Dual Evolution forGeometric Reconstruction

Huaiping Yang(FSP Project S09202)

Johannes Kepler University of Linz

1st FSP-Meeting in Graz, Nov. 23-25, 2005

Overview

Introduction Outline of our method Evolution equation Synchronization of dual representations Refine the evolution result Experimental Results Conclusions

Introduction

Geometric reconstruction from discrete point data sets has various applications:

Two types of representations: Parametric curves/surfaces. Implicit curves/surfaces

We use a combination of both representaions. Improved handling of both topology changes

and shape constraints

Outline of our method

We restrict our discussion to 2D cases: B-spline curves T-spline level sets

Outline of our dual evolution: Initialization (pre-compute the evolution speed function)

Evolution and synchronization (until some stopping criterion is satisfied)

Refinement

Evolution equation We want to move the active curve (parametric or implicit)

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:

From evolution equation:

we get , a discretized version of each evolution

step can be formulated as a least squares problem:

Implicit curve evolution

We use implicit T-spline curves [Sederberg2003],

and

is the T-spline function,

where , are cubic B-spline basis functions associated with

knot vectors ,

Implicit curve evolution During the evolution, the following condition always holds:

which implies

Combine it with and , we get

which also can be formulated as a least squares problem:

Solve the evolution equation In order to prevent the linear system from being ill-posed,

we add a damping term , then we get

We use the Levenberg-Marquardt (L-M) method to choose . The same strategy is also used for parametric curve evolution.

Parametric curve synchronization

Detect self-intersections: if there is any conflict of dual normal directions,

then there may be some self-intersections happening around .

In practice, we choose

Parametric curve synchronization

Change the topology (eliminate self-intersections) Split the B-spline curve

Remove those curves with wrong direction

Project to zero level set

Parametric curve synchronization

This strategy also works for elimination of local self-intersections.

Implicit curve synchronization

Approximate the signed distance field of B-spline curve

Through discretization, it can be formulated as a least squares problem:

- Singed distance field

- Weight coefficient

- Sampling points (adaptive to the distribution of T-spline control points)

Implicit curve synchronization

This synchronization has three purposes: Make the implicit curve close to the coupled parametric curve. Remove additional branches (Topology constraint). Level set reinitialization.

Implicit curve synchronization

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

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

Experimental results Fitting unorganized data points without noise

Experimental results

Fitting unorganized data points with noise

Experimental results Image contour detection

Conclusions and future work

Dual evolution combines advantages of both parametric and implicit representations.

The same evolution law and the synchronization step can produce the dual representations simultaneously and efficiently.

Future work More complex topological changes (splitting + merging) Adaptive redistribution of control points during the evolution More intelligent and robust evolution speed function Other shape constraints (symmetries, convexity) Extend to 3D

References V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic active

contours”, International Journal of Computer Vision, 22(1), 1997, pp. 61-79

H. Pottmann, S. Leopoldseder and M. Hofer, “Approximation with Active B-spline curves and surfaces”, Proc. Pacific Graphics, 2002, pp. 8-25

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

Thanks!