1. INFORMATIK A Multi-scale Approach to 3D Scattered Data Interpolation with Compactly Supported Basis Functions Yutaka Ohtake Yutaka Ohtake Alexander Belyaev Alexander Belyaev Hans-Peter Seidel Hans-Peter Seidel

2. INFORMATIK Objective Convert scattered points into implicit representations f(x,y,z)=0. f(x,y,z)=0 that interpolates points Scattered points Convert

3. INFORMATIK Implicit Representation Surface: f(x,y,z)=0 (implicit surface) Surface: f(x,y,z)=0 (implicit surface) Inside: f(x,y,z)>0 Outside: f(x,y,z) 0 Outside: f(x,y,z)<0 A cross-section of f(x,y,z) A polygonization of f(x,y,z)=0

4. INFORMATIK Advantages of Implicits Constructive Solid Geometry Union, intersection, difference, blending, embossing, …Union, intersection, difference, blending, embossing, … Constructive Solid Geometry Union, intersection, difference, blending, embossing, …Union, intersection, difference, blending, embossing, … =/ (blending =

5. INFORMATIK Advantages of Implicits Filling missing part of the objects Zero sets of f(x,y,z) represents a closed surface.Zero sets of f(x,y,z) represents a closed surface. Filling missing part of the objects Zero sets of f(x,y,z) represents a closed surface.Zero sets of f(x,y,z) represents a closed surface.

6. INFORMATIK Previous Works Using Radial Basis Functions (RBF) Muraki et al. 1991Muraki et al. 1991 –Blobby model Savchenko et al. 1995, Turk et al. 1999Savchenko et al. 1995, Turk et al. 1999 –Thin-plate splines Morse et al. 2001Morse et al. 2001 –Compactly supported piecewise polynomial RBF Carr et al. 2001Carr et al. 2001 –Biharmonic splines and truncated series expansions Using Radial Basis Functions (RBF) Muraki et al. 1991Muraki et al. 1991 –Blobby model Savchenko et al. 1995, Turk et al. 1999Savchenko et al. 1995, Turk et al. 1999 –Thin-plate splines Morse et al. 2001Morse et al. 2001 –Compactly supported piecewise polynomial RBF Carr et al. 2001Carr et al. 2001 –Biharmonic splines and truncated series expansions Can process large point sets

7. INFORMATIK Compactly Supported RBFs Fast, but have several drawbacks. Require uniform samplingRequire uniform sampling Fail to fill holesFail to fill holes It can be defined in narrow band of original data. (not solid)It can be defined in narrow band of original data. (not solid) Fast, but have several drawbacks. Require uniform samplingRequire uniform sampling Fail to fill holesFail to fill holes It can be defined in narrow band of original data. (not solid)It can be defined in narrow band of original data. (not solid) Irregular sampling Narrow band holes

8. INFORMATIK Problem of CSRBFs We can recognize inside/outside information only near the surface. ??? (Out of support) Inside Outside

9. INFORMATIK Our Approach Multi-scale approach Points many Support size small few large

10. INFORMATIK Contents Single-scale InterpolationSingle-scale Interpolation Polynomial Basis RBFPolynomial Basis RBF Multi-scale InterpolationMulti-scale Interpolation Results and ProblemsResults and Problems Single-scale InterpolationSingle-scale Interpolation Polynomial Basis RBFPolynomial Basis RBF Multi-scale InterpolationMulti-scale Interpolation Results and ProblemsResults and Problems

11. INFORMATIK On-surface point Standard RBF Interpolations Solve linear equations about unknown coefficients Off-surface point

12. INFORMATIK Basic Idea of Interpolation 1.Define local shape implicit functions 2.Blend the functions (weighted sum) Solving a sparse linear system.Solving a sparse linear system. 1.Define local shape implicit functions 2.Blend the functions (weighted sum) Solving a sparse linear system.Solving a sparse linear system.

13. INFORMATIK Local Shape Function Height function in implicit form Least square fitting to near points

14. INFORMATIK Formulation Local shape function in implicit form Compactly supported radial basis (blending) function Introduced by Wendland 1995 2D Graph of Unknown (Shift amount)

15. INFORMATIK Results of single-level interpolation 35K points 5 sec. 134K points 47 sec. Holes remain Narrow band domain

16. INFORMATIK Results for Irregular Sampling Irregularly sampled points Many holes remain because of small support of basis functions, but large support leads to inefficient computation procedure.

