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

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