Adaptive Mesh Extraction using Simplification and Refinement

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1 Adaptive Mesh Extraction using Simplification and Refinement Adelailson Peixoto and Luiz Velho Abstract. This work presents a method for multiresolution mesh extraction with important mathematical properties. The generated mesh represents a regular surface from 3D Euclidian space. The input surface may be specified either implicitly or directly as a volumetric object. The method applies simplification operations to obtain a low resolution initial mesh and applies refinement operations to obtain a multiresolution representation from initial mesh. These combined operations provide nice properties to the mesh. 1. Introduction In several areas like medical visualization, geosciences and others, mesh extraction is crucial for many applications. In these areas, initial data may be represented as a volume data that encodes samples of a function over a grid. Each function value defines an iso-surface inside the volume that may be extracted as a mesh. Most existing algorithms for mesh extraction are classified as voxel based and as cross section based algorithms. The former generates the mesh by creating triangles inside voxels that intersect the iso-surface [14]. The latter generates triangles between consecutive pairs of parallel slices, where each slice contains a set of contours that represent the iso-surface [1,2,3,5,7,8,11]. In general, both methods generate meshes with constant resolution and uniform sampling. Because large volume are very common, these methods may result in meshes with millions of triangles. This may encumber several applications like rendering, denoising, network transmission and others. Multiresolution mesh representation helps to overcome constant resolution drawbacks [6]. It is defined as a set of meshes {M 0,..., M n } such that the number of vertices, edges and triangles of mesh M i increases Curve and Surface Fitting: Saint-Malo XXX (eds.), pp Copyright oc 2002 by Nashboro Press, Nashville, TN. ISBN XXX. All rights of reproduction in any form reserved.

2 2 A. Peixoto and L. Velho monotonically with i and there is a dependency relation between triangles at two subsequent levels i and i + 1. In general multiresolution mesh is obtained by simplification or by refinement methods. Simplification methods act over a supersampled initial mesh and coarsify it, creating lower resolution meshes [9,12,13,17]. Initial mesh may arise from 3D scanning techniques or from iso-surface extraction, like Marching Cubes (MC) [14]. One way to obtain the multiresolution is from a repeated decimation process applied to initial mesh, where vertices are iteratively removed. The problem is that most decimation algorithms produce highly irregular meshes, that are undesirable to many applications. An alternative to building multiresolution through decimation is remeshing, where the problem of an oversampled mesh is repaired through remeshing techniques. Although these methods are compelling, in many applications they seem to be inefficient and inelegant in the settings of iso-surface extraction from volumes. If one desires to extract an adaptively sampled hierarchical mesh, e.g., it is more natural that samples are adaptively added to the mesh in accord with the specific features of the underlying iso-surface. Refinement is applied over an initial coarse mesh and progressively computes higher resolution meshes [10,20]. In many applications it is more suitable than simplification: e.g. in network transmission it is more natural transmit first low resolution information, followed by details. So, combining simplification and refinement seems to be a good strategy to generate meshes with nice properties and available to more aplications. This work presents a combined simplification and refinement method such that the generated mesh presents nice properties (Section 2). 2. Algorithm Overview In our approach, the mesh generation is performed in two main steps: simplification and refinement. The simplification step generates a coarse mesh, called base mesh, containing the correct topology of the iso-surface. First, we generate a supersampled and structured set of points G, called the connectivity graph, which contains the geometry and topology of the iso-surface. Now we construct a set of disks over G such that the centers of the disks define vertices of the base mesh. From this set of disks we generate a set, called the Voronoi covering, that will be used to triangulate the base mesh. The simplification step is described in details in Section 3. The refinement step is applied to the base mesh to generate a mesh hierarchy. It consists of two operators that are applied simultaneously to the mesh: subdivision and adaptation of triangles. The former subdivides triangles, adding new vertices, edges and triangles to the mesh. The latter computes positions of new vertices and adjust them. Refinement is described in Section 4. Figure 1 shows a general scheme of our method. The proposed method presents a set of important properties: gener-

