Fractional calculus extends traditional, integer-based calculus to include non-integer orders, offering a powerful tool for a range of engineering applications, including image processing. This work delves into the utility of fractional calculus in two crucial aspects of image processing: image enhancement and denoising. We explore the foundational theories of fractional calculus together with its amplitude–frequency characteristics. Our focus is on the effectiveness of fractional differential operators in enhancing image features and reducing noise. Experimental results reveal that fractional calculus offers unique benefits for image enhancement and denoising. Specifically, fractional-order differential operators outperform their integer-order counterparts in accentuating details such as weak edges and strong textures in images. Moreover, fractional integral operators excel in denoising images, not only improving the signal-to-noise ratio but also better preserving essential features such as edges and textures when compared to traditional denoising techniques. Our empirical results affirm the effectiveness of the fractional-order calculus-based image-processing approach in yielding optimal results for low-level image processing.

Sparse representation (SR)-based denoising method attenuates random seismic noise trace-by-trace, which may not be able to employ the coherency between neighboring traces. To address this, we propose a multiple measurement vector (MMV)-based algorithm for robust denoising of multichannel seismic gathers. Seismic reflectors in common offset gather (COG) is characterized by horizontal events, which satisfy the common sparsity assumption, making it ideal for the MMV approach. ℓ2,1-norm regularization, which enforces two constraints on source matrix, i.e., temporal sparsity and the horizontal continuity, is then adopted in the MMV model to stabilize the lateral variation between channels. Besides, ℓ2,1-norm regularization provides a reasonable intrinsic structure to reduce the multisolution of the algorithm. Celebrating the strengths of iteratively reweighted (IR), we present a novel MMV algorithm, IR ℓ 2,1 norm minimization (IR-ℓ2,1), that further improves the performance. The formulated IR-ℓ2,1 can be minimized by the alternating direction method of multipliers. Both synthetic and field data applications confirm the effectiveness of the proposed denoising method.

The theory of fractional calculus extends the order of classical integer calculus from integer to non-integer. As a new engineering application tool, it has made many important research achievements in many fields, including image processing. This paper mainly studies the application of fractional calculus theory in image enhancement and denoising, including the basic theory of fractional calculus and its amplitude frequency characteristics, the application of fractional differential operator in image enhancement, and the application of fractional integral operator in image denoising. The experimental results show that the fractional calculus theory has more special advantages in image enhancement and denoising. Compared with the existing integer order image enhancement operators, the fractional differential operator can more effectively enhance the "weak edge" and "strong texture" details of the image. The fractional order integral image denoising operator can not only improve the signal-to-noise ratio of the image compared to traditional denoising methods, but also better preserve detailed information such as edges and textures of the image.

Spatial irregular sampling and random noise are two important factors that restrict the accuracy of seismic imaging. Seismic wavefield reconstruction and denoising based on sparse representation are two popular antidotes to these two inevitable issues, respectively. This article presents a non-parametric simultaneous reconstruction and denoising via sparse and low-rank regularization that dealt with the prestack gathers efficiently and automatically. The proposed method makes no additional prior assumptions on original data other than that the seismic signal is compressible. The key parameters estimation adopts a data-driven framework without person-dependent intervention. The basic idea of the approach is to combine the two related algorithms. Thus, the sparse decomposition needs to be performed only once. We first extract the solution matrix via Fourier dictionary and then perform the reconstruction and denoising successively in the sparse domain. Obtaining a perfect interpolation result requires that the seismic data satisfy the Shannon–Nyquist sampling theorem. However, data with steep-dip events or gaps, which cannot be adequate for the procedure, are a challenge that must be faced. This work proposes to deal with the common-offset gathers, which is characterized by flat, even approximate horizontal events, to handle the under-sampling obstacle. Another excellent property of the common-offset gathers is the simple and periodic repetitive texture structure, which can be represented sparsely and accurately by the Fourier dictionary. Thus, the computational complexity of the sparse representation is reduced. Both synthetic and practical applications indicate that our algorithm is efficient and effective.