Related papers: Properties of Farey Sequence and their Application…
In the recent advancement of multimedia technologies, it becomes a major concern of detecting visual attention regions in the field of image processing. The popularity of the terminal devices in a heterogeneous environment of the multimedia…
Fundamental to the theory of continued fractions is the fact that every infinite continued fraction with positive integer coefficients converges; however, it is unknown precisely which continued fractions with integer coefficients (not…
The {Farey fractions} $F_n$ of order $n$ consist of all fractions $\frac{h}{k}$ in lowest terms lying in the closed unit interval and having denominator at most $n$. This paper considers the products $F_n$ of all nonzero Farey fractions of…
Fourier transformations of pseudo-Boolean functions are popular tools for analyzing functions of binary sequences. Real-world functions often have structures that manifest in a sparse Fourier transform, and previous works have shown that…
This version corrects minor inaccuracies and missprints. One drawing is changed. We continue to study some arithmetical properties of Farey sequences by the method introduced by F.Boca, C.Cobeli and A.Zaharescu (2001). Let $\Phi_{Q}$ be the…
Fourier ptychography is a recently explored imaging method for overcoming the diffraction limit of conventional cameras with applications in microscopy and yielding high-resolution images. In order to splice together low-resolution images…
This study reexamines diffusive representations for fractional integrals with the goal of pioneering new variants of such representations. These variants aim to offer highly efficient numerical algorithms for the approximate computation of…
The nonlinear Fourier transform (NFT) has recently gained significant attention in fiber optic communications and other engineering fields. Although several numerical algorithms for computing the NFT have been published, the design of…
In this note, we study a family of subgraphs of the Farey graph, denoted as $\mathcal{F}_N$ for every $N\in\mathbb{N}.$ We show that $\mathcal{F}_N$ is connected if and only if $N$ is either equal to one or a prime power. We introduce a…
Generalized Fourier series with orthogonal polynomial bases have useful applications in several fields, including differential equations, pattern recognition, and image and signal processing. However, computing the generalized Fourier…
Ptychography is a popular imaging technique that combines diffractive imaging with scanning microscopy. The technique consists of a coherent beam that is scanned across an object in a series of overlapping positions, leading to reliable and…
The following document presents some novel numerical methods valid for one and several variables, which using the fractional derivative, allow to find solutions for some non-linear systems in the complex space using real initial conditions.…
The methods of extracting image features are the key to many image processing tasks. At present, the most popular method is the deep neural network which can automatically extract robust features through end-to-end training instead of…
The substantial computational costs of diffusion models, especially due to the repeated denoising steps necessary for high-quality image generation, present a major obstacle to their widespread adoption. While several studies have attempted…
Phase retrieval, i.e., the problem of recovering a function from the squared magnitude of its Fourier transform, arises in many applications such as X-ray crystallography, diffraction imaging, optics, quantum mechanics, and astronomy. This…
This paper proposes fractional order graph neural networks (FGNNs), optimized by the approximation strategy to address the challenges of local optimum of classic and fractional graph neural networks which are specialised at aggregating…
Kernel methods are powerful and flexible approach to solve many problems in machine learning. Due to the pairwise evaluations in kernel methods, the complexity of kernel computation grows as the data size increases; thus the applicability…
For a fixed positive integer d, we show the existence of the limiting gap distribution measure for the sets of Farey fractions a/q of order Q with a not divisible by d, and respectively with q relatively prime with d, as Q tends to…
The division operation is important for many areas of data processing. Especially considering today's demand for hardware accelerators for machine learning algorithms, there is a high demand for an efficient calculation of the division…
Fourier methods have a long and proven track record as an excellent tool in data processing. As memory and computational constraints gain importance in embedded and mobile applications, we propose to combine Fourier methods and recurrent…