Skip to main content

Research Repository

Advanced Search

All Outputs (11)

A Tight Smooth Approximation of the Maximum Function and its Applications (2022)
Journal Article
Zhang, K., & Yin, K. (2022). A Tight Smooth Approximation of the Maximum Function and its Applications. Journal of Convex Analysis, 29(4),

We analyse the C1,1 tight approximations of the finite maximum function defined by the upper compensated convex transform introduced in a previous paper of the second author [ Compensated convexity and its applications, Ann. Inst. H. Poincaré (C), No... Read More about A Tight Smooth Approximation of the Maximum Function and its Applications.

Some computable quasiconvex multiwell models in linear subspaces without rank-one matrices (2022)
Journal Article
Zhang, K., & Yin, K. (2022). Some computable quasiconvex multiwell models in linear subspaces without rank-one matrices. Electronic Research Archive, 30(5), 1632-1652. https://doi.org/10.3934/era.2022082

In this paper we apply a smoothing technique for the maximum function, based on the compensated convex transforms, originally proposed by Zhang in [1] to construct some computable multiwell non-negative quasiconvex functions in the calculus of variat... Read More about Some computable quasiconvex multiwell models in linear subspaces without rank-one matrices.

Compensated Convex-Based Transforms for Image Processing and Shape Interrogation (2022)
Book Chapter
Orlando, A., Crooks, E., & Zhang, K. (2022). Compensated Convex-Based Transforms for Image Processing and Shape Interrogation. In K. Chen, C. Schönlieb, X. Tai, & L. Younces (Eds.), Handbook of mathematical models and algorithms in computer vision and imaging (1-60). Cham: Springer. https://doi.org/10.1007/978-3-030-03009-4_106-1

This paper reviews some recent applications of the theory of the compensated convex trans-forms or of the proximity hull as developed by the authors to image processing and shape inter-rogation with special attention given to the Hausdorff stability... Read More about Compensated Convex-Based Transforms for Image Processing and Shape Interrogation.

Compensated Convexity on Bounded Domains, Mixed Moreau Envelopes and Computational Methods (2021)
Journal Article
Zhang, K., Orlando, A., & Crooks, E. (2021). Compensated Convexity on Bounded Domains, Mixed Moreau Envelopes and Computational Methods. Applied Mathematical Modelling, 94, 688-720. https://doi.org/10.1016/j.apm.2021.01.040

Compensated convex transforms have been introduced for extended real valued functions defined over Rn. In their application to image processing, interpolation and shape interrogation, where one deals with functions defined over a bounded domain, one... Read More about Compensated Convexity on Bounded Domains, Mixed Moreau Envelopes and Computational Methods.

Hausdorff Stability and Error Estimates for Compensated Convexity Based Methods for Approximation and Interpolation for Functions in Rn (2020)
Journal Article
Alatawi, M., & Zhang, K. (2020). Hausdorff Stability and Error Estimates for Compensated Convexity Based Methods for Approximation and Interpolation for Functions in Rn. Journal of Convex Analysis, 27(4),

We establish error estimates and Hausdorff stability for approximations and interpolations for sampled functions in Rn by using compensated convex transforms introduced previously by K. Zhang [Compensated convexity and its applications, Ann. l’Instit... Read More about Hausdorff Stability and Error Estimates for Compensated Convexity Based Methods for Approximation and Interpolation for Functions in Rn.

Compensated convexity methods for approximations and interpolations of sampled functions in Euclidean spaces: applications to contour lines, sparse data and inpainting (2018)
Journal Article
Zhang, K., Crooks, E., & Orlando, A. (2018). Compensated convexity methods for approximations and interpolations of sampled functions in Euclidean spaces: applications to contour lines, sparse data and inpainting. SIAM Journal on Imaging Sciences, 11(4), 2368-2428. doi:10.1137/17M116152X

This paper is concerned with applications of the theory of approximation and interpolation based on compensated convex transforms developed in [55]. We apply our methods to (i) surface reconstruction starting from the knowledge of finitely many level... Read More about Compensated convexity methods for approximations and interpolations of sampled functions in Euclidean spaces: applications to contour lines, sparse data and inpainting.

Compensated convexity methods for approximations and interpolations of sampled functions in Euclidean spaces: theoretical foundations (2016)
Journal Article
Zhang, K., Crooks, E., & Orlando, A. (in press). Compensated convexity methods for approximations and interpolations of sampled functions in Euclidean spaces: theoretical foundations. SIAM Journal on Mathematical Analysis, 48(6), https://doi.org/10.1137/15M1045673

We introduce Lipschitz continuous and C¹,¹ geometric approximation and interpolation methods for sampled bounded uniformly continuous functions over compact sets and over complements of bounded open sets in Rn by using compensated convex transforms.... Read More about Compensated convexity methods for approximations and interpolations of sampled functions in Euclidean spaces: theoretical foundations.

Compensated convex transforms and geometric singularity extraction from semiconvex functions (2016)
Journal Article
Zhang, K., Crooks, E., & Orlando, A. (in press). Compensated convex transforms and geometric singularity extraction from semiconvex functions. https://doi.org/10.1360/N012015-00339

The upper and lower compensated convex transforms are `tight' one-sided approximations for a given function. We apply these transforms to the extraction of fine geometric singularities from general semiconvex/semiconcave functions and DC-functions in... Read More about Compensated convex transforms and geometric singularity extraction from semiconvex functions.

Compensated convexity, multiscale medial axis maps and sharp regularity of the squared-distance function (2015)
Journal Article
Zhang, K., Crooks, E., & Orlando, A. (in press). Compensated convexity, multiscale medial axis maps and sharp regularity of the squared-distance function. SIAM Journal on Mathematical Analysis, 47(6), https://doi.org/10.1137/140993223

In this paper we introduce a new stable mathematical model for locating and measuring the medial axis of geometric objects, called the quadratic multiscale medial axis map of scale λ, and provide a sharp regularity result for the squared-distance fun... Read More about Compensated convexity, multiscale medial axis maps and sharp regularity of the squared-distance function.

Compensated convexity and Hausdorff stable extraction of intersections for smooth manifolds (2014)
Journal Article
Zhang, K., Orlando, A., & Crooks, E. (2015). Compensated convexity and Hausdorff stable extraction of intersections for smooth manifolds. Mathematical Models and Methods in Applied Sciences, 25(5), https://doi.org/10.1142/S0218202515500207

We apply compensated convex transforms to define a multiscale Hausdorff stable method to extract intersections between smooth compact manifolds represented by their characteristic functions or as point clouds embedded in Rn. We prove extraction resul... Read More about Compensated convexity and Hausdorff stable extraction of intersections for smooth manifolds.

Compensated convexity and Hausdorff stable geometric singularity extractions (2014)
Journal Article
Zhang, K., Orlando, A., & Crooks, E. (2014). Compensated convexity and Hausdorff stable geometric singularity extractions. Mathematical Models and Methods in Applied Sciences, 25(4), https://doi.org/10.1142/S0218202515500189

We develop and apply the theory of lower and upper compensated convex transforms introduced in [K. Zhang, Compensated convexity and its applications, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008) 743–771] to define multiscale, parametrized, geo... Read More about Compensated convexity and Hausdorff stable geometric singularity extractions.