Identification problem and image reconstruction of the exponential X-ray transform

The emission-type CT which utilizes the radiation from the radioisotope (RI) injected into the body can reconstruct the biochemical information inside the body as an image. At present, the multicross-sectional method is employed in reconstructing the three-dimensional distribution of RI, as in the case of X-ray CT. In this method, a spatial nonuniform resolution is produced in the reconstructed RI image. It should be noted, on the other hand, that RI injected into the body emits radiations in all directions in the space. By utilizing all those radiations, a three-dimensional image can be reconstructed without a nonuniformity in the resolution. This paper shows that the three-dimensional distribution of RI can directly be reconstructed from the radiations emitted spatially from inside the body. It is shown first that the problem to determine the three-dimensional distribution of RI from the radiations is the inverse problem with the exponential line integral as the measured value. The relation between the Fourier transform of the projections by the exponential line integral and the Fourier transform of the original three-dimensional image is analyzed. Then it is shown that the radiation attenuation coefficient of RI can be determined only from the projection data. Furthermore, based on the relation between the Fourier and Hankel transformations, a method is derived for the direct three-dimensional image reconstruction from the radiation projection using the series expansion.