By E. (Edmond) Ramis, C. (Claude) Deschamps, J Odoux
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4 Goals (revisited) We formulate an abstract set of goal properties on a metric h1 , h2 of curves. G|c on spaces 1. [rescaling invariance] For any λ > 0, if we rescale c to λc, then h1 , h2 G|λc = λa h1 , h2 G|c (where a ∈ lR is an universal constant); 2. [Euclidean invariance] Suppose that A is an Euclidean transformation, and R is its rotational part; if we apply A to c and R to h0 , h1 , then Rh1 , Rh2 G|Ac = h1 , h2 G|c ; 3. [parameterization invariance] the metric does not depend on the ˜ c˜ = h c when c˜(t) = c(ϕ(t)) parameterization of the curve, that is h ˜ and h(t) = h(ϕ(t)).
1). We may then define a topology of shapes by deciding that An → A when uAn → uA uniformly on compact sets. This convergence coincides with the Kuratowski topology of closed sets; if we limit shapes to be compact sets, the Kuratowski topology is induced by the Hausdorff distance. 2. 2 Trouvé-Younes et al (see Glaunès et al. t to all diffeomorphisms G0 that do not move a template shape. 3 (Representation by angle function) In the work of Klassen et al. , Srivastava et al. , Mio and Srivastava , smooth planar closed curves c : S 1 → lR2 of length 2π are represented by a pair of angle-velocity functions c(u) ˙ = exp(φ(u)+iα(u)) (identifying lR2 = C) I then (φ, α) are embedded as a subset N in L2 (0, 2π) or W 1,2 (0, 2π).
1 gradient need not be computed as a convolution; In the end we obtain that the H 1 ˜ gradient enjoys nearly the same computational speed as the H 0 so the H gradient; moreover the resulting gradient flow is more stable, so it works fine in numerical implementations for a larger choice of time step discretization. For ˜ 1 Sobolev active contours are very fast. 21) with Dc M := h : S 1 → lRn | avgc (h) = 0 . If we assign to lRn its usual Euclidean norm, and to Dc M the H0j norm, then h 2 ˜j H ˜ = |avgc (h)|2lRn + λL2j h 2 H0j .
Analyse 1: exercises avec solutions by E. (Edmond) Ramis, C. (Claude) Deschamps, J Odoux