This project has focused on two different reconstruction techniques for SPECT data. The first method, known as the filtered back projection method, is based on the Radon transform and therefore requires that the data are collected as parallel beam projections. It is a fast algorithm and the most widely used in today's commercial systems. Noise problems can be successfully dealt with through Wiener filtering, which is the optimal trade-off between the ramp filtering required to reconstruct the image and the desire to avoid noise-related artifacts. Attenuation is difficult to handle in an exact way for the filtered back projection method. In this project, we used a post-processing operation known as Chang's method to correct for attenuation. There are, however, pre-processing techniques as well for attenuation compensation in this algorithm.
The second method used in this project was an algebraic technique which is more general and can handle a large class of geometries, including fan beam projections. This method formulates the reconstruction problem in terms of a set of linear equations, and is closely related to image restoration techniques. In fact, modifying only the constraints under which the reconstruction equations are solved, one can implement, e.g., an expectation maximization algorithm. Attenuation correction in the algebraic method can easily be included into the actual reconstruction process. Algebraic methods, however, are very computationally intensive, and in particular requires large amounts of memory to store the coefficient matrix that arises. At present, algebraic methods are primarily a research topic, but with the development of faster computers with more memory we anticipate that algorithms will begin to be more widely used.
Finally, it deserves to be mentioned that there exists a third way of image reconstruction from projected data. These methods, called direct Fourier methods, involve constructing the 2-D Fourier transform of the image from the 1-D Fourier transforms of the projections according to the Fourier slice theorem. This 2-D transform can then be inverted using the FFT algorithm, potentially resulting in an extremely efficient reconstruction method. The problem in this method is interpolation in the Fourier domain since the 2-D transform is available on a polar grid. The question of how to do this with sufficient accuracy is still an active research topic.