3 Mesh Extraction using Simplification and Refinement 3 Fig. 1. General Scheme. ated mesh exhibits correct topology; triangles have a good aspect ratios; meshes have a semi-regular structure; the method applies to surfaces with boundary and with multiple connected components; the mesh has a multiresolutionrepresentation; the mesh converges correctly to the iso-surface; refinement is adaptive; triangles subdivision is consistent. 3. Simplification This step is based on [10], where extracted multiresolution meshes produce triangles with a good aspect ratio. We employ a similiar process to extracted a corse mesh with constant resolution, instead of a multiresolution representation. Connectivity Graph A surfel is a surface patch formed by the intersection of the iso-surface with a voxel [10,20]. A node is the intersection of a surfel with a voxel edge. A c-vertex (candidate vertex) v is associated to each surfel such that the coordinates of v are the average of the node coordinates of the surfel associated to v (Figure 2a). Two c-vertices are neighbors if their associated surfels share at least one node. The connectivity graph G is defined by all c-vertices, and its edges are defined by pairs of neighbors c-vertices. The weight of each edge is the Euclidian distance between the c-vertices it connects. G represents the highest resolution of the iso-surface. Disk Covering Given a point p S, wheres is a regular surface, there exist an open neighborhood of p homeomorphic to a planar disk [4]. This neighorhood D S is called a disk over S. If for all p S there is a disk D over S with p D, we say that there is a disk covering C S associated to S. The goal is to construct a disk covering C G over the graph G. Toachievethe

4 4 A. Peixoto and L. Velho Fig. 2. a) Surfel and c-vertex. b) Disk constraints. c) Region classification. mesh properties, two topological constraints are imposed to C G (Figure 2b): first, C G is an independent disk covering, what means that no disk may contain the center of another disk. The second is that the radii of any two intersecting disks don t differ by more than a factor of two. The C G construction is made from discrete definitions of geodesic and disk. Ageodesicgeod u,v, u, v G, isacurveoverg composed by all c- vertices that minimize the distance between u and v. The length of geod u,v is denoted by leng u,v. A disk Dv, r with center v and radius r, isasetof c-vertices w G such that leng w,v r. The computation of a disk Dv r begins at the c-vertex v, and goes on, by a propagation process, until reaching all c-vertices w, withleng w,v r. The implementation of disk generation may be found in [10,16]. In order to assume that disk Dv r is homeomorphic to a planar disk, it is necessary to verify the occurrence of topological events during disk generation. A topological event occurs if the disk boundary self-intersects, or if the disk boundary vanishes. Let B G be the boundary of disk Dv r,andletu B be the closest c-vertex to v. Consider the sets V u = {x Dv x r is neighbor of u} and B u = {x B x is neighbor of u}. There is a boundary self-intersection if the set V u is not connected, and there is a boundary vanishing if B u = B. Inorderto maintain radius constraint when a topological event occurs, the radius of the disk must be halved. Let r max be the maximum radius allowed in disk covering C G. C G is constructed by classifying c-vertices of G in two types: forbidden or F, indicates c-vertices that cannot be taken as center of any disk, and allowable i or A i, indicates c-vertices that can be taken as center of a disk of radius r max /2 i (Figure 2c). Initially all c-vertices from G are classified as A 0. Suppose v is a A i c-vertex (v may be the center of a disk with radius r = r max /2 i ). But during generation of Dv r, topological event may occur, and the resulting disk Dv r will be halved until the event disappears, which means that the disk radius r r max /2 i. In order to maintain radius constraints, the region that surrounds a disk of radius r max /2 i must be classified as A i. An implementation of C G is shown in [10,16].

5 Mesh Extraction using Simplification and Refinement 5 Fig. 3. a) Incorrect Cylinder. b) Disk covering and Voronoi covering subdivision. Voronoi Covering Given a disk Dv Rv C G,aVoronoi cell VC G (v) is the set of c-vertices w G, such that for any disk Du Ru C G, leng w,v <leng w,u.avoronoi Covering VC G is the set of all Voronoi cells defined in C G. Two cells VC G(v) andv C G (u) areneighbors if there exist c-vertices x VC G(v) andy V C G (u) such that x and y are neighbors in G. The set of c-vertices x VC G(v) with the property that there exist y V C G(u) such that x and y are neighbors in G is called the u-boundary of VC G(v). Ac-vertexx VC G (v) is called a Voronoi vertex if it belongs to at least two separated boundaries of VC G (v). Voronoi vertices are very important to generating dual of Voronoi covering VC G. Sometimes, because of the amount of cells in VC G, it is not possible to reconstruct the base mesh correctly. In Figure 3a, the amount of disks (and cells) aren t sufficient to reconstruct a cylinder base mesh correctly. This problem didn t appear in [10] because C G was sufficiently refined to generate a final adapted mesh. To solve this problem here, we have not permitted the occurrence of violated cells in the Voronoi covering, i.e., cells with disconnected boundaries and cells with less than three boundaries. If they occur, we subdivide them: first we subdivide the disk covering, by halving disks that correspond to the violated cells and to their neighbor cells. When disks are halved, some forbidden c-vertices become A i and new disks must be created. After updating C G,weupdateVC G (Figure 3b). All this refinement process is applied locally to a neighborhood of violated cells, it means we don t need to recompute all cells of VC G. The subdivision proceeds until there are no more violated cells. [16] shows an implementation of VC G. Triangulation From the definition of a regular surface [4], it follows that the VC G has the same structure of a planar Voronoi diagram, so its dual is equivalent to the Delaunay triangulation. A triangulation may be computed by traversing all neighbors of each cell VC G (v) and generating all triangles sharing c-vertex v. To do this, neighbor cells of VC G (v) must be sorted. The simple projection of centers at a tangent plane is not a good way to sort cells: it may not reflect the correct order.

