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