Total Variation Denoising on Hexagonal Grids

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.