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

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[57] Ganesh Sundaramoorthi, Anthony Yezzi, and Andrea Mennucci. Coarseto-finesegmentation and tracking using Sobolev Active Contours. IEEETransactions on Pattern Analysis and Machine Intelligence (TPAMI), 2008.doi: 10.1109/TPAMI.2007.70751.[58] Ganesh Sundaramoorthi, Anthony Yezzi, Andrea Mennucci, and GuillermoSapiro. New possibilities with Sobolev active contours. Intn. Journ. ComputerVision, 2008. doi: 10.1007/s11263-008-0133-9.[59] Alain Trouvé and Laurent Younes. Local geometry of deformable templates.SIAM J. Math. Anal., 37(1):17–59 (electronic), 2005. ISSN 0036-1410.[60] A. Tsai, A. Yezzi, W. Wells, C. Tempany, D. Tucker, A. Fan, E. Grimson, andA. Willsky. Model-based curve evolution technique for image segmentation.In Proc. of IEEE Conf. on Computer Vision and Pattern Recognition,volume 1, pages 463–468, Dec. 2001.[61] A. Tsai, A. Yezzi, and A. S. Willsky. Curve evolution implementation of themumford-shah functional for image segmentation, denoising, interpolation,and magnification. IEEE Transactions on Image Processing, 10(8):1169–1186, Aug 2001.[62] Andy Tsai, Anthony J. Yezzi, William M. Wells III, Clare Tempany, DeweyTucker, Ayres Fan, W. Eric L. Grimson, and Alan S. Willsky. Model-basedcurve evolution technique for image segmentation. In CVPR (1), pages463–468, 2001.[63] L. A. Vese and T. F. Chan. A multiphase level set framework for imagesegmentation using the mumford and shah model. Int. J. Computer Vision,50(3):271–293, 2002.[64] A. Yezzi, A. Tsai, and A. Willsky. A statistical approach to snakes forbimodal and trimodal imagery. In Int. Conf. on Computer Vision, pages898–903, October 1999.[65] Anthony Yezzi and Andrea Mennucci. Conformal metrics and true “gradientflows” for curves. In International Conference on Computer Vision(ICCV05), pages 913–919, 2005. doi: 10.1109/ICCV.2005.60. URLhttp://research.microsoft.com/iccv2005/.[66] Anthony Yezzi and Andrea Mennucci. Geodesic homotopies. In EU-SIPCO04, 2004. URL http://www.eurasip.org/content/Eusipco/2004/defevent/papers/cr1925.pdf.[67] Anthony Yezzi and Andrea Mennucci. Metrics in the space of curves. arXiv,2004.[68] Laurent Younes. Computable elastic distances between shapes. SIAMJournal of Applied Mathematics, 58(2):565–586, 1998.114
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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
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The proof is by direct computation.
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The term preshape space is sometime
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5 Representation/embedding/quotient
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6.1.1 Length induced by a distanceI
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0101200000000000011111111111110000
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6.2.4 Applications in computer visi
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of a small ball from A. The motion
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CutΩ(The arrows represent the dist
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7.3 L 1 metric and Plateau problemI
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Definition 8.3 (Flat curves) Let Z
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Proof. Fix α 0 ∈ S \ Z. Let T =
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8.2.2 RepresentationThe Stiefel man
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9.3 Conformal metricsYezzi and Menn
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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 86 and 87: We already presented all the calcul
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
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