Tomographic image reconstruction by eigenvector decomposition: its limitations and areas of applicability

This paper analyzes in detail the process of tomographic image reconstruction by pseudo-inversion of the blurring matrix of a PET imaging system. Eigenvector and eigenvalue decomposition is used as a method to evaluate the physical reasons for the ill-conditioned nature of the problem. It is shown that finding an accurate pseudo-inverse for even a modest PET array of 8 x 8 pixels is a difficult task for a computer with 48-bit mantissa. The problem is caused by the strong ambiguity with which the detector system measures the activity at each pixel. For a problem in which imaging with a complete detector ring is not possible, and in which invariance of the point response function cannot be maintained, the pseudo-inverse method of reconstruction is, however, shown to be very useful. Advantage is taken of the fact that the activitiy to be measured is localized in a single plane, without over-or underlying activity. A planar camera configuration yields very well conditioned matrices that are separable for a large number of useful cases. It is even possible to define pixel sizes which are considerably smaller than the detector size and solve the problem without a substantial increase in the noise magnification factor. Recognizing that the above application is equivalent to a case of very well defined time-of-flight (TOF) measurement, the simple initial PET study is reevaluated by inclusion of TOF information.