Linear Convolution Operations Research Articles

Applications of Some Subclasses of Meromorphic Functions Associated with the q-Derivatives of the q-Binomials

In this article, we make use of the q-binomial theorem to introduce and study two new subclasses ℵ(αq,q) and ℵ(α,q) of meromorphic functions in the open unit disk U, that is, analytic functions in the punctured unit disk U∗=U\{0}={z:z∈Cand0<z<1}. We derive inclusion relations and investigate an integral operator that preserves functions which belong to these function classes. In addition, we establish a strict inequality involving a certain linear convolution operator which we introduce in this article. Several special cases and corollaries of our main results are also considered.

Asymptotic Spectral Representation of Linear Convolutional Layers

By stacking a number of convolutional layers, convolutional neural networks (CNNs) have made remarkable performance boosts in many artificial intelligence applications. While the convolution operation is well-understood, it is still a mystery why repeated convolutions yield so good expressive power and generalization performance. Noting that the linear convolution operation can be represented as a matrix-vector product with the matrix being of a Toeplitz structure, we propose to inspect the individual convolutional layer through its asymptotic spectral representation - the spectral density matrix - by leveraging Toeplitz matrix theory. Thanks to such spectral representation, we are able to develop a simple singular value approximation method with improved accuracy, and spectral norm upper bounds with reduced computational complexity, compared with the state-of-the-art methods. Both the improved approximation and upper bounds can be employed as regularization techniques to further enhance the generalization performance of CNNs. By extensive experiments on well-deployed CNN models (e.g., ResNets), we also demonstrate that the approximation approach achieves higher accuracy and the upper bounds are effective spectral regularizers for generalization.

On Multivalent Analytic Functions Considered by a Multi-Arbitrary Differential Operator in a Complex Domain

(1) Background: There is an increasing amount of information in complex domains, which necessitates the development of various kinds of operators, such as differential, integral, and linear convolution operators. Few investigations of the fractional differential and integral operators of a complex variable have been undertaken. (2) Methods: In this effort, we aim to present a generalization of a class of analytic functions based on a complex fractional differential operator. This class is defined by utilizing the subordination and superordination theory. (3) Results: We illustrate different fractional inequalities of starlike and convex formulas. Moreover, we discuss the main conditions to obtain sandwich inequalities involving the fractional operator. (4) Conclusion: We indicate that the suggested class is a generalization of recent works and can be applied to discuss the upper and lower bounds of a special case of fractional differential equations.

Analytic Normalized Solutions of 2D Fractional Saint-Venant Equations of a Complex Variable

Saint-Venant equations describe the flow below a pressure surface in a fluid. We aim to generalize this class of equations using fractional calculus of a complex variable. We deal with a fractional integral operator type Prabhakar operator in the open unit disk. We formulate the extended operator in a linear convolution operator with a normalized function to study some important geometric behaviors. A class of integral inequalities is investigated involving special functions. The upper bound of the suggested operator is computed by using the Fox-Wright function, for a class of convex functions and univalent functions. Moreover, as an application, we determine the upper bound of the generalized fractional 2-dimensional Saint-Venant equations (2D-SVE) of diffusive wave including the difference of bed slope.

Performance analysis of Vedic mathematics algorithms on re-configurable hardware platform

For the overall performance of systems like microprocessors and digital signal processors (DSPs) platforms, arithmetic units, all must be efficient in terms of speed, power, and area. Multipliers and dividers are inevitable hardware employed in such systems. This paper focuses on Vedic mathematics algorithms for multiplication and division for power-efficient, faster, and area-efficient design. For four- and eight-bit Vedic multiplication algorithms, Urdhva Tiryagbhyam and Nikhilam Sutras are employed in this paper. For eight-bit Vedic division algorithms, Nikhilam and Dhwajank Sutras are used. The Vedic mathematics algorithms are also compared to conventional methods of multiplication (like Array multiplier) and division (using Booth multiplication algorithm). As an application of DSP, the linear convolution operation is implemented using both conventional and Vedic algorithms. It has been observed that the Vedic algorithms operate faster, consume less power, and occupy less area on a targeted hardware platform. The implementations were carried out using the Verilog HDL language and Xilinx’s Vivado EDA tool. To measure various performance parameters, Cadence simvision (using 180-nm GPDK CMOS Technology) and Xilinx’s ISE tool were also used.

