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A major challenge in single particle reconstruction methods usingcryo-electron microscopy is to attain a resolution sufficient to interpret finedetails in three-dimensional (3D) macromolecular structures. Obtaining highresolution 3D reconstructions is difficult due to unknown orientations andpositions of the imaged particles, possible incomplete coverage of the viewingdirections, high level of noise in the projection images, and limiting effectsof the contrast transfer function of the electron microscope. In this paper, wefocus on the 3D reconstruction problem from projection images assuming anexisting estimate for their orientations and positions. We propose a fast andaccurate Fourier-based Iterative Reconstruction Method (FIRM) that exploits theToeplitz structure of the operator ${\bf A}^{*}{\bf A}$, where $\bf A$ is theforward projector and ${\bf A}^{*}$ is the back projector. The operator ${\bfA}^{*}{\bf A}$ is equivalent to a convolution with a kernel. The kernel ispre-computed using the non-uniform Fast Fourier Transform and is efficientlyapplied in each iteration step. The iterations by FIRM are thereforeconsiderably faster than those of traditional iterative algebraic approaches,while maintaining the same accuracy even when the viewing directions areunevenly distributed. The time complexity of FIRM is comparable to the directFourier inversion method. Moreover, FIRM combines images from different defocusgroups simultaneously and can handle a wide range of regularization terms. Weprovide experimental results on simulated data that demonstrate the speed andaccuracy of FIRM in comparison with current methods.

A Fourier-based Approach for Iterative 3D Reconstruction from Cryo-EM Images