Возможности обращения обобщенного преобразования Радона и рентгеновская томография [Итоговый научный отчет по междисциплинарному интеграционному проекту СО РАН: “Разработка теории и вычислительной технологии решения обратных и экстремальных задач с приложением в математической физике и гравимагниторазведке”]

This paper is a review of the authors articles [1–3]. They are devoted to a new problem of integral geometry in which the known data are integrals along all straight lines in plane. The integrand is an unknown function depending of space points and parameters characterizing straight lines. The surfaces of integrand discontinuity are desired quantity. It is connected with certain problems of X-Ray tomography solved as inverse problems for transport equation [4,5]. The obtained results [1,2] differ from the corresponding fragments in [4,5] by higher generality and simplicity. Also a question of incomplete data has been studied [3]. Probably the statements in this article have its own value and can be used in development of tomography.