Does truncation always result in an under-determined problem? An SVD study

In a single photon emission computed tomography (SPECT) study, the projections may be truncated due to the relatively small detector size. The truncation problem is severe in a transmission study especially when a fan-beam collimator is used. It is commonly believed that truncated projections always result in an under-determined reconstruction problem and cause artifacts in the reconstruction. In fact, when the projections are truncated, an exact image can be reconstructed under certain circumstances such as that the image is band-limited and the sinogram is continuously measured. We use the singular value decomposition (SVD) to analyze the imaging system. It is demonstrated that the reconstruction with truncated projections is not always an under-determined problem. However, a fully determined problem can be ill-conditioned, which is characterized by its condition number. One should try to reduce the condition number as much as possible. In practice if we have a situation where truncation is not avoidable, we recommend the following: (1) increase the number of views while acquiring data, (2) decrease the sampling interval on the detector, (3) increase the pixel size of the image, sacrificing resolution for smaller condition number, (4) use constrains such as supports, non-negativity, smoothness, and so on, and (5) use a good projection model with accurate image basis functions to reduce aliasing artifacts