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 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.
Inferring functional connectivity, or statistical dependencies between activity in different regions of the brain, is of great interest in the study of neurocognitive conditions. For example, studies ...1-3 indicate that patterns in connectivity might yield potential biomarkers for conditions such as Alzheimer's and autism. We model functional connectivity using Markov Networks, which use conditional dependence to determine when brain regions are directly connected. In this paper, we show that standard large-scale two-sample testing that compares graphs from distinct populations using subject level estimates of functional connectivity, fails to detect differences in functional connections. We propose a novel procedure to conduct two-sample inference via resampling and randomized edge selection to detect differential connections, with substantial improvement in statistical power and error control.
Gaussian Graphical Models (GGM) are popularly used in neuroimaging studies based on fMRI, EEG or MEG to estimate functional connectivity, or relationships between remote brain regions. In ...multi-subject studies, scientists seek to identify the functional brain connections that are different between two groups of subjects, i.e. connections present in a diseased group but absent in controls or vice versa. This amounts to conducting two-sample large scale inference over network edges post graphical model selection, a novel problem we call Population Post Selection Inference. Current approaches to this problem include estimating a network for each subject, and then assuming the subject networks are fixed, conducting two-sample inference for each edge. These approaches, however, fail to account for the variability associated with estimating each subject's graph, thus resulting in high numbers of false positives and low statistical power. By using resampling and random penalization to estimate the post selection variability together with proper random effects test statistics, we develop a new procedure we call \(R^{3}\) that solves these problems. Through simulation studies we show that \(R^{3}\) offers major improvements over current approaches in terms of error control and statistical power. We apply our method to identify functional connections present or absent in autistic subjects using the ABIDE multi-subject fMRI study.
Using Gaussian graphical models as the basis for functional connectivity, we propose new models and test statistics to detect whether subject covariates predict differences in network metrics in a ...population of subjects. Our approach emphasizes the need to account for errors in estimating subject level networks when conducting inference at the population level. Using simulations, we show that failure to do so reduces statistical power in detecting covariate effects for realistic graph structures. We illustrate the benefits of our procedure for clinical neuroimaging using a resting-state fMRI study of neurofibromatosis-I.
Anisotropic Nonlocal Means Denoising Maleki, Arian; Narayan, Manjari; Baraniuk, Richard G
arXiv (Cornell University),
12/2012
Paper, Journal Article
Odprti dostop
It has recently been proved that the popular nonlocal means (NLM) denoising algorithm does not optimally denoise images with sharp edges. Its weakness lies in the isotropic nature of the ...neighborhoods it uses to set its smoothing weights. In response, in this paper we introduce several theoretical and practical anisotropic nonlocal means (ANLM) algorithms and prove that they are near minimax optimal for edge-dominated images from the Horizon class. On real-world test images, an ANLM algorithm that adapts to the underlying image gradients outperforms NLM by a significant margin.
Suboptimality of nonlocal means on images with sharp edges Maleki, Arian; Narayan, Manjari; Baraniuk, Richard
2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton),
2011-Sept.
Conference Proceeding
Odprti dostop
We conduct an asymptotic risk analysis of the non-local means image denoising algorithm for Horizon class images that are piecewise constant with a sharp edge discontinuity. We prove that the ...mean-square risk of nonlocal means is suboptimal and in fact is within a log factor of the mean square risk of wavelet thresholding.
We conduct an asymptotic risk analysis of the nonlocal means image denoising algorithm for the Horizon class of images that are piecewise constant with a sharp edge discontinuity. We prove that the ...mean square risk of an optimally tuned nonlocal means algorithm decays according to \(n^{-1}\log^{1/2+\epsilon} n\), for an \(n\)-pixel image with \(\epsilon>0\). This decay rate is an improvement over some of the predecessors of this algorithm, including the linear convolution filter, median filter, and the SUSAN filter, each of which provides a rate of only \(n^{-2/3}\). It is also within a logarithmic factor from optimally tuned wavelet thresholding. However, it is still substantially lower than the the optimal minimax rate of \(n^{-4/3}\).
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