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In this article, the author presents a practical and accessible framework to understand some of the basic underpinnings of these methods, with the intention of leading the reader to a broad understanding of how they interrelate. The author also illustrates connections between these techniques and more classical (empirical) Bayesian approaches. The proposed framework is used to arrive at new insights and methods, both practical and theoretical. In particular, several novel optimality properties of algorithms in wide use such as block-matching and three-dimensional (3-D) filtering (BM3D), and methods for their iterative improvement (or nonexistence thereof) are discussed. A general approach is laid out to enable the performance analysis and subsequent improvement of many existing filtering algorithms. While much of the material discussed is applicable to the wider class of linear degradation models beyond noise (e.g., blur,) to keep matters focused, we consider the problem of denoising here.