<p>We prove that the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> distance between the minimizer of the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\ell ^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>ℓ</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-anisotropic Rudin-Osher-Fatemi (ROF) functional and its minimizer over the space of piecewise constant functions on a rectilinear grid is <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {O}(h^{{1}/{2} - {q'}/{2q}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo>-</mo> <msup> <mi>q</mi> <mo>′</mo> </msup> <mo stretchy="false">/</mo> <mrow> <mn>2</mn> <mi>q</mi> </mrow> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <i>h</i> is the grid’s mesh size and the datum belongs to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^q$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>q</mi> </msup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$q \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. These convergence rates are valid in any dimension <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$d\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. However, in dimension <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$d = 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> they can be further improved to <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {O}(h^{{1}/{2} - {1}/{2q}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mrow> <mn>2</mn> <mi>q</mi> </mrow> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. To establish the error bounds, <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$L^q$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>q</mi> </msup> </math></EquationSource> </InlineEquation> 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.</p>

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

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2026年8月4日

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Clemens Kirisits,
Eric Setterqvist

摘要

We prove that the $L^2$ $L^{2}$ distance between the minimizer of the $\ell ^1$ $ℓ^{1}$ -anisotropic Rudin-Osher-Fatemi (ROF) functional and its minimizer over the space of piecewise constant functions on a rectilinear grid is $\mathcal {O}(h^{{1}/{2} - {q'}/{2q}})$ $O (h^{1 / 2 - q^{'} / 2 q})$ , where h is the grid’s mesh size and the datum belongs to $L^q$ $L^{q}$ , $q \ge 2$ $q \geq 2$ . These convergence rates are valid in any dimension $d\ge 1$ $d \geq 1$ . However, in dimension $d = 1$ $d = 1$ they can be further improved to $\mathcal {O}(h^{{1}/{2} - {1}/{2q}})$ $O (h^{1 / 2 - 1 / 2 q})$ . To establish the error bounds, $L^q$ $L^{q}$ 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.