Multi-modal dictionary learning for image separation with application in art investigation

In support of art investigation, we propose a new source separation method that unmixes a single X-ray scan acquired from double-sided paintings. In this problem, the X-ray signals to be separated have similar morphological characteristics, which brings previous source separation methods to their limits. Our solution is to use photographs taken from the front and back-side of the panel to drive the separation process. The crux of our approach relies on the coupling of the two imaging modalities (photographs and X-rays) using a novel coupled dictionary learning framework able to capture both common and disparate features across the modalities using parsimonious representations; the common component models features shared by the multi-modal images, whereas the innovation component captures modality-specific information. As such, our model enables the formulation of appropriately regularized convex optimization procedures that lead to the accurate separation of the X-rays. Our dictionary learning framework can be tailored both to a single- and a multi-scale framework, with the latter leading to a significant performance improvement. Moreover, to improve further on the visual quality of the separated images, we propose to train coupled dictionaries that ignore certain parts of the painting corresponding to craquelure. Experimentation on synthetic and real data - taken from digital acquisition of the Ghent Altarpiece (1432) - confirms the superiority of our method against the state-of-the-art morphological component analysis technique that uses either fixed or trained dictionaries to perform image separation.Comment: submitted to IEEE Transactions on Images Processin

Rectified Gaussian Scale Mixtures and the Sparse Non-Negative Least Squares Problem

In this paper, we develop a Bayesian evidence maximization framework to solve the sparse non-negative least squares (S-NNLS) problem. We introduce a family of probability densities referred to as the Rectified Gaussian Scale Mixture (R- GSM) to model the sparsity enforcing prior distribution for the solution. The R-GSM prior encompasses a variety of heavy-tailed densities such as the rectified Laplacian and rectified Student- t distributions with a proper choice of the mixing density. We utilize the hierarchical representation induced by the R-GSM prior and develop an evidence maximization framework based on the Expectation-Maximization (EM) algorithm. Using the EM based method, we estimate the hyper-parameters and obtain a point estimate for the solution. We refer to the proposed method as rectified sparse Bayesian learning (R-SBL). We provide four R- SBL variants that offer a range of options for computational complexity and the quality of the E-step computation. These methods include the Markov chain Monte Carlo EM, linear minimum mean-square-error estimation, approximate message passing and a diagonal approximation. Using numerical experiments, we show that the proposed R-SBL method outperforms existing S-NNLS solvers in terms of both signal and support recovery performance, and is also very robust against the structure of the design matrix.Comment: Under Review by IEEE Transactions on Signal Processin

A Sparse Bayesian Estimation Framework for Conditioning Prior Geologic Models to Nonlinear Flow Measurements

We present a Bayesian framework for reconstruction of subsurface hydraulic properties from nonlinear dynamic flow data by imposing sparsity on the distribution of the solution coefficients in a compression transform domain

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

A general framework for solving image inverse problems is introduced in this paper. The approach is based on Gaussian mixture models, estimated via a computationally efficient MAP-EM algorithm. A dual mathematical interpretation of the proposed framework with structured sparse estimation is described, which shows that the resulting piecewise linear estimate stabilizes the estimation when compared to traditional sparse inverse problem techniques. This interpretation also suggests an effective dictionary motivated initialization for the MAP-EM algorithm. We demonstrate that in a number of image inverse problems, including inpainting, zooming, and deblurring, the same algorithm produces either equal, often significantly better, or very small margin worse results than the best published ones, at a lower computational cost.Comment: 30 page

ISTA-Net: Interpretable Optimization-Inspired Deep Network for Image Compressive Sensing

