Graph Laplacian Regularization with Sparse Coding for Image Restoration and Representation

Abstract

Sparse coding is widely used in image denoising, deblurring, clustering, and classification. However, most existing approaches to sparse coding failed to consider the fact that high dimensional data naturally reside on geometrical structure of the data space. It has been shown that geometric information of the data is important for both inversion and discrimination. In this paper, we proposed a generalized framework for image restoration and representation by combining sparse coding and graph based algorithms. In image denoising and deblurring problems, an image is first decomposed into cartoon layer (piecewise-smooth contents) and texture layer (textures and sharp edges) using morphological component analysis (MCA); then optimal graph Laplacian regularizer (OGLR) algorithm and simultaneous sparse coding with Gaussian scale mixture prior (SSC-GSM) algorithm are applied to cartoon layer and texture layer, respectively; final restored image is generated by aggregating the outcomes from two algorithms. The proposed hybrid image restoration algorithm outperforms state-of-the-art image denoising algorithms, such as BM3D on natural images, measured in PSNR, and performs comparatively in image deblurring. In image clustering and classification problems, we convert our generalized framework into a novel dual graph regularized sparse coding method to transform the nonlinear data space and feature space into linear space, two efficient optimization algorithms are provided for the numerical implementation. The experimental results show that our generalized graph Laplacian and sparse coding framework performs competitively with popular denoising, deblurring, clustering, and classification methods.