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In this paper we study an anisotropic variant of the Rudin-Osher-Fatemifunctional with $L^1$ fidelity term of the form \[ E(u) = \int_{\mathbb{R}^n}\phi(\nabla u) + \lambda \| u -f \|_{L^1(\mathbb{R}^n)}. \] We willcharacterize the minimizers of $E$ in terms of the Wulff shape of $\phi$ andthe dual anisotropy. In particular we will calculate the subdifferential of$E$. We will apply this characterization to the special case $\phi = |\cdot|_1$and $n=2$, which has been used in the denoising of 2D bar codes. In this case,we determine the shape of a minimizer $u$ when $f$ is the characteristicfunction of a circle.

Characterization of minimizers of an anisotropic variant of the Rudin-Osher-Fatemi functional with $L^1$ fidelity term