Shape Analysis via Oriented Distance Functions

Delfour, Michel C.; Zolésio, Jean-Paul

doi:10.1006/jfan.1994.1086

Cited by 139 publications

(80 citation statements)

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“…If ∂Ω is a manifold of class C 1 , then there is some ε > 0 such that for all points y ∈ Ω such that d(y, Ω) < ε there is z ∈ ∂Ω and t ∈ (0, ε) such that y = z − tν(z), ν(z) being the outer unit normal to ∂Ω at z ( [11]). In other words, Ω ε := {x ∈ Ω :…”

Section: Appendixmentioning

confidence: 99%

A Parabolic Quasilinear Problem for Linear Growth Functionals

Andreu¹,

Caselles²,

Mazón³

2002

Rev. Mat. Iberoam.

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We prove existence and uniqueness of solutions for the Dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth. A typical example of energy functional we consider is the one given by the nonparametric area integrand f (x, ξ) = 1 + ξ 2 , which corresponds with the time-dependent minimal surface equation. We also study the asymptotic behaviour of the solutions.

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Section: Appendixmentioning

confidence: 99%

A Parabolic Quasilinear Problem for Linear Growth Functionals

Andreu¹,

Caselles²,

Mazón³

2002

Rev. Mat. Iberoam.

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“…There is a neighborhood M of L 0 in U , in which ρ is smooth and F is a foliation on M , see [7]. Moreover, |∇ρ| = 1 on M [5]. By Corollary 11 the extremal function for the p-modulus of F is of the form…”

Section: Remarkmentioning

confidence: 98%

On the extremal function of the modulus of a foliation

Ciska-Niedziałomska

2016

Arch. Math.

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Abstract.We investigate the properties of the modulus of a foliation on a Riemannian manifold. We give necessary and sufficient conditions for the existence of the extremal function and state some of its properties. We obtain an integral formula which, in a sense, combines the integral over the manifold with the integral over the leaves. We state a relation between the extremal function and the geometry of the distribution orthogonal to a foliation.Mathematics Subject Classification. 53C12, 58C35.

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“…4 This is a large set (too large for our applications) which we use as a "frame of reference." DZ was introduced by Delfour and Zolésio [9], [10] and contains the sets F and C 2 introduced below. For technical reasons related to compactness properties (see Section 3.2) we consider the following subset of DZ: Definition 7 (Set DZ 0 ).…”

Section: The Set Of All Shapesmentioning

confidence: 99%

Approximations of Shape Metrics and Application to Shape Warping and Empirical Shape Statistics

Charpiat

Faugeras

Keriven³

et al. 2006

Statistics and Analysis of Shapes

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Abstract. This paper proposes a framework for dealing with several problems related to the analysis of shapes. Two related such problems are the definition of the relevant set of shapes and that of defining a metric on it. Following a recent research monograph by Delfour and Zolésio [11], we consider the characteristic functions of the subsets of R 2 and their distance functions. The L 2 norm of the difference of characteristic functions, the L ∞ and the W 1,2 norms of the difference of distance functions define interesting topologies, in particular the well-known Hausdorff distance. Because of practical considerations arising from the fact that we deal with image shapes defined on finite grids of pixels, we restrict our attention to subsets of R 2 of positive reach in the sense of Federer [16], with smooth boundaries of bounded curvature. For this particular set of shapes we show that the three previous topologies are equivalent. The next problem we consider is that of warping a shape onto another by infinitesimal gradient descent, minimizing the corresponding distance. Because the distance function involves an inf, it is not differentiable with respect to the shape. We propose a family of smooth approximations of the distance function which are continuous with respect to the Hausdorff topology, and hence with respect to the other two topologies. We compute the corresponding Gâteaux derivatives. They define deformation flows that can be used to warp a shape onto another by solving an initial value problem. We show several examples of this warping and prove properties of our approximations that relate to the existence of local minima. We then use this tool to produce computational definitions of the empirical mean and covariance of a set of shape examples. They yield an analog of the notion of principal modes of variation. We illustrate them on a variety of examples.

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Shape Analysis via Oriented Distance Functions

Cited by 139 publications

References 0 publications

A Parabolic Quasilinear Problem for Linear Growth Functionals

A Parabolic Quasilinear Problem for Linear Growth Functionals

On the extremal function of the modulus of a foliation

Approximations of Shape Metrics and Application to Shape Warping and Empirical Shape Statistics

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