New Mathematical Model and Measurement Scheme of Electrical Tomography and Its Fast Resolution by General Ray Method

摘要

Computer Tomography consists in the image reconstruction of an interior of a body using the measurements on its surface of characteristics of some external field. Very often this reconstruction can be posed mathematically as a coefficient inverse problem for a differential equation describing the distribution of the field in considered region. Coefficients are functions of the space variables and characterize properties of a media. Well known are X-rays, ultrasonic, radioisotopes, infrared and electrical impedance tomography. For example, X-rays tomography leads to the linear mathematical model and for its resolving can be used corresponding linear fast algorithms, but the know methods for the electrical impedance tomography imaging are non-linear, that leads to algorithms which require a large time of calculations on computer. We propose here another approach for the mathematical modelling and the measurement scheme for the external data. We explain it for the plane case and describe adaptation for space case. We construct the mathematical model on the basis of General Ray Principle, proposed by the author for distribution of different, in particular electromagnetic, fields. Proposed model leads to the classic Radon transformation that appears as specific element in new General Ray Method, constructed and realized by the author as a simple linear fast numerical algorithm. This approach and algorithms are justified with the numerical experiments on the simulated model examples in MATLAB system.