Analysis And Synthesis Sparse Modeling Methods Image Processing PDF Download

Natural images are usually either sparse by themselves or sparse after a transform. For instance, images of stars from telescope are sparse by themselves and images of humans are sparse under the wavelet transform. Sparsity plays an important role in image processing and machine learning. How to build appropriate sparse models, how to numerically find solutions from the sparse models and how to derive theoretical guarantees of the correctness of the solutions are essential for the success of sparsity in applications. In this thesis, we study sparse modeling in image processing and machine learning. We propose a novel two-stage (projection followed by correction) modeling framework for image reconstruction from nonuniform Fourier measurements. We design domain adaptive sparsity regularization schemes that are able to maintain image smoothness as well as edge details. Various numerical results in both one-dimensional signals and two-dimensional images are also presented to demonstrate the superior performance of the proposed model compared with other methods. We study the problem of learning data adaptive over complete bases so that each data example can be linearly represented by few basis vectors. In the specific case when the input data size is large, we design a novel distributed dictionary learning algorithm which builds on the classical consensus approach. Most importantly, the proposed algorithm imposes a structured constraint to ensure that the derived dictionary has low coherence, that is, the columns of the dictionaries are nearly orthogonal. We analyze the solution obtained from the well known Least Absolute Shrinkage and Selection Operator (Lasso) model in the feature selection scenario. A substantial challenge in feature selection to specify the relation between the factors/features and the observations (e.g., linear or nonlinear). The relation is however often unknown. We prove that under certain conditions, Lasso is able to find the correct features (asymptotically) even though the model is misspecified. The contributions of this thesis are summarized below: 1. We propose a new two-stage model with domain adaptive sparsity constraints for image reconstruction from nonuniform Fourier measurements. 2. We design a new distributed dictionary learning algorithm. 3. We prove a new theorem on the correctness of Lasso model in sparse feature selection when the relationship between observations and features is misspecified.