Statistical parameterization of inverse problems

Abstract

Image restoration, computerized tomography, and other similar problems are considered as a unified class of stochastic inverse problems. The conventional approach to these problems that proceeds from some integral or functional equations suffers from three main shortcomings: (i) subjectivity, (ii) inability to account for the inner (radiational) noise, and (iii) inability to include the fundamental concept of the natural limit of solution accuracy. A general approach is developed, the Statistical Parameterization of Inverse Problems (SPIPR), that takes into account both the inner and external random noise and gives an explicit form of the above-mentioned natural limit. Applications of the SPIPR to various problems show that the maximum likelihood method as the concrete way to obtain an object estimate has practically limiting efficiency. Two new fields of applications of the SPIPR are outlined along with the image restoration problem: the elimination of blurring due to atmosphere turbulence and reconstruction of an object structure in the computerized tomography. The expressions for the main distribution function in all these problems are found. The corresponding real examples and model cases are considered as well.