Technical Report MSC-2009-16

Title: Automatic Parameter Tuning for Inverse Problems
Authors: Raja Giryes
Supervisors: Michael Elad and Yonina Eldar
Abstract: Linear inverse problems are very common in signal and image processing. In these problems, an original image x is deteriorated by a (known) linear operator H, followed by additive white Gaussian noise w. Given the measurement y=Hx+w, the goal is to reconstruct x. When the image x is known to have a sparse representation over a dictionary D, an effective way to recover x is to use an iterative shrinkage algorithm. This algorithm provides an iterative estimator for x denoted by hlambda,K(y) where K is the number of iterations and lambda is a parameter of the method, and are to be tuned for best performance. We would desire to choose these parameters so as to minimize the mean-squared-error (MSE) of the resulting estimate. However, since the original signal is unknown, the MSE will depend on x in general, and therefore cannot be minimized. The generalized Stein Unbiased Risk Estimator (GSURE), proposed in a recent work, provides an unbiased MSE estimation for a candidate solution of the problem posed above.

In our work we use this estimator to drive an automatic tuning of the estimator hlambda,K(y), bypassing the need for the ideal image x. This idea of tuning the algorithm parameters using GSURE has been proposed recently by Vonesh et. al.. However, their work is restricted to square and invertible operators H. Furthermore, their method, referred to as the global method, sets one of the parameters, given the other, by minimizing the overall estimated MSE.

Our work differs from this contribution in several important ways. First, our proposed algorithms for parameter selection handle any matrix H, including ill-posed and even rectangular operators. Second, we put forward two additional parameter setting schemes, both based on a greedy approach that tunes the lambda per iteration with and without a look-ahead option. This way, each iteration leads to a different value of lambda, and this gives a natural way to set the number of iterations simultaneously – the iterations can be stopped when the estimated MSE improvement is below a certain threshold. Third, we offer a way to adopt the parameter lambda to the scale of the atoms in the representation, thus enabling different shrinkage per scale. Fourth and last, we provide extensive comparisons to conventional methods for parameter selection showing the superiority of the use of GUSRE.

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