# Skeleton extraction 2011

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<ul><li> 1. Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution Extraction and Tracking of a body Skeleton from Multiview Alexander Pinzn Fernandez Advisor: Eduardo Romero Universidad Nacional de Colombia March 24, 2011Alexander Pinzn Fernandez | Bioingenium Research Group 1/18 | </li> <li> 2. Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution Outline 1 Introduction 2 Skeleton Extraction 3 sjSkeletonizer 4 The Problem 5 The Proposed SolutionAlexander Pinzn Fernandez | Bioingenium Research Group 2/18 | </li> <li> 3. Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution Skeleton Extraction System Skeletonization reduces dimensionality and represents a body as a 1D structure [Cornea2007]. The method converges to a contracted mesh using a Mesh smoothing. Figure: From left to right: Contracted meshAlexander Pinzn Fernandez | Bioingenium Research Group 3/18 | </li> <li> 4. Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution Mesh Smoothing Methods Denition Mesh smoothing is a method to remove noise [Zhang2002]. Figure: (1) Original Shape, (2 3, 4) after (10, 50, 200) smoothing steps respectively, algorithm used Taubins Smoothing [Taubin2000]Alexander Pinzn Fernandez | Bioingenium Research Group 4/18 | </li> <li> 5. Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution Mesh Smoothing Methods Mesh Smoothing Methods. Laplacian methods [1]. Taubin method [Taubin2005]. Statistical methods [Yagou2002].Alexander Pinzn Fernandez | Bioingenium Research Group 5/18 | </li> <li> 6. Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution Laplacian Smoothing The basic idea is that a vertex of a mesh is incrementally moved towards the Laplacian direction [Bray2004]. X t = L (X ), which is implemented as the forward dierence equation: Xt+1 = (I + L) Xt where X is the set of vertices, L is the Laplacian, and R is a diusion speed. and the discrete Approximation reads as: L (xi ) = wij (xj xi ) , xj Neighbor (xi )Alexander Pinzn Fernandez | Bioingenium Research Group 6/18 | </li> <li> 7. Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution The Laplacian can be approximated as 1 Simple Laplacian wij = m 1 Scale-dependent Laplacian wij = xi xj Normal Curvature wij = cot j + cot jAlexander Pinzn Fernandez | Bioingenium Research Group 7/18 | </li> <li> 8. Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution Skeleton can be obtained by smoothing BUT under two restrictions, WL which strengths the Laplacian and WH which maintains the vertices in the original location. WL L 0 Skeleton Extraction Xt+1 = Where WH W H Xt L (X ) =Laplacian smoothing and WL , WH must be iterated t+1 t WL = SL WL :Contraction constraints, SL = 2.0 t+1 0 A0 i WH,i = WH,i At : Attraction constraints i At and A0 are the current and original areas of a ring (neighbors) of vertex xi i i t =represents each iterationAlexander Pinzn Fernandez | Bioingenium Research Group 8/18 | </li> <li> 9. Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution Linear system WL L 0 Xt+1 = WH 2NxN W H Xt 2NxN Where N is number of vertices (X 30.000) We need to solve the system Ax = B The system is over-determined (more equations than unknowns). We solve it in the least-squares sense 2 minimize Ax b which is equivalent to minimizing the following quadratic energy. 2 2 2 WL LXt+1 + WH,i xt+1 xt iAlexander Pinzn Fernandez | Bioingenium Research Group 9/18 | </li> <li> 10. Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution sjSkeletonizer . sjSkeletonizer is our prototype application. Use CGAL (Computational Geometry Algorithms Library) Use Graphite (Numerical Geometry and Computer Graphics) Integrate numerical libraries: ACE, AMD, ARPACK, ARPACK_UTIL, CBLAS, CCOLAMD, CHOLMOD, CLAPACK, COLAMD, F2CLIBS, METIS, MISC, NL, SUPERLU, TAUCS Use QT (Toolkit for creating graphical user interfaces) OpenGL (Open Graphics Library)Alexander Pinzn Fernandez | Bioingenium Research Group 10/18 | </li> <li> 11. Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution Problems in Isometric transformations Isometric Change Two nodes appear where there should be a single oneAlexander Pinzn Fernandez | Bioingenium Research Group 11/18 | </li> <li> 12. Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution The Proposed Solution Add a new constraint Trying to smooth the vertices along the lineAlexander Pinzn Fernandez | Bioingenium Research Group 12/18 | </li> <li> 13. Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution Not so fast, life is not that easy!!! An additional diculty is to introduce the distance to a line into the traditional AX=B The distance from a Point to a Line |(P2 P1 )(P1 P0 )| line = P1 + t P2 , point P0 =distance(line, P0 ) = |P2 P1 | Every point in a plane satises this equation P0 : ax + b0 y + c0 z + d0 = 0. Points from a line belong simulaneaously to two planes whose equations can be completely determined from the line equation.Alexander Pinzn Fernandez | Bioingenium Research Group 13/18 | </li> <li> 14. Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution Line Distance Constraint two additional constraint matrix WD1 , WD2 contain [a, b, c, d ] plane parameters for every vertex [xi , yi , zi , 1] and new sparse linear system WL L 0 WH W H Xt WD1 Xt+1 = 0 WD2 0Alexander Pinzn Fernandez | Bioingenium Research Group 14/18 | </li> <li> 15. Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution ResultsAlexander Pinzn Fernandez | Bioingenium Research Group 15/18 | </li> <li> 16. Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution Particular Diculties The point can be anywhere on the line. The skeleton has many branches More equations than unknowns. The solution should be restricted to a particular region of the line.Alexander Pinzn Fernandez | Bioingenium Research Group 16/18 | </li> <li> 17. Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution Discussion How to add a new constraint on the system of equations. Any ideas to eliminate the error of the two nodes. What kind of numerical method to use?Alexander Pinzn Fernandez | Bioingenium Research Group 17/18 | </li> <li> 18. Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution Bibliography Mathieu Desbrun, Mark Meyer, Peter Schrder, and Alan H. Barr. Implicit fairing of irregular meshes using diusion and curvature ow. In SIGGRAPH 99: Proceedings of the 26th annual conference on Computer graphics and interactive techniques, pages 317324, New York, NY, USA, 1999. ACM Press/Addison-Wesley Publishing Co. Skeleton Extraction.Alexander Pinzn Fernandez | Bioingenium Research Group 18/18 | </li> </ul>