Convex Approximation Technique for Interacting Line Elements Deblurring: a New Approach

The problem of image restoration from blur and noise is studied. A solution of the problem is understood as the minimum of an energy function composed by two terms. The first is the data fidelity term, while the latter is related to the smoothness constraints. The discontinuities of the ideal image are unknown and must be estimated. In particular, the involved images are supposed to be piecewise continuous and with thin and continuous edges. In this paper we assume that the smoothness constraints can be either of the first order, or the second order, or the third order. The energy function that implicitly refers to discontinuities is called dual energy function. To minimize the non--convex dual energy, a GNC (Graduated Non-Convexity) technique is used. The GNC algorithm proposed in this paper is indicated as CATILED, short for Convex Approximation Technique for Interacting Line Elements Deblurring. We also prove in the Appendix the new duality Theorem 3 stated in Section 3. Theorem 3 shows that the first convex approximation defined in CATILED has good qualities for the reconstruction. The experimental results, given in Section 10, confirm the applicability of the technique.