Filter design for filtered back-projection guided by the interpolation model

We consider using spline interpolation to improve the standard filtered back-projection (FBP) tomographic reconstruction algorithm. In particular, we propose to link the design of the filtering operator with the interpolation model that is applied to the sinogram. The key idea is to combine the ramp filtering and the spline fitting process into a single filtering operation. We consider three different approaches. In the first, we simply adapt the standard FBP for spline interpolation. In the second approach, we replace the interpolation by an oblique projection onto the same spline space; this increases the peak signal-noise ratio by up to 2.5 dB. In the third approach, we perform an explicit discretization by observing that the ramp filter is equivalent to a fractional derivative operator that can be evaluated analytically for splines. This allows for an exact implementation of the ramp filter and improves the image quality by an additional 0.2 dB. This comparison is unique as the first method has been published only for degree n=0, whereas the two other methods are novel. We stress that the modification of the filter improve the reconstruction quality especially at low (faster) interpolation degrees (n = 1); the difference between the methods becomes marginal for cubic or higher degrees (n ≥ 3).