Evolving CurvesSurfaces for Geometric Reconstruction and Image Segmentation - PowerPoint PPT Presentation

Title: Evolving CurvesSurfaces for Geometric Reconstruction and Image Segmentation


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Evolving Curves/Surfaces forGeometric
Reconstruction and Image Segmentation

• Huaiping Yang
• (Joint work with Bert Juettler)
• Johannes Kepler University of Linz
• Workshop on Algebraic Spline Curves and Surfaces,
  May 17-18, 2006, Eger, Hungary

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• B-spline curve evolution

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• T-spline level-set evolution

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Overview

• Introduction
• Outline of our method
• B-spline curve evolution (2D)
• T-spline level-set evolution (2D 3D)
• Refine the evolution result
• Experimental Results
• Conclusions

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Introduction

• Geometric reconstruction from discrete point data
  sets has various applications
• We consider two types of representations
• Parametric curves
• Implicit curves/surfaces (level-sets)
• We provide a unified framework for both
• shape reconstruction from unorganized points and
• image segmentation.

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Outline of our method

• We call the evolutionary curves/surfaces
• active curves/surfaces or
• active shape (to fit the target shape)
• Outline of our algorithm
• Initialization (pre-compute the evolution speed
  function)
• Evolution (which generates time-dependant
  families of curves/surfaces, until some stopping
  criterion is satisfied)
• Refinement

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Evolution equation

• We want to move the active curve/surface along
  its normal directions

-
- Points on the curve
- Time variable
- Unit normal vector
- Evolution speed function
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Evolution speed function

• For image contour detection, we use a modified
  version of that proposed by Caselles et al.
  Caselles1997

• For unorganized data points fitting, we use

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Parametric curve evolution

• B-spline curve representation

• B-spline curve evolution
• Evolution with normal velocity

From evolution equation
, we get
Then we choose
by solving
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Parametric curve evolution
through discretization, we replace with

• Smoothness constraint

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Parametric curve evolution

• Solve the evolution equation
• To minimize the object function

by solving a sparse linear system
depends on the noise level of the input
data.
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Level-sets evolution

• T-spline level sets
• Implicit T-spline curves

and
is the T-spline function, (cubic in our case)
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Level-sets evolution

• Implicit T-spline surfaces

and

• T-spline level sets evolution
• Evolution with normal velocity

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Level-sets evolution

• The definition of level-sets

implies
Combine it with
and
, we get
Then we choose
by solving
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Level-sets evolution
through discretization, we replace with

• Distance field constraint
• Why distance field constraint?
• To avoid the time-consuming re-initialization
  steps, which has to be frequently applied to
  restore the signed distance field property of the
  level-set function for most existing level-set
  evolutions.

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Level-sets evolution
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Level-sets evolution
Since an ideal signed distance function
satisfies , we propose
Again, through discretization, we replace
with
where
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Level-sets evolution

• Smoothness constraint

• Solve the evolution equation
• To minimize the object function

by solving a sparse linear system
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Influence of different weights
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Refine the evolution result

• For the given data points, the evolution result
  is refined by solving a non-linear least squares
  problem,

- Given data points
- Closest point of , on the active
curve/surface

• For the given image data, using detected edge
  points around the active curve as target data
  points.

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Experimental results

• Parametric curve evolution (without noise)

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Experimental results

• Parametric curve evolution (with noise)

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Experimental results

• Implicit curve evolution (image segmentation)

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Experimental results

• Implicit curve evolution (2D)

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Experimental results

• Implicit curve evolution (3D)

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Conclusions and future work

• Evolution process can be reduced to a (sparse)
  system of linear equations.
• Distance field constraints can avoid additional
  branches of the level-sets without using
  re-initialization steps.
• Future work
• Adaptive redistribution of control points during
  the evolution
• More intelligent and robust evolution speed
  function
• Other shape constraints (symmetries, convexity)
• Use dual evolution to combine advantages of both
  parametric and implicit representations

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References

• V. Caselles, R. Kimmel, and G. Sapiro, Geodesic
  active contours, International Journal of
  Computer Vision, 22(1), 1997, pp. 61-79
• B. Juettler and A. Felis, Least-squares fitting
  of algebraic spline surfaces , Advances in
  Computational Mathematics , 17, 2002, pp. 135-152
• W. Wang, H. Pottmann and Y. Liu, Fitting
  B-spline curves to point clouds by squared
  distance minimization, ACM Transactions on
  Graphics, to appear, 2005
• T. W. Sederberg, J. Zheng, A. Bakenov and A.
  Nasri, T-splines and T-NURCCS, ACM Transactions
  on Graphics, 22(3), 2003, pp. 477-484
• J. Nocedal and S. J. Wright, Numerical
  optimization, Springer Verlag, 1999