Total Variation Denoising on Hexagonal Grids

TL;DR

This paper explores total variation denoising on hexagonal grids, demonstrating that such discretization reduces artifacts and staircasing compared to Cartesian grids, using graph cut optimization techniques.

Contribution

It introduces a novel approach combining total variation denoising with hexagonal lattice discretization and graph cut algorithms, improving image quality and artifact reduction.

Findings

Hexagonal grids produce closer approximations to original images.

Hexagonal discretization reduces metrication artifacts.

Staircasing effects are mitigated on hexagonal lattices.

Abstract

This work combines three paradigms of image processing: i) the total variation approach to denoising, ii) the superior structure of hexagonal lattices, and iii) fast and exact graph cut optimization techniques. Although isotropic in theory, numerical implementations of the $BV$ seminorm invariably show anisotropic behaviour. Discretization of the image domain into a hexagonal grid seems perfectly suitable to mitigate this undesirable effect. To this end, we recast the continuous problem as a finite-dimensional one on an arbitrary lattice, before focussing on the comparison of Cartesian and hexagonal structures. Minimization is performed with well-established graph cut algorithms, which are easily adapted to new spatial discretizations. Apart from producing minimizers that are closer in the $\ell^1$ sense to the clean image for sufficiently high degrees of regularization, our experiments suggest that the hexagonal lattice also allows for a more effective reduction of two major drawbacks of existing techniques: metrication artefacts and staircasing. For the sake of practical relevance we address the difficulties that naturally arise when dealing with non-standard images.