Analysis of planar shapes using geodesic paths on shape spaces

By Eric Klassen, Anuj Srivastava, Washington Mio, Shantanu H. Joshi. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2004

532 citations recorded by the source corpus.

Subfields: Computer Vision and Pattern Recognition, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0d949618d3c9dded12644687420b62e16b7274f1

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Citations in the corpus

Shape representation and classification using the Poisson equation (2004, background): “Various abstract shape spaces are proposed based, e.g., on conformal maps [48], harmonic embedding [17], or suitable parameterization of closed curves [ 31 ].”
A Metric on Shape Space with Explicit Geodesics (2007, background): “The definition and study of spaces of plane shapes has recently met a large amount of interest [2, 5, 7, 10, 18, 15], and has important applications, in object recognition, for the analysis of shape databases or in medical imaging.”
Metrics in the space of curves (2004, methodology): “M, that is, there exists a∈ lR3,a6= 0 s.t. a1 cos(θ(s)) +a2 sin(θ(s)) +a3 = 0 foralmostalls; then, by integrating,a3 = 0, thereforea1 cos(θ(s))+a2 sin(θ(s)) = 0 that means that θ∈ Z. See also §3.1 in [KSMJ03]. This is the proof of 2.15: (48)in writing ξ= γ◦ φ, we, in a sense, deﬁne by a pullback of ”
Statistical shape analysis: clustering, learning, and testing (2005, methodology): “Klassen et al. [15] approximate geodesics on S by successively drawing infinitesimal line segments in L L 2 and projecting them onto S, as depicted in Fig.”
Shape Analysis of Elastic Curves in Euclidean Spaces (2011, methodology): “In each case the top row shows a non-elastic geodesic ( [14]) while the bottom rows the elastic geodesic between the same shapes.”
Geometric modeling in shape space (2007, methodology): “A computational approach to spaces of curves was presented in [Klassen et al. 2004] but has no natural extension to surfaces.”
Principal component analysis for Riemannian manifolds, with an application to triangular shape spaces (2006, background): “In fact, by using arbitrary geodesics instead of projections of straight lines in the tangent space of a mean (the projections are usually also geodesics) we obtain a better fit and find an increase in the amount of variance explained by the first principal component geodesic.”
INTRINSIC SHAPE ANALYSIS: GEODESIC PCA FOR RIEMANNIAN MANIFOLDS MODULO ISOMETRIC LIE GROUP ACTIONS (2007, methodology): “For the shape space of closed planar curves of Klassen et al. (2004) using direction functions θ, H is the space of Fourier series, n = 3, and φ(θ) = (∫ 2π 0 cos θ(s)ds, ∫ 2π 0 sin θ(s)ds, ∫ 2π 0 θ(s)ds ) defines a subspace of codimension 3.”
Kernel Density Estimation and Intrinsic Alignment for Shape Priors in Level Set Segmentation (2006, background): “Some more recent advances were done by Trouvé (1998) and Younes (1998) and by Klassen et al. (2004).”
Integral Invariants for Shape Matching (2006, background): “Differential invariants to actions of various Lie groups have been addressed thoroughly [44] , [36], [17], [58], [76], [30], [48], [64], [91], [85], [37].”
Properties of Sobolev-type metrics in the space of curves (2006, background): “Let then f : [0, 2π ]→ R be smooth and with support in [1, 3] and f (2) = 1; let moreover C(θ, t) := (c1(θ), tf (θ)) so |∂θC| = √ 1+ (tf ′(θ))2 > 1.”