Methods of converting multidimensional measuring information to a number by means of differential equations in partial derivatives

Abstract

We have proved that finite difference representation of differential equations in partial derivatives is the most effective method of converting multidimensional measuring information to a single number. The devised conversion methods use Laplace’s elliptic equations and Fourier’s parabolic equations. To convert information,we have solved a number of tasks. We have devised a method of elliptic conversion of static images of the measured object by means of finite difference representation of Laplace’s differential equation in the second-order partial derivatives, which means a change of the current value of the pixel brightness to a new one that has two gradations of brightness––black and white. Pixel summation turns the relative number of black pixels into a single number as a result of elliptic conversion. We have improved the method of parabolic conversion of dynamic images of the measured object by means of finite difference representation of Fourier’s differential equation in the second-order partial derivatives,which means a change of the current value of the pixel brightness to a new one that has two gradations of brightness––black and white. Pixel summation turns the relative number of black pixels into a single number as a result of parabolic conversion. We have used the devised methods of industrial testing of metrological support for the technological process of continuous casting of copper ingots. The testing has proved technical and economic effectiveness of the approach. The paper contains examples of technical and economic effectiveness of the devised methods in metallurgical production.