1. visualization and graphics research group CIPIC January 30, 2003Multiresolution (ECS 289L) - Winter 20031 MAPS – Multiresolution Adaptive Parameterization of Surfaces (SIGGRAPH ’98) By Aaron W. F. Lee; Wim Sweldens; Peter Schröder; Lawrence Cowsar; David Dobkin Presented by Nameeta Shah and Yong J. Kil

2. visualization and graphics research group CIPIC January 30, 2003Multiresolution (ECS 289L) - Winter 20032 Introduction (1/2) What does MAPS do? Construct smooth parameterizations of irregular connectivity triangulations of arbitrary genus 2-manifolds. Construct hierarchy of models of different fineness in O(N log N) time and space complexity. Irregular Original Mesh (M)Smooth Parameterized Mesh (M J )

3. visualization and graphics research group CIPIC January 30, 2003Multiresolution (ECS 289L) - Winter 20033 Introduction (2/2) What is a smooth parameterization? A “nice” parameterization. E.g. having subdivision connectivity, that is, a mesh produced by 4-to-1 splitting. Advantages: Texture mapping, morphing, adaptive remeshing, and other classical multiresolution analysis.

4. visualization and graphics research group CIPIC January 30, 2003Multiresolution (ECS 289L) - Winter 20034 General Overview M M0M0 MJMJ 1.Take the original mesh (M) 2.Define a base (coarse) mesh (M 0 ) with a mapping (parameterization) of all the original points. 3.Subdivide (e.g. 4-to-1) the faces of the base mesh and do a inverse mapping.

5. visualization and graphics research group CIPIC January 30, 2003Multiresolution (ECS 289L) - Winter 20035 Previous Work Multiresolution Analysis of Arbitrary Meshes (SIGGRAPH ’95) by Matthias Eck and Hoppe. 1.Base domain by Voronoi tiling. 2.Parameterization by sequence of local harmonic maps. Cons: Time complexity. No explicit control over the base domain. E.g. feature edges.

6. visualization and graphics research group CIPIC January 30, 2003Multiresolution (ECS 289L) - Winter 20036 Overview of MAPS Constructs the base mesh (M 0 ) by using ideas based on mesh simplification. I.e. Dobkin-Kirkpatrick (DK) algorithm: Iterative vertex removal with guarantees on number of intermediate mesh levels. Constructs the parameterization iteratively along with the vertex removal strategy.

7. visualization and graphics research group CIPIC January 30, 2003Multiresolution (ECS 289L) - Winter 20037 Features of MAPS Features of MAPS: Fast coarsification strategy to define the base domain (M 0 ), avoiding difficulties of finding Voronoi tiles. Vertex and edge tags to constrain the parameterization to align with selected features. Adaptive subdivision connectivity.

8. visualization and graphics research group CIPIC January 30, 2003Multiresolution (ECS 289L) - Winter 20038 Notations Triangular mesh is represented as a pair ( P, K ) where P is a set of N point positions p i = (x i,y i,z i ), 1  I  N and K contains the adjacency information. (( P L, K L ) is the original mesh and ( P 0, K 0 ) is the base mesh)  (| K |) is the polyhedron consisting of points, edges and triangles in R 3 A set of vertices is independent if no two vertices are neighbors A set of vertices is maximally independent if no larger independent set contains it. 1-ring neighborhood of a vertex {i} is the set N (i) = { j | {i, j}  K } The star of a vertex {i} is the set of simplices containing i. The curvature estimate at a vertex {i} is K (i) = | k 1| + | k 2|

9. visualization and graphics research group CIPIC January 30, 2003Multiresolution (ECS 289L) - Winter 20039 Hierarchical Representation Vertex Removal –Based on the Dobkin – Kirkpatrick (DK) algorithm Basic idea for going from level l to l -1 –Take any maximal independent set among the vertices of degree atmost b (b=11) in P l ( l is the level in the hierarchy) –Remove the star of all the vertices in the set –Retriangulate the hole Advantage – guarantees L i.e. the number of levels to be O(logN) Drawback - Randomly chosen vertices, not using the geometric information Solution –Put vertices in priority queue based on their weights calculated as follows: w(, i) = * a(i) / a max + (1- ) * k (i) / k max –Intuitively, vertices with small and flat star area will be weighed less

