Total Generalized Variation for Piecewise Constant Functions on Triangular Meshes with Applications in Imaging

2022/06/24 by Lukas P. Baumgartner, Ronny Bergmann, Baumgärtner, Lukas +7 · 2 citations
Computer Science · Medicine · #FOS: Mathematics #Image and Signal Denoising Methods #Medical Image Segmentation Techniques #Medical Imaging Techniques and Applications #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2206.12331

openalex publication_date 2022/06/24 · openalex created_date 2022/09/28 · openalex updated_date 2026/07/28

Abstract

We propose a novel discrete concept for the total generalized variation (TGV), which has originally been derived to reduce the staircasing effect in classical total variation (TV) regularization, in image denoising problems. We describe discrete, second-order TGV for piecewise constant functions on triangular meshes, thus allowing the TGV functional to be applied to more general data structures than pixel images, and in particular in the context of finite element discretizations. Particular attention is given to the description of the kernel of the TGV functional, which, in the continuous setting, consists of linear polynomials. We discuss how to take advantage of this kernel structure using piecewise constant functions on triangular meshes. Numerical experiments include denoising and inpainting problems for images defined on non-standard grids, including data from a 3D scanner.

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