Guoshen Yu

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

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.

Image modeling and enhancement via structured sparse model selection

An image representation framework based on structured sparse model selection is introduced in this work. The corresponding modeling dictionary is comprised of a family of learned orthogonal bases. For an image patch, a model is first selected from this dictionary through linear approximation in a best basis, and the signal estimation is then calculated with the selected model. The model selection leads to a guaranteed near optimal denoising estimator. The degree of freedom in the model selection is equal to the number of the bases, typically about 10 for natural images, and is significantly lower than with traditional overcomplete dictionary approaches, stabilizing the representation. For an image patch of size √N × √N, the computational complexity of the proposed framework is O (N2), typically 2 to 3 orders of magnitude faster than estimation in an overcomplete dictionary. The orthogonal bases are adapted to the image of interest and are computed with a simple and fast procedure. State-of-the-art results are shown in image denoising, deblurring, and inpainting.

Statistical Compressed Sensing of Gaussian Mixture Models

A novel framework of compressed sensing, namely statistical compressed sensing (SCS), that aims at efficiently sampling a collection of signals that follow a statistical distribution, and achieving accurate reconstruction on average, is introduced. SCS based on Gaussian models is investigated in depth. For signals that follow a single Gaussian model, with Gaussian or Bernoulli sensing matrices of O(k) measurements, considerably smaller than the O(k log(N/k)) required by conventional CS based on sparse models, where N is the signal dimension, and with an optimal decoder implemented via linear filtering, significantly faster than the pursuit decoders applied in conventional CS, the error of SCS is shown tightly upper bounded by a constant times the best k-term approximation error, with overwhelming probability. The failure probability is also significantly smaller than that of conventional sparsity-oriented CS. Stronger yet simpler results further show that for any sensing matrix, the error of Gaussian SCS is upper bounded by a constant times the best k-term approximation with probability one, and the bound constant can be efficiently calculated. For Gaussian mixture models (GMMs), that assume multiple Gaussian distributions and that each signal follows one of them with an unknown index, a piecewise linear estimator is introduced to decode SCS. The accuracy of model selection, at the heart of the piecewise linear decoder, is analyzed in terms of the properties of the Gaussian distributions and the number of sensing measurements. A maximization-maximization (Max-Max) algorithm that iteratively estimates the Gaussian models parameters, the signals model selection, and decodes the signals, is presented for GMM-based SCS. In real image sensing applications, GMM-based SCS is shown to lead to improved results compared to conventional CS, at a considerably lower computational cost.

Task-Driven Adaptive Statistical Compressive Sensing of Gaussian Mixture Models

A framework for adaptive and non-adaptive statistical compressive sensing is developed, where a statistical model replaces the standard sparsity model of classical compressive sensing. We propose within this framework optimal task-specific sensing protocols specifically and jointly designed for classification and reconstruction. A two-step adaptive sensing paradigm is developed, where online sensing is applied to detect the signal class in the first step, followed by a reconstruction step adapted to the detected class and the observed samples. The approach is based on information theory, here tailored for Gaussian mixture models (GMMs), where an information-theoretic objective relationship between the sensed signals and a representation of the specific task of interest is maximized. Experimental results using synthetic signals, Landsat satellite attributes, and natural images of different sizes and with different noise levels show the improvements achieved using the proposed framework when compared to more standard sensing protocols. The underlying formulation can be applied beyond GMMs, at the price of higher mathematical and computational complexity.

Efficient matrix completion with Gaussian models

A general framework based on Gaussian models and a MAPEM algorithm is introduced in this paper for solving matrix/ table completion problems. The numerical experiments with the standard and challenging movie ratings data show that the proposed approach, based on probably one of the simplest probabilistic models, leads to the results in the same ballpark as the state-of-the-art, at a lower computational cost.

Statistical compressive sensing of Gaussian mixture models

A new framework of compressive sensing (CS), namely statistical compressive sensing (SCS), that aims at efficiently sampling a collection of signals that follow a statistical distribution and achieving accurate reconstruction on average, is introduced. For signals following a Gaussian distribution, with Gaussian or Bernoulli sensing matrices of O(k) measurements, considerably smaller than the O(k log(N/k)) required by conventional CS, where N is the signal dimension, and with an optimal decoder implemented with linear filtering, significantly faster than the pursuit decoders applied in conventional CS, the error of SCS is shown tightly upper bounded by a constant times the best k-term approximation error, with overwhelming probability. The failure probability is also significantly smaller than that of conventional CS. Stronger yet simpler results further show that for any sensing matrix, the error of Gaussian SCS is upper bounded by a constant times the best k-term approximation with probability one, and the bound constant can be efficiently calculated. For signals following Gaussian mixture models, SCS with a piecewise linear decoder is introduced and shown to produce for real images better results than conventional CS based on sparse models.

Adapted statistical compressive sensing: Learning to sense gaussian mixture models

A framework for learning sensing kernels adapted to signals that follow a Gaussian mixture model (GMM) is introduced in this paper. This follows the paradigm of statistical compressive sensing (SCS), where a statistical model, a GMM in particular, replaces the standard sparsity model of classical compressive sensing (CS), leading to both theoretical and practical improvements. We show that the optimized sensing matrix outperforms random sampling matrices originally exploited both in CS and SCS.