Geometric Curve Evolution and Image Processing

2003
ISBN: 978-3-540-00402-8
Verlag: Springer Berlin Heidelberg

Buch, Englisch, Band 1805, 194 Seiten, Paperback, Format (B × H): 155 mm x 235 mm, Gewicht: 318 g

Reihe: Lecture Notes in Mathematics

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.

Zielgruppe

Research

Autoren/Hrsg.

Cao, Frédéric

Fachgebiete

• Mathematik | Informatik Mathematik Geometrie Nicht-Euklidische Geometrie
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik EDV | Informatik Programmierung | Softwareentwicklung Grafikprogrammierung

Inhaltsverzeichnis

Preface.- Part I. The curve smoothing problem: 1. Curve evolution and image processing; 2. Rudimentary bases of curve geometry.- Part II. Theoretical curve evolution: 3. Geometric curve shortening flow; 4. Curve evolution and level sets.- Part III. Numerical curve evolution: 5. Classical numerical methods for curve evolution; 6. A geometrical scheme for curve evolution.- Conclusion and perspectives.- A. Proof of Thm. 4.3.4.- References.- Index.