Filtered Back-Projection

We will now describe how we can invert the Radon transform, so that we can—in principle—reconstruct the image of the object from the measured data. Details of the implementation of analytic inversion methods are widely available, such as the self-contained treatment including necessary mathematical background in [43] and the more detailed treatment in [129], so we will not reproduce them here. The first part of the chapter gives an intuitive introduction to the filtered back-projection (FBP) method and the underlying Fourier slice theorem. Our emphasis in the second part of the chapter is on what we can learn from analytical theory about what constitutes sufficient data for a reconstruction, how unstable that reconstruction process is with respect to errors in the data, and how we can recognize valid data. Finally, we go on to briefly discuss the sufficiency of data and analytical reconstruction methods for the 3D case.