A rank selection of MV-PURE with an unbiased predicted-MSE criterion and its efficient implementation in image restoration

The Minimum-Variance Pseudo-Unbiased Reduced-rank Estimator (MV-PURE) is designed, as a natural reduced-rank extension of the Gauss-Markov estimator, for the unknown deterministic vector in ill-conditioned linear regression model. In this paper, we propose a novel rank-selection for the MV-PURE to achieve a small Mean Square Error (MSE). The proposed rank-selection is realized by minimizing an unbiased estimate of the predicted-MSE, not of the MSE. Our unbiased estimate can be applicable to any noise distribution with zero mean and a finite covariance matrix, while Stein-type unbiased criteria cannot in general. We apply the proposed selection to an image restoration problem and introduce its efficient O(m log m) implementation by using a special structure found in typical blur matrices, where the blur matrix is of size m×m. A numerical example demonstrates that the MV-PURE with the proposed rank-selection achieves a MSE comparable with the minimal MSE for the unknown vector among all possible ranks.