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

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[30] Serge Lang. Fundamentals of differential geometry. Springer–Verlag, 1999.[31] M. Leventon, E. Grimson, and O. Faugeras. Statistical shape influence ingeodesic active contours. In IEEE Conf. on Comp. Vision and Patt. Recog.,volume 1, pages 316–323, 2000.[32] YanYan Li and Louis Nirenberg. The distance function to the boundary,Finsler geometry and the singular set of viscosity solutions of some Hamilton-Jacobi equations. Comm. Pure Appl. Math., LVIII, 2005. (first received as apersonal communication in June 2003).[33] R. Malladi, J. Sethian, and B. Vemuri. Shape modeling with front propagation:a level set approach. IEEE Transactions on Pattern Analysis andMachine Intelligence, (17):158–175, 1995.[34] A. C. G. Mennucci. On asymmetric distances. 2nd version, preprint, 2004.URL http://cvgmt.sns.it/papers/and04/.[35] Andrea Mennucci, Anthony Yezzi, and Ganesh Sundaramoorthi. Propertiesof Sobolev Active Contours. Interf. Free Bound., 10:423–445, 2008.[36] Peter W. Michor and David Mumford. An overview of the Riemannian metricson spaces of curves using the Hamiltonian approach. Applied and ComputationalHarmonic Analysis, 2007. doi: 10.1016/j.acha.2006.07.004. URLhttp://www.mat.univie.ac.at/~michor/curves-hamiltonian.pdf.[37] Peter W. Michor and David Mumford. Riemannian geometris of space ofplane curves. J. Eur. Math. Soc. (JEMS), 8:1–48, 2006.[38] Peter W. Michor and David Mumford. Vanishing geodesic distance onspaces of submanifolds and diffeomorphisms. Documenta Math., 10:217–245,2005. URL http://www.univie.ac.at/EMIS/journals/DMJDMV/vol-10/05.pdf.[39] Peter W. Michor, David Mumford, Jayant Shah, and Laurent Younes. Ametric on shape space with explicit geodesics. Atti Accad. Naz. Lincei Cl.Sci. Fis. Mat. Natur. Rend. Lincei, 9, 2008.[40] Washington Mio and Anuj Srivastava. Elastic-string models for representationand analysis of planar shapes. In CVPR (2), pages 10–15, 2004.[41] Washington Mio and Anuj Srivastava. Elastic-string models for representationand analysis of planar shapes. In Conference on Computer Visionand Pattern Recognition (CVPR), June 2004. URL http://stat.fsu.edu/~anuj/pdf/papers/CVPR_Paper_04.pdf.[42] D. Mumford and J. Shah. Boundary detection by minimizing functionals.In Proc. IEEE Conf. Computer Vision Pattern Recognition, 1985.112
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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
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 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 114 and 115: [57] Ganesh Sundaramoorthi, Anthony
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