6 6 A. Peixoto and L. Velho Given a cell C cur with center c, we want to construct all triangles sharing c. For this, we must take a cell N cur with center n 1 and neighbor of C cur. We must compute a cell N nxt,neighborofc cur and of N cur simultaneously, with center n 2. The first triangle is c, n 1,n 2. The second triangle is computed by making N cur equals to N nxt and then computing anewn nxt, as before. We repeat this process until all triangles sharing c are created. It is important to take into account in the computation of N nxt the cases where C cur is on the boundary. Another important observation concerns the computation of holes: besides requiring N nxt to be simultaneous neighbor of C cur and N cur, we need the existence of neighboring Voronoi vertices c C cur,n 1 N cur and n 2 N nxt ; if there are no such Voronoi vertices, then C cur,n cur and N nxt form a hole. [16] shows an implementation of the triangulation. 4. Refinement Refinement is applied to a triangle T if its variance V T (d) >δ[20]. V T is computed over a discrete set of distances between T and the iso-surface: V T (d) = A(d 2 ) A(d), where A denotes the average of its argument. Refinement is applied simultaneously in two steps: subdivison (changes the mesh connectivity), and adaptation (changes the mesh geometry). Subdivision Triangle subdivision is based on variable resolution 4-8 meshes [19], where mesh connectivity is equivalent to [4.8 2 ] Laves tilings [18,19]. These tilings are defined by right isoscele triangles with vertices of valence 4 and 8, alternately. Its basic structure is a pair of such triangles that form a square as the basic block. The crucial advantage of this scheme is that it uses bisection as the basic refinement operation, rather than the more complex and commonly used face and vertex split. A bisection operation is applied to the internal edge of a basic block, subdividing the block into 4 triangles. The structure of 4-8 meshes is suitable for adaptive refinement because, even for non-uniform refinement, the mesh is still guaranteed to be conforming. A general framework for 4-8 meshes may be found in [18,19]. This refinement process permits the creation of semi-regular meshes. Adaptation When an edge e =(v 1,v 2 ) is subdivided by bisection, a new vertex v is added to the mesh. The coordinates of v must be computed such that triangles adapt to the iso-surface. The operations we ve applied to the mesh geometry are: moving vertices toward iso-surface, and reparameterizing vertices to improve mesh smoothness.

