Matrix-value Linear Regression for Image Denoising

Since images (or patches) can be inherently represented by matrices, in this paper, the image denosing task is addressed into learning global linear operator by using matrix-value linear regression strategy. From given training noisy images (or patches) and their corresponding clear ones, a series of linear mappings are potentially defined by those image pairs, and then integrated as a global linear operator on training set. Matrix-value linear regression can be employed to learn the global linear operator. The proposed algorithm can commendably overcome the disadvantages of the vector-based methods. Empirical experiments illustrate that the proposed algorithm is feasible and efficient, as well as easy to be implemented. Introduction Image denoising is an important image processing task whether as a process itself or a preprocessing in image process. In most cases, it is assumed that the noisy image is the summation of original image and a noise component. The goal of image denoising is to remove the noise component while retaining the important details as much as possible [1]. Over the past few decades, numerous image denoising algorithms have been proposed for the quality improvement of images corrupted with some kind of noise model, and in general can be categorized as spatial domain, transform domain, and dictionary learning based according to the image representation [2]. Spatial domain method take advantage of the correlations among distinct pixels or patches in the image to reduce noise level. Many spatial domain based algorithms have been developed for noise reduction, such as adaptive median filter, Wiener filter, steering kernel regression, nonlocal means filter, nonlocal median filter, etc. Transform domain methods are based on the assumption that the images (or image patches) can be well represented by the few transform basis with a series of coefficients. The basic idea is to remove or shrink smaller coefficients as far as possible because they are related to image details and noise. Within this category, a few typical algorithms are discrete cosine transform, wavelets transform, curvelets transform, contourlets transform, etc [1,2]. The dictionary learning based methods attempt to construct or learn an appropriate over-complete dictionary that can accurately fit the local structures of images, such that the estimated image can be expressed as a linear combination of only few atoms chosen out from this dictionary. Several representative algorithms in this category are the K-clustering with singular value decomposition (K-SVD), learned simultaneous sparse coding (LSSC), clustering-based sparse representation (CSR), and nonlocally centralized sparse representation (NCSR). The dictionary learning based methods have now become a trend for image denoising [3]. The existing image denoising methods are mostly implemented by vectorizing the images (or image patches). Two-dimensional images (or image patches) are held in these models as one-dimensional column vectors after lexicographic ordering. However, this would inevitably cause damage to the inherent structural information hidden in original images, and come at the expense of omitting much image-level information. Moreover, vectorization tends to result in the overfitting problem because the dimension of the vectorized images (or image patches) may be larger than that of