In this paper, we provide a sparse image restoration algorithm with a SSIM-based objective function. The proposed technique is a modification to the SSIM-inspired OMP (iOMP) and, and it has two parallel sparse restoration paths. One of them is ${\mathcal{L}}_2$-sense OMP and the other is $\text{SSIM}$-sense OMP (iOMP). Both paths intersects only at the starting point and gives different quality levels after each iteration. This distinction enables us to select the coefficients of the best quality and to avoid the uncertainty issue of iOMP. From the point of view of the SSIM levels, the conducted experiments proved that, the proposed methodology works better than iOMP and OMP. Also, the performance of this method is checked for significance by the t-test, and the obtained results proved that the method works well especially for large images and the data-independent based dictionary.

This work comprises the development of a quality enhancement technique for image encoders that use compressive sensing. The recommended solution seeks to maximize the perceptual quality based objective function, unlike other sparse representation algorithms that minimizes the error-based objective function. The key idea behind this work is to develop an iterative methodology that works as a modifier for the sparse coefficients. The modification procedure is SSIM-based and has been carried out in an iterative and linear manner. The conducted experiments revealed that the recommended technique works better than another SSIM-based modifier termed the SSIM-inspired OMP (iOMP) in terms of SSIM levels gained. The t-test is also utilized to examine our performance for significance, and the results show that the method works well for any type of image and any size, especially when a data-independent based dictionary is used.

This paper introduces two novel self-induced basis strategies, interior cevian initialization for orthogonal matching pursuit (ICe) and its $$l_2$$l 2 -sense optimized version, optimized interior cevian initialization (OICe). These methods address the complexity and convergence issues of the previously proposed self-inspired bases (SIBS) approach. Both algorithms enhance sparse image representation by generating adaptive auxiliary atoms tailored to the image, thereby improving reconstruction quality and computational efficiency. Experimental results show that ICe achieves up to a 35.6% improvement in Normalized Mean Squared Error (NMSE) and a 16.4% increase in Structural Similarity Index (SSIM) compared to conventional OMP. OICe, as the $$l_2$$l 2 -sense optimized version of ICe, delivers even greater performance, with improvements of up to 40% in NMSE and 18.5% in SSIM. Additionally, both ICe and OICe significantly reduce bits per pixel (BPP) for image encoding, outperforming SIBS when paired with state-of-the-art dictionaries such as KSVD, MOD, RBDL, and ODL. These results highlight the effectiveness of ICe and OICe in addressing key challenges in sparse representation, offering enhanced accuracy and efficiency across a variety of applications.

In the past years, Sparse Modeling (SM) has been used in a variety of applications, such as data representation. In this type of data modeling, the data is divided into small segments, and each one is processed independently using SM techniques to find the best few coefficients that capture the most important features of the segment. This paper proposes a backward stage for SM to process all segments at the same time. The main idea is to select significant coefficients that capture the global features of the data, not just the features of the small segments. To evaluate the effectiveness of the proposed algorithm, extensive experiments were conducted on different time series data. The results showed that the proposed algorithm works better with time series data that has localized features, such as speech data. The study also compared the proposed algorithm to other recent backward techniques and found that it outperforms the others significantly, especially when using learned bases.

In this paper, we present two variants of the backward replacement approach, T-BRe and H-BRe, aiming to enhance the compression capabilities of forward greedy pursuit algorithms. We assess the effectiveness of these methods using three greedy pursuit algorithms (OMP, OOMP, and FBP) and conduct simulations on image and speech data with a mathematically generated dictionary. Additionally, we provide a mathematical analysis of the H-BRe algorithm, proving its efficacy and addressing its limitations. Both analysis and simulations demonstrate that the H-BRe algorithm significantly boosts the compression capabilities of forward greedy pursuit algorithms. The improvements vary depending on factors such as the forward modelling technique, stopping criterion, and the random behaviour of the backward replacements. The high-performance (H) variants perform best in various modelling approaches across PESQ and SSIM values, while the B and T variants also offer efficiency in some cases. Despite the promising results, our methods have limitations, including time complexity, local error minimization, dependency on atom types, and the assumption of linear distribution.