History

The adaptive-additive algorithm was originally created to reconstruct the spatial frequency phase of light intensity in the study of stellar interferometry. Since then, the AA algorithm has been adapted to work in the fields of Fourier Optics by Soifer and Dr. Hill, soft matter and optical tweezers by Dr. Grier, and sound synthesis by Röbel.

Algorithm

1. Define input amplitude and random phase
2. Forward Fourier Transform
3. Separate transformed amplitude and phase
4. Compare transformed amplitude/intensity to desired output amplitude/intensity
5. Check convergence conditions
6. Mix transformed amplitude with desired output amplitude and combine with transformed phase
7. Inverse Fourier Transform
8. Separate new amplitude and new phase
9. Combine new phase with original input amplitude
10. Loop back to Forward Fourier Transform

Example

For the problem of reconstructing the spatial frequency phase (k-space) for a desired intensity in the image plane (x-space). Assume the amplitude and the starting phase of the wave in k-space is ${\displaystyle A_{0}}$ and ${\displaystyle \phi _{n}^{k}}$ respectively. Fourier transform the wave in k-space to x space.

${\displaystyle A_{0}e^{i\phi _{n}^{k}}{\xrightarrow {FFT}}A_{n}^{f}e^{i\phi _{n}^{f}}}$

Then compare the transformed intensity ${\displaystyle I_{n}^{f}}$ with the desired intensity ${\displaystyle I_{0}^{f}}$, where

${\displaystyle I_{n}^{f}=\left(A_{n}^{f}\right)^{2},}$

${\displaystyle \varepsilon ={\sqrt {\left(I_{n}^{f}\right)^{2}-\left(I_{0}\right)^{2}}}.}$

Check ${\displaystyle \varepsilon }$ against the convergence requirements. If the requirements are not met then mix the transformed amplitude ${\displaystyle A_{n}^{f}}$ with desired amplitude ${\displaystyle A^{f}}$.

${\displaystyle {\bar {A}}_{n}^{f}=\left[aA^{f}+(1-a)A_{n}^{f}\right],}$

where a is mixing ratio and

${\displaystyle A^{f}={\sqrt {I_{0}}}}$.

Note that a is a percentage, defined on the interval 0 ≤ a ≤ 1.

Combine mixed amplitude with the x-space phase and inverse Fourier transform.

${\displaystyle {\bar {A}}^{f}e^{i\phi _{n}^{f}}{\xrightarrow {iFFT}}{\bar {A}}_{n}^{k}e^{i\phi _{n}^{k}}.}$

Separate ${\displaystyle {\bar {A}}_{n}^{k}}$ and ${\displaystyle \phi _{n}^{k}}$ and combine ${\displaystyle A_{0}}$ with ${\displaystyle \phi _{n}^{k}}$. Increase loop by one ${\displaystyle n\to n+1}$ and repeat.