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

More documents

Recommendations

Info

normal vector, 6objective function, 5open sets, 20operator, 35orbit, 32, 94orthogonaldecomposition of deformation, 68parameterizationinvariance, see “reparameterizationinvariant”curve-wise, 92, 93homotopy-wise, 93parametric or landmark averaging, 9path-metric, 42, 43pathological, 18, 91paths, 3PCA, 9Picard–Lindelöf theorem, 78planar curves, 3polar change of coordinates, 48pose, 52preshape space, 38, 53primitive operator, 69principal component analysis, 9, 30principal variation, 10prior information, 18probability measure, 7, 32projection, 32projection operator, 69, 74quotient, 32quotient distance ˆd, 55quotient space, 4random processes, 10random vector, 9region based energies, 12region-based, 11, 12, 83removing, 47reparameterization, 4, 100reparameterization invariant, 4, 6, 36,37, 39, 61, 92, 95–97rescaling, 37, 82, 86, 100invariance, 39, 61Riemannian metric, 28rigidified norm, 60rigidified norms, 58Rinow, 30, 43, 44rotation, 37, 100invariance, 39, 61rotation index, 53run-length function, 64, 75SAC, 61scalar product, 5, 28scale invariant, 61, 86, 88segmentation, 83semidistance, 20, 93semimetric, 93seminorm, 22, 93set symmetric difference, 8set symmetric distance, 8shape, 18shape analysis, 3, 7, 62shape optimization, 2, 19, 62shape space, 2, 37, 40, 41, 46of curves up to pose, 52categories of —, 37shooting geodesics, 29short length bias, 15, 85signed distance function, 8signed distance level set averaging, 9signed scalar curvature, 6smooth functions, 24Sobolev active contour, 61Sobolev space, 23, 92Sobolev-type metrics, 61space of non-translating deformations,68space of translations, 68special rotation, 37square root lifting, 57standard distance, 93Stiefel manifold, 57stochastic processes, 10submanifold, 26supess, 22tangent bundle, 26tangent space, 3, 26108
tangent vector, 5topological hyperspace, 43topological space, 20topology, 20total variation length, 42translation, 37, 100invariance, 39, 61up to pose, 52vertical space, 94visual tracking, 62, 86winding number, 53109
Page 1 and 2:
Metrics of curves in shape optimiza
Page 3 and 4:
shape analysis where we study a fam
Page 5 and 6:
• F (c) = F (c ◦ φ) for all cu
Page 7 and 8:
κ > 0HNNHNHκ < 0Figure 1: Example
Page 9 and 10:
In the case of planar curves c 1 ,
Page 11 and 12:
(a) (b) (c) (d) (e)Figure 3: Segmen
Page 13 and 14:
where φ may be chosen to beφ(x) =
Page 15 and 16:
2.4.1 Example: geometric heat flowW
Page 17 and 18:
2.4.5 Centroid energyWe will now pr
Page 20 and 21:
shapes. Unfortunately, H 0 does not
Page 22 and 23:
• If the second request is waived
Page 24 and 25:
• The Fréchet space of smooth fu
Page 26 and 27:
Since φ k are homeomorphisms, then
Page 28 and 29:
3.6.1 Riemann metric, lengthDefinit
Page 30 and 31:
Theorem 3.30 Suppose that M is a sm
Page 32 and 33:
Example 3.38 Let M = C ∞ ([−1,
Page 34 and 35:
• 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 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 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
show all

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

Create successful ePaper yourself

Delete template?

Save as template?