A Tour of Modern Image Filtering⋆

Recent developments in computational imaging and restoration have heralded the arrival and convergence of several powerful methods for adaptive processing of multidimensional data. Examples include Moving Least Square (from Graphics), the Bilateral Filter and Anisotropic Diffusion (from Machine Vision), Boosting, Kernel, and Spectral Methods (from Machine Learning), Non-local Means and its variants (from Signal Processing), Bregman Iterations (from Applied Math), Kernel Regression and Iterative Scaling (from Statistics). While these approaches found their inspirations in diverse fields of nascence, they are deeply connected. In this paper: • I present 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. I also illustrate connections between these techniques and more classical (and 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 BM3D, and methods for their iterative improvement (or non-existence 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,) in order to keep matters focused, we consider the problem of denoising here. ∗Electrical Engineering Department, University of California, Santa Cruz CA, 95064 USA. e-mail: [email protected], Phone:(650) 322-2311, Fax: (410) 322-2312 .... with Some New Insights. An earlier version of this manuscript was titled “A Tour of Modern Image Processing”. To learn more, view my talk based on this paper, available at http://tinyurl.com/moderntour September 19, 2011 DRAFT