The problem of reconstructing the 3D structure from 2D projection images with known orientatations is a classical inverse problem in computerized tomography. Our reconstruction algorithm is based on the Fourier slice theorem. The algorithm exploits the Toeplitz structure of the operator A*A, where A is the forward projector and A* is the back projector. The operator A*A is equivalent to a convolution with a kernel. The kernel is pre-computed using the non-uniform Fast Fourier Transform and is efficiently applied in each iteration step of the algorithm. The iterations are therefore considerably faster than those of traditional iterative algebraic approaches, while maintaining the same accuracy even when the viewing directions are unevenly distributed. The time complexity of the algorithm is comparable to the gridding-based direct Fourier inversion method. Moreover, it can combine images from different defocus groups simultaneously and can handle a wide range of regularization terms.
L. Wang, Y. Shkolnisky, A. Singer, A Fourier-Based Approach for Iterative 3D Reconstruction from Cryo-EM Images, Princeton University Technical Report (2012). arXiv:1307.5824 [math.NA]
C. Vonesch, L. Wang, Y. Shkolnisky, A. Singer, Fast wavelet-based single-particle reconstruction in cryo-EM, 8th IEEE International Symposium on Biomedical Imaging (ISBI’11), March 30 – April 2, 2011. PMID: 22536462 [PubMed] PMCID: PMC3334313.