Null-space function estimation for the interior problem

Abstract

In single-photon emission computed tomography (SPECT), projection data can be truncated when the camera's field of view is smaller than the object to be imaged. Using truncated projections to reconstruct a region of interest (ROI) is a reality we must face if small detectors are used. The truncated data result in an underdetermined system of imaging equations, which may lead to non-unique solutions. Data sampling and photon attenuation may also affect the solution uniqueness and stability. The uniqueness of the solutions in the ROI can be investigated by studying the null-space functions in the ROI. This paper uses an iterative algorithm to estimate the null-space image, to determine the sampling conditions under which a stable ROI reconstruction is possible with truncated data and to investigate whether attenuation can influence the ROI reconstruction bias. This iterative algorithm is validated by the singular value decomposition method. We show that if the ROI is sufficiently sampled, the null-space image is close to zero inside the ROI, and any almost-zero offset is insignificant in SPECT, because the noise is a much more dominating degradation factor.