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Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation

Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation. Yukie Nagai. Yutaka Ohtake. Kiwamu Kase. Hiromasa Suzuki. THE U NIVERSITY OF T OKYO RIKEN. Introduction Algorithm Details Result and Discussion.

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Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation

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  1. Extraction of Skeletal Meshesfrom Volumetric Databy Sparse Polynomial Approximation Yukie Nagai Yutaka Ohtake Kiwamu Kase Hiromasa Suzuki THE UNIVERSITY OF TOKYO RIKEN

  2. Introduction • Algorithm Details • Result and Discussion ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS 2008

  3. mesh model manufacture CT scanned data Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Objectives • Extracting a skeletal sheet of a thin plate structure • Input: CT-scanned data of an object • Output: a skeletal mesh • Applications • Reverse engineering, defect analysis etc. Direct extraction ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  4. skeletal sheet maxima of intensity field density intensity object CT scanned data Polygonized Skeletal Structure (Maxima) Scalar Field (Partition of Unity) Volumetric Data • Noise-robust • Desired boundary Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Our Approach ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  5. 2. Weighted Delaunay tetrahedrization 1. Approximation of the intensity based on partition of unity intensity 3. Finding the maxima and make polygons supports of polynomials Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Algorithm Overview ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  6. Introduction • Algorithm Details • Result and Discussion ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS 2008

  7. 2. Weighted Delaunay tetrahedrization 1. Approximation of the intensity based on partition of unity intensity 3. Finding the maxima and make polygons supports of polynomials Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Algorithm Overview ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  8. Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion 1. Approximation • Extension of [Ohtake et al. 2003] to volumetric data • Approximate intensity by a quadric • Local approximation function • Quadric • Size of support is adapted to geometrical complexity • Weight function • Object covering with the set of supports ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  9. intensity Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Decision of and Support • Support center • Local approximation function • Support size • Intuition ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  10. intensity Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Decision of and Support • Support center • A randomly selected object point • Local approximation function • Support size • Intuition ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  11. Large error → Shrink support intensity Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Decision of and Support • Support center • A randomly selected object point • Local approximation function • Approximating the intensity of points inside the support • Support size • Intuition • Least square fitting • Bisection method ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  12. Small error → Expand support Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Decision of and Support • Support center • A randomly selected object point • Local approximation function • Approximating the intensity of points inside the support • Support size • Intuition • Least square fitting • Bisection method intensity ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  13. Error = Tolerance Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Decision of and Support • Support center • A randomly selected object point • Local approximation function • Approximating the intensity of points inside the support • Support size • Intuition • Least square fitting • Bisection method intensity ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  14. : points in support Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Decision of and Support • Support center • A randomly selected object point • Local approximation function • Approximating the intensity of points inside the support • The minimizer of approximation error • Support size • A solution of • Error function ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  15. : background : object Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Covering • Extension of [Wu and Kobbelt 2000] to 3D : background : object (not covered ) ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  16. : background : object (not covered ) : object (covered) : background : object Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Covering • Extension of [Wu and Kobbelt 2000] to 3D ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  17. : background : object (not covered ) : object (covered) : background : object Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Covering • Extension of [Wu and Kobbelt 2000] to 3D ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  18. : background : object (not covered ) : object (covered) : background : object Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Covering • Extension of [Wu and Kobbelt 2000] to 3D ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  19. : background : object (not covered ) : object (covered) : background : object Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Covering • Extension of [Wu and Kobbelt 2000] to 3D ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  20. : background : object (not covered ) : object (covered) : background : object Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Covering • Extension of [Wu and Kobbelt 2000] to 3D ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  21. : background : object (not covered ) : object (covered) : background : object Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Covering • Extension of [Wu and Kobbelt 2000] to 3D ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  22. : background : object (not covered ) : object (covered) : background : object Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Covering • Extension of [Wu and Kobbelt 2000] to 3D ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  23. : background : object (not covered ) : object (covered) : background : object Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Covering • Extension of [Wu and Kobbelt 2000] to 3D ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  24. : background : object (not covered ) : object (covered) : background : object Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Covering • Extension of [Wu and Kobbelt 2000] to 3D ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  25. : background : object (not covered ) : object (covered) : background : object Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Covering • Extension of [Wu and Kobbelt 2000] to 3D ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  26. : background : object (not covered ) : object (covered) : background : object Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Covering • Extension of [Wu and Kobbelt 2000] to 3D ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  27. 2. Weighted Delaunay tetrahedrization 1. Approximation of the intensity based on partition of unity intensity supports of polynomials Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Algorithm Overview ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  28. Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion 2. Tetrahedral Mesh Generation • Weighted Delaunay tetrahedrization • Only for support-covered area • Generators • Support centers • Weights • Squared support radii ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  29. 2. Weighted Delaunay tetrahedrization 1. Approximation of the intensity based on partition of unity intensity 3. Finding the maxima and make polygons supports of polynomials Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Algorithm Overview ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  30. Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion 3. Skeletal Structure Extraction • Derivatives: analytical evaluation • Maximality test[Ohtake et al. 2004] • Examine every edge of tetrahedral mesh • Extremality • Maximality ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  31. Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion 3. Skeletal Structure Extraction • Maximality test[Ohtake et al. 2004] • Extremality: • Maximality: • Calculate the coordinate of maximal point • Assumption • changes linearly along the edge is the inner division point ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  32. Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion 3. Skeletal Structure Extraction • Maximality test[Ohtake et al. 2004] • Extremality: • Maximality: • Calculate the coordinate of maximal point • is the inner division point • Generate small triangles around ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  33. Introduction • Algorithm Details • Result and Discussion ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS 2008

  34. Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Results • Pros • Noise-robust • Desired boundary • No branch near boundaries ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  35. Under-smoothing Over-smoothing Ours Canny Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Results • Comparison • Canny edge detector • Object with a non-uniform thickness ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  36. Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation 1. Introduction 2. Algorithm Details 3. Result and Discussion Conclusion and Discussion • Conclusion • Thin plate objects • Direct extraction of skeletal mesh from volumetric data • Adaptive polynomial approximation • Pros • Noise-robust • User-desired boundary • Cons • Non-manifold parts • and are unstable ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS2008

  37. Thank You for Your Attention. Extraction of Skeletal Meshes from Volumetric Data by Sparse Polynomial Approximation Yukie Nagai: nagai@den.rcast.u-tokyo.ac.jp Yutaka Ohtake: yu-ohtake@den.rcast.u-tokyo.ac.jp Kiwamu Kase: kiwamu@riken.jp Hiromasa Suzuki: suzuki@den.rcast.u-tokyo.ac.jp ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS 2008

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