An asymptotic minimax analysis of nonlocal means on edges
Baraniuk, Richard G.
Master of Music
This 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  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  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.
Applied sciences; Applied mathematics; Electrical engineering