Attenuation corrected tensor tomography - attenuation helps in the case of insufficient measurements

Abstract

This study was performed to investigate whether knowledge of the attenuation distribution is helpful in reconstruction of a symmetric second order tensor field (three unknown components for every point in two-dimensional (2D) Euclidean space and six unknown components in three-dimensional (3D) Euclidean space ) from fewer than three directional measurements of attenuated projections in 2D space and fewer than six directional measurements in 3D space. A least-squares method was adopted to reconstruct the tensor field in 2D space from one directional measurement instead of three. With the prior information of the attenuation distribution a better reconstruction could be expected, compared to the situation where attenuation was assumed to be absent. As a second example, diffusion tensor tomography MRI was simulated to show that knowledge of the attenuation distribution helps in the reconstruction of a tensor field from fewer projections of attenuated directional measurements. If attenuation is known, the algorithm was able to accurately reconstruct all six unknown components of the tensor field from six directional projections. For the reconstruction of only five directional measurements involving insufficient measurements, a better reconstruction of the components of the tensor field was obtained for the case of a constant attenuator greater than zero than for the case with a constant attenuator equal to zero. This presents an intriguing inverse problem where attenuation seems to help in the solution of an underdetermined inverse problem.