By Frédéric Cao

In picture processing, "motions by way of curvature" supply an effective method to delicate curves representing the bounds of items. In any such movement, every one aspect of the curve strikes, at any rapid, with a standard speed equivalent to a functionality of the curvature at this element. This ebook is a rigorous and self-contained exposition of the strategies of "motion by means of curvature". The method is axiomatic and formulated when it comes to geometric invariance with admire to the location of the observer. this is often translated into mathematical phrases, and the writer develops the technique of Olver, Sapiro and Tannenbaum, which classifies all curve evolution equations. He then attracts a whole parallel with one other axiomatic technique utilizing level-set equipment: this results in generalized curvature motions. eventually, novel, and intensely exact, 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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**Sample text**

It is a third order equation, but it is equivalent to a second order and simple equation which is ∂C = κ1/3 N. 2) ∂t This equation, discovered by Sapiro and Tannenbaum [152, 153], is called the afﬁne intrinsic heat equation or the afﬁne curve shortening. The mean curvature F. Cao: LNM 1805, pp. 31–53, 2003. 5) and the afﬁne heat equation are certainly the most important geometric plane curve evolution equations. First because they are among the simplest in their respective group of invariance. ) Next, they are still the only cases for which a complete theory of existence, uniqueness, regularity and asymptotic state is known.

There exists t1 < T such that C(t) is convex on (t1 , T ). The renormalized curve with constant area converges to an ellipse. 1 If A and B are subsets of R2 , their Hausdorff distance is dH (A, B) = max(sup d(x, B), sup d(y, A)). 2 Short-time existence in the general case When the normal velocity G is an increasing and smooth function of the curvature, results of short-time existence come from the theory of parabolic equations. If G(κ) = κγ with 0 < γ < 1, the theory does not apply directly since G is singular for κ = 0.

The result makes appear a privileged invariant form, the group arc-length. Let us ﬁrst deﬁne what an invariant form is. 17. An invariant form ω is a differential form such that for any g ∈ G and (x, u) ∈ M, we have ω|g·(x,u) ◦ dg|(x,u) = ω|(x,u) , where dg denotes the differential of the application (x, u) → g · (x, u). 16) means that the inﬁnitesimal volume given by the one-form is unchanged under the group action. For instance, if ω = dr is the Euclidean length, a rotation of a small segment changes its orientation but not its length.