skeleton extraction 2011

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Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution Extraction and Tracking of a body Skeleton from Multiview Alexander Pinzón Fernandez Advisor: Eduardo Romero Universidad Nacional de Colombia March 24, 2011 Alexander Pinzón Fernandez | Bioingenium Research Group 1/18 |

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Page 1: Skeleton extraction 2011

Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution

Extraction and Tracking of a body Skeletonfrom Multiview

Alexander Pinzón FernandezAdvisor: Eduardo Romero

Universidad Nacional de Colombia

March 24, 2011

Alexander Pinzón Fernandez | Bioingenium Research Group 1/18 |

Page 2: Skeleton extraction 2011

Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution

Outline

1 Introduction

2 Skeleton Extraction

3 sjSkeletonizer

4 The Problem

5 The Proposed Solution

Alexander Pinzón Fernandez | Bioingenium Research Group 2/18 |

Page 3: Skeleton extraction 2011

Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution

Skeleton Extraction

SystemSkeletonization reduces dimensionality and represents a body as a1D structure [Cornea2007].The method converges to a contracted mesh using a Meshsmoothing.

Figure: From left to right: Contracted meshAlexander Pinzón Fernandez | Bioingenium Research Group 3/18 |

Page 4: Skeleton extraction 2011

Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution

Mesh Smoothing Methods

DefinitionMesh smoothing is a method to remove noise [Zhang2002].

Figure: (1) Original Shape, (2 3, 4) after (10, 50, 200) smoothing stepsrespectively, algorithm used Taubin’s Smoothing [Taubin2000]

Alexander Pinzón Fernandez | Bioingenium Research Group 4/18 |

Page 5: Skeleton extraction 2011

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 Pinzón Fernandez | Bioingenium Research Group 5/18 |

Page 6: Skeleton extraction 2011

Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution

Laplacian Smoothing

The basic idea is that a vertex of a mesh is incrementally movedtowards the Laplacian direction [Bray2004].

∂X∂t = λL (X ),

which is implemented as the forward difference equation:Xt+1 = (I + λL) Xtwhere X is the set of vertices, L is the Laplacian, and λ ∈ R is adiffusion speed.and the discrete Approximation reads as:L (xi ) =

∑wij (xj − xi ) , xj ∈ Neighbor (xi )

Alexander Pinzón Fernandez | Bioingenium Research Group 6/18 |

Page 7: Skeleton extraction 2011

Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution

The Laplacian can be approximated as

Simple Laplacian wij = 1m

Scale-dependent Laplacian wij = 1‖xi−xj‖

Normal Curvature wij = cotαj + cotβj

Alexander Pinzón Fernandez | Bioingenium Research Group 7/18 |

Page 8: Skeleton extraction 2011

Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution

Skeleton can be obtained by smoothing

BUT under two restrictions, WL which strengths the Laplacianand WH which maintains the vertices in the original location.

Skeleton Extraction[

WLLWH

]Xt+1 =

[0

WHXt

]Where

L (X ) =Laplacian smoothing and WL, WH must be iteratedW t+1

L = SLW tL :Contraction constraints, SL = 2.0

W t+1H,i = W 0

H,i

√A0

iAt

i: Attraction constraints

Ati and A0

i are the current and original areas of a ring (neighbors) of vertex xit =represents each iteration

Alexander Pinzón Fernandez | Bioingenium Research Group 8/18 |

Page 9: Skeleton extraction 2011

Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution

Linear system

[WLLWH

]2NxN

Xt+1 =

[0

WHXt

]2NxN

Where N is number of vertices (X ≈ 30.000)We need to solve the system Ax = BThe system is over-determined (more equations than unknowns).We solve it in the least-squares sense

minimize ‖Ax − b‖2

which is equivalent to minimizing the following quadratic energy.

‖WLLXt+1‖2 +∑iW 2

H,i ‖xt+1 − xt‖2

Alexander Pinzón Fernandez | Bioingenium Research Group 9/18 |

Page 10: Skeleton extraction 2011

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 Pinzón Fernandez | Bioingenium Research Group 10/18 |

Page 11: Skeleton extraction 2011

Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution

Problems in Isometric transformations

Isometric Change

Two nodes appear where there should be a single one

Alexander Pinzón Fernandez | Bioingenium Research Group 11/18 |

Page 12: Skeleton extraction 2011

Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution

The Proposed Solution

Add a new constraintTrying to smooth the vertices along the line

Alexander Pinzón Fernandez | Bioingenium Research Group 12/18 |

Page 13: Skeleton extraction 2011

Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution

Not so fast, life is not that easy!!!

An additional difficulty is to introduce the distance to a line intothe traditional AX=B

The distance from a Point to a Line

line =−→P1 + t

−→P2, point P0=⇒distance(line, P0) = |(P2−P1)×(P1−P0)|

|P2−P1|

Every point in a plane satisfies this equation P0 : ax + b0y + c0z + d0 = 0.Points from a line belong simulaneaously to two planes whose equations can becompletely determined from the line equation.

Alexander Pinzón Fernandez | Bioingenium Research Group 13/18 |

Page 14: Skeleton extraction 2011

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

WLLWHWD1WD2

Xt+1 =

0

WHXt00

Alexander Pinzón Fernandez | Bioingenium Research Group 14/18 |

Page 15: Skeleton extraction 2011

Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution

Results

Alexander Pinzón Fernandez | Bioingenium Research Group 15/18 |

Page 16: Skeleton extraction 2011

Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution

Particular Difficulties

The point can be anywhere on the line.The skeleton has many branchesMore equations than unknowns.The solution should be restricted to a particular region of theline.

Alexander Pinzón Fernandez | Bioingenium Research Group 16/18 |

Page 17: Skeleton extraction 2011

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 Pinzón Fernandez | Bioingenium Research Group 17/18 |

Page 18: Skeleton extraction 2011

Introduction Skeleton Extraction sjSkeletonizer The Problem The Proposed Solution

Bibliography

Mathieu Desbrun, Mark Meyer, Peter Schröder, and Alan H. Barr.Implicit fairing of irregular meshes using diffusion and curvature flow.In SIGGRAPH ’99: Proceedings of the 26th annual conference on Computergraphics and interactive techniques, pages 317–324, New York, NY, USA, 1999.ACM Press/Addison-Wesley Publishing Co.Skeleton Extraction.

Alexander Pinzón Fernandez | Bioingenium Research Group 18/18 |