A statistical analysis of the kernel-based MMSE estimator with application to image reconstruction

In this paper we carry out a statistical analysis of a multivariate minimum mean square error (MMSE) estimator developed from a nonparametric kernel-based probability density function. This kernel-based MMSE (KMMSE) estimation has been recently proposed by the authors and successfully applied to image and video reconstruction. The statistical analysis that we present here allows us to understand the utility and limitations of this estimator. Thus, we propose a couple of estimation error measures intended for locally linear signals and show how KMMSE can approximate a sparse estimator. Also, we study how the estimation error propagates when signals are reconstructed recursively, that is, when already-reconstructed samples are used for estimating new samples. As an application, we focus on the problem of the filling ordering (FO) associated to the reconstruction of heterogeneous image blocks. Thus, borrowing the concept of soft data from missing data theory, the error measures established in the first part of the paper can be transformed into reliability measures from which a novel FO procedure is developed. We show that the proposed FO outperforms other state-of-the-art procedures. We develop a statistical analysis of the kernel-based minimum mean square error (KMMSE) estimator with emphasis in locally linear functions.Two error risk measures are proposed for this case.We show that KMMSE with a suitable bandwidth estimation procedure provides an approximation to sparse linear prediction with a bias correction term.For the case of recursive estimation (as it is commonly used in image reconstruction), the propagation error is also assessed through a vector Taylor series expansion.The MSE and propagation error measures previously proposed are applied to a novel filling ordering procedure.