A Tour of Modern Image Processing

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 and Spectral Methods (from Machine Learning), Non-local Means (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 unified 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 Bayesian approaches. • The proposed framework is used to arrive at new insights, methods, and both practical and theoretical results. 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. • Several theoretical results are discussed which will enable the performance analysis and subsequent improvement of any existing restoration algorithm. While much of the material discussed is applicable to wider class of linear degradation models (e.g. noise, blur, etc.,) 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 This work was supported in part by the US Air Force Grant FA9550-07-1-0365 and the National Science Foundation Grant CCF-1016018 February 18, 2011 DRAFT