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We solve the image denoising problem with a dictionary learning technique bywriting a convex functional of a new form. This functional contains beside theusual sparsity inducing term and fidelity term, a new term which inducessimilarity between overlapping patches in the overlap regions. The functionaldepends on two free regularization parameters: a coefficient multiplying thesparsity-inducing $L_{1}$ norm of the patch basis functions coefficients, and acoefficient multiplying the $L_{2}$ norm of the differences between patches inthe overlapping regions. The solution is found by applying the iterativeproximal gradient descent method with FISTA acceleration. In the case oftomography reconstruction we calculate the gradient by applying projection ofthe solution and its error backprojection at each iterative step. We study thequality of the solution, as a function of the regularization parameters andnoise, on synthetic datas for which the solution is a-priori known. We applythe method on experimental data in the case of Differential Phase Tomography.For this case we use an original approach which consists in using vectorialpatches, each patch having two components: one per each gradient component. Theresulting algorithm, implemented in the ESRF tomography reconstruction codePyHST, results to be robust, efficient, and well adapted to strongly reduce therequired dose and the number of projections in medical tomography.

plos one

A Convex Functional for Image Denoising based on Patches with  Constrained Overlaps and its vectorial application to Low Dose Differential  Phase Tomography