Discontinuity preserving regularization of inverse visual problems

The method of Tikhonov regularization has been widely used to form well-posed inverse problems in low-level vision. The application of this technique usually results in a least squares approximation or a spline fitting of the parameter of interest. This is often adequate for estimating smooth parameter fields. However, when the parameter of interest has discontinuities the estimate formed by this technique will smooth over the discontinuities. Several techniques have been introduced to modify the regularization process to incorporate discontinuities. Many of these approaches however, will themselves be ill-posed or ill-conditioned. This paper presents a technique for incorporating discontinuities into the reconstruction problem while maintaining a well-posed and well-conditioned problem statement. The resulting computational problem is a convex functional minimization problem. This method is compared to previous approaches and examples are presented for the problems of reconstructing curves and surfaces with discontinuities and for estimating image data. Computational issues arising in both analog and digital implementations are also discussed.>