1. Hierarchical Error-Driven Approximation of Implicit Surfaces from Polygonal Meshes Takashi Kanai Yutaka Ohtake Kiwamu Kase University of Tokyo RIKEN, VCAD Modeling Team 4 th Symposium on Geometry Processing, Cagliari, Sardinia, Italy 26 th June, 2006

2. Outline  Introduction  Related Work  Purpose of Our Research  Our Approach Polygon-Implicit Error Metric Hierarchical Approximation Algorithm Preserving Sharp Features  Results and Discussions  Conclusions and Future Work

3. Outline  Introduction  Related Work  Purpose of Our Research  Our Approach Polygon-Implicit Error Metric Hierarchical Approximation Algorithm Preserving Sharp Features  Results and Discussions  Conclusions and Future Work

4. Implicit Surface Modeling Points with Normals (typical output of range scanners) Implicit Solid f (x,y,z) > 0 … inside f (x,y,z) < 0 … outside f (x,y,z) = 0 … approximates points Surface Reconstruction by Implicit Surfaces

5. Related Work  Points → Implicit Surfaces Blobby model [Muraki91] Globally-supported RBF [Savchenko95, Carr01, Turk02] … Compactly-supported RBF [Morse01, Ohtake03] MLS [Alexa01], MPU [Ohtake03], SLIM [Ohtake05]  Polygons → Implicit Surfaces [Shen 04] Constructs a MLS surface from neighbor polygons Coarse-to-fine (or non-hierarchical) approach Difficult to control shape according to errors … needs re-computation for different error thresholds

6. Purpose and Approach  Implicit Surface Approximation from Polygonal Meshes Approximation control by errors → Implicit surfaces with different resolutions can be extracted quickly  Fine-to-Coarse Approach Originated from the mesh simplification method Can construct a hierarchical structure of implicit surfaces  Ordered in the decrease of errors QEM-like error functions  Fast computation of implicit surface fitting  Compact storage of parameters Can preserve sharp features

7. Outline  Introduction  Related Work  Purpose of Our Research  Our Approach Polygon-Implicit Error Metric Hierarchical Approximation Algorithm Preserving Sharp Features  Results and Discussions  Conclusions and Future Work

8. SLIM （ Sparse Low-Degree IMplicit ） [Ohtake et al. 2005]  Hierarchical tree structure of nodes  Low-degree Implicit Polynomial （ We use only Quadratic here ）  An implicit polynomial / a node parent node child node

9. Error Functions Between a Polygon and an Implicit Surface A barycentric coordinate of a triangle T  Distance error function  Gradient error function v0v0 v1v1 v2v2 x s t x

10. Geometric Meanings of Error Functions → works so as to restrain the movements of each other.

11. Error Function as Quadric Form  Two error functions can also be represented as the Quadric Form: : 10-dim. coefficient vector : 10×10 symmetric matrix : 10-dim. vector : a scalar … PIEM (Polygon-Implicit Error Metric)

12. Property of PIEMs  An analogy of QEM (Quadric Error Metric) [Garland97]  Addition of two PIEMs:

13. Error Function for Polygonal Mesh λ … A parameter for adjusting the scale of two functions E dis (M) … two-dimensional quantity (squared distance) E nrm (M) … dimensionless quantity (normalized gradient) （ A i … the area of a triangle ）

14. Error function of two meshes  Does not satisfy the linearity → add λ to PIEM 122 floating points per a node

15. Implicit Surface Optimization  Solving p by least squares fitting  10-dimensional linear equation （ non-sparse matrix ） → SVD (Singular Value Decomposition) min

16. Hierarchical Approximation Algorithm  Hierarchical Face Clustering [Garland 01] Dual Graph of a mesh  Face … node  Face connectivity … edge Edge collapse operations … sequentially applied by using the same scheme of [Garland 97] Each edge collapse … apply implicit surface fitting using PIEM for a combined node A combined node … a parent node of two end nodes Leaf nodes … created by Implicitization

17. Implicitization of a Mesh mesh (340K Faces ） Implicit surface (340K nodes)  A plane of a polygon: a 1 … a 6 of an implicit polynomial f(x) are zero

18. Management of Error Metrics  Management by PIEMs Fast computation, but requires large memory space (200K polygons → 325MB)  “on-the-fly” management Saves memory space, but relatively slow (especially before the end of the algorithm)  A hybrid scheme （◎） on-the-fly (90% of the whole process) → PIEMs

19. Preserving Sharp Features  PU (Partition of Unity) [Ohtake03] Sharp edges are not preserved  Definition of Sharp features in MPU More than two implicit polynomials for a node Apply max/min Boolean operation → The algorithm is more complicated

20. A New PU Evaluation Method  Cluster faces on-the-fly Judgement by using a gradient on a point x Execute PU for each cluster max/min Boolean operation between clusters  A slight modification of the approximation algorithm Delete DG edges of sharp features Multiply large weights (e.g. 1,000) to the boundary nodes (same as [Garland97]) × Up to three clusters × Can not perform large simplification Future work

21. Results  Fandisk (12K nodes) → simplified to 5K nodes Original PU New PU

22. Outline  Introduction  Related Work  Purpose of Our Research  Our Approach Polygon-Implicit Error Metric Hierarchical Approximation Algorithm Preserving Sharp Features  Results and Discussions  Conclusions and Future Work

23. LOD Control Based on Errors 7212,2587,07622,31770,273 #nodes ： Approximation of Implicit Surfaces from 1M polygons

24. Adaptive LOD 1K Nodes 90K Nodes 9.4K Nodes

25. Approximation from Polygon Soup Need to construct edges of a dual graph (ex. [Barequet98]) 67K polygon soup 67K nodes 2.5K nodes

26. Conclusion  Hierarchical Approximation of Implicit Surfaces Surface fitting using QEM-like error functions Construction of hierarchical structure LOD control based on errors Preservation of Sharp Features  Future Work: Less-memory construction based on out-of-core strategy Avoid “two-sheet” problem polygons “two sheet” implicit surface (hyperboloid … )

27. Thank You!

29. Partition of Unity distance weight Support of f (x) f (x)=0 (local approx.) f (x)>0 f (x)<0 Weighted average of local approximations

30. Evaluation of Error Function  Evaluation in 2D  Measure [Taubin94] → Our metric yields the same effects as fitting with a highly dense point set Uniform sampling (5 point / an edge) Dense spatial Uniform sampling Our function Sparse spatial Uniform sampling

31. Updating Support Spheres  Two approaches: [James04]  We adopt layered hierarchy Center point and radius for a parent node can be computed by only using those of child nodes wrapped hierarchylayered hierarchy

32. Statistical Summary Tested on Athlon 64 3500+ CPU and 2GB RAM PC