J Math Imaging Vis (2012) 44:168–184 DOI 10.1007/s10851-011-0319-6 Convex Approximation Technique for Interacting Line Elements Deblurring: a New Approach Antonio Boccuto · Ivan Gerace · Patrizia Pucci Published online: 27 September 2011 © Springer Science+Business Media, LLC 2011 Abstract 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 as- sume 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 en- ergy, 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 Sect. 3. The- orem 3 shows that the first convex approximation defined in CATILED has good qualities for the reconstruction. The experimental results, given in Sect. 10, confirm the applica- bility of the technique. Keywords Image restoration problem · Regularization · Dual energy functions · Graduated non-convexity A. Boccuto · I. Gerace · P. Pucci ( ) Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy e-mail: pucci@dmi.unipg.it A. Boccuto e-mail: boccuto@dmi.unipg.it I. Gerace e-mail: gerace@dmi.unipg.it 1 Introduction Techniques for restoring digital images have applications in several scientific fields, like biomedicine, astronomy, robotics, and so on. Indeed, in these branches the image is a fundamental tool of investigation. However, sometimes the bad quality of the available images does not allow us to use them immediately. In these cases it is necessary to proceed to a restoration of the involved image, in order to eliminate the presence of noise and the effects of the blur. In this paper the problem of defining suitable models and efficient algo- rithms for restoring piecewise continuous regular images is investigated. The problem of restoring images deals with es- timating the original image, by starting from the observed image and by the characteristic of the blur. In this situation we assume to know the blur mask, while when the mask is unknown the problem is called blind restoration. Techniques to solve the blind problem [13, 17, 18] have to refer to the algorithms for the unblind case. The restoration problem is ill-posed in the sense of Hadamard (cf. [6, 12]), that is, in some cases, the solution neither exists, nor is unique, nor can be stable in presence of noise. Thus, regularization techniques (cf. [5, 6, 11, 12, 20, 21]) are useful tools to transform this problem in a well- posed one. The solution is the minimum of a suitable energy function, which is called primal energy function and is the sum of two terms. The first measures the data consistency and the latter the faithfulness to the regularity properties of the solution. In particular, concerning the data consistency we use models based on Euclidean norms and the Gaussian regularization. In order to obtain more realistic restored images, we have to take into account the discontinuities which appear in the intensity field. Edges of the objects arising in the image pro- duce parts of these discontinuities.