their geometry is de- tioned goal is explained in the next section. termined by the coordinates of their vertices; never- theless, the number of vertices is different between a simplex meshe VS and a triangulation VT . The Euler’s characteristic for a triangulation without holes and its 3 INTERPOLATION BASED ON dual simplex mesh states: TANGENT PLANES VS VT − = 2(1 − g); (1) The equation of a plane can be denoted as A · p = 0, T 2 where A = [a;b;c;d] and p = [xp;yp;zp;1] is a point where g is the genus of the mesh. As the sets of coor- lying on this plane. The coefficients a;b;c are the −! dinates have different sizes for a triangulation and its components of the unit vector N normal to the plane, −! dual simplex mesh (different number of vertices), no accordingly a2 + b2 + c2 = 1, and d = − N · p. For q homeomorphism can be constructed between them. an arbitrary point in the space, jA · qj is the distance to the plane. Considering now a set of planes pi represented by Ai · p = 0 (i = 1;:::;L), the distance between T any point q = [x;y;z;1] and each plane pi is jAi · qj. On the other hand, consider a set of points p j ( j = 1;:::;M). If we want to find the point q minimizing its distance to planes pi and points p j, the function to be considered follows: L M 2 2 D(q) = ∑ ai jAi · qj + ∑ b j q − p j (2) i=1 j=1 Figure 1: Simplex meshes and triangulations are topolog- where ai and b j are positive weights for the distance ical but not geometrical duals. White dots: triangulation to the planes (in order to respect geometry and cur- vertices; Black dots: simplex mesh vertices. A vertex of the vature) and points (controlling shape regularity), re- simplex mesh pi and its three neighbors pN1; pN2; pN3 are spectively. Equation (2) can be rewritten in matrix shown. form as: D(q) = qT Qq (3) Simplex meshes can be used to efficiently im- plement segmentation methods based on deformable where L M models. Each vertex of a simplex mesh has three T Q = ∑ aiAi Ai + ∑ b jQ j (4) neighbors pN1(i); pN2(i); pN3(i) (Fig. 1); between them, i=1 j=1 152 FROM TRIANGULATION TO SIMPLEX MESH AND VICE-VERSA - A Simple and Efficient Conversion and ʌʌ⃗ ͈ͥ pα ͈ʌʌ⃗ 2 3 ͦ 1 0 0 −x j π p p π2 6 0 1 0 −y j 7 1 1 pβ 2 Q j = 6 7 (5) 4 0 0 1 −z j 5 −x −y −z x2 + y2 + z2 j j j j j j Figure 2: Solution of equation (2) as the affine combination Since Q and AT A are symmetric matrices, then of the generalized intersection of planes pi (pa) and the av- j i i erage of all points p (p , here for = ). Q is also symmetric and can be written as: i b bi b 2q q q q 3 11 12 13 14 where w is a free positive parameter controlling the q q q q Q = 6 12 22 23 247 (6) smoothness of the interpolation, and c is the average 4q q q q 5 u 13 23 33 34 position of the vertices p . Replacing q by its estimate q q q q i 14 24 34 44 q¯ = [x¯;y¯;z¯]T in eq.
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