10. visualization and graphics research group CIPIC January 30, 2003Multiresolution (ECS 289L) - Winter 200310 Flattening and Retriangulation Conformal map z a which minimizes metric distortion to map the neighborhood of a removed vertex into the plane. ** Piecewise linear approximation of z a is denoted by  i for the removed vertex {i}. Vertex {i} is at the origin and its 1-ring neighbors are mapped as follows:  i (p j k ) = r k a exp(i  k a), where k i = # of 1-ring neighbors a = 2  /  ki r k = || p i - p j k || Retriangulate using Constrained Delaunay triangulation

11. visualization and graphics research group CIPIC January 30, 2003Multiresolution (ECS 289L) - Winter 200311 Mesh levels

12. visualization and graphics research group CIPIC January 30, 2003Multiresolution (ECS 289L) - Winter 200312 Initial Parameterization Construction of a bijection  from  (| K L |) to  (| K 0 |) Want to have a mapping  l from the top level L to mesh level l which will allow us to map points between meshes at any level of the hierarchy Barycentric coordinates are used for parameterization Constructing  l - 1 for each vertex {i} at level L Case 1. {i} is in the current level, nothing to do  l - 1 (p i ) =  l (p i ) = p i Case 2. {i} just got removed in the current level  l - 1 (p i ) =  p j +  p k +  p m where p i is in {j, k, m} in the new level

13. visualization and graphics research group CIPIC January 30, 2003Multiresolution (ECS 289L) - Winter 200313 Parameterization contd. {i} was removed before previous level –If the triangle that contained {i} at the previous level is still in the new level, do nothing. –Otherwise, assign barycentric coordinates based on the new triangle that {i} is in.

14. visualization and graphics research group CIPIC January 30, 2003Multiresolution (ECS 289L) - Winter 200314 Tagging and Feature Lines Mark important vertices Mark important paths

15. visualization and graphics research group CIPIC January 30, 2003Multiresolution (ECS 289L) - Winter 200315 Remeshing Uniform remeshing Smoothing the parameterization Adaptive Remeshing

16. visualization and graphics research group CIPIC January 30, 2003Multiresolution (ECS 289L) - Winter 200316 Xianfeng Gu and Shing-tung Yau, Computing Conformal Structures of Surfaces, Communications in Information and Systems vol. 2, no. 2, pp. 121-146, december 2002 Conformal Structures of Surfaces

17. visualization and graphics research group CIPIC January 30, 2003Multiresolution (ECS 289L) - Winter 200317 Results(1/3)

18. visualization and graphics research group CIPIC January 30, 2003Multiresolution (ECS 289L) - Winter 200318 Results(2/3)

19. visualization and graphics research group CIPIC January 30, 2003Multiresolution (ECS 289L) - Winter 200319 Results(3/3)

20. visualization and graphics research group CIPIC January 30, 2003Multiresolution (ECS 289L) - Winter 200320 Statistics

21. visualization and graphics research group CIPIC January 30, 2003Multiresolution (ECS 289L) - Winter 200321 Conclusion (Pros vs. Cons) Contribution Describe an O(N log N) time and storage algorithm to construct a logarithmic level hierarchy of arbitrary topology. Construct a smooth parameterization of the original mesh within an error tolerance. Using the smooth parameterization, it can do adaptive, hierarchical remeshing of arbitrary meshes into subdivision connectivity meshes. Allows feature preservation of vertices and edges Useful Multiresolution editing and compression, morphing, texture mapping.

22. visualization and graphics research group CIPIC January 30, 2003Multiresolution (ECS 289L) - Winter 200322 Misc2