Learning Tomography: Direct Fourier Reconstruction

After the back-projection [Link] and iterative [Link] methods, the Direct Fourier Reconstruction technique allows to compute a reconstruction directly in frequency space.
>Here is an introduction of the principle of this algorithm.

1- Central Slice theorem

When a series of back-projections are added during the reconstruction process (Fig. 1AB),  the corresponding 2D Fourier space (FFT of Fig. 1B) is filled by radial lines (Fig. 1C). Thus, there is a direct correspondence between the back-projections and these lines: this is the central slice theorem.

Fig.1: During the back-projection method corresponds in

2- Implementation

According to the central slice theorem, it is possible to calculate a 2D reconstruction  by inserting in a 2D Fourier space, 1D Fourier transforms of each row of the sinogram (Fig.1).

Fig.1: Steps required to compute a 2D reconstruction

 As shown in Fig. 1, the Direct Fourier Reconstruction is composed of the following steps:

1. Computing the Fourier transforms of the sinogram rows.
2. Filling a 2D Fourier space with central lines.
3. Computing the inverse Fourier transform to get the final 2D reconstruction. 

This can be summarized by the following pseudo-code:

  // Part I: FT
  for each row of sinogram
    FT_sinogram[row] = compute 1D_FFT(row)
  end for

  // Part II: Filling
  for y=0 to height_of_FFT2D
    for x=0 to width_of_FFT2D
      convert (x,y) to polar_coord (radius, theta)
      FFT2D[x,y] ← FT_sinogram[radius, theta]
    end for
  end for
  // Part III: Fourier -> Real space
  compute Inverse FFT2D


Now, it is time to see how to implement the first part of our Direct Fourier Reconstruction...

>> Next: Calculating the stack of 1D-Fourier Transforms

Other crazybiocomputing posts

Further readings are available in ...

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