PSF extraction from movies, and iterated deconvolution
Sky Coyote and Eliot Young, Jan 2005

Short exposure movies of a point source produce interferometric "speckles". Each frame of data is the convolution of a point source first with the instantaneous atmosphere PSF, and then with the constant telescope PSF. ```   I(t) = (O * A(t)) * P
= A(t) * P

where I(t) = data image frame of movie
O = object, a point source
A(t) = atmosphere PSF of each frame
P = constant PSF of telescope
* = convolution
```

The sum of all frames is the long-exposure "blur", while the sum of all or some shifted frames produces an approximation of the telescope PSF. A better PSF approximation can be computed from the average Fourier magnitude and phase of each frame of data. The average real and imaginary components produce the "blur"; the average magnitude and sum of phases produces "garbage"; while the average magnitude and zero phase produce a PSF. The Fourier PