“Direct and Statistical Approaches to Tomographic Image Reconstruction”

by Xiao-Hong Yan

August 1992

The problem addressed in this research is three dimensional (3-D) tomographic image reconstruction using direct and statistical approaches. In the former approach, our focus is the reconstruction of a 3-D object from its cone beam projections with potential applications in cone beam 3-D X-ray computed tomography (CT) and single photon emission computed tomography (SPECT). In the latter approach, we restrict our attention to the applications of positron emission tomographic (PET) image reconstruction, although many of the techniques described here can also be applied to SPECT. A general formula for image reconstruction from cone beam data is derived by modifying a result due to Kirillov. Applying this formula to various cone beam geometries results in a class of filtered backprojection algorithms. This formula is known to lead to exact reconstructions in cases in which the cone vertices form certain unbounded curves. An example of such a curve is an infinite straight line. In the case where the curve is a circle, this formula leads to the well known Feldkamp algorithm, for which the reconstructions are only approximations to the true image. We apply this general formula to the cases where the curve is an ellipse and a spiral and new algorithms are derived. For the approximate inverse, we derive the spatially varying point spread function (PSF), which should be useful in the design of cone beam imaging systems. The properties of these algorithms are investigated through studies of the system point spread function and reconstructions of computer generated phantom data.

The statistical approach to PET image reconstruction described here follows a Bayesian formulation. The PET image is defined on a two dimensional lattice as a collection of random variables representing the mean positron emission rate from the elemental volume (pixel) surrounding each lattice site. Between each pair of pixels in the image, a binary line process is introduced to model the presence or absence of a discontinuity in the image. The image and its associated line process are jointly modeled as a Markov random field (MRF) with a joint Gibbs distribution chosen to favor the formation of images consisting of locally smooth, connected regions. We describe a maximum a posteriori (MAP) estimation algorithm based on the generalized EM algorithm for reconstructing the PET image using the above model. The incorporation of a line process in the image model also provides a useful mechanism for the introduction of strong a priori information obtained from high resolution registered anatomical magnetic resonance (MR) images. Through a boundary finding process, we can detect anatomical boundaries from these MR images and introduce them as fixed prior line sites in the PET estimation algorithm. The potential performance of the method is tested using a 3-D brain phantom and patient data.