Direct spline-based inversion of the three-dimensional Radon transform with application to cardiac phantom data

While the exact inverse three-dimensional Radon transform is a continuous integral equation, the discrete nature of the data output by tomographic imaging systems demands that images be reconstructed using a discrete approximation to the transform. However, by fitting an analytic function to the planar-integral 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 3D Radon transform in which cubic spline functions are fit to the planar-integral data, allowing exact computation of the second derivative that appears in the inversion formula and also eliminating the need for interpolation upon backprojection. This approach is theoretically intriguing and has the advantage of directness when one wishes to smooth noisy data prior to reconstruction. In this case, a smoothing spline can be fit to the data and reconstruction can proceed directly from the spline coefficients. The authors find that the 3D direct-spline algorithm has superior resolution to 3D filtered backprojection, albeit with higher noise, and that it has a slightly lower ideal-observer signal-to-noise ratio for the detection of a 1-cm, spherical lesion with a 6:1 lesion-background concentration ratio