Half--Quadratic Image Restoration with a Non-Parallelism Constraint

The problem of image restoration from blur and noise is studied. By regularization techniques, a solution of the problem is found as the minimum of a primal energy function, which is formed by two terms. The former deals with faithfulness to the data, and the latter is associated with the smoothness constraints. We impose that the obtained results are images piecewise continuous and with thin edges. In correspondence with the primal energy function, there is a dual energy function, which deals with discontinuities implicitly. We present a unified approach of the duality theory, also to consider the non-parallelism constraint. We construct a dual energy function, which is convex and imposes such a constraint. To reconstruct images with Boolean discontinuities, the proposed energy function can be used as an initial approximation in a Graduated Non-Convexity algorithm. The experimental results confirm that such a technique inhibits the formation of parallel lines.