Exact formulation of stackgram filters in sinogram domain

An idea to filter positron emission tomography (PET) data in an intermediate state between sinogram and image domains, called stackgram domain, has been shown to have advantages over common radial sinogram domain filtering. This method has been approximated with sinogram filter design that utilizes sinusoidal trajectories. However, the performance of the approximation tested with different filter types is not as stable as the performance of these same filter types in the stackgram domain. The reason for this is that the samples used in the stackgram domain are interpolated samples between the radial bins of the sinogram and not exact radial sinogram bins as in the approximation. Additionally, the output samples of the stackgram filter are usually also between two neighboring radial sinogram samples. The main motivation is to understand better how the stackgram filters work by showing what they exactly look like in the sinogram domain. This is accomplished by introducing extended sinogram, i.e. interpolating the radial direction enough for it to contain all the values equivalent to the stackgram domain values. This way the equivalent filters can be shown for different radial locations of the sinogram. Finally, the decimation is performed in order to obtain the same output as the one obtained when the filtered stackgram is transformed back into the sinogram domain.