An Algorithm for Incomplete Range of Views Reconstruction

To reconstruct a cross-section of a 3D object, most algorithms require knowledge of the projection data in a full range of views [1]. In some practical situations [2, 3], reconstruction from an incomplete range of views is inevitable although it is not desirable from a mathematical point of view. Objects to be reconstructed are of compact support. Their Fourier transforms can be extended to an entire function of exponential growth (band-limited function). Consequently, there is a unique solution to the incomplete range of views reconstruction problem. On the other hand, the problem is an ill-posed problem. For example, it has been indicated [4] that the spectrum of the singular values of the Radon transform for limited range of views is split up into_two parts. One part consists of singular values near one, and the other part consists of singular values near zero. The recovery of small singular values is necessary for a process to reconstruct objects with good quality. This, however, exacerbates instability in the process, i.e., a small error in the projection data might lead to an undesirable large difference in reconstructed images. Making use of a priori information on image and projection data is being examined to reduce this instability.