As part of tomography with a limited number of angles, the resulting inverse problem is ill-conditioned. The algorithms “classic”, analytic and algebraic produce many artifacts. The optimization methods solve this problem by using a regularization. In my work, I guess the parsimonious or sparse images in a given representation. A sparse image is an image that has few non-zero components. To promote this parsimony, I will rectify the problem with a standard and solve one of the two equivalent problems below.
My job is to assess the theoretical limits of these problems while being the closest to reality. Recent studies  have highlighted sufficient conditions ensuring the reconstruction of the image by minimizing and provide an error of this reconstruction . These conditions require the existence of a particular vector called dual certificate . Depending on the application, it may be possible to calculate a candidate who will be a conditional dual certificate [2,4].
This work primarily cover the construction of such candidates. I propose in my work, a greedy approach to calculate a candidate who is a dual certificate for almost all images “rebuildable” and minimizing the theoretical reconstruction error.
 Charles Dossal and Remi Tesson. Consistency of l1 recovery from noisy deterministic measurements. Applied and Computational Harmonic Analysis, 2013.
 Jean-Jacques Fuchs. Recovery of exact sparse representations in the presence of bounded noise. Information Theory, IEEE Transactions on, 51(10) :3601–3608, 2005.
 Markus Grasmair, Otmar Scherzer, and Markus Haltmeier. Necessary and suﬃcient conditions for linear convergence of l1-regularization. Commun. Pure and Appl. Math., 64(2) :161–182, 2011.
 Samuel Vaiter, Gabriel Peyré, Charles Dossal, and Jalal Fadili. Robust sparse analysis regularization. 2011.