Metrics of curves in shape optimization and analysis - Andrea Carlo ...
Metrics of curves in shape optimization and analysis - Andrea Carlo ... Metrics of curves in shape optimization and analysis - Andrea Carlo ...
6.2.4 Applications in computer visionCharpiat et al. [9] propose an approximation method to compute len H (ξ) bymeans of a family of energies defined using a smooth integrand; the approximationis mainly based on the property ‖f‖ L p → p ‖f‖ L ∞, for any measurable functionf defined on a bounded domain; they successively devise a method to findapproximation of geodesics.6.3 A Hausdorff-like distance of compact setsIn Duci and Mennucci [15] a L p -like distance on the compact subsets of lR N wasproposed. (Here p ∈ [1, ∞).)To this end, we fix ϕ : [0, ∞) → (0, ∞) , decreasing, C 1 , with ϕ(|x|) ∈ L p . Wethen define v A (x) := ϕ(u A (x)), where u A is the distance function. We eventuallydefine the distanced(A, B) := ‖v A − v B ‖ L p . (6.2)Remark 6.15 This shape space is a perfect example of the representation/embedding/quotient scheme. Indeed, this shape space is represented as N c ={v A | A compact} and embedded in L p . Given v ∈ N c , we recover the shapeA = {v = ϕ −1 (0)} (that is a level set of v).Example 6.16 A simple example (that works for all p) is given by ϕ(t) = e −t ,so that v A (x) = exp (−u A (x)); in this case, A = {v = 1}.The distance d of eqn. (6.2) enjoys the following properties.Properties 6.17• It is Euclidean invariant;• it is locally compact but not path-metric;• the topology induced is the same as that induced by d H ;• minimal geodesics do exist, since D g := {B | d g (A, B) ≤ ρ} is compact.The proofs are in [15]. We present a numerical computation of a minimal geodesic(by A. Duci) in Figure 7 on the next page.6.3.1 Analogy with the Hausdorff metric, L p vs L ∞We recall that d H (A, B) = ‖u A (x) − u B (x)‖ L ∞ ; whereas instead now we areproposing d(A, B) := ‖v A −v B ‖ L p . The idea being that this distance of compactsets is modeled on L p , whereas the Hausdorff distance is “modeled” on L ∞ .(Note that the Hausdorff distance is not really obtained by embedding, sinceu A ∉ L ∞ ). L p is more regular than L ∞ , as shown by this remark.Remark 6.18 Given any f, g ∈ L p with p ∈ (1, ∞), the segment connectingf to g is the unique minimal geodesic connecting them. Suppose now that thedimension of L ∞ (Ω, A, µ) is greater than 1. Given generically f, g ∈ L ∞ , thereis an uncountable number of minimal geodesics connecting them. 99 “Generically” is meant in the Baire sense: the set of exceptions is of first category.46
Figure 7: Example of a minimal geodesic(For the proof see 2.11 & 2.13 in [15]). The above result suggests that it maybe possible to shoot geodesics in the “Riemannian metric” associated to thisdistance.6.3.2 Contingent coneLet again N c = {v A | A compact} be the family of all representations. Given av ∈ N c , let T v N c ⊂ L p be the contingent coneT v N c := {lim t n (v n − v) | t n > 0, v n ∈ N c , v n → v}n{}v n − v= λ lim | λ ≥ 0, v n → v ,n ‖v n − v‖ L pwhere it is intended that the above limits are in the sense of strong convergencein L p .T v N c contains all directions in which it is possible “in the L p sense” toinfinitesimally deform a compact set. For example, if Φ(x, t) : lR N × (−ε, ε) →lR N is smooth diffeomorphical motion of lR N , and A t := Φ(A, t), then v At is(locally) Lipschitz, so it is differentiable for almost all t, and the derivativeis in T N c . This includes all perspective, affine, and Euclidean deformations.Unfortunately the contingent cone is not capable of expressing some shapedeformations.Example 6.19 We consider the removing motion; let A be compact, andsuppose that x is in the internal part of A; let A t := A \ B(x, t) be the removal47
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- Page 7 and 8: κ > 0HNNHNHκ < 0Figure 1: Example
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- Page 15 and 16: 2.4.1 Example: geometric heat flowW
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- Page 26 and 27: Since φ k are homeomorphisms, then
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- Page 32 and 33: Example 3.38 Let M = C ∞ ([−1,
- Page 34 and 35: • sometimes S 1 will be identifie
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- Page 38 and 39: The term preshape space is sometime
- Page 40 and 41: 5 Representation/embedding/quotient
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- Page 48 and 49: of a small ball from A. The motion
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- Page 66 and 67: and̂∇˜HjE(0) = ̂∇ H 0E(0),̂
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Figure 7: Example <strong>of</strong> a m<strong>in</strong>imal geodesic(For the pro<strong>of</strong> see 2.11 & 2.13 <strong>in</strong> [15]). The above result suggests that it maybe possible to shoot geodesics <strong>in</strong> the “Riemannian metric” associated to thisdistance.6.3.2 Cont<strong>in</strong>gent coneLet aga<strong>in</strong> N c = {v A | A compact} be the family <strong>of</strong> all representations. Given av ∈ N c , let T v N c ⊂ L p be the cont<strong>in</strong>gent coneT v N c := {lim t n (v n − v) | t n > 0, v n ∈ N c , v n → v}n{}v n − v= λ lim | λ ≥ 0, v n → v ,n ‖v n − v‖ L pwhere it is <strong>in</strong>tended that the above limits are <strong>in</strong> the sense <strong>of</strong> strong convergence<strong>in</strong> L p .T v N c conta<strong>in</strong>s all directions <strong>in</strong> which it is possible “<strong>in</strong> the L p sense” to<strong>in</strong>f<strong>in</strong>itesimally deform a compact set. For example, if Φ(x, t) : lR N × (−ε, ε) →lR N is smooth diffeomorphical motion <strong>of</strong> lR N , <strong>and</strong> A t := Φ(A, t), then v At is(locally) Lipschitz, so it is differentiable for almost all t, <strong>and</strong> the derivativeis <strong>in</strong> T N c . This <strong>in</strong>cludes all perspective, aff<strong>in</strong>e, <strong>and</strong> Euclidean deformations.Unfortunately the cont<strong>in</strong>gent cone is not capable <strong>of</strong> express<strong>in</strong>g some <strong>shape</strong>deformations.Example 6.19 We consider the remov<strong>in</strong>g motion; let A be compact, <strong>and</strong>suppose that x is <strong>in</strong> the <strong>in</strong>ternal part <strong>of</strong> A; let A t := A \ B(x, t) be the removal47
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