An Existence Theorem and Lattice Approximations for a Variational Problem Arising in Computer Vision

Kulkarni, Sanjeev R.; Mitter, Sanjoy K.; Richardon, Thomas J.

doi:10.1007/978-1-4684-6393-4_14

Cited by 8 publications

(4 citation statements)

References 10 publications

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“…This behavior is similar to the one observed for the Mumford-Shah functional (see [39]). (2)- (4) is in general ill-posed, in the sense that the infimum in (2) is zero, and there is no solution under constraint (4).…”

Section: Analysis Of the Modelsupporting

confidence: 87%

How to Choose Interpolation Data in Images

Belhachmi¹,

Bucur²,

Burgeth³

et al. 2009

SIAM J. Appl. Math.

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We introduce and discuss shape based models for finding the best interpolation data when reconstructing missing regions in images by means of solving the Laplace equation. The shape analysis is done in the framework of Γ-convergence, from two different points of view. First, we propose a continuous PDE model and get pointwise information on the "importance" of each pixel by a topological asymptotic method. Second, we introduce a finite dimensional setting into the continuous model based on fat pixels (balls with positive radius), and study by Γ-convergence the asymptotics when the radius vanishes. In this way, we obtain relevant information about the optimal distribution of the best interpolation pixels. We show that the resulting optimal data sets are identical to sets that can also be motivated using level set ideas and approximation theoretic considerations. Numerical computations are presented that confirm the usefulness of our theoretical findings for PDE-based image compression.

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Section: Analysis Of the Modelsupporting

confidence: 87%

How to Choose Interpolation Data in Images

Belhachmi¹,

Bucur²,

Burgeth³

et al. 2009

SIAM J. Appl. Math.

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show abstract

“…A main unconvenient in this approach is the irregularity of the mapping u Ä S u . Our talk, is in a way similar to that of [11], where u and 1 are independent variables situation which would also allow to impose some extra constraints on 1.…”

Section: An Existence Results In Computer Visionmentioning

confidence: 98%

“…A relaxed bi-dimensional problem is considered in [11] where in a first step is taken a length penalty term (m(1 = ))Â2= for which is proved the existence of the minimizing term. In this case, a minimizing boundary may have a nonzero Lebesgue measure.…”

Section: An Existence Results In Computer Visionmentioning

confidence: 99%

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Free Boundary Problems and Density Perimeter

Bucur

Zolésio

1996

Journal of Differential Equations

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We introduce a new concept``the density perimeter,'' which substitutes the measure of length of the perimeter of a set in free boundary problems. We deduce some links between the Hausdorff convergence and the char convergence for a family of domains with bounded density perimeter. If this perimeter is considered as a penalty term, we give existence results for the variational problem which appears in computer vision and in a Bernoulli-like free boundary problem. We also make some considerations concerning a 1-convergence property of the density perimeter.

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Oriented Distance Functions in Shape Analysis and Optimization

Delfour

Zolésio

1995

Control and Optimal Design of Distributed Parameter Systems

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An Existence Theorem and Lattice Approximations for a Variational Problem Arising in Computer Vision

Cited by 8 publications

References 10 publications

How to Choose Interpolation Data in Images

How to Choose Interpolation Data in Images

Free Boundary Problems and Density Perimeter

Oriented Distance Functions in Shape Analysis and Optimization

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