Statistical Tests for Total Variation Regularization Parameter Selection

Total Variation (TV) is an effective method of removing noise in digital image processing while preserving edges [23]. The choice of scaling or regularization parameter in the TV process defines the amount of denoising, with value of zero giving a result equivalent to the input signal. Here we explore three algorithms for specifying this parameter based on the statistics of the signal in the total variation process. The Discrepancy Principle, a new algorithm based on the χ method for Tikhonov regularization [17]–[21], and an ”empirically Bayesian” approach suggested in [9]. In all three algorithms TV regularization is viewed as an M-estimator [3] and it is assumed to converge to a well defined limit even if the probability model is not correctly specified. These regularization parameter selection algorithms are implemented in such a way that they can supplement any TV optimization algorithm. The algorithms are useful for computationally large problems because a single regularization parameter is found that satisfies an appropriate statistical test, and the regularization parameter does not need to be manually adjusted, or iterated to zero. This is especially useful for nonlinear problems where an underlying linear problem is solved iteratively, taking the guesswork out of choosing the regularization parameter in each iterate.