{"ID":2864414,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.23995","arxiv_id":"2509.23995","title":"Mixed-Derivative Total Variation","abstract":"The formulation of norms on continuous-domain Banach spaces with exact pixel-based discretization is advantageous for solving inverse problems (IPs). In this paper, we investigate a new regularization that is a convex combination of a TV term and the $\\M(\\R^2)$ norm of mixed derivatives. We show that the extreme points of the corresponding unit ball are indicator functions of polygons whose edges are aligned with either the $x_1$- or $x_2$-axis. We then apply this result to construct a new regularization for IPs, which can be discretized exactly by tensor products of first-order B-splines, or equivalently, pixels. Furthermore, we exactly discretize the loss of the denoising problem on its canonical pixel basis and prove that it admits a unique solution, which is also a solution to the underlying continuous-domain IP.","short_abstract":"The formulation of norms on continuous-domain Banach spaces with exact pixel-based discretization is advantageous for solving inverse problems (IPs). In this paper, we investigate a new regularization that is a convex combination of a TV term and the $\\M(\\R^2)$ norm of mixed derivatives. We show that the extreme points...","url_abs":"https://arxiv.org/abs/2509.23995","url_pdf":"https://arxiv.org/pdf/2509.23995v1","authors":"[\"Vincent Guillemet\",\"Michael Unser\"]","published":"2025-09-28T17:37:31Z","proceeding":"math.NA","tasks":"[\"math.NA\",\"math.OC\"]","methods":"[]","has_code":false}