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The article is dedicated to the problems of using multidimensional numbers for mathematical and computer modeling of complex physical processes and the design of knowledge-intensive devices, including digital image processing. The emphasis is on the issues of building the methods for processing three-dimensional signals. It is proposed to use three-dimensional variables presented in the form of hypercomplex numbers to formulate the three-dimensional Fourier transformation forms, which allows to analyze and process three-dimensional signals.

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The paper proposes a method for taking into account the effect of space charge on the quality of focusing of charged particles and investigates the problem of the formation of an intense flux of charged particles using a transaxial cathode lens, which is a special case of cathode lenses with two planes of symmetry. The article solves a self-consistent problem by representing the distribution functions of potentials and space charge in the form of power series in small parameters. The author uses an improved and proven method for studying cathode lenses, based on taking into account the longitudinal component of the total aberration. The necessary equations and formulas are obtained for calculating the parameters and designing a source of charged particles based on a transaxial cathode lens.