On the ROF Model in Rectilinear Anisotropy: Piecewise Constant Approximation and Universal Minimality

We prove that the L² L 2 distance between the minimizer of the ¹ ℓ 1 -anisotropic Rudin-Osher-Fatemi (ROF) functional and its minimizer over the space of piecewise constant functions on a rectilinear grid is O(h^1/2 - q'/2q) O ( h 1 / 2 - q ′ / 2 q ) , where h is the grid’s mesh size and the datum belongs to Lq L q , q ≥ 2 q ≥ 2 . These convergence rates are valid in any dimension d≥ 1 d ≥ 1 . However, in dimension $$d = 1$$ d = 1 they can be further improved to O(h^1/2 - 1/2q) O ( h 1 / 2 - 1 / 2 q ) . To establish the error bounds, Lq L q estimates of the ROF minimizer in terms of the datum are critical. Such estimates are particular cases of a universal minimality property of the ROF minimizer derived in the second part of the paper. There it is shown, in both the finite-dimensional and infinite-dimensional settings, that the minimizer simultaneously minimizes a broad class of convex functionals over a neighbourhood of the datum arising in the convex dual of the ROF problem. This extends previous results of similar type about taut strings and the ROF problem.