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

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References[1] Luigi Ambrosio, Giuseppe Da Prato, and Andrea Mennucci. An introductionto measure theory and probability. Scuola Normale Superiore, 2007. URLhttp//dida.sns.it/dida2/cl/07-08/folde2/pdf0.[2] T. M. Apostol. Mathematical Analysis. Addison - Wesley, 1974.[3] C. J. Atkin. The Hopf-Rinow theorem is false in infinite dimensions. Bull.London Math. Soc., 7(3):261–266, 1975.[4] H. Brezis. Analisi Funzionale. Liguori Editore, Napoli, 1986. (italiantranslation of Analyse fonctionelle, Masson, 1983, Paris).[5] J Canny. A computational approach to edge detection. IEEE Trans. PatternAnal. Mach. Intell., 8(6):679–698, 1986. ISSN 0162-8828.[6] V. Caselles, F. Catte, T. Coll, and F. Dibos. A geometric model for edgedetection. Num. Mathematik, 66:1–31, 1993.[7] V. Caselles, R. Kimmel, and G. Sapiro. Geodesic active contours. InProceedings of the IEEE Int. Conf. on Computer Vision, pages 694–699,Cambridge, MA, USA, June 1995.[8] T. Chan and L. Vese. Active contours without edges. IEEE Transactionson Image Processing, 10(2):266–277, February 2001.[9] G. Charpiat, O. Faugeras, and R. Keriven. Approximations of shape metricsand application to shape warping and empirical shape statistics. Foundationsof Computational Mathematics, 2004. doi: 10.1007/s10208-003-0094-xgg819.INRIA report 4820 (2003).[10] G. Charpiat, R. Keriven, J.P. Pons, and O. Faugeras. Designing spatiallycoherent minimizing flows for variational problems based on active contours.In ICCV, 2005. doi: 10.1109/ICCV.2005.69.[11] G. Charpiat, P. Maurel, J.-P. Pons, R. Keriven, and O. Faugeras. Generalizedgradients: Priors on minimization flows. International Journal of ComputerVision, 2007. doi: 10.1007/s11263-006-9966-2.[12] Y. Chen, H. Tagare, S. Thiruvenkadam, F. Huang, D. Wilson, K. Gopinath,R. Briggs, and E. Geiser. Using prior shapes in geometric active contoursin a variational framework. International Journal of Computer Vision, 50(3):315–328, Dec 2002.[13] D. Cremers and S. Soatto. A pseuso distance for shape priors in level setsegmentation. In IEEE Int. Workshop on Variational, Geometric and LevelSet Methods, pages 169–176, 2003.[14] Manfredo Perdigão do Carmo. Riemannian Geometry. Birkhäuser, 1992.110
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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
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 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 112 and 113: [30] Serge Lang. Fundamentals of di
Page 114 and 115: [57] Ganesh Sundaramoorthi, Anthony
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