Spline-based inverse Radon transform in two and three dimensions

While the exact inverse Radon transform is a continuous integral equation, the discrete nature of the data output by tomographic imaging systems generally demands that images be reconstructed using a discrete approximation to the transform. However, by fitting an analytic function to the projection data prior to reconstruction, one can avoid such approximations and preserve and exploit the continuous nature of the inverse transform. The authors present methods for the evaluation of the inverse Radon transform in two and three dimensions in which cubic spline functions are fit to the projection data, allowing the integrals that represent the filtration of the sinogram to be carried out in closed form and also eliminating the need for interpolation upon backprojection. Moreover, in the presence of noise, the algorithm can be used to reconstruct directly from the coefficients of smoothing splines, which are the minimizers of a popular curve-fitting measure. The authors find that the 2D and 3D direct-spline algorithms have superior resolution to their 2D and 3D FBP counterparts, albeit with higher noise levels, and that they have slightly lower ideal-observer signal-to-noise ratios for the detection of a 1-cm, spherical lesion with a 6:1 lesion-background concentration ratio