Examining the Function of Meromorphic with Using the Linear Convolution Operator

In this study, it is mentioned that meromorphic functions are univalent functions that are analytical everywhere. Complex analytical transformations were investigated by mentioning the necessary form for f (z) to have meromorphic function. It is a function that satisfies the condition   0 hz  . For analytic functions of f and g in the D unit disk, ()fz shows the meromorphic function class with P and subclasses of the P meromorphic analytical function class using the subordination principle between functions with the help of Hadamard product and linear operators. In this way proves is provided.

Operators of Basic (or q-) Calculus and Fractional q-Calculus and Their Applications in Geometric Function Theory of Complex Analysis

Basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and basic (or q-) hypergeometric polynomials, are known to have widespread applications, particularly in several areas of number theory and combinatorial analysis such as (for example) the theory of partitions. Our usages here, in this survey-cum-expository article, of the q-calculus and the fractional q-calculus in geometric function theory of complex analysis are believed to encourage and motivate significant further developments on these and other related topics. By applying a fractional q-calculus operator, we define the subclasses $${\mathcal{S}}_{n}^{\alpha }(\lambda ,\beta ,b,q)$$ and $${\mathcal{G}}_{n}^{\alpha }(\lambda ,\beta ,b,q)$$ of normalized analytic functions with complex order and negative coefficients. Among the results investigated for each of these function classes, we derive their associated coefficient estimates, radii of close-to-convexity, starlikeness and convexity, extreme points and growth and distortion theorems. Our investigation here is motivated essentially by the fact that basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several areas of number theory such as the theory of partitions. In fact, basic (or q-) hypergeometric functions are useful also in a wide variety of fields including, for example, combinatorial analysis, finite vector spaces, lie theory, particle physics, nonlinear electric circuit theory, mechanical engineering, theory of heat conduction, quantum mechanics, cosmology and statistics (see also (Srivastava and Karlsson in Multiple Gaussian hypergeometric series. pp 350–351, 1985) and the references cited thereon). In the last section on conclusion, we choose to point out the fact that the results for the q-analogues, which we consider in this article for $$0< q < 1$$, can easily (and possibly trivially) be translated into the corresponding results for the (p, q)-analogues (with $$0< q < p \leqq 1$$) by applying some obvious parametric and argument variations, the additional parameter p being redundant. Several other families of such extensively- and widely-investigated linear convolution operators as (for example) the Dziok–Srivastava, Srivastava–Wright and Srivastava–Attiya linear convolution operators, together with their extended and generalized versions, are also briefly considered.

Direct prediction of petrophysical and petroelastic reservoir properties from seismic and well-log data using nonlinear machine learning algorithms

An analytical comparison of seismic inversion with several multivariate predictive techniques is made. Statistical data reduction techniques are examined that incorporate various machine learning algorithms, such as linear regression, alternating conditional expectation regression, random forest, and neural network. Seismic and well-log data are combined to estimate petrophysical or petroelastic properties, like bulk density. Currently, spatial distribution and estimation of reservoir properties is leveraged by inverting 3D seismic data calibrated to elastic properties (VP, VS, and bulk density) obtained from well-log data. Most commercial seismic inversions are based on linear convolution, i.e., one-dimensional models that involve a simplified plane-parallel medium. However, in cases that are geophysically more complex, such as fractured and/or fluid-rich layers, the conventional straightforward prediction relationship breaks down. This is because linear convolution operators no longer adequately describe seismic wavefield propagation due to nonlinear energy absorption. Such nonlinearity is also suggested by the seismic nonstationarity phenomenon, expressed by vertical and horizontal changes in the shape of the seismic wavelet (amplitude and frequency variations). The nonlinear predictive operator, extracted by machine learning algorithms, makes it possible in certain cases to estimate petrophysical reservoir properties more accurately and with less influence of interpretational bias.

