Two dimensional image restoration using Linear Programming

Abstract

In this paper we address the issues involved in implementing the Linear Programming (LP) method of image restoration in two dimensions. A modified pivot strategy is introduced in order to reduce the number of iterations. This approach is necessary due to the number of arithmetic operations required per iteration for two dimensional data. We shall also discuss the effects of additive noise, such as that due to quantization. The effects of noise on the performance of restorations with both separable and nonseparable degradations will be presented. The performance of the LP approach has previously been seen to show a preference for sparse images C13, i.e. images with many zero valued pixels. This preference is also found in the two dimensional case. The error in the LP restored image is shown to be less for sparse images than it is for dense images with the same signal-to-noise ratio (SNR). Linear Programming, despite its name, is nonlinear. That is, the LP solution of the sum of two images is not necessarily the sum of the two solutions obtained for each image separately. The LP method also allows for inequality as well as equality constraints to be imposed on the solution. The advantage in performance of the constrained nonlinear approach taken here over linear non-constrained methods is also demonstrated. This is accompli shed by illustrating the pseudo-inverse solution for each LP restoration. The pseudo-inverse method represents the optimal non-constrained linear restoration.