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Yutaka Ohtake

A Multi-scale Approach to 3D Scattered Data Interpolation with Compactly Supported Basis Functions. Alexander Belyaev. Hans-Peter Seidel. Yutaka Ohtake. Objective. Convert scattered points into implicit representations f(x,y,z)=0. Convert.

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Yutaka Ohtake

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  1. AMulti-scale Approach to 3D Scattered Data Interpolation with Compactly Supported Basis Functions Alexander Belyaev Hans-PeterSeidel Yutaka Ohtake

  2. Objective • Convert scattered points into implicit representations f(x,y,z)=0. Convert f(x,y,z)=0 thatinterpolates points Scattered points

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

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

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

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

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

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

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

  10. Contents • Single-scale Interpolation • Polynomial Basis RBF • Multi-scale Interpolation • Results and Problems

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

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

  13. Local Shape Function Height function in implicit form Least square fittingto near points

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

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

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

  17. Contents • Single-scale Interpolation • Multi-scale Interpolation • Results and Problems

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

  19. Construction of Point Hierarchy • Uniform octree based down sampling. • Coordinates and normals are the average of leaf nodes. • Final level is decided according to density of points. Appendedto hierarchy Given points Level 1 (23 cells) Level 3 Level 4 Level 5 Level 6 Level 2

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

  21. Contents • Single-scale Interpolation • Multi-scale Interpolation • Results and Problems

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

  23. Comparison with method by Carr[SIG01] (FastRBF) Original13K points FastRBF 30 sec. Our method 7 sec.

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

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

  26. Feature Based Shape Reconstruction Inter-polation Features(ridges and ravines) Only feature pointsare kept Reconstructionresult

  27. Points with normalsfrom mesh Points with noisy normals Polygonizationf=0

  28. ComplicatedTopological Object Point set surface Level2 Level3 Level1 Level4 Level5 Level6

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

  30. Sharp Features Original meshwith sharp features FastRBF(bi-harmonic) The proposedmethod

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

  32. Summary • Multi-scale approach to CS-RBFs • Simple and fast. • Robust to • Irregular sampling • Quality of normals • Future Work • Avoiding extra zero-sets • Sharp features • Shape texture reconstruction

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