Deconvolution and sparsity based image restoration

Deconvolution and sparse representation are the two key areas in image and signal processing. In this thesis the classical image restoration problem is addressed using these two modalities. Image restoration, such as deblurring, dnoising, and in-painting belongs to the class of ill-posed linear inverse problems, which requires a proper regularization for a credible solution. The aim is to develop techniques that are stable, practical and require a minimum amount of prior knowledge. The two main approaches that we focused upon in this thesis are image deconvolution for blurred image restoration and dictionary learning algorithms for sparse image denoising and in-painting. In the first approach, iterative least square and maximum likelihood based deconvolution methods are derived for image deblurring application. Three novel methods are presented i) a hybrid Fourier-wavelet deblurring (HFW) method based on expectation maximization (EM) approach, ii) sparse non-negative matrix approximation (SNMA), and iii) Kullback-Leibler divergence minimization (KLD). For HFW, the main objective function was split into Fourier domain deconvolution and wavelet domain denoising, to avoid the computational burden of handling blurring matrix in wavelet domain. Further, the wavelet coefficients were modelled using the class of Gaussian scale mixture (GSM) model, which represent the heavy tailed distribution. The SMNA and KLD are designed for a more challenging task of blind image deconvolution (BID), where either no or very little prior information about the original image and blurring operator is provided. In SMNA an explicit blur estimation and strict positive constraint on the observed and original image, are utilized to retrieve the latent original image. The third method is derived using successive minimization of KLD between a model and a desired family of probability distributions. This algorithm can be viewed as cascaded EM in information geometric terms. In the second approach, dictionary learning methods with sparsity constraint on original image are designed to address the image denoising and in-painting problem. Recently, the sparse representation emerged as a useful regularization in ill-posed linear inverse problems. The main assumption in this direction is that the original image has a sparse representation over some dictionary. Three novel dictionary learning algorithms are outlined. In the first method, an orthogonal dictionary based on the profile-likelihood estimate is derived with single Eigen decomposition. Most of the dictionary learning algorithms confined the sparsity constraint to the sparse coding stage. In the second method, we look at the impact of enforcing the sparsity constraint also in the dictionary update stage. Within this framework different constraints such as smoothness of dictionary atoms and other can also be enforced to enhance the dictionary strength. In the last method, we looked into the double sparsity constraint, where the strength of explicit and implicit dictionaries is combined for efficient data representation. This model is based on a sparse representation of dictionary atoms over an implicit base dictionary. The advantage of this dictionary structure is evident from the experimental simulations