A Random Walk on Image Patches

Taylor, Kye; Meyer, François G.

doi:10.1137/110839370

Cited by 11 publications

(15 citation statements)

References 40 publications

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“…By analyzing the E-step and the M-step of the EM algorithm, we find that the actions of the symmetrization is a type of data clustering. This observation echoes with a number of recent work that shows ordering and grouping of non-local patches are key to high-quality image denoising [24]- [26]. There are two contributions of this paper, described as follows.…”

Section: Contributionssupporting

confidence: 66%

Understanding Symmetric Smoothing Filters: A Gaussian Mixture Model Perspective

Chan

Zickler

2017

IEEE Trans. on Image Process.

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Abstract-Many patch-based image denoising algorithms can be formulated as applying a smoothing filter to the noisy image. Expressed as matrices, the smoothing filters must be row normalized so that each row sums to unity. Surprisingly, if we apply a column normalization before the row normalization, the performance of the smoothing filter can often be significantly improved. Prior works showed that such performance gain is related to the Sinkhorn-Knopp balancing algorithm, an iterative procedure that symmetrizes a row-stochastic matrix to a doublystochastic matrix. However, a complete understanding of the performance gain phenomenon is still lacking.In this paper, we study the performance gain phenomenon from a statistical learning perspective. We show that SinkhornKnopp is equivalent to an Expectation-Maximization (EM) algorithm of learning a Gaussian mixture model of the image patches. By establishing the correspondence between the steps of Sinkhorn-Knopp and the EM algorithm, we provide a geometrical interpretation of the symmetrization process. This observation allows us to develop a new denoising algorithm called Gaussian mixture model symmetric smoothing filter (GSF). GSF is an extension of the Sinkhorn-Knopp and is a generalization of the original smoothing filters. Despite its simple formulation, GSF outperforms many existing smoothing filters and has a similar performance compared to several state-of-the-art denoising algorithms.

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Section: Contributionssupporting

confidence: 66%

Understanding Symmetric Smoothing Filters: A Gaussian Mixture Model Perspective

Chan

Zickler

2017

IEEE Trans. on Image Process.

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“…The value of s E, ℓ;2 controls the number of locations in { g j ∈ 𝒢: D ( g k , g j ) ≤ s E, ℓ;2 }, which is a patch set at g k (Taylor and Meyer, 2012), whereas s E, ℓ;1 is used to shrink small | w E, g k g j | to zero.…”

Section: Multiscale Weighted Principal Component Regressionmentioning

confidence: 99%

“…It has been shown that algorithms based on patch information have led to state-of-the art techniques for classification and denoising (Taylor and Meyer, 2012; Li et al, 2011; Polzehl and Spokoiny, 2006; Arias-Castro et al, 2012). …”

Section: Multiscale Weighted Principal Component Regressionmentioning

confidence: 99%

MWPCR: Multiscale Weighted Principal Component Regression for High-Dimensional Prediction

Zhu

Shen

Peng

et al. 2017

Journal of the American Statistical Association

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We propose a multiscale weighted principal component regression (MWPCR) framework for the use of high dimensional features with strong spatial features (e.g., smoothness and correlation) to predict an outcome variable, such as disease status. This development is motivated by identifying imaging biomarkers that could potentially aid detection, diagnosis, assessment of prognosis, prediction of response to treatment, and monitoring of disease status, among many others. The MWPCR can be regarded as a novel integration of principal components analysis (PCA), kernel methods, and regression models. In MWPCR, we introduce various weight matrices to prewhitten high dimensional feature vectors, perform matrix decomposition for both dimension reduction and feature extraction, and build a prediction model by using the extracted features. Examples of such weight matrices include an importance score weight matrix for the selection of individual features at each location and a spatial weight matrix for the incorporation of the spatial pattern of feature vectors. We integrate the importance score weights with the spatial weights in order to recover the low dimensional structure of high dimensional features. We demonstrate the utility of our methods through extensive simulations and real data analyses of the Alzheimer’s disease neuroimaging initiative (ADNI) data set.

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“…The value of s E ,ℓ;2 controls the number of vertexes in { g ′ ∈ : D ( g , g ′) ≤ s E ,ℓ;2 }, which is a patch set at vertex g [18], whereas s E ,ℓ;1 is used to shrink small | w E , gg ′ |s into zero.…”

Section: Spatially Weighted Principal Component Regressionmentioning

confidence: 99%

“…It has been shown that for graph data, algorithms based on patch information have led to state-of-the art techniques for classification and denoising. See for example, [18] for overviews of imaging patches.…”

Section: Spatially Weighted Principal Component Regressionmentioning

confidence: 99%

Spatially Weighted Principal Component Regression for High-Dimensional Prediction

Shen

Zhu

2015

Lecture Notes in Computer Science

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We consider the problem of using high dimensional data residing on graphs to predict a low-dimensional outcome variable, such as disease status. Examples of data include time series and genetic data measured on linear graphs and imaging data measured on triangulated graphs (or lattices), among many others. Many of these data have two key features including spatial smoothness and intrinsically low dimensional structure. We propose a simple solution based on a general statistical framework, called spatially weighted principal component regression (SWPCR). In SWPCR, we introduce two sets of weights including importance score weights for the selection of individual features at each node and spatial weights for the incorporation of the neighboring pattern on the graph. We integrate the importance score weights with the spatial weights in order to recover the low dimensional structure of high dimensional data. We demonstrate the utility of our methods through extensive simulations and a real data analysis based on Alzheimer’s disease neuroimaging initiative data.

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A Random Walk on Image Patches

Cited by 11 publications

References 40 publications

Understanding Symmetric Smoothing Filters: A Gaussian Mixture Model Perspective

Understanding Symmetric Smoothing Filters: A Gaussian Mixture Model Perspective

MWPCR: Multiscale Weighted Principal Component Regression for High-Dimensional Prediction

Spatially Weighted Principal Component Regression for High-Dimensional Prediction

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