Cao F. Geometric Curve Evolution and Image Processing

Springer, 2003. — 187 p.These lectures intend to give a self-contained exposure of some techniques for computing the evolution of plane curves. The motions of interest are the so-called motions by curvature. They mean that, at any instant, each point of the curve moves with a normal velocity equal to a function of the curvature at this point. This kind of evolution is of some interest in differential geometry, for instance in the problem of minimal surfaces. The interest is not only theoretical since the motions by curvature appear in the modeling of various phenomena as crystal growth, flame propagation and interfaces between phases. More recently, these equations have also appeared in the young field of image processing where they provide an efficient way to smooth curves representing the contours of the objects. This smoothing is a necessary step for image analysis as soon as the analysis uses some local characteristics of the contours. Indeed, natural images are very noisy and differential features are unreliable if one is not careful before computing them. A solution consists in smoothing the curves to eliminate the small oscillations without changing the global shape of the contours. What kind of smoothing is suitable for such a task? The answer shall be given by an axiomatic approach whose conclusions are that the class of admissible motions is reduced to the motions by curvature. Once this is established, the wellposedness of these equations has to be examined. For certain particular motions, this turns to be true but no complete results are available for the general existence of these motions. This problem shall be turned around by introducing a weak notion of solution using the theory of viscosity solutions of partial differential equations (PDE). A complete theory of existence and uniqueness of those equations will be presented, as self-contained as possible. (Only a technical, though important, lemma will be skipped.) The numerical resolution of the motions by curvature is the next topic of interest. After a rapid review of the most commonly used algorithms, a completely different numerical scheme is presented. Its originality is that it satisfies exactly the same invariance properties as the equations of motion by curvature. It is also inconditionally stable and its convergence can be proved in the sense of viscosity solutions. Moreover, it allows to precisely compute motions by curvature, when the normal velocity is a power of the curvature more than 3, or even 10 in some cases, which seems a priori nearly impossible in a numerical point of view. Many numerical experiments are presented.The curve smoothing problem
Curve evolution and image processing
Rudimentary bases of curve geometry
Theoretical curve evolution
Geometric curve shortening flow
Curve evolution and level sets
Numerical curve evolution
Classical numerical methods for curve evolution
A geometrical scheme for curve evolution
Conclusion and perspectives