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Title: Sparse and low-rank models for image restoration
Authors: Gu, Shuhang
Degree: Ph.D.
Issue Date: 2017
Abstract: Image restoration aims to recover the latent high quality image from its degraded observation. As one of the most classical and fundamental topics in image processing and low-level vision, image restoration has been widely studied in the community, and a variety of approaches have been proposed, including filtering-based approaches, transformation-based approaches and variational approaches. The image sparsity priors have been used, either explicitly or implicitly, in many of these approaches, and have been playing a crucial role to improve the image restoration performance. In particular, the sparse representation based methods have achieved a great success in image restoration in the last decade. Based on the representation scheme of a signal vector, sparse representation models can be generally categorized into analysis sparse representation (ASR) models and synthesis sparse representation (SSR) models. Low-rank minimization models have also been proposed to exploit the sparsity (i.e., low-rankness) of a matrix of correlated vectors. Different models will have respectively their merits and drawbacks. The ASR based methods regularize the projection coeffcients of a signal over an analysis dictionary, and they are able to supply robust priors for image large scale structures. However, the projective coding mechanism of ASR based methods restrict their capacity of benefting from highly redundant dictionaries, limiting their fexibility in modeling image complex texture structures. The SSR based methods represent a signal as the linear combination of a few atoms in an over-complete dictionary. To model image local structures, most SSR methods partition an image into over-lapped patches to process, which brings the inconsistency issue of over-lapped patches as well as the heavy computation burden. The low-rank methods regularize the number of independent subspaces of a matrix. Since directly minimizing the rank of a matrix is an NP hard problem, many recently developed low-rank methods adopt the nuclear norm (i.e., the l1 norm of singular values) for low-rank approximation. However, the nuclear norm shrinks the singular values equally, ignoring the different importance of different singular values. In this thesis, we address the above-mentioned problems of ASR, SSR and low-rank methods, and develop new sparsity-based models for image restoration. We first investigate the ASR models for guided image restoration, and present a weighted ASR model learning scheme for RGB image guided depth image restoration. By introducing a guidance weight function, we largely improve the flexibility of ASR models and make it be able to deal with guided image enhancement tasks. Having the objective function of weighted ASR model, we utilize a task-driven training strategy to learn stage-wise dynamic parameters from training data. As a result, the proposed algorithm is able to generate high quality output efficiently. Experiments on guided depth image upsampling and noisy depth image restoration validate the effectiveness of the proposed method.
To address the inconsistency issues in previous patch-based SSR models, we propose a convolutional sparse coding (CSC) scheme for image super-resolution. By working directly on the whole image, the proposed CSC algorithm does not need to partition the image into overlapped patches, and it can exploit the image global correlation to reconstruct more robustly image local structures. State-of-the-art super-resolution results demonstrate the advantage of the proposed CSC method. Instead of investigating the ASR and SSR models individually, we propose to integrate the two models to exploit their complementary representation mechanisms. In the proposed joint convolutional analysis and synthesis (JCAS) model, a single image is adaptively decomposed into two layers, one is used for SSR and the other for ASR. The intrinsic complementarity of ASR and SSR allows them cooperate to separate the input image into a structure layer and a texture layer. We adaptively train the synthesis dictionary to learn the required texture pattern for specifc tasks. The proposed JCAS method shows very competitive performance in single image layer separation tasks, such as texture-cartoon decomposition and rain streak removal. Besides vector based one dimensional sparse representation models, we also investigate matrix based two dimensional low-rank models for image restoration. Specifcally, we extend the nuclear norm minimization (NNM) to weighted nuclear norm minimization (WNNM) by introducing a weight vector to weight different singular values of the data ma-trix. Although the WNNM model is nonconvex, we prove that the corresponding weighted nuclear norm proximal (WNNP) operator is equivalent to a standard quadratic programming problem with linear constraint. Very importantly, we show that the WNNM problem has closed-form optimal solution when the weights are of non-descending order. With WNNP, several extensions of the WNNM problem, including robust PCA and matrix completion, can be readily derived with the ADMM (alternating direction method of multipliers) paradigm. The proposed WNNM methods achieve state-of-the-art performance in typical low level vision tasks, including image denoising, background subtraction and image in painting. In summary, in this thesis we investigate in-depth the sparse representation and low-rank minimization based image restoration methods, and develop several new models for different image restoration applications. Our models not only enrich the understanding of sparsity based statistical image modeling, but also demonstrate state-of-the-art performance in image restoration and other low level vision problems.
Subjects: Hong Kong Polytechnic University -- Dissertations
Image reconstruction
High resolution imaging
Image processing -- Digital techniques
Pages: xvii, 144 pages : color illustrations
Appears in Collections:Thesis

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