Minimizers of anisotropic Rudin-Osher-Fatemi models

Abstract

We give the explicit formulas of the minimizers of the anisotropic Rudin-Osher-Fatemi models

\begin{array}{l}{E}_{\mathrm{1}}^{\phi }(u)={\displaystyle {\int }_{\mathrm{\Omega }}{\phi }^{\mathrm{o}}(\mathrm{D}u)\mathrm{d}x+\lambda {\displaystyle {\int }_{\mathrm{\Omega }}\left|u-f\right|\mathrm{d}x,}}\hspace{1em}u\in \mathrm{B}\mathrm{V}(\mathrm{\Omega }),\hfill \\ E_{\mathrm{2}}^{\phi }(u)={\displaystyle {\int }_{\mathrm{\Omega }}{\phi }^{\mathrm{o}}(\mathrm{D}u)\mathrm{d}x+\lambda {\displaystyle {\int }_{\mathrm{\Omega }}{(u-f)}^{\mathrm{2}}\mathrm{d}x,\hspace{1em}u\in \mathrm{B}\mathrm{V}(\mathrm{\Omega }),}}\hfill \end{array}

where \mathrm{\Omega }\subset {ℝ}^{\mathrm{2}} is a domain, {\phi }^{\mathrm{o}} is an anisotropic norm on {ℝ}^{\mathrm{2}}, and f is a solution of the anisotropic 1-Laplacian equations.

Key words： Anisotropic Rudin-Osher-Fatemi (ROF) model; Euler-Lagrange equation; \phi-curvature

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