Digital image restoration is the process of taking a noisy, blurred image and restoring it to its unblurred form. Performing this task typically requires some kind of mathematical model of the blurring process. One way is for image restoration to be thought of as a 2D deconvolution problem, where

• f is the original, unblurred image
• g is some noise function modeling the blur that has taken place
• h is the blurred image

and we are trying to solve the convolution equation f*g = h for f, given h. One of the primary difficulties in the deconvolution process is actually regularizing the data for deconvolution.

Regularization algorithms such as (Tikhonov regularization or truncated SVD) are typically used in discretizations of deconvolution problems. The main drawback to traditional regularization techniques is that they tend to produce smooth solutions – making it difficult to restore sharp edges in f.

The discretization problem can be represented in the form of a least-squares system, min ||A*z – b||_2. The truncated SVD algorithm (TSVD) attempts to minimize the 2-norm of z subject to ||A_k*z – b||_2 = min. By replacing the 2-norm ||z||_2 with ||L_p*z||_2 (where L_p is the pth derivative operator), we can get solutions with maximum flatness or minimum roughness, but the solutions still generally remain continuous and smooth. An alternative method, the partial-polynomial truncated SVD algorithm (PP-TSVD for short), replaces the 2-norm ||z||_2 in TSVD by the seminorm ||L_p*z||_1. The solution to this new formulation takes on the form of partial polynomials (hence the name) and thus can have a number of non-smooth “break points.” This allows us to obtain the sharp edge in image restoration we were seeking.

A research project at the Technical University of Denmark has developed a 2D extension to the PP-TSVD algorithm that performs digital image restoration and scales well to large data sets.

http://www2.imm.dtu.dk/~pch/PPTSVD/pptsvd.html

http://citeseer.ist.psu.edu/hansen99pptsvd.html