7 Mesh Extraction using Simplification and Refinement 7 Movement of vertex v (with coordinates p) towards the iso-surface S is performed by appling a local operator defined by F S (p) = T S (p) T S (p). This operator was inspired by [15], where T S is an implicit function. In our definition of F S, T S (p) is the signed distance between p and the isosurface, and T S (p) is the gradient that points in the normal direction of the iso-surface. If the subdivision edge e intersects the medial axis of the iso-surface, it is possible that the new vertex v will move in an incorrect direction. In the general case, this F S formulation is not able to decide the correct direction in which v moves. In [15], F S works well because it is guaranteed that vertices are in a neighborhood of the iso-surface, and subdivision edges don t intersect the medial axis. In [15] F S is applied to a constant resolution mesh. To move new vertices correctly we ve applied the geodesic geod v1,v2, instead of the F S operator: when the edge (v 1,v 2 ) is refined, the new vertex v is moved to the middle point of geod v1,v2. Points of geod v1,v2 may be computed like a disk generation process, where propagation begins at v 1 and proceeds until reaching v 2. If v 1 (or v 2 ) is on the boundary of the base mesh, it must be moved towards the boundary of the isosurface (connectivity graph). If v 1 and v 2 are both on the boundary, besides moving them towards the boundary of the iso-surface, we must compute geod v1,v2 on points restricted to the boundary. This process is implemented in [16]. After moving vertex v towards the iso-surface, a reparameterization is applied to it. This intends to maintain the mesh smoothness during the refinement process. Reparameterization may be defined as a tangential component F T of the Laplacian operator U, defined as [15,20]: U(p) = 1 n n i=1 (p i p), where p represents v coordinates and p i is are v neighbor is coordinates. The tangential component F T is defined as F T (p) =τ[u(p) (U(p). n) n], where τ is a constant and n is the normal to v. Application of F T to v optimizes triangles that share v, but may move v away from the iso-surface. As v is close to the iso-surface S, itmaybemovedback toward S by using F S operator. 5. Results, Conclusions and Future Work We have employed two error estimation functions. The first is an edge 1 deviation function, defined as ɛ e = U(e), where A(T )isthearea T A(T ) of each triangle T, U(e) =D(p) 2 (A(T 1 )+A(T 2 ))/2 is the error computed on edge e, D(p) is the distance from medial point p (of e) to the iso-surface and, T 1 and T 2 are triangles that share e. The second function estimates the deviation of the triangle normals 1 from T S. It is defined as [15] ɛ n = [A(T )(1 n(t ).m(t ) )], T e A(T ) T

8 8 A. Peixoto and L. Velho Fig. 4. Multiresolution mesh: base mesh (left), refined mesh (right). Fig. 5. Object Top: Multiresolution Method (left), Marching Cubes (right). Fig. 6. Normal and edge estimation and aspect ratio histogram. where m(t ) is the gradient computed at the centroid of T, and n(t ) is the unit normal to T. Figure 4 shows a base mesh generated from a synthetic volume and the multiresolution mesh. Figure 5 shows details of the top of the mesh, comparing our method with MC. Figure 6 shows the graphs of e and n applied to the above refined meshes in different resolutions: the graphs show that the mesh converges to the iso-surface during the refinement. Figure 6 (right) shows the aspect ratio histogram comparing our method with MC. We ve presented a method based on combined simplification and refinement operations for generating adaptive multiresolution meshes, with good properties. The method is mainly based on [10,20], and used important definitions and results of [15,19]. In [10] the main structures (DG, VCG and dual) are computed for

9 Mesh Extraction using Simplification and Refinement 9 every mesh extracted in a given resolution. Besides this disadvantage, there is no dependency relation between two generated meshes (there is no mesh hierarchy), and the mesh isn t semi-regular. Our method creates the main structures only once (base mesh), and creates a mesh hierarchy with a semi-regular structure. Some advantages of our work in relation to [20] are the capabilities of extracting meshes with boundary and with multiple connected components. Furthermore, we maintain efficiently and directly the consistence of meshes during the subdivision, due to bisection [19], while in [20] it is necessary to take into account some operations to maintain the mesh conformity. The combined simplification/refinement operations used by our method provide a set of important properties to the mesh (Section 2). This set of properties isn t provided by any other method. Future work will include the following: Definition of smooth operators that keep sharp features during refinement. Definition of criteria to decide when vertex displacement need not be computed by geodesics. After some refinement steps, displacement would be computed by the F S operator. Definition of criteria to choose disk centers. References 1. Bajaj, C., E. Coyle, and K. Link, Arbitrary topology shape reconstruction from planar cross sections, Graph. Models Image Process 58 (1996), Barequet, G., and M. Sharir, Piecewise-linear interpolation between polygonal slices,computer Vision and Image Undestanding 6 (1996), Boissonnat, J. D., Shape reconstruction from planar cross sections, Computer Vision, Graphics and Image Processing 44 (1988), Carmo,M.P.,Differential Geometry of Curves and Surfaces, Englewood Cliffs, N. J.: Prentice-Hall, Cong, G., and B. Parvin, An algebraic solution to surface recovery from cross-sectional contours, Graphical Models and Image Processing 61 (1999), Eck, M., T. DeRose, T. Duchamp, H. Hoppe, M. Lounsbery, and W. Stuertzle, Multiresolution analysis of arbitrary meshes, Proceedings of SIGGRAPH, 1995, Ekoule, A. B., F. C. Peyrin, and C. L. Odet, A triangulation algorithm from arbitrary shaped multiple planar contours, ACM Trans. on Graph 10 (1991),

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