17. INFORMATIK Contents Single-scale InterpolationSingle-scale Interpolation Multi-scale InterpolationMulti-scale Interpolation Results and ProblemsResults and Problems Single-scale InterpolationSingle-scale Interpolation Multi-scale InterpolationMulti-scale Interpolation Results and ProblemsResults and Problems

18. INFORMATIK Algorithm 1. Construction of a point hierarchy. 2. Coarse-to-fine interpolations. 1. Construction of a point hierarchy. 2. Coarse-to-fine interpolations.

19. INFORMATIK Construction of Point Hierarchy Uniform octree based down sampling.Uniform octree based down sampling. Coordinates and normals are the average of leaf nodes.Coordinates and normals are the average of leaf nodes. Final level is decided according to density of points.Final level is decided according to density of points. Uniform octree based down sampling.Uniform octree based down sampling. Coordinates and normals are the average of leaf nodes.Coordinates and normals are the average of leaf nodes. Final level is decided according to density of points.Final level is decided according to density of points. Level 1 (2 3 cells) Level 2 Level 3 Level 4 Level 5 Level 6 Given points Given points Appended to hierarchy

20. INFORMATIK Coarse-to-fine interpolation Level k-1 Level k Same form f ( x ) as in the single scale Diameter of object

21. INFORMATIK Contents Single-scale InterpolationSingle-scale Interpolation Multi-scale InterpolationMulti-scale Interpolation Results and ProblemsResults and Problems Single-scale InterpolationSingle-scale Interpolation Multi-scale InterpolationMulti-scale Interpolation Results and ProblemsResults and Problems

22. Level 9(final level) Level 8 Level 8 Approximation (error < 2 -8 ) 544K points 901K functions 901K functions 19 min. 332Mbyte Pentium 4 1.6 GHz 7.5 min. 198Mbyte 363 K functions

23. INFORMATIK Comparison with method by Carr[SIG01] (FastRBF) Our method 7 sec. FastRBF 30 sec. Original 13K points

24. Points with normals form a merged mesh by VRIP (Stand scan only) Noise come from noisy boundary

25. INFORMATIK Irregular Sampling Data 90% decimated Joint parts are smooth

26. INFORMATIK Feature Based Shape Reconstruction Features (ridges and ravines) Only feature points are kept Reconstruction result Inter- polation

27. Points with normals from mesh Points with noisy normals Polygonization f=0

28. INFORMATIK Complicated Topological Object Point set surface Level1 Level6Level5Level4 Level3Level2

29. INFORMATIK Extra Zero-set If the object has very thin parts, extra zero-sets may appear. Octree based down-sampling is not sensitive topological changes.Octree based down-sampling is not sensitive topological changes. A smart down-sampling procedure is required.A smart down-sampling procedure is required. If the object has very thin parts, extra zero-sets may appear. Octree based down-sampling is not sensitive topological changes.Octree based down-sampling is not sensitive topological changes. A smart down-sampling procedure is required.A smart down-sampling procedure is required. No extra zero-set inside the bounding box Extra zero-sets appear near thin parts.

30. INFORMATIK Sharp Features Original mesh with sharp features The proposed method FastRBF (bi-harmonic)

31. INFORMATIK Shape Textures From two bunny’s range data Too smooth Holes are filled, but

32. INFORMATIK Summary Multi-scale approach to CS-RBFsMulti-scale approach to CS-RBFs Simple and fast.Simple and fast. Robust toRobust to –Irregular sampling –Quality of normals Future WorkFuture Work Avoiding extra zero-setsAvoiding extra zero-sets Sharp featuresSharp features Shape texture reconstructionShape texture reconstruction Multi-scale approach to CS-RBFsMulti-scale approach to CS-RBFs Simple and fast.Simple and fast. Robust toRobust to –Irregular sampling –Quality of normals Future WorkFuture Work Avoiding extra zero-setsAvoiding extra zero-sets Sharp featuresSharp features Shape texture reconstructionShape texture reconstruction