Solving Inverse Problems With Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity

By G. Yu, Guillermo Sapiro, Stéphane Mallat. IEEE Transactions on Image Processing, 2010

619 citations recorded by the source corpus.

A general framework for solving image inverse problems with piecewise linear estimations is introduced in this paper. The approach is based on Gaussian mixture models, which are estimated via a maximum a posteriori expectation-maximization algorithm. A dual mathematical interpretation of the proposed framework with a structured sparse estimation is described, which shows that the resulting piecewise linear estimate stabilizes the estimation when compared with traditional sparse inverse problem techniques. We demonstrate that, in a number of image inverse problems, including interpolation, zooming, and deblurring of narrow kernels, the same simple and computationally efficient algorithm yields results in the same ballpark as that of the state of the art.

Subfields: Computational Mechanics, Computer Vision and Pattern Recognition

View Paper: https://www.semanticscholar.org/paper/88a3843a9973e95ae5171d020d803478fe55e217

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Citations in the corpus

A Statistical Prediction Model Based on Sparse Representations for Single Image Super-Resolution (2014, methodology): “Models that are commonly exploited in single image SR methods include image smoothness and geometric regularity of image structures [3]), gradient profile priors [4], self-similarity of image patches within and across different scales in the same image [6], [7], [24], and sparsity – either in the wa”
Denoising Prior Driven Deep Neural Network for Image Restoration (2018, methodology): “Generally, existing IR methods can be classiﬁed into two main categories, i.e., model-based methods [1], [8], [9], [13]– [18] and learning-based methods [19]–[24].”
Sparse Modeling for Image and Vision Processing (2014, methodology): “…video processing [Protter and Elad, 2009], and finally, we conclude this section with a presentation of other patch-modeling approaches [Buades et al., 2005, Awate and Whitaker, 2006, Dabov et al., 2007a, Takeda et al., 2007, Zoran and Weiss, 2011, Yu et al., 2012, Chatterjee and Milanfar, 2012].”
Fast and Robust Archetypal Analysis for Representation Learning (2014, background): “For example, natural image patches have been modeled with sparse coding [18] or mixture of Gaussians [25], yielding powerful representations for image restoration.”
Tradeoffs Between Convergence Speed and Reconstruction Accuracy in Inverse Problems (2016, positive): “A related study [5] showed that it is enough to use a small number of Gaussians to represent all the patches in natural images instead of using a dictionary that spans a much larger”
Patch Group Based Nonlocal Self-Similarity Prior Learning for Image Denoising (2015, methodology): “Considering that Gaussian Mixture Model (GMM) has been successfully used to model the image patch priors in EPLL [26] and PLE [28], we propose to extend patch based GMM to patch group based GMM (PG-GMM) for NSS prior learning.”
Image Denoising Using the Higher Order Singular Value Decomposition (2013, positive): “Recently, other successful denoising techniques, such as [46] and [41] (which use Gaussian mixture models) or [19], [44], and [45] (which learn overcomplete dictionaries), have emerged.”
The Little Engine That Could: Regularization by Denoising (RED) (2016, positive): “A recently released paper extends the above by using a speciﬁc GMM-based denoiser, showing that these two conditions are met, thus guaranteeing global convergence of their ADMM scheme [38].”
A Nonlocal Bayesian Image Denoising Algorithm (2013, methodology): “Formally, this method [49] is an almost literal application of the PLE method [46]; see section 3.”
Sliced Wasserstein Distance for Learning Gaussian Mixture Models (2017, methodology): “GMMs have been utilized for: image representation [5, 16] to generate feature signatures, point set registration [23], adaptive contrast enhancement [9], inverse problems including super-resolution and deblurring [18, 54], time series classification [8], texture segmentation [42], and robotic visuom”
From learning models of natural image patches to whole image restoration (2011, methodology): “For the GMM prior, we optimized (by hand) the values for β on the 5 images from the Berkeley training set - these were set to β = 1 σ2 · [1, 4, 8, 16, 32, 64].”
The student-t mixture as a natural image patch prior with application to image compression (2014, methodology): “Aside from being another unsupervised learning technique, sparse coding has been related with GMM in another way: Some authors (Yu et al., 2012; Zoran and Weiss, 2011) have suggested the interpretation of a GMM as a structured sparse coding model.”