Igor Vasilievich Prokhorov

The issues of numerical determination of a function defined on a plane by its integrals along strips of relatively small width are considered. An algorithm for solving the problem is proposed, based on extrapolation of several images obtained as a result of approximate inversion of the Radon transform for strips of different widths. The results of numerical calculations are presented, limitations and prospects for using the extrapolation approach are indicated.

The inverse problem for the nonstationary integro-differential equation of high-frequency acoustic radiative transfer is considered. It consists in determining the surfaces of discontinuity of the volume scattering coefficient from the time-angular distribution of the flux density at a given point in three-dimensional space. Numerical algorithms for solving the inverse problem are proposed. They are based on the introduction of special indicator functions that explicitly indicate the location of the lines of discontinuity of the scattering coefficient in a given plane. The process of ultrasonic sounding in the marine environment was simulated by the Monte Carlo methods, the efficiency of the algorithms for localizing scattering inhomogeneities was demonstrated, and the effect of incompleteness of the initial data on the quality of tomographic images was analyzed numerically.

This paper proposes a new approach to improving image quality in pulsed X-ray tomography. The method is based on establishing a functional dependence of the reconstructed images on the duration of the probing pulses and applying an extrapolation procedure. The numerical experiments demonstrated that the developed algorithm effectively suppresses the influence of scattered radiation and significantly increases image contrast. The proposed alternative approach allows substantially increasing the stability of the method even for media containing strong scattering inhomogeneities and with a significant level of noise in the projection data. In addition, the algorithm has greater stability to errors in the source data caused by an increase in the duration of the probing pulses. The numerical experiments confirmed the high efficiency of the extrapolation tomography algorithm for recovering the internal structure of the test object.

This paper investigates the inverse problem for the non-stationary radiation transfer equation, which involves finding the attenuation coefficient using the data of serial irradiation of the medium with pulses of various durations. In the framework of single and double scattering approximations, we obtain asymptotic estimates of the scattered radiation flux density for a short duration of the probing pulse. We propose extrapolation procedures for the ballistic component of the radiation transfer equation solution using the data of multiple irradiations of the medium by pulsed radiation sources, which allows us to obtain approximate formulas for finding the attenuation coefficient. The results of numerical experiments with a well-known digital phantom confirm the effectiveness of the extrapolation algorithm for improving the quality of tomographic images of scattering media.

The inverse problem for the non-stationary radiative transfer equation is considered, which consists in finding the attenuation coefficient according to the pulsed multi-energy X-ray exposure. For a short duration of the probing pulse, the asymptotic solution of the inverse problem is found. The problem of identifying an unknown substance by attenuation coefficients approximately found on a finite set of energy values is formulated. Algorithms for solving identification problems are proposed. The results of the numerical simulation are presented for a wide range of substances of interest in medical computed tomography.

Within the framework of the nonstationary radiative transfer equation, we propose a new extrapolation method to improve the quality of tomographic images of the attenuation coefficient. The method is based on irradiation of the medium under study with a series of pulses of different duration. A numerical analysis of the proposed algorithm is carried out using a well-known test phantom. The results show that the algorithm effectively suppresses the negative influence of scattering in X-ray tomography of inhomogeneous media.

The paper considers the mathematical problems of constructing sonar images of the seabed from the data of measurements of a multibeam side-scan sonar. For the nonstationary radiative transfer equation describing the process of acoustic sounding in the ocean, we investigate the inverse problem of finding the discontinuity lines of the bottom scattering coefficient. A numerical algorithm for solving the inverse problem is developed, and the analysis of the quality of localization of the boundaries of inhomogeneities of the seabed depending on the number of angles and the sounding range is carried out.

The inverse problem for the nonstationary radiative transfer equation is considered, which consists in finding the scattering coefficient for a given time-angular distribution of the solution to the equation at a certain point. To solve this problem, the single scattering approximation in the pulsed sounding mode is used. A comparative analysis of the error in solving the inverse problem in the single scattering approximation for two-dimensional and three-dimensional models describing the process of high-frequency acoustic sounding in a fluctuating ocean is carried out. It is shown that in the two-dimensional case the error of the approximate solution significantly exceeds the error in the three-dimensional model.