Baraniuk, Richard G.2013-03-082013-03-082012Narayan, Manjari. "An asymptotic minimax analysis of nonlocal means on edges." (2012) Master’s Thesis, Rice University. <a href="https://hdl.handle.net/1911/70368">https://hdl.handle.net/1911/70368</a>.https://hdl.handle.net/1911/70368This thesis analyzes the non-local means denoising algorithm using the criterion of minimax optimality from statistical decision theory. We show that nonlocal means is minimax suboptimal on images with smooth discontinuities [1] with a rate of convergence of [Special characters omitted.] ( n -1 ) comparable to that of wavelet thresholding. The suboptimality is a consequence of the isotropic nature of the algorithm, and its inability to adapt to the smoothness of the discontinuity. However, all is not lost for nonlocal methods. We also propose an anisotropic nonlocal means algorithm [2] that can attain the optimal rate of [Special characters omitted.] ( n -4/3 ) as well as deliver superior denoising performance using image gradients on synthetic and empirical images, respectively. Nonlocal means is an instance of exemplar based image processing methods. This result broadly implies that exemplar methods that respect anisotropy can yield superior performance in estimating edges in both theory and practice.76 p.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.Applied sciencesApplied mathematicsElectrical engineeringThesisNarayanMTHESIS E.E. 2012 NARAYAN