Any Dimensional Reconstruction from Hyperplanar Projections
In this paper we examine the reconstruction of functions of any dimension from hyperplanar projections. This is a generalization of a problem that has generated much interest recently, especially in the field of medical imaging. Computed Axial Tomography (CAT) and Nuclear Magnetic Resonance (NMR) are two medical techniques that fall in this framework. CAT scans measure the hydrogen density along planes through the body. Here we will examine reconstruction methods that involve backprojecting the projection data and summing this over the entire region of interest. There are two methods for doing this. One method is to filter the projection data first, and then backproject this filtered data and sum over all projection directions. The other method is to backproject and sum the projection data first, and then filter. The two methods are mathematically equivalent, producing very similar equations. We will derive the reconstruction formulas for both methods for any number of dimensions. We will examine the cases of two and three dimensions, since these are the only ones encountered in practice. The equations are very different for these cases. In general, the equations are very different for even and odd dimensionality. We will discuss why this is so, and show that the equations for even and odd dimensionality are related by the Hilbert Transform.