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

B-spline curve evolution

T-spline level-set evolution

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

Parametric curve evolution

through discretization, we replace with

Smoothness constraint

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

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

Influence of different weights

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.

Experimental results Parametric curve evolution (without noise)

Experimental results Parametric curve evolution (with noise)

Experimental results Implicit curve evolution (image segmentation)

Experimental results Implicit curve evolution (2D)

Experimental results

Implicit curve evolution (3D)

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