Χ Test for Total Variation Regularization

Total Variation (TV) is an effective method of removing noise in digital image processing while preserving edges [27]. 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 χ2 method for Tikhonov regularization [20, 21, 22, 23, 24], and an “empirically Bayesian” approach suggested in [9]. All three algorithms view the TV problem statistically and consequently, TV regularization is viewed as an M-estimator [3] that 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. We determine that there is χ2 test for TV regularization based on the statistics of the TV functional. The degrees of freedom for this test are estimated numerically. The performance of the algorithm is evaluated on 96 test images with four different images and four blurring operators. Using a χ2 test to find a regularization parameter is advantageous for nonlinear or computationally large problems because it automates selection of the parameter, and gives a statistical justification for it that takes away the guesswork when manually adjusting or iterating it to zero.