With the aim of developing a fast yet accurate algorithm for compressive sensing (CS) reconstruction of natural images, we combine in this paper the merits of two existing categories of CS methods: the structure insights of traditional optimization-based methods and the speed of recent network-based ones. Specifically, we propose a novel structured deep network, dubbed ISTA-Net, which is inspired by the Iterative Shrinkage-Thresholding Algorithm (ISTA) for optimizing a general $\ell_1$ norm CS reconstruction model. To cast ISTA into deep network form, we develop an effective strategy to solve the proximal mapping associated with the sparsity-inducing regularizer using nonlinear transforms. All the parameters in ISTA-Net (\eg nonlinear transforms, shrinkage thresholds, step sizes, etc.) are learned end-to-end, rather than being hand-crafted. Moreover, considering that the residuals of natural images are more compressible, an enhanced version of ISTA-Net in the residual domain, dubbed {ISTA-Net}$^+$, is derived to further improve CS reconstruction. Extensive CS experiments demonstrate that the proposed ISTA-Nets outperform existing state-of-the-art optimization-based and network-based CS methods by large margins, while maintaining fast computational speed. Our source codes are available: \textsl{http://jianzhang.tech/projects/ISTA-Net}.Comment: 10 pages, 6 figures, 4 Tables. To appear in CVPR 201

PEAR: PEriodic And fixed Rank separation for fast fMRI

In functional MRI (fMRI), faster acquisition via undersampling of data can improve the spatial-temporal resolution trade-off and increase statistical robustness through increased degrees-of-freedom. High quality reconstruction of fMRI data from undersampled measurements requires proper modeling of the data. We present an fMRI reconstruction approach based on modeling the fMRI signal as a sum of periodic and fixed rank components, for improved reconstruction from undersampled measurements. We decompose the fMRI signal into a component which a has fixed rank and a component consisting of a sum of periodic signals which is sparse in the temporal Fourier domain. Data reconstruction is performed by solving a constrained problem that enforces a fixed, moderate rank on one of the components, and a limited number of temporal frequencies on the other. Our approach is coined PEAR - PEriodic And fixed Rank separation for fast fMRI. Experimental results include purely synthetic simulation, a simulation with real timecourses and retrospective undersampling of a real fMRI dataset. Evaluation was performed both quantitatively and visually versus ground truth, comparing PEAR to two additional recent methods for fMRI reconstruction from undersampled measurements. Results demonstrate PEAR's improvement in estimating the timecourses and activation maps versus the methods compared against at acceleration ratios of R=8,16 (for simulated data) and R=6.66,10 (for real data). PEAR results in reconstruction with higher fidelity than when using a fixed-rank based model or a conventional Low-rank+Sparse algorithm. We have shown that splitting the functional information between the components leads to better modeling of fMRI, over state-of-the-art methods

Tensor Decompositions for Signal Processing Applications From Two-way to Multiway Component Analysis

The widespread use of multi-sensor technology and the emergence of big datasets has highlighted the limitations of standard flat-view matrix models and the necessity to move towards more versatile data analysis tools. We show that higher-order tensors (i.e., multiway arrays) enable such a fundamental paradigm shift towards models that are essentially polynomial and whose uniqueness, unlike the matrix methods, is guaranteed under verymild and natural conditions. Benefiting fromthe power ofmultilinear algebra as theirmathematical backbone, data analysis techniques using tensor decompositions are shown to have great flexibility in the choice of constraints that match data properties, and to find more general latent components in the data than matrix-based methods. A comprehensive introduction to tensor decompositions is provided from a signal processing perspective, starting from the algebraic foundations, via basic Canonical Polyadic and Tucker models, through to advanced cause-effect and multi-view data analysis schemes. We show that tensor decompositions enable natural generalizations of some commonly used signal processing paradigms, such as canonical correlation and subspace techniques, signal separation, linear regression, feature extraction and classification. We also cover computational aspects, and point out how ideas from compressed sensing and scientific computing may be used for addressing the otherwise unmanageable storage and manipulation problems associated with big datasets. The concepts are supported by illustrative real world case studies illuminating the benefits of the tensor framework, as efficient and promising tools for modern signal processing, data analysis and machine learning applications; these benefits also extend to vector/matrix data through tensorization. Keywords: ICA, NMF, CPD, Tucker decomposition, HOSVD, tensor networks, Tensor Train