Equally sloped tomography with oversampling reconstruction

We report the development of equally sloped tomography for the reconstruction of a 3D object from a number of 2D projections. In a combination of pseudopolar fast Fourier transform and the oversampling method with an iterative algorithm, equally sloped tomography makes superior 3D reconstruction to conventional tomography that has an intrinsic drawback due to the use of equally angled 2D projections. We believe this general approach will find applications in x-ray imaging, electron microscopy, coherent diffraction microscopy, and other tomographic imaging fields.

Figure 1

Geometric construction for 2D PPFFT and its inversion. The object is in Cartesian coordinates with $N\times N$ points where $N=4$ in this example. The grid points in Fourier space are on a set of concentric squares (i.e., dashed lines) with the number of squares equal to $N$. There are a total of $2N$ solid lines with each having $2N$ grid points, whereas each line represents a Fourier slice. The grid interval on each Fourier slice varies with $\theta$ and is characterized by $\alpha$ in Eq. (1).

Figure 2

Missing grid points due to a resolution circle. The grid points denoted by circles, which are within the resolution circle, represent the Fourier slices calculated from a limited number of “measured” projections. The grid points denoted by squares (outside the resolution circle) are inaccessible from the “measured” projections due to the resolution limitation. The grid points denoted by triangles represent a missing Fourier slice.

Figure 3

Tomographic reconstruction of a simulated 3D biological vesicle. (a), (b), and (c) are the central slices in $XY$, $XZ$, and $ZY$ plane of the 3D vesicle. (d), (e), and (f) are the central slices of the vesicle reconstructed from 27 projections by using oversampling reconstruction. (g), (h), and (i) are the central slices reconstructed by using weighted backprojection of conventional tomography.

Figure 4

A quantitative comparison of the reconstructions by using Fourier-ring correlation. Curves I and II correspond to the noise-free reconstructions by oversampling reconstruction and weighted back-projection. Curves III and IV correspond to the reconstructions with $\mathrm{SNR}=1$ by oversampling reconstruction and weighted backprojection.

Figure 5

(a) Geometric construction of specimen unboundedness where $Z$ is the rotation axis. The region of solid lines is imaged by all the projections and the region of dotted lines by some of the projections. (b) A Fourier-ring correlation comparison of the reconstructions for an unbounded specimen by using oversampling reconstruction (curve I) and weighted back-projection (curve II).