Image restoration remains a core problem in image processing. The objective is to reduce noise while preserving the structural integrity and content of the original image. Noise encountered in images generally falls into two main categories: additive and multiplicative. Both types can often be effectively modeled in practical scenarios as additive white Gaussian noise (AWGN). In this thesis, several denoising methods for natural images are investigated. Natural images typically contain patterns and structures that are repeated across different regions. This characteristic, known as non-local self-similarity, enables the identification of similar patches even in spatially distant areas of the image. For each patch, similar patches can be found and converted into column vectors to form a matrix of similar patches, whose columns exhibit a high degree of similarity. The rank of a matrix is defined as the number of linearly indepen- dent vectors it contains. When the columns or rows of a matrix are similar or can be expressed as linear combinations of one another, they are considered linearly dependent, leading to a reduction in the matrix rank. Therefore, due to the strong correlation among similar patches, the resulting matrix is typically of low rank. Motivated by this property, we investigated the Weighted Nuclear Norm Minimization (WNNM) method. In the WNNM model, the denoising process is formulated as an optimization problem, where the objective is to minimize the weighted nuclear norm of a matrix. The nuclear norm, defined as the sum of the singular values of a matrix, serves as a convex relaxation of the matrix rank. Unlike conventional approaches, WNNM assigns a distinct weight to each singular value, with smaller singular values being penalized more heavily. This weighting strategy allows the principal components of the matrix to be better preserved while noise is more effectively suppressed. However, the similarity of nonlocal image patches is influenced by both the noise level and the presence of irregular structures. As the noise level increases or when more irregular structures are present, the underlying similarity between patches decreases, which leads to a reduction in the performance of the WNNM approach. These limitations pro‎mp‎ted us to use a global spatial constraint. To address these challenges, we explore the Fields of Experts (FoE) model. FoE is a high-order Markov Random Field (MRF) model that captures richer structures in natural images. It utilizes filters trained on natural images and defines an energy-based framework to retain detailed textures and edges. However, due to its local nature, FoE models do not exploit nonlocal redundancies, which limits their effectiveness in highly noisy data. To overcome these indi- vidual shortcomings, this thesis presents a novel denoising framework named RFoE (Regularized Field of Experts), which combines a nonlocal low-rank approximation with a global FoE prior and a data fidelity term. The low-rank structure is enforced using a weighted nuclear norm, while the FoE prior captures higher-order structural information. The resulting optimization problem is solved using an alternating direction minimization strategy with half-quadratic splitting. Finally, extensive experiments on benchmark images with varying levels of AWGN demonstrate that the ex- amined RFoE model outperforms traditional denoising methods such as NLM, BM3D and KSVD in terms of PSNR, SSIM, and MAE. Additionally, numerical convergence analysis confirms the model’s computational applicability and efficiency.

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زهرا صابري , هادي روحاني