FPGA based Edge Detection using Sobel Filter

In image processing, Sobel operator is utilised especially inside algorithms of edge-detection. It is a discreet differentiation operator which calculates the gradient approximation of the function picture intensity. The outcome of the Sobel operation at each location of the image is either the appropriate gradient vector or the vector standard. The Sobel operator relies on the image being converted into horizontal and vertical with a tiny, separable and integrated valued filter. This means that the computation is quite inexpensive. PAN Poanta satellite image was used for this work using Java, Core Java in GDAL package. As compared to in built Sobel operator, the image generated for this work is very fine and sharp as a result of noise suppression to a considerable extent. Inorder to do edge detection efficiently with minimal amount of false results, a correct form of Sobel filter ( I’=√(I*X)²+(I*Y)2 ) was used instead of the approximation(I’=I*X+I*Y) for the sake of computation.

On nonlocal perturbation of the problem on eigenvalues of differentiation operator on a segment

This work is devoted to the construction of a characteristic polynomial of the spectral problem of a first-order differential equation on an interval with a spectral parameter in a boundary value condition with integral perturbation which is an entire analytic function of the spectral parameter. Based on the characteristic polynomial formula, conclusions about the asymptotics of the spectrum of the perturbed spectral problem are established.

Pseudodifferential Equation of Fluctuations of Nonstationary Gravitational Fields

Developing Holtzmark’s idea, the distribution of nonstationary fluctuations of local interaction of moving objects of the system with gravitational influence, which is characterized by the Riesz potential, is constructed. A pseudodifferential equation with the Riesz fractional differentiation operator is found, which corresponds to this process. The general nature of symmetric stable random Lévy processes is determined.

THE NON-LOCAL TIME PROBLEM FOR ONE CLASS OF PSEUDODIFFERENTIAL EQUATIONS WITH SMOOTH SYMBOLS

In this paper we investigate the differential-operator equation $$ \partial u (t, x) / \partial t + \varphi (i \partial / \partial x) u (t, x) = 0, \quad (t, x) \in (0, + \infty) \times \mathbb {R} \equiv \Omega, $$ where the function $ \varphi \in C ^ {\infty} (\mathbb {R}) $ and satisfies certain conditions. Using the explicit form of the spectral function of the self-adjoint operator $ i \partial / \partial x $, in $ L_2 (\mathbb {R}) $ it is established that the operator $ \varphi (i \partial / \partial x) $ can be understood as a pseudodifferential operator in a certain space of type $ S $. The evolution equation $ \partial u / \partial t + \sqrt {I- \Delta} u = 0 $, $ \Delta = D_x ^ 2 $, with the fractionation differentiation operator $ \sqrt { I- \Delta} = \varphi (i \partial / \partial x) $, where $ \varphi (\sigma) = (1+ \sigma ^ 2) ^ {1/2} $, $ \sigma \in \mathbb {R} $ is attributed to the considered equation. Considered equation is a nonlocal multipoint problem with the initial function $ f $, which is an element of a space of type $ S $ or type $ S '$ which is a topologically conjugate with a space of type $ S $ space. The properties of the fundamental solution of such a problem are established, the correct solvability of the problem in the half-space $ t> 0 $ is proved, the representation of the solution in the form of a convolution of the fundamental solution with the initial function is found, the behavior of the solution $ u (t, \cdot) $ for $ t \to + \infty $ (solution stabilization) in spaces of type $ S '$.

Dirichlet problem for a second-order ordinary differential equation with a distributed differentiation operator

For an ordinary second-order differential equation with an operator of continuously distributed differentiation with variable coefficients, a solution to the Dirichlet problem is constructed using the Green’s function method.

On a Problem that Does not Have Basis Property of Root Vectors, Associated with a Perturbed Regular Operator of Multiple Differentiation

A spectral problem for a multiple differentiation operator with integral perturbation of boundary value conditions which are regular but not strongly regular is considered in the paper. The feature of the problem is the absence of the basis property of the system of root vectors. A characteristic determinant of the spectral problem is constructed. It is shown that absence of the basis property of the system of root functions of the problem is unstable with respect to the integral perturbation of the boundary value condition