Adaptive nonlocal and structured sparse signal modeling and applications

Features based on sparse representation, especially using the synthesis dictionary model, have been heavily exploited in signal processing and computer vision. Many applications such as image and video denoising, inpainting, demosaicing, super-resolution, magnetic resonance imaging (MRI), and computed tomography (CT) reconstruction have been shown to benefit from adaptive sparse signal modeling. However, synthesis dictionary learning typically involves expensive sparse coding and learning steps. Recently, sparsifying transform learning received interest for its cheap computation and its optimal updates in the alternating algorithms. Prior works on transform learning have certain limitations, including (1) limited model richness and structure for handling diverse data, (2) lack of non-local structure, and (3) lack of effective extension to high-dimensional or streaming data. This dissertation focuses on advanced data-driven sparse modeling techniques, especially with nonlocal and structured sparse signal modeling. In the first work of this dissertation, we propose a methodology for learning, dubbed Flipping and Rotation Invariant Sparsifying Transforms (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 image sparse representation, segmentation, denoising, robust inpainting, and compressed sensing-based magnetic resonance image reconstruction. Next, we present an online high-dimensional sparsifying transform learning method for spatio-temporal data, and demonstrate its usefulness with a novel video denoising framework, dubbed VIDOSAT. The proposed method is based on our previous work on online sparsifying transform learning, which has low computational and memory costs, and can potentially handle streaming video. By combining with a block matching (BM) technique, the learned model can effectively adapt to video data with various motions. The patches are constructed either from corresponding 2D patches in successive frames or using an online block matching technique. The proposed online video denoising requires little memory and others efficient processing. Numerical experiments are used to analyze the contribution of the various components of the proposed video denoising scheme by "switching off" these components - for example, fixing the transform to be 3D DCT, rather than a learned transform. Other experiments compare to the performance of prior schemes such as dictionary learning-based schemes, and the state-of-the-art VBM3D and VBM4D on several video data sets, demonstrating the promising performance of the proposed methods. In the third part of the dissertation, we propose a joint sparse and low-rank model, dubbed STROLLR, to better represent natural images. Patch-based methods exploit local patch sparsity, whereas other works apply low-rankness of grouped patches to exploit image non-local structures. However, using either approach alone usually limits performance in image restoration applications. In order to fully utilize both the local and non-local image properties, we develop an image restoration framework using a transform learning scheme with joint low-rank regularization. The approach owes some of its computational efficiency and good performance to the use of transform learning for adaptive sparse representation rather than the popular synthesis dictionary learning algorithms, which involve approximation of NP-hard sparse coding and expensive learning steps. We demonstrate the proposed framework in various applications to image denoising, inpainting, and compressed sensing based magnetic resonance imaging. Results show promising performance compared to state-of-the-art competing methods. Last, we extend the effective joint sparsity and low-rankness model from image to video applications. We propose a novel video denoising method, based on an online tensor reconstruction scheme with a joint adaptive sparse and low-rank model, dubbed SALT. An efficient and unsupervised online unitary sparsifying transform learning method is introduced to impose adaptive sparsity on the fly. We develop an efficient 3D spatio-temporal data reconstruction framework based on the proposed online learning method, which exhibits low latency and can potentially handle streaming videos. To the best of our knowledge, this is the first work that combines adaptive sparsity and low-rankness for video denoising, and the first work that solves the proposed problem in an online fashion. We demonstrate video denoising results over commonly used videos from public datasets. Numerical experiments show that the proposed video denoising method outperforms competing methods

From Rank Estimation to Rank Approximation: Rank Residual Constraint for Image Restoration

In this paper, we propose a novel approach to the rank minimization problem, termed rank residual constraint (RRC) model. Different from existing low-rank based approaches, such as the well-known nuclear norm minimization (NNM) and the weighted nuclear norm minimization (WNNM), which estimate the underlying low-rank matrix directly from the corrupted observations, we progressively approximate the underlying low-rank matrix via minimizing the rank residual. Through integrating the image nonlocal self-similarity (NSS) prior with the proposed RRC model, we apply it to image restoration tasks, including image denoising and image compression artifacts reduction. Towards this end, we first obtain a good reference of the original image groups by using the image NSS prior, and then the rank residual of the image groups between this reference and the degraded image is minimized to achieve a better estimate to the desired image. In this manner, both the reference and the estimated image are updated gradually and jointly in each iteration. Based on the group-based sparse representation model, we further provide a theoretical analysis on the feasibility of the proposed RRC model. Experimental results demonstrate that the proposed RRC model outperforms many state-of-the-art schemes in both the objective and perceptual quality

A Study on Clustering for Clustering Based Image De-Noising

In this paper, the problem of de-noising of an image contaminated with Additive White Gaussian Noise (AWGN) is studied. This subject is an open problem in signal processing for more than 50 years. Local methods suggested in recent years, have obtained better results than global methods. However by more intelligent training in such a way that first, important data is more effective for training, second, clustering in such way that training blocks lie in low-rank subspaces, we can design a dictionary applicable for image de-noising and obtain results near the state of the art local methods. In the present paper, we suggest a method based on global clustering of image constructing blocks. As the type of clustering plays an important role in clustering-based de-noising methods, we address two questions about the clustering. The first, which parts of the data should be considered for clustering? and the second, what data clustering method is suitable for de-noising.? Then clustering is exploited to learn an over complete dictionary. By obtaining sparse decomposition of the noisy image blocks in terms of the dictionary atoms, the de-noised version is achieved. In addition to our framework, 7 popular dictionary learning methods are simulated and compared. The results are compared based on two major factors: (1) de-noising performance and (2) execution time. Experimental results show that our dictionary learning framework outperforms its competitors in terms of both factors.Comment: 9 pages, 8 figures, Journal of Information Systems and Telecommunications (JIST