On the ROF Model in Rectilinear Anisotropy: Piecewise Constant Approximation and Universal Minimality
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| Format: | Preprint |
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2019
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| author | Kirisits, Clemens Setterqvist, Eric |
| author_facet | Kirisits, Clemens Setterqvist, Eric |
| contents | We prove that the $L^2$ distance between the minimizer of the $\ell^1$-anisotropic Rudin-Osher-Fatemi (ROF) functional and its minimizer over the space of piecewise constant functions on a rectilinear grid is $\mathcal{O}(h^{\frac12 - \frac{q'}{2q}})$, where $h$ is the grid's mesh size and the datum belongs to $L^q$, $q \ge 2$. These convergence rates are valid in any dimension $d\ge 1$. However, in dimension $d = 1$ they can be further improved to $\mathcal{O}(h^{\frac12 - \frac{1}{2q}})$. To establish the error bounds, $L^q$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1910_05186 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | On the ROF Model in Rectilinear Anisotropy: Piecewise Constant Approximation and Universal Minimality Kirisits, Clemens Setterqvist, Eric Optimization and Control 46N10, 65D18, 65K99, 65N15, 68U10, 92C55 We prove that the $L^2$ distance between the minimizer of the $\ell^1$-anisotropic Rudin-Osher-Fatemi (ROF) functional and its minimizer over the space of piecewise constant functions on a rectilinear grid is $\mathcal{O}(h^{\frac12 - \frac{q'}{2q}})$, where $h$ is the grid's mesh size and the datum belongs to $L^q$, $q \ge 2$. These convergence rates are valid in any dimension $d\ge 1$. However, in dimension $d = 1$ they can be further improved to $\mathcal{O}(h^{\frac12 - \frac{1}{2q}})$. To establish the error bounds, $L^q$ 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. |
| title | On the ROF Model in Rectilinear Anisotropy: Piecewise Constant Approximation and Universal Minimality |
| topic | Optimization and Control 46N10, 65D18, 65K99, 65N15, 68U10, 92C55 |
| url | https://arxiv.org/abs/1910.05186 |