  A major challenge in single particle reconstruction methods using
cryo-electron microscopy is to attain a resolution sufficient to interpret fine
details in three-dimensional (3D) macromolecular structures. Obtaining high
resolution 3D reconstructions is difficult due to unknown orientations and
positions of the imaged particles, possible incomplete coverage of the viewing
directions, high level of noise in the projection images, and limiting effects
of the contrast transfer function of the electron microscope. In this paper, we
focus on the 3D reconstruction problem from projection images assuming an
existing estimate for their orientations and positions. We propose a fast and
accurate Fourier-based Iterative Reconstruction Method (FIRM) that exploits the
Toeplitz structure of the operator ${\bf A}^{*}{\bf A}$, where $\bf A$ is the
forward projector and ${\bf A}^{*}$ is the back projector. The operator ${\bf
A}^{*}{\bf A}$ 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. The iterations by FIRM 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 FIRM is comparable to the direct
Fourier inversion method. Moreover, FIRM combines images from different defocus
groups simultaneously and can handle a wide range of regularization terms. We
provide experimental results on simulated data that demonstrate the speed and
accuracy of FIRM in comparison with current methods.