Analysis of the Image Spectrum for Distortion Diagnostics

The diagnostic of possible distortions in given visual data is a significant problem in signal and image processing. In the paper, we discuss the task of automatic determination of the type and parameters of expected distortion on the analysis of the obtained image. In solving the problem, we restrict the research by the identification of linear homogeneous convolution operators of the following three types: circular form with rectangular profile, circular form with Gaussian profile, and linear motion blur ones. The diagnostics identification procedures base on the image spectrum analysis and use the mean radial profile of the image spectrum amplitude. For every distortion type, we elaborate corresponding diagnostics procedures, that when combined into one common process, form a unique recognition algorithm. The studies and experiments demonstrate the reliability of distortion identification algorithms against signal precision and noise influence.

A theoretical insight into morphological operations in surface measurement by introducing the slope transform

As one of the tools for surface analysis, morphological operations, although not as popular as linear convolution operations (e.g., the Gaussian filter), are really useful in mechanical surface reconstruction, surface filtration, functional simulation, etc. By introducing the slope transform originally developed for signal processing into the field of surface metrology, an analytic capability is gained for morphological operations, paralleling that of the Fourier transform in the context of linear convolution. Using the slope transform, the tangential dilation is converted into the addition in the slope domain, just as by the Fourier transform, the convolution switches into the multiplication in the frequency domain. Under the theory of the slope transform, the slope and curvature changes of the structuring element to the operated surface can be obtained, offering a deeper understanding of morphological operations in surface measurement. The analytical solutions to the tangential dilation of a sine wave and a disk by a disk are derived respectively. An example of the discretized tangential dilation of a sine wave by the disks with two different radii is illustrated to show the consistency and distinction between the tangential dilation and the classical dilation.

Local saturation of a positive linear convolution operator

Let { H n (t)} be a sequence of non-negative, even, and continuous functions on ℝ. In this paper, we consider a convolution operator J n (f;x)= ∫ 0 ∞ f(t) H n (t−x)dt, f∈ L p ( R + ), and then investigate the local saturation of J n (f;x).MSC:44A35.

ON PRESENTATION OF LINEAR OPERATORS COMMUTING WITH DIFFERENTIATION IN SIMPLY-CONNECTED DOMAIN

Let   H G be a space of analytic functions of one variable in simply - connected domain G of the complex plane. It is known that a linear complex convolution operator is generated by a one - variable analytic function, a multivalued one in general. A known problem when all such functions are single - valued is solved. As it turned out, the solution to the problem is connected with the geometry of G domain. Set   s G with property   s G G G   is termed re sidue of G domain. A class of simply connected regions whose residue is a connected set is described. Let the linear operator be continuous in function space, analytical in simply - connected domain G, and let it commute with diffe rentiation. Then it can be reduced to a complex convoluti on operator. It is proved that the function generating such an operator will always be single - valued for regions with a connected residue. When the residue of region G is not connected, there is always a complex convolution operator with a multivalued func tion generating a kernel.

Super-resolution non-parametric deconvolution in modelling the radial response function of a parallel plate ionization chamber

The signal of the dosimetric detector is generally dependent on the shape and size of the sensitive volume of the detector. In order to optimize the performance of the detector and reliability of the output signal the effect of the detector size should be corrected or, at least, taken into account. The response of the detector can be modelled using the convolution theorem that connects the system input (actual dose), output (measured result) and the effect of the detector (response function) by a linear convolution operator. We have developed the super-resolution and non-parametric deconvolution method for determination of the cylinder symmetric ionization chamber radial response function. We have demonstrated that the presented deconvolution method is able to determine the radial response for the Roos parallel plate ionization chamber with a better than 0.5 mm correspondence with the physical measures of the chamber. In addition, the performance of the method was proved by the excellent agreement between the output factors of the stereotactic conical collimators (4–20 mm diameter) measured by the Roos chamber, where the detector size is larger than the measured field, and the reference detector (diode). The presented deconvolution method has a potential in providing reference data for more accurate physical models of the ionization chamber as well as for improving and enhancing the performance of the detectors in specific dosimetric problems.

Texture Deviation Mapping Based on Detail Enhancement

To provide with an effective technique for non-photorealistic rendering for computer generated images with artistic appearances from 2D images motivates our work in this paper. The methods proposed in this paper are inspired by the image deviation mapping constructed from a single texture background image. We establish our method for obtaining artistic appearances taking the deviation mapping as the underlying basis. Based on the simple linear filtering convolution operation, which is well suited of progressive coarsening of images and for detail extraction, and the image’s detail, such as edge and tone can be preserved in the final artistic appearance. This method has the exact computational complexity, and this technique is easily to implement and the rendering speed is fast.

An estimate for some positive linear operators in L p spaces

In this note we prove that an estimate due to Ivanov and Pichugov, for positive linear convolution operators in Lp spaces, can be extended to a class of positive linear operators which are not of convolution type.

Maritime radar display unit based on PC for safe ship navigation

A prototype radar display unit was implemented using inexpensive off-the-shelf components, including a nonlinear estimation algorithm for the target tracking in a clutter environment. Two custom designed boards; an analog signal processing board and a DSP board, can be plugged into an expansion slot of a personal computer (PC) to form a maritime radar display unit. Our system provided all the functionality specified in the International Maritime Organization (IMO) resolution A422(XI). The analog signal processing board was used for A/D conversion as well as rain and sea clutter suppression. The main functions of the DSP board were scan conversion and video overlay operations. A host PC was used to run the tracking algorithm of targets in clutter, using the discrete-time Bayes optimal (nonlinear, and non-Gaussian) estimation method, and the graphic user interface (GUI) software for Automatic Radar Plotting Aid (ARPA). The proposed tracking method recursively found the entire probability density function of the target position and velocity by converting into linear convolution operations.

잔향 생성기에서 실시간 마스킹 효과를 이용한 고속 컨벌루션 방법

With the advent of sound field simulator, many sound fields have been reproduced by obtaining the impulse responses of specific acoustic spaces like famous concert hall, opera house. This sound field reproduction has been done by the linear convolution operation between the sound input signal and the impulse response of certain acoustic space. However, the conventional finite impulse response based linear convolution operation always makes real-time implementation of sound field generator impossible due to the large amount of computational burden. This paper introduces the fast convolution method using perceptual redundancy in the processed signals, input audio signal and room impulse response. Temporal and spectral real-time masking blocks are implemented in the proposed convolution structure. It reduces the computational burden of convolution methods for real-time implementation of a sound field generator. The conventional convolutions are compared with the proposed one in views of computational burden and sound quality. In the proposed method, a considerable reduction in the computational burden was realized with acceptable changes in sound quality.

A Nonlinear Estimation Algorithm and Its Optical Implementation for Target Tracking in Clutter Environment

The two-dimensional tracking problem in a clutter environment is solved in the discrete-time Bayes optimal (nonlinear, and non-Gaussian) estimation framework. The proposed method recursively finds the probability density functions of the target position and velocity. With our approach, the nonlinear estimation problem is converted into simpler linear convolution operations that can be implemented with optical devices efficiently. We present a possible optical implementation architecture, and its functionality is verified through computer simulation.

When is a linear convolution system stabilizable?

Lack of operator closedness and closability arguments are used to discuss the impossibility of finite ℓ p / L p gain stabilization of unstable finite-dimensional linear (convolution operator) systems in the doubly infinite time-axis case for any 1⩽ p⩽∞. The presented analysis generalizes and refines a result due to Georgiou and Smith (IEEE Trans. Automat. Control 40 (1995) 516) for the L 2(−∞,∞) setting.

Extracting interaction forces and complementary observables in dynamic probe microscopy

We address the problem of interaction sensing in dynamic probe microscopy. An iterative method is presented for extracting the laws of interaction from dynamically measured observables. The method exploits the fact that in the limit of negligible anharmonicity of the tip motion, the observables are related to corresponding force laws via linear convolution operators involving weakly divergent kernels, which represent weighted monopole or dipole moments of the interaction probed by the tip.