Definition Of Convolution Research Articles

CCNet-5D: 5D convolutional neural network for seismic data interpolation

Convolutional neural network (CNN)-based seismic interpolation methods recover missing traces by feeding corrupted data to a trained neural network, whose parameters are obtained by training pairs of corrupted data and their complete labels. Compared with traditional reconstruction approaches, these methods require less human-computer interaction and computation time; therefore, CNN approaches have been popular for 2D/3D seismic interpolation. Five-dimensional seismic data interpolation methods recover missing traces simultaneously in five dimensions, considering all the physical coordinates with high accuracy. Unfortunately, existing deep-learning frameworks (such as TensorFlow and PyTorch) only provide low-dimensional convolution operators (no more than three dimensions), which makes it difficult to generalize to 5D seismic reconstruction directly. To this end, we develop an effective 5D-CNN by cascading low-dimensional convolution to deal with 5D seismic interpolation, referred to as the CCNet-5D. First, based on the definition of convolution and the theory of tensor decomposition, the 5D convolution operator is approximated by the sum of multiple cascading of low-dimensional convolution operators. Then, we determine the CNN architecture for the 5D convolutional operators by cascading the 3D and 2D convolutional layers, called the CC-3D2D module. The final CCNet-5D is constructed by stacking the four resulting CC-3D2D modules. In our setup, 5D-CNN outperforms the existing 5D traditional method and 3D CNN-based method.

Some New Applications of the q-Analogous of Differential and Integral Operators for New Subclasses of q-Starlike and q-Convex Functions

In the geometric function theory of complex analysis, the investigation of the geometric properties of analytic functions using q-analogues of differential and integral operators is an important area of study, offering powerful tools for applications in numerical analysis and the solution of differential equations. Many topics, including complex analysis, hypergeometric series, and particle physics, have been generalized in q-calculus. In this study, first of all, we define the q-analogues of a differential operator (DRλ,qm,n) by using the basic idea of q-calculus and the definition of convolution. Additionally, using the newly constructed operator (DRλ,qm,n), we establish the q-analogues of two new integral operators (Fλ,γ1,γ2,…γlm,n,q and Gλ,γ1,γ2,…γlm,n,q), and by employing these operators, new subclasses of the q-starlike and q-convex functions are defined. Sufficient conditions for the functions (f) that belong to the newly defined classes are investigated. Additionally, certain subordination findings for the differential operator (DRλ,qm,n) and novel geometric characteristics of the q-analogues of the integral operators in these classes are also obtained. Our results are generalizations of results that were previously proven in the literature.

The Definition of Convolution in Deep Learning by using Matrix

The Definition of Convolution in Deep Learning by using Matrix

Commutative Anisotropic Convolution on the 2-Sphere

We develop a new type of convolution between two signals on the 2-sphere. This is the first type of convolution on the 2-sphere which is commutative. Two other advantages, in comparison with existing definitions in the literature, are that 1) the new convolution admits anisotropic filters and signals and 2) the domain of the output remains on the sphere. Therefore, the new convolution well emulates the conventional Euclidean convolution. In addition to providing the new definition of convolution and discussing its properties, we provide the spectral analysis of the convolution output. This convolutional framework can be useful in filtering applications for signals defined on the 2-sphere.

Fractional Fourier transform of Lorentz beams

This paper introduces Lorentz beams to describe certain laser sources that produce highly divergent fields. The fractional Fourier transform (FRFT) is applied to treat the propagation of Lorentz beams. Based on the definition of convolution and the convolution theorem of the Fourier transform, an analytical expression for a Lorentz beam passing through a FRFT system has been derived. By using the derived formula, the properties of a Lorentz beam in the FRFT plane are illustrated numerically.

Fractional Fourier transform of Lorentz-Gauss beams

Lorentz-Gauss beams are introduced to describe certain laser sources that produce highly divergent beams. The fractional Fourier transform (FRFT) is applied to treat the propagation of Lorentz-Gauss beams. Based on the definition of convolution and the convolution theorem of the Fourier transform, an analytical expression for a Lorentz-Gauss beam passing through an FRFT system has been derived. By using the derived expression, the properties of a Lorentz-Gauss beam in the FRFT plane are graphically illustrated with numerical examples.

On the Definition of Convolutions for Distributions

On the Definition of Convolutions for Distributions