Exploiting Restricted Boltzmann Machines and Deep Belief Networks in Compressed Sensing

This paper proposes a CS scheme that exploits the representational power of restricted Boltzmann machines and deep learning architectures to model the prior distribution of the sparsity pattern of signals belonging to the same class. The determined probability distribution is then used in a maximum a posteriori (MAP) approach for the reconstruction. The parameters of the prior distribution are learned from training data. The motivation behind this approach is to model the higher-order statistical dependencies between the coefficients of the sparse representation, with the final goal of improving the reconstruction. The performance of the proposed method is validated on the Berkeley Segmentation Dataset and the MNIST Database of handwritten digits.Comment: Accepted for publication at IEEE Transactions on Signal Processin

Cross-scale predictive dictionaries

Sparse representations using data dictionaries provide an efficient model particularly for signals that do not enjoy alternate analytic sparsifying transformations. However, solving inverse problems with sparsifying dictionaries can be computationally expensive, especially when the dictionary under consideration has a large number of atoms. In this paper, we incorporate additional structure on to dictionary-based sparse representations for visual signals to enable speedups when solving sparse approximation problems. The specific structure that we endow onto sparse models is that of a multi-scale modeling where the sparse representation at each scale is constrained by the sparse representation at coarser scales. We show that this cross-scale predictive model delivers significant speedups, often in the range of 10-60$\times$, with little loss in accuracy for linear inverse problems associated with images, videos, and light fields.Comment: 12 page

Analysis of Fast Alternating Minimization for Structured Dictionary Learning

Methods exploiting sparsity have been popular in imaging and signal processing applications including compression, denoising, and imaging inverse problems. Data-driven approaches such as dictionary learning and transform learning enable one to discover complex image features from datasets and provide promising performance over analytical models. Alternating minimization algorithms have been particularly popular in dictionary or transform learning. In this work, we study the properties of alternating minimization for structured (unitary) sparsifying operator learning. While the algorithm converges to the stationary points of the non-convex problem in general, we prove rapid local linear convergence to the underlying generative model under mild assumptions. Our experiments show that the unitary operator learning algorithm is robust to initialization

Learning Multi-Layer Transform Models

Learned data models based on sparsity are widely used in signal processing and imaging applications. A variety of methods for learning synthesis dictionaries, sparsifying transforms, etc., have been proposed in recent years, often imposing useful structures or properties on the models. In this work, we focus on sparsifying transform learning, which enjoys a number of advantages. We consider multi-layer or nested extensions of the transform model, and propose efficient learning algorithms. Numerical experiments with image data illustrate the behavior of the multi-layer transform learning algorithm and its usefulness for image denoising. Multi-layer models provide better denoising quality than single layer schemes.Comment: In Proceedings of the Annual Allerton Conference on Communication, Control, and Computing, 201

FRIST - Flipping and Rotation Invariant Sparsifying Transform Learning and Applications

Features based on sparse representation, especially using the synthesis dictionary model, have been heavily exploited in signal processing and computer vision. However, synthesis dictionary learning typically involves NP-hard sparse coding and expensive learning steps. Recently, sparsifying transform learning received interest for its cheap computation and its optimal updates in the alternating algorithms. In this work, we develop a methodology for learning Flipping and Rotation Invariant Sparsifying Transforms, dubbed FRIST, to better represent natural images that contain textures with various geometrical directions. The proposed alternating FRIST learning algorithm involves efficient optimal updates. We provide a convergence guarantee, and demonstrate the empirical convergence behavior of the proposed FRIST learning approach. Preliminary experiments show the promising performance of FRIST learning for sparse image representation, segmentation, denoising, robust inpainting, and compressed sensing-based magnetic resonance image reconstruction.Comment: Published in Inverse Problem

Data-Driven Learning of a Union of Sparsifying Transforms Model for Blind Compressed Sensing

Compressed sensing is a powerful tool in applications such as magnetic resonance imaging (MRI). It enables accurate recovery of images from highly undersampled measurements by exploiting the sparsity of the images or image patches in a transform domain or dictionary. In this work, we focus on blind compressed sensing (BCS), where the underlying sparse signal model is a priori unknown, and propose a framework to simultaneously reconstruct the underlying image as well as the unknown model from highly undersampled measurements. Specifically, our model is that the patches of the underlying image(s) are approximately sparse in a transform domain. We also extend this model to a union of transforms model that better captures the diversity of features in natural images. The proposed block coordinate descent type algorithms for blind compressed sensing are highly efficient, and are guaranteed to converge to at least the partial global and partial local minimizers of the highly non-convex BCS problems. Our numerical experiments show that the proposed framework usually leads to better quality of image reconstructions in MRI compared to several recent image reconstruction methods. Importantly, the learning of a union of sparsifying transforms leads to better image reconstructions than a single adaptive transform.Comment: Appears in IEEE Transactions on Computational Imaging, 201

PWLS-ULTRA: An Efficient Clustering and Learning-Based Approach for Low-Dose 3D CT Image Reconstruction

The development of computed tomography (CT) image reconstruction methods that significantly reduce patient radiation exposure while maintaining high image quality is an important area of research in low-dose CT (LDCT) imaging. We propose a new penalized weighted least squares (PWLS) reconstruction method that exploits regularization based on an efficient Union of Learned TRAnsforms (PWLS-ULTRA). The union of square transforms is pre-learned from numerous image patches extracted from a dataset of CT images or volumes. The proposed PWLS-based cost function is optimized by alternating between a CT image reconstruction step, and a sparse coding and clustering step. The CT image reconstruction step is accelerated by a relaxed linearized augmented Lagrangian method with ordered-subsets that reduces the number of forward and back projections. Simulations with 2-D and 3-D axial CT scans of the extended cardiac-torso phantom and 3D helical chest and abdomen scans show that for both normal-dose and low-dose levels, the proposed method significantly improves the quality of reconstructed images compared to PWLS reconstruction with a nonadaptive edge-preserving regularizer (PWLS-EP). PWLS with regularization based on a union of learned transforms leads to better image reconstructions than using a single learned square transform. We also incorporate patch-based weights in PWLS-ULTRA that enhance image quality and help improve image resolution uniformity. The proposed approach achieves comparable or better image quality compared to learned overcomplete synthesis dictionaries, but importantly, is much faster (computationally more efficient).Comment: Accepted to IEEE Transaction on Medical Imagin

Fast Multi-class Dictionaries Learning with Geometrical Directions in MRI Reconstruction

Objective: Improve the reconstructed image with fast and multi-class dictionaries learning when magnetic resonance imaging is accelerated by undersampling the k-space data. Methods: A fast orthogonal dictionary learning method is introduced into magnetic resonance image reconstruction to providing adaptive sparse representation of images. To enhance the sparsity, image is divided into classified patches according to the same geometrical direction and dictionary is trained within each class. A new sparse reconstruction model with the multi-class dictionaries is proposed and solved using a fast alternating direction method of multipliers. Results: Experiments on phantom and brain imaging data with acceleration factor up to 10 and various undersampling patterns are conducted. The proposed method is compared with state-of-the-art magnetic resonance image reconstruction methods. Conclusion: Artifacts are better suppressed and image edges are better preserved than the compared methods. Besides, the computation of the proposed approach is much faster than the typical K-SVD dictionary learning method in magnetic resonance image reconstruction. Significance: The proposed method can be exploited in undersapmled magnetic resonance imaging to reduce data acquisition time and reconstruct images with better image quality.Comment: 13 pages, 15 figures, 5 table

Compressive hyperspectral imaging via adaptive sampling and dictionary learning

In this paper, we propose a new sampling strategy for hyperspectral signals that is based on dictionary learning and singular value decomposition (SVD). Specifically, we first learn a sparsifying dictionary from training spectral data using dictionary learning. We then perform an SVD on the dictionary and use the first few left singular vectors as the rows of the measurement matrix to obtain the compressive measurements for reconstruction. The proposed method provides significant improvement over the conventional compressive sensing approaches. The reconstruction performance is further improved by reconditioning the sensing matrix using matrix balancing. We also demonstrate that the combination of dictionary learning and SVD is robust by applying them to different datasets

Learning a collaborative multiscale dictionary based on robust empirical mode decomposition

Dictionary learning is a challenge topic in many image processing areas. The basic goal is to learn a sparse representation from an overcomplete basis set. Due to combining the advantages of generic multiscale representations with learning based adaptivity, multiscale dictionary representation approaches have the power in capturing structural characteristics of natural images. However, existing multiscale learning approaches still suffer from three main weaknesses: inadaptability to diverse scales of image data, sensitivity to noise and outliers, difficulty to determine optimal dictionary structure. In this paper, we present a novel multiscale dictionary learning paradigm for sparse image representations based on an improved empirical mode decomposition. This powerful data-driven analysis tool for multi-dimensional signal can fully adaptively decompose the image into multiscale oscillating components according to intrinsic modes of data self. This treatment can obtain a robust and effective sparse representation, and meanwhile generates a raw base dictionary at multiple geometric scales and spatial frequency bands. This dictionary is refined by selecting optimal oscillating atoms based on frequency clustering. In order to further enhance sparsity and generalization, a tolerance dictionary is learned using a coherence regularized model. A fast proximal scheme is developed to optimize this model. The multiscale dictionary is considered as the product of oscillating dictionary and tolerance dictionary. Experimental results demonstrate that the proposed learning approach has the superior performance in sparse image representations as compared with several competing methods. We also show the promising results in image denoising application.Comment: to be published in Neurocomputin