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

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We already presented all the calculus needed to compute the H 0 and ˜H 1gradients of E 1 (see Section 2.4.4 and Example 10.21); let us summarize all theresults.• In the case of H 0 gradient descent flow, the second term is ill posed , sinceL φ /L 2 > 0 and ∇ H 0L is the driving term in the geometric heat flow.This is also clear from the explicit formula∇ H 0E 1 (c) = [∇φ · N − κ(φ − avg c (φ))]N (10.41)since avg c (φ) is the average value of φ, therefore φ − avg c (φ) is negativeon roughly half of the curve; so the second term tries to increase length onhalf of curve using a geometric heat flow (and this is ill posed); whereasthe first term ∇φ does not stabilize the flow.• In the case of the Sobolev gradient, we saw in the proof of Theorem 10.31that the gradients ∇˜H1L φ and ∇˜H1L are locally Lipschitz in C 1 , so, by usingLemma 10.25 we conclude that the gradient ∇˜H1E 1 is locally Lipschitz aswell, hence the gradient flow is well defined.10.8.2 New edge-based active contour modelsThe average weighted length idea can be used to build new energies with interestingapplications in tracking.Let us call∫E old (c) = φ(c(s)) dsthe traditional edge-based active contour energy.When using a Sobolev–type norm, we can consider non-shrinking edge-basedmodels, as in the following examples.Example 10.36 Let us consider the energy∫E new (c) = φ(c(s)) ( L −1 + αLκ 2 (s) ) dswherecc• the first term is edge-detection without shrinking bias (nor regularity),• and the second term is a kind of elastic regularization (that will bediscussed also in the next section) that is moreover relaxed near edges;the length terms render the whole energy scale invariant.The frames in Figure 14 on the following page show initialization and finalresults obtained with different norms and energies.86
Example 10.37 Let us consider a length increasing edge-based model. Inthis case, we maximize the energy∫∫E inc (c) = φ(c(s)) ds − α κ 2 (s) dscwhere φ > 0 is high near edges. We compare numerical results with a typicaledge-based balloon model∫∫E bal (c) = φ(c(s)) ds − α φ(x) dxcRwhere R is the area enclosed by c. See figure 15.cInitial E old , H 0 E old , Sobolev E new , SobolevFigure 14: Comparison of segmentations obtained with different energies, asexplained in Example 10.36. (From [58] c○ 2008 IEEE. Reproduced with permission).Figure 15: Left to right we see the initial contour, the minima of E bal forα = 0.2, 0.25, 0.4 using H 0 ; the maximum for E inc with α = 0.1 using ˜H 1flowing. See Example 10.37. (From [58] c○ 2008 IEEE. Reproduced with permission).87
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Metrics of curves in shape optimiza
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shape analysis where we study a fam
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• F (c) = F (c ◦ φ) for all cu
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κ > 0HNNHNHκ < 0Figure 1: Example
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In the case of planar curves c 1 ,
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(a) (b) (c) (d) (e)Figure 3: Segmen
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where φ may be chosen to beφ(x) =
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2.4.1 Example: geometric heat flowW
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2.4.5 Centroid energyWe will now pr
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shapes. Unfortunately, H 0 does not
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• If the second request is waived
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• The Fréchet space of smooth fu
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Since φ k are homeomorphisms, then
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3.6.1 Riemann metric, lengthDefinit
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Theorem 3.30 Suppose that M is a sm
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Example 3.38 Let M = C ∞ ([−1,
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• sometimes S 1 will be identifie
Page 36 and 37: The proof is by direct computation.
Page 38 and 39: The term preshape space is sometime
Page 40 and 41: 5 Representation/embedding/quotient
Page 42 and 43: 6.1.1 Length induced by a distanceI
Page 44 and 45: 0101200000000000011111111111110000
Page 46 and 47: 6.2.4 Applications in computer visi
Page 48 and 49: of a small ball from A. The motion
Page 50 and 51: CutΩ(The arrows represent the dist
Page 52 and 53: 7.3 L 1 metric and Plateau problemI
Page 54 and 55: Definition 8.3 (Flat curves) Let Z
Page 56 and 57: Proof. Fix α 0 ∈ S \ Z. Let T =
Page 58 and 59: 8.2.2 RepresentationThe Stiefel man
Page 60 and 61: 9.3 Conformal metricsYezzi and Menn
Page 62 and 63: 10.1.1 Related worksA family of met
Page 64 and 65: Definition 10.7 (Convolution) A arc
Page 66 and 67: and̂∇˜HjE(0) = ̂∇ H 0E(0),̂
Page 68 and 69: and we simply integrate twice! More
Page 70 and 71: Corollary 10.18 In particular, the
Page 72 and 73: 10.6 Existence of gradient flowsWe
Page 74 and 75: • The length functional (from C 1
Page 76 and 77: We can eventually estimate the diff
Page 78 and 79: 7.552.5-10 -7.5 -5 -2.5 2.5 5 7.5 1
Page 80 and 81: By using (i) and (ii) from lemma 10
Page 82 and 83: 10.6.3 Existence of flow for geodes
Page 84 and 85: 10.7.1 Robustness w.r.to local mini
Page 88 and 89: 10.9 New regularization methodsTypi
Page 90 and 91: can be solved for k (this is not so
Page 92 and 93: 2. Now• at t = 1/2 it achieves th
Page 94 and 95: Definition 11.10 • The orbit is O
Page 96 and 97: y eqn. (11.3), where the terms RHS
Page 98 and 99: So Prop. 11.19 guarantees that the
Page 100 and 101: Theorem 11.23 Suppose that the Riem
Page 102 and 103: 2.2.3 Examples of geometric energy
Page 104 and 105: 9 Riemannian metrics of immersed cu
Page 106 and 107: contour continuation, 11convolution
Page 108 and 109: normal vector, 6objective function,
Page 110 and 111: References[1] Luigi Ambrosio, Giuse
Page 112 and 113: [30] Serge Lang. Fundamentals of di
Page 114 and 115: [57] Ganesh Sundaramoorthi, Anthony
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