Learning Tomography: Angular coverage

After studying the impact of the number of projections [Link], it is time to explore the ins and outs of the angular coverage and its consequences on the quality of the reconstructed image.



2- Which angular coverage ?

2-1- Why not from 0 to 360°?

In the previous posts, sinograms were calculated from 0 to 180°, why? ... simply because projections calculated at an angle of α and of (α + 180°) are identical and thus are redundant for the reconstruction.
Explanation: If we choose two projections respectively calculated at 30° and 210° (=30°+180°) from the Lena image (Fig.1).

Fig.1: Lena respectively rotated by 30° and 210° (=30+180°)

Calculating projections from Fig.1A and Fig.1B yields two rigorously identical images (see difference in last row of Fig. 2) except the horizontal flip ... and it isn't a flip but rather a 180° Y-rotation.

Fig.2: Projections (two first rows) calculated from Fig.1. Third row: Flipped 210° projection. Fourth row: Image difference between the 30° projection and the horizontally flipped 210°projection. The two images are rigorously identical (all the pixel of the difference have a zero value). For sake of convenience, the projections were vertically stretched.

Now, when we compute the back-projection. The first projection is back-projected and rotated by 30° leading to image of Fig.3A. The second back-projection of 210° yields ...  the same exact pattern (Fig. 3B).

Fig.3: Comparison between 30° and 210° back-projections.

Note: Keep in mind that in Fourier Space, a (back-) projection corresponds to a 1D-central slice. Thus, to cover the whole 2D-Fourier space, we only need an angular range of 0° to 180° as shown in the following scheme.

Fig. 3bis: Scheme of the 2D-Fourier space filled by the 0° to 180° projections. There is no need to more than 180° to fully cover the space.

This confirms that these extra data (180° to 360° projections) are redundant and are just slowing down the reconstruction process without any improvement. In conclusion, we only need a maximum angular coverage of 180°.

Note: Depending of the experimental device used for the data collection, it's sometimes interesting to collect over several turns as explained in this post [Link].

2-2- Is it possible to reduce the angular coverage?

W O R K    I N   P R O G R E S S

Example calculated from 30° to 150°

Fig.4: Reconstruction calculated from 1D-projections between 30° and 150°. The contours of Lena's face have clearly disappear. Right: FFT of Lena reconstruction. The missing data appears as a bow tie in the power spectrum (FFT image).

Note: In electron microscopy, due to mechanical constraints, the data collection is usually done between -55° to +55° yielding in 3D the so-called missing wedge.

Other crazybiocomputing posts

Further readings are available in ...

• Learning Tomography Series  [Link]
• Image Processing TOC [Link]