Grid filters for local nonlinear image restoration

A new approach to local nonlinear image restoration is described, based on approximating functions using a regular grid of points in a many-dimensional space. Symmetry reductions and compression of the sparse grid make it feasible to work with twelve-dimensional grids as large as 2212. Unlike polynomials and neural networks whose ltering complexity per pixel is linear in the number of lter coe cients, grid lters have O(1) complexity per pixel. Grid lters require only a single presentation of the training samples, are numerically stable, leave unusual image features unchanged, and are a superset of order statistic lters. Results are presented for additive noise, blurring, and superresolution. iv Acknowledgements I thank Dr. Ed Jernigan, my supervisor, for his encouraging words and for allowing me freedom to explore. I am grateful to the Vision and Image Processing Laboratory folks for providing a friendly and relaxing environment. My readers, Dr. Paul Fieguth and Dr. Glenn Heppler, provided many useful suggestions. Special thanks are due to Dr. Gregory V. Wilson for being a tireless mentor and friend. My partner, Lindsay Patten, provided a context of love, support, and happiness which have seen me through this degree. Finally, I acknowledge with gratitude the nancial support I have received from the National Science and Engineering Research Council of Canada (NSERC), and the Department of Systems Design Engineering, University of Waterloo. v