A fast reconstruction algorithm for electron microscope tomography.

We have implemented a Fast Fourier Summation algorithm for tomographic reconstruction of three-dimensional biological data sets obtained via transmission electron microscopy. We designed the fast algorithm to reproduce results obtained by the direct summation algorithm (also known as filtered or R-weighted backprojection). For two-dimensional images, the new algorithm scales as O(N(theta)M log M)+O(MN log N) operations, where N(theta) is the number of projection angles and M x N is the size of the reconstructed image. Three-dimensional reconstructions are constructed from sequences of two-dimensional reconstructions. We demonstrate the algorithm on real data sets. For typical sizes of data sets, the new algorithm is 1.5-2.5 times faster than using direct summation in the space domain. The speed advantage is even greater as the size of the data sets grows. The new algorithm allows us to use higher order spline interpolation of the data without additional computational cost. The algorithm has been incorporated into a commonly used package for tomographic reconstruction.