Minimax solution of inverse problems and deconvolution by mirror wavelet thresholding

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Abstract
We consider ill-posed inverse problems where inverting the distortion of signals and images in presence of additive noise is numerically unstable. The properties of linear and non-linear diagonal estimators in an orthogonal basis lead to general conditions to build nearly minimax optimal thresholding estimators. The deconvolution of bounded variation signals and images is studied in further details, with an application to the deblurring of satellite images. Besides their optimality properties, a competition set by the French spatial agency (CNES) showed that this type of algorithms gives the best numerical results among all competing algorithms.

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