Shape-from-Shading with discontinuous image brightness

Camilli, Fabio; Prados, Emmanuel

doi:10.1016/j.apnum.2006.03.007

Cited by 11 publications

(2 citation statements)

References 18 publications

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“…In the standard reference on optimal control [3], it is assumed that x → c(x, a) is uniformly continuous. This assumption is then used to prove regularity of the value function v. There is relatively little research devoted to relaxing the regularity condition on c. There are some results for the optimal control problem associated with the Eikonal equation [36,8,14], which allow c to have discontinuities. These results assume that A = R d and make essential use of either Lipschitzness of v, or uniform continuity and/or coercivity of p → H(x, p), none of which hold for the variational problem (1.4).…”

Section: Analysis Of Variational Problemmentioning

confidence: 99%

A Hamilton--Jacobi Equation for the Continuum Limit of Nondominated Sorting

Calder¹,

Esedoḡlu²,

Hero³

2014

SIAM J. Math. Anal.

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We show that nondominated sorting of a sequence X 1 , . . . , Xn of independent and identically distributed random variables in R d has a continuum limit that corresponds to solving a Hamilton-Jacobi equation involving the probability density function f of X i . Nondominated sorting is a fundamental problem in multiobjective optimization and is equivalent to finding the canonical antichain partition and to problems involving the longest chain among Euclidean points. As an application of this result, we show that nondominated sorting is asymptotically stable under bounded random perturbations in X 1 , . . . , Xn. We give a numerical scheme for computing the viscosity solution of this Hamilton-Jacobi equation and present some numerical simulations for various density functions.

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Section: Analysis Of Variational Problemmentioning

confidence: 99%

A Hamilton--Jacobi Equation for the Continuum Limit of Nondominated Sorting

Calder¹,

Esedoḡlu²,

Hero³

2014

SIAM J. Math. Anal.

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“…What we are looking for is very much linked to viscosity solution of the Hamilton-Jacobi equation |∇v| = ξ, when ξ is only L q * (see [6] for a definition via W 1,q test functions and local a.e. inequality and [7] for some other notions and equivalences in the case ξ ∈ L ∞ ). Yet, we are here interested only in the following two notions: the one presented in [8] (and in Section 2.2) and the maximal a.e.…”

Section: γ−Convergence Of the Discrete Functionals To The Continuous Onementioning

confidence: 99%

Numerical approximation of continuous traffic congestion equilibria

Benmansour

Carlier

Peyré

et al. 2009

Networks &Amp; Heterogeneous Media

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Starting from a continuous congested traffic framework recently introduced in [8], we present a consistent numerical scheme to compute equilibrium metrics. We show that equilibrium metric is the solution of a variational problem involving geodesic distances. Our discretization scheme is based on the Fast Marching Method. Convergence is proved via a Γ-convergence result and numerical results are given.

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Shape from Shading

Tankus

Sochen

Yeshurun

2009

Wiley Encyclopedia of Computer Science and Engineering

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Shape‐from‐shading (SfS) is a fundamental problem in computer vision. Its goal is reconstruction of surface depth (i.e., distance from camera plane) based on a single image of the surface. The problem was introduced in the early 1970s by Horn. A very common assumption in this field is that image projection is orthographic. We will present the orthographic shape‐from‐shading problem and an algorithm for its solution: the fast marching method of Kimmel and Sethian. We shall than reexamine the basis of SfS, which is the image irradiance equation, under a perspective projection assumption. The resultant equation does not depend on the depth function directly, but on its natural logarithm, and as such it is invariant to scale changes of the depth function. A reconstruction method based on the perspective formula is then described; it is a modification of the aforementioned orthographic fast marching method. Then, a comparison of the orthographic fast marching, perspective fast marching, and the perspective algorithm of Prados and Faugeras on synthetic images is are presented. The two perspective methods equate with each other and show better reconstruction results than the orthographic. We then compare the orthographic and perspective versions of the fast marching method on endoscopic images. The perspective algorithm outperformed the orthographic one. These findings suggest that the more realistic set of assumptions of perspective SfS improves reconstruction significantly with respect to orthographic SfS. The findings also provide evidence that perspective SfS can be used for real‐life applications in fields such as endoscopy.

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Shape-from-Shading with discontinuous image brightness

Cited by 11 publications

References 18 publications

A Hamilton--Jacobi Equation for the Continuum Limit of Nondominated Sorting

A Hamilton--Jacobi Equation for the Continuum Limit of Nondominated Sorting

Numerical approximation of continuous traffic congestion equilibria

Shape from Shading

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