A reliable and efficient computational algorithm for restoring blurred and noisy images is proposed. The restoration process is based on the minimal total variation principle introduced by Rudin et al. For discrete images, the proposed algorithm minimizes a piecewise linear l1 function (a measure of total variation) subject to a single 2-norm inequality constraint (a measure of data fit). The algorithm starts by finding a feasible point for the inequality constraint using a (partial) conjugate gradient method. This corresponds to a deblurring process. Noise and other artifacts are removed by a subsequent total variation minimization process. The use of the linear l1 objective function for the total variation measurement leads to a simplier computational algorithm. Both the steepest descent and an affine scaling Newton method are considered to solve this constrained piecewise linear l1 minimization problem. The resulting algorithm, when viewed as an image restoration and enhancement process, has the feature that it can be used in an adaptive/interactive manner in situations when knowledge of the noise variance is either unavailable or unreliable. Numerical examples are presented to demonstrate the effectiveness of the proposed iterative image restoration and enhancement process.
Bibliographical noteFunding Information:
Manuscript received December 1, 1994; revised October 17, 1995. This work was supported in part by the Applied Mathematical Sciences Research Program (KC-04-02) of the Office of Energy Rcsearch of the US. Department of Energy under Grant DE-FG02-90ER25013.AO00, by NSF, AFOSR, and ONR through Grant DMS-8920550, by the Comell Theory Center, which receives major funding from the National Science Foundation and IBM corporation, with additional support from New York State and members of its Corporate Research Institute, by the Air Force Office of Scientific Research under Grant F49620-93-1-0500, the Department of Energy under Grant DE-FG02-94ER25225, and the National Science Foundation under Grant DMS-9210489.