CLEAR: Covariant LEAst-Square Refitting with Applications to Image Restoration

Abstract : In this paper, we propose a new framework to remove parts of the systematic errors affecting popular restoration algorithms, with a special focus for image processing tasks. Generalizing ideas that emerged for $\ell_1$ regularization, we develop an approach re-fitting the results of standard methods towards the input data. Total variation regularizations and non-local means are special cases of interest. We identify important covariant information that should be preserved by the re-fitting method, and emphasize the importance of preserving the Jacobian (w.r.t. the observed signal) of the original estimator. Then, we provide an approach that has a ``twicing'' flavor and allows re-fitting the restored signal by adding back a local affine transformation of the residual term. We illustrate the benefits of our method on numerical simulations for image restoration tasks.
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Charles-Alban Deledalle, Nicolas Papadakis, Joseph Salmon, Samuel Vaiter. CLEAR: Covariant LEAst-Square Refitting with Applications to Image Restoration . SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2017, 10 (1), pp.243-284. ⟨10.1137/16M1080318⟩. ⟨hal-01333295v3⟩

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