Evolving Curves/Surfaces for Geometric Reconstruction and Image Segmentation Huaiping Yang (Joint...
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Transcript of Evolving Curves/Surfaces for Geometric Reconstruction and Image Segmentation Huaiping Yang (Joint...
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
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
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.
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
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
-
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:
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
Solve the evolution equationTo minimize the object function
Parametric curve evolution
by solving a sparse linear system
depends on the noise level of the input data.
Level-sets evolution
T-spline level sets Implicit T-spline curves
and
is the T-spline function, (cubic in our case)
Level-sets evolution
Implicit T-spline surfaces
and
T-spline level sets evolution Evolution with normal velocity
Level-sets evolution
The definition of level-sets
implies
Combine it with and , we get
Then we choose by solving
through discretization, we replace with
Level-sets evolution
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.
Level-sets evolution
Since an ideal signed distance function satisfies ,
we propose
Again, through discretization, we replace with
where
Solve the evolution equationTo minimize the object function
Level-sets evolution
by solving a sparse linear system
Smoothness constraint
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.
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
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