Geometric Curve Evolution and Image Processing

Frédéric Cao (auth.)9783540004028, 3-540-00402-5

In image processing, “motions by curvature” provide an efficient way to smooth curves representing the boundaries of objects. In such a motion, each point of the curve moves, at any instant, with a normal velocity equal to a function of the curvature at this point. This book is a rigorous and self-contained exposition of the techniques of “motion by curvature”. The approach is axiomatic and formulated in terms of geometric invariance with respect to the position of the observer. This is translated into mathematical terms, and the author develops the approach of Olver, Sapiro and Tannenbaum, which classifies all curve evolution equations. He then draws a complete parallel with another axiomatic approach using level-set methods: this leads to generalized curvature motions. Finally, novel, and very accurate, numerical schemes are proposed allowing one to compute the solution of highly degenerate evolution equations in a completely invariant way. The convergence of this scheme is also proved.

Table of contents :
1. Curve evolution and image processing….Pages 3-21
2. Rudimentary bases of curve geometry….Pages 23-28
3. Geometric curve shortening flow….Pages 31-53
4. Curve evolution and level sets….Pages 55-103
5. Classical numerical methods for curve evolution….Pages 107-110
6. A geometrical scheme for curve evolution….Pages 111-166
Conclusion and perspectives….Pages 167-169
A. Proof of Thm. 4.34….Pages 171-176
References….Pages 177-184
Index….Pages 185-187