Metrics of curves in shape optimization and analysis - Andrea Carlo ...

10.6 Existence of gradient flowsWe recall this definition (that was already presented informally in the introduction).Definition 10.22 (Gradient descent flow) Given a differentiable energy E :M → lR, and a metric 〈, 〉 c , let ∇E(c) be the gradient. Let us fix moreoverc 0 : S 1 → lR n , c 0 ∈ M. The gradient descent flow of E is the solutionC = C(t, θ) of the initial value P.D.E.{∂t C = −∇E(C)C(0, θ) = c 0 (θ)We present an example computation for the geodesic active contour modelon a radially symmetric “image”.Example 10.23 LetC(t, θ) = r(t) (cos θ, sin θ) ,φ(x) = d(|x|)with r, d : lR → lR + ; the energy takes the form∫E(C) = φ(C(t, s)) ds = 2π r(t) d(r(t))Then C is the ˜H 1 gradient descent iffCr ′ (t) = − 1 (d(r(t)) + r(t) d ′ (r(t)) ) .λ2πwhereas C is the H 0 gradient descent iffr ′ (t) = −2π ( d(r(t)) + r(t)d ′ (r(t)) ) ,where we use the definition (10.4) of H 0 .Note also thatProof. Indeed,∂ t E(C) = r ′ (t) ( d(r(t)) + r(t)d ′ (r(t)) ) .ds = r(t) dθD s =1r(t) D θlen(C) = 2πr(t)φ(C) = d(r(t))∇φ(x) = x|x| d′ (|x|) for x ≠ 0∇φ(C) = d ′ (r(t))(cos θ, sin θ)avg c (∇φ(C)) = 072