[57] Ganesh Sundaramoorthi, Anthony Yezzi, <strong>and</strong> <strong>Andrea</strong> Mennucci. Coarseto-f<strong>in</strong>esegmentation <strong>and</strong> track<strong>in</strong>g us<strong>in</strong>g Sobolev Active Contours. IEEETransactions on Pattern Analysis <strong>and</strong> Mach<strong>in</strong>e Intelligence (TPAMI), 2008.doi: 10.1109/TPAMI.2007.70751.[58] Ganesh Sundaramoorthi, Anthony Yezzi, <strong>Andrea</strong> Mennucci, <strong>and</strong> GuillermoSapiro. New possibilities with Sobolev active contours. Intn. Journ. ComputerVision, 2008. doi: 10.1007/s11263-008-0133-9.[59] Ala<strong>in</strong> Trouvé <strong>and</strong> Laurent Younes. Local geometry <strong>of</strong> 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, <strong>and</strong>A. Willsky. Model-based curve evolution technique for image segmentation.In Proc. <strong>of</strong> IEEE Conf. on Computer Vision <strong>and</strong> Pattern Recognition,volume 1, pages 463–468, Dec. 2001.[61] A. Tsai, A. Yezzi, <strong>and</strong> A. S. Willsky. Curve evolution implementation <strong>of</strong> themumford-shah functional for image segmentation, denois<strong>in</strong>g, <strong>in</strong>terpolation,<strong>and</strong> magnification. IEEE Transactions on Image Process<strong>in</strong>g, 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, <strong>and</strong> Alan S. Willsky. Model-basedcurve evolution technique for image segmentation. In CVPR (1), pages463–468, 2001.[63] L. A. Vese <strong>and</strong> T. F. Chan. A multiphase level set framework for imagesegmentation us<strong>in</strong>g the mumford <strong>and</strong> shah model. Int. J. Computer Vision,50(3):271–293, 2002.[64] A. Yezzi, A. Tsai, <strong>and</strong> A. Willsky. A statistical approach to snakes forbimodal <strong>and</strong> trimodal imagery. In Int. Conf. on Computer Vision, pages898–903, October 1999.[65] Anthony Yezzi <strong>and</strong> <strong>Andrea</strong> Mennucci. Conformal metrics <strong>and</strong> true “gradientflows” for <strong>curves</strong>. In International Conference on Computer Vision(ICCV05), pages 913–919, 2005. doi: 10.1109/ICCV.2005.60. URLhttp://research.micros<strong>of</strong>t.com/iccv2005/.[66] Anthony Yezzi <strong>and</strong> <strong>Andrea</strong> Mennucci. Geodesic homotopies. In EU-SIPCO04, 2004. URL http://www.eurasip.org/content/Eusipco/2004/defevent/papers/cr1925.pdf.[67] Anthony Yezzi <strong>and</strong> <strong>Andrea</strong> Mennucci. <strong>Metrics</strong> <strong>in</strong> the space <strong>of</strong> <strong>curves</strong>. arXiv,2004.[68] Laurent Younes. Computable elastic distances between <strong>shape</strong>s. SIAMJournal <strong>of</strong> 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
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- Page 110 and 111: References[1] Luigi Ambrosio, Giuse
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