Image Compression and Enlargement Algorithms

In some cases, there are problems associated with the compression and enlargement of images. The use of splines is quite effective in some cases. In this paper, a new image compression algorithm is presented. The features of increasing the size of an image when using local polynomial or non-polynomial splines are considered. The main method for enlarging an image is based on the use of splines of the second and third order of approximation. Polynomial and trigonometric splines are considered. To speed up the process of enlarging the image, we used the parallelization techniques

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Approximation by the Third-Order Splines on Uniform and Non-uniform Grids and Image Processing

WSEAS TRANSACTIONS ON MATHEMATICS ◽

10.37394/23206.2020.19.7 ◽

2020 ◽

Vol 19 ◽

Keyword(s):

Order Approximation ◽

Approximation Error ◽

Polynomial Splines ◽

Third Order ◽

The Third ◽

Uniform Grids ◽

Trigonometric Splines ◽

Image Enlargement ◽

Third Order Approximation ◽

Approximation Polynomial

This work is one of a series of papers that is devoted to the further investigation of polynomial splines and trigonometric splines of the third order approximation. Polynomial basis splines are better known and therefore more commonly used. However, the use of trigonometric basis splines often provides a smaller approximation error. In some cases, the use of the trigonometric approximations is preferable to the polynomial approximations. Here we continue to compare these two types of approximation. The Lebesgue functions and constants are discussed for the polynomial splines and the trigonometric splines. The examples of the applications of the splines to image enlargement are given.

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Application Local Polynomial and Non-polynomial Splines of the Third Order of Approximation for the Construction of the Numerical Solution of the Volterra Integral Equation of the Second Kind

WSEAS TRANSACTIONS ON MATHEMATICS ◽

10.37394/23206.2021.20.2 ◽

2021 ◽

Vol 20 ◽

Author(s):

I. G. Burova

Keyword(s):

Integral Equation ◽

Numerical Solution ◽

Volterra Integral Equation ◽

Numerical Experiments ◽

Polynomial Interpolation ◽

Order Of Approximation ◽

Polynomial Splines ◽

Third Order ◽

The Third ◽

Local Polynomial

The present paper is devoted to the application of local polynomial and non-polynomial interpolation splines of the third order of approximation for the numerical solution of the Volterra integral equation of the second kind. Computational schemes based on the use of the splines include the ability to calculate the integrals over the kernel multiplied by the basis function which are present in the computational methods. The application of polynomial and nonpolynomial splines to the solution of nonlinear Volterra integral equations is also discussed. The results of the numerical experiments are presented.

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Application of Non-polynomial Splines to Solving Differential Equations

WSEAS TRANSACTIONS ON MATHEMATICS ◽

10.37394/23206.2020.19.58 ◽

2020 ◽

Vol 19 ◽

Keyword(s):

Heat Conduction Problem ◽

Numerical Differentiation ◽

Exponential Polynomial ◽

Order Of Approximation ◽

Polynomial Splines ◽

Implicit Difference Scheme ◽

Local Polynomial ◽

Second Derivatives ◽

Implicit Difference Schemes ◽

Derivatives Of

The application of the local polynomial and non-polynomial to the construction of methods for numerically solving the heat conduction problem is discussed. The non-polynomial splines are used here to approximate the partial derivatives. Formulas for numerical differentiation based on the application of the nonpolynomial splines of the fourth order of approximation are constructed. Particular attention is paid to polynomial, trigonometric, exponential, polynomial-trigonometric and polynomial-exponential splines. This approach allows us to construct explicit and implicit difference schemes. The main focus of the paper is on implicit difference scheme. New approximations with splines of the Lagrange and Hermite type with new properties are obtained. These approximations take into account the first and second derivatives of the function being approximated. Numerical examples are given.

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Approximations With Polynomial, Trigonometric, Exponential Splines of the Third Order and Boundary Value Problem

International Journal of Circuits, Systems and Signal Processing ◽

10.46300/9106.2020.14.61 ◽

2020 ◽

Vol 14 ◽

Keyword(s):

Boundary Value Problem ◽

Parabolic Problem ◽

Order Approximation ◽

Boundary Value ◽

Numerical Differentiation ◽

Polynomial Splines ◽

Third Order ◽

Numerical Examples ◽

The Third ◽

Trigonometric Splines

This paper is devoted to the construction of localapproximations of functions of one and two variables using thepolynomial, the trigonometric, and the exponential splines. Thesesplines are useful for visualizing flows of graphic information.Here, we also discuss the parallelization of computations. Someattention is paid to obtaining two-sided estimates of theapproximations using interval analysis methods. Particularattention is paid to solving the boundary value problem by usingthe polynomial splines and the trigonometric splines of the thirdand fourth order approximation. Using the considered splines,formulas for a numerical differentiation are constructed. Theseformulas are used to construct computational schemes for solvinga parabolic problem. Questions of approximation and stability ofthe obtained schemes are considered. Numerical examples arepresented.

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