  We solve the image denoising problem with a dictionary learning technique by
writing a convex functional of a new form. This functional contains beside the
usual sparsity inducing term and fidelity term, a new term which induces
similarity between overlapping patches in the overlap regions. The functional
depends on two free regularization parameters: a coefficient multiplying the
sparsity-inducing $L_{1}$ norm of the patch basis functions coefficients, and a
coefficient multiplying the $L_{2}$ norm of the differences between patches in
the overlapping regions. The solution is found by applying the iterative
proximal gradient descent method with FISTA acceleration. In the case of
tomography reconstruction we calculate the gradient by applying projection of
the solution and its error backprojection at each iterative step. We study the
quality of the solution, as a function of the regularization parameters and
noise, on synthetic datas for which the solution is a-priori known. We apply
the method on experimental data in the case of Differential Phase Tomography.
For this case we use an original approach which consists in using vectorial
patches, each patch having two components: one per each gradient component. The
resulting algorithm, implemented in the ESRF tomography reconstruction code
PyHST, results to be robust, efficient, and well adapted to strongly reduce the
required dose and the number of projections in medical tomography.
