Compensated convexity on bounded domains, mixed Moreau envelopes and computational methods

Zhang, Kewei; Orlando, Antonio; Crooks, Elaine

doi:10.1016/j.apm.2021.01.040

Cited by 2 publications

(5 citation statements)

References 33 publications

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“…| and (iii) the execution time t c in seconds by a PC with processor Intel® Core™ i7-4510U CPU@2.00 GHz and 8GB of memory RAM. λ (χ K ) computed by Algorithm 2 [124] which coincides with the one computed using the parabola envelope scheme [49].…”

Section: Prototype Examplementioning

confidence: 80%

“…Such reduction is achieved, one way or another, by a dimensional reduction. The fundamental idea of the scheme presented in [124], for instance, is the generalization of the Euclidean distance transform of binary images, by replacing the binary image by an arbitrary function on a grid. The decomposition of the structuring element which yields the exact Euclidean distance transform [99] into basic ones, leads to a simple and fast algorithm where the discrete lower Moreau envelope can be computed by a sequence of local operations, using one-dimensional neighborhoods.…”

Section: A Moreau Envelope Based Algorithmmentioning

confidence: 99%

“…By taking the infimum in (4.5) over a finite number m ≥ 1 of directions, we obtain the m−th approximation of the discrete Moreau lower envelope M h λ (f )(x k ) which can be evaluated by taking the values f m (x k ) given by Algorithm 2. For the convergence analysis and converegence rate we refer to [124] where it is shown that the scheme has a linear convergence rate with respect to h.…”

Section: A Moreau Envelope Based Algorithmmentioning

confidence: 99%

“…From the mathematical/computer science perspective, however, feature identification from sampled domains is challenging and usually methods are justified only by either ad hoc arguments or numerical experiments. Figure 4 Via fast and robust numerical implementations of the transforms [124], this theory also gives rise to a highly-effective computational toolbox for applications. The efficiency of the numerical computations benefits greatly from the locality property, which holds despite the global nature of the convex envelope itself.…”

Section: Introductionmentioning

confidence: 99%

“…Figure 19 Supports of the exact and computed upper compensated transform of the characteristic function of a singleton set of R 2 by the different numerical schemes. (a) Exact support given by B(0; 1/ √ λ) for λ = 0.01; (b) Support of C u,h λ (χ K ) computed by Algorithm 1 [80]; (c) Support of C u,h λ (χ K ) computed by the biconjugate based scheme [70, 57] for h d = 0.001; (d) Support of C u,hλ (χ K ) computed by Algorithm 2[124] which coincides with the one computed using the parabola envelope scheme[49].…”

mentioning

confidence: 99%

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Compensated Convex Based Transforms for Image Processing and Shape Interrogation

Orlando¹,

Crooks²,

Zhang³

2020

Preprint

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This paper reviews some recent applications of the theory of the compensated convex transforms or of the proximity hull as developed by the authors to image processing and shape interrogation with special attention given to the Hausdorff stability and multiscale properties. The paper contains also numerical experiments that demonstrate the performance of our methods compared to the state-of-art ones.

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Section: Prototype Examplementioning

confidence: 80%

Section: A Moreau Envelope Based Algorithmmentioning

confidence: 99%

Section: A Moreau Envelope Based Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Compensated Convex Based Transforms for Image Processing and Shape Interrogation

Orlando¹,

Crooks²,

Zhang³

2020

Preprint

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Some computable quasiconvex multiwell models in linear subspaces without rank-one matrices

Yin

Zhang

2022

era

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<abstract>In this paper we apply a smoothing technique for the maximum function, based on the compensated convex transforms, originally proposed by Zhang in [<xref ref-type="bibr" rid="b1">1</xref>] to construct some computable multiwell non-negative quasiconvex functions in the calculus of variations. Let $ K\subseteq E\subseteq M^{m\times n} $ with $ K $ a finite set in a linear subspace $ E $ without rank-one matrices of the space $ M^{m\times n} $ of real $ m\times n $ matrices. Our main aim is to construct computable quasiconvex lower bounds for the following two multiwell models with possibly uneven wells: i) Let $ f:K\subseteq E\to E^\perp $ be an $ L $-Lipschitz mapping with $ 0\leq L\leq 1/\alpha $ and $ H_2(X) = \min\{ |P_EX-A_i|^2+\alpha|P_{E^\perp}X-f(A_i)|^2+\beta_i:\, i = 1, 2, \dots, k\} $, where $ \alpha > 0 $ is a control parameter, and ii) $ H_1(X) = \alpha|P_{E^\perp}X|^2+\min\{\sqrt{|\mathcal{U}_i(P_EX-A_i)|^2+\gamma_i}: i = 1, 2, \dots, k\} $, where $ A_i\in E $ with $ U_i:E\to E $ invertible linear transforms for $ i = 1, 2, \dots, k $. If the control paramenter $ \alpha > 0 $ is sufficiently large, our quasiconvex lower bounds are 'tight' in the sense that near each 'well' the lower bound agrees with the original function, and our lower bound are of $ C^{1, 1} $. We also consider generalisations of our constructions to other simple geometrical multiwell models and discuss the implications of our constructions to the corresponding variational problems.</abstract>

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Compensated convexity on bounded domains, mixed Moreau envelopes and computational methods

Cited by 2 publications

References 33 publications

Compensated Convex Based Transforms for Image Processing and Shape Interrogation

Compensated Convex Based Transforms for Image Processing and Shape Interrogation

Some computable quasiconvex multiwell models in linear subspaces without rank-one matrices

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