By Joachim Weickert
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Schemes of this type, however, renounce consistency with the original equation as well as rotational invariance: round shapes evolve into polygonal structures. A consistent semi-implicit approximation of MCM which discretizes the firstorder spatial derivatives explicitly and the second-order derivatives implicitly has been proposed by Alvarez [10]. In order to solve the resulting linear system of equations he applies symmetric Gauß–Seidel iterations. Nicolet and Sp¨ uhler [301] investigate a consistent fully implicit scheme for MCM leading to a nonlinear system of equations.
However, the reader should be aware of the fact that – although this equation looks like a linear one-dimensional heat equation – it is in fact nonlinear, since the arc-length v is again a function of the curve. Theoretical results For the evolution of a smooth curve under its curvature, it has been shown in [188, 151, 169] that a smooth solution exists for some finite time interval [0, T ). e. a point with a circle as limiting shape. 99) is called Euclidean shortening flow [150]. 100) T = 21 σ 2 .
64) We observe that for this type of morphological operations only grey-level sets matter. As a consequence, they are invariant under monotone grey-level rescalings. 6 and some modifications 32 CHAPTER 1. 5. It is a very desirable property in all cases where brightness changes of the illumination occur or where one wants to be independent of the specific contrast range of the camera. On the other hand, for applications like edge detection or image restoration, contrast provides important information which should be taken into account.