By Frédéric Cao
In photo processing, "motions by way of curvature" supply a good method to delicate curves representing the bounds of items. In this sort of movement, every one element of the curve strikes, at any immediate, with an ordinary pace equivalent to a functionality of the curvature at this aspect. This publication is a rigorous and self-contained exposition of the options of "motion by way of curvature". The process is axiomatic and formulated when it comes to geometric invariance with admire to the location of the observer. this can be translated into mathematical phrases, and the writer develops the procedure of Olver, Sapiro and Tannenbaum, which classifies all curve evolution equations. He then attracts an entire parallel with one other axiomatic strategy utilizing level-set tools: this ends up in generalized curvature motions. eventually, novel, and intensely actual, numerical schemes are proposed permitting one to compute the answer of hugely degenerate evolution equations in a totally invariant means. The convergence of this scheme can be proved.
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Extra info for Geometric Curve Evolution and Image Processing
Let a, b ∈ R with a < b. A Jordan curve is a continuous mapping from [a, b] to R2 which is one-to-one on (a, b). In fact, the range Im(C) of a curve C will be of more interest than the curve itself, that is, we do not mind how the curve is described. Some authors thus prefer to deﬁne a plane curve as the range of a mapping from an interval to R2 . Nevertheless, when we want to operate on the curve, we need at least some local coordinates. As usual in geometry, we would like to describe phenomena that do not depend upon any particular system of coordinates.
A curve C1 will be called an admissible parameterization of C if and only if there exists an homeomorphism ψ : I → J such that C1 = C ◦ ψ. The theories we shall develop will focus on characteristics of curves that will not depend on any particular parameterization; in this case, these values will be called intrinsic. An intrinsic characteristic may either be a scalar or a vector, it may be local or global. In the case of a global characteristic, we say that I is intrinsic to C if for F. Cao: LNM 1805, pp.
12) becomes ux ut = (1 + ux )2 ∂ux F + 3ux uxx ∂uxx F. We now replace ut by F (ux , uxx ) and get the equation ux F = (1 + ux )2 ∂ux F + 3ux uxx ∂uxx F, where ux and uxx must be considered as two independent variables. We can check that uxx F1 (ux , uxx ) = 1 + u2x and F2 (ux , uxx ) = 1 + u2x are solutions. The equation ut = F1 (ux , uxx ) is the motion of the graph with a constant normal velocity, whereas ut = F2 (ux , uxx ) corresponds to the mean curvature ﬂow. Both admit the Euclidean group as a symmetry group.