The influence of the regularization parameter and the first estimate on the performance of Tikhonov regularized non‐linear image restoration algorithms

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Abstract
This paper reports studies on the influence of the regularization parameter and the first estimate on the performance of iterative image restoration algorithms. We discuss regularization parameter estimation methods that have been developed for the linear Tikhonov–Miller filter to restore images distorted by additive Gaussian noise. We have performed experiments on synthetic data to show that these methods can be used to determine the regularization parameter of non‐linear iterative image restoration algorithms, which we use to restore images contaminated by Poisson noise. We conclude that the generalized cross‐validation method is an efficient method to determine a value of the regularization parameter close to the optimal value. We have also derived a method to estimate the regularization parameter of a Tikhonov regularized version of the Richardson–Lucy algorithm. These iterative image restoration algorithms need a first estimate to start their iteration. An obvious and frequently used choice for the first estimate is the acquired image. However, the restoration algorithm could be sensitive to the noise present in this image, which may hamper the convergence of the algorithm. We have therefore compared various choices of first estimates and tested the convergence of various iterative restoration algorithms. We found that most algorithms converged for most choices, but that smoothed first estimates resulted in a faster convergence.