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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 Meshesfrom Volumetric Databy Sparse Polynomial Approximation Yukie Nagai Yutaka Ohtake Kiwamu Kase Hiromasa Suzuki THE UNIVERSITY OF TOKYO RIKEN
Introduction • Algorithm Details • Result and Discussion ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS 2008
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
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
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
Introduction • Algorithm Details • Result and Discussion ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS 2008
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
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
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
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
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
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
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
: 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
: 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
: 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
: 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
: 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
: 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
: 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
: 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
: 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
: 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
: 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
: 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
: 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
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
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
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
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
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
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
Introduction • Algorithm Details • Result and Discussion ICCSA COMPUTATIONAL GEOMETRY AND APPLICATIONS 2008
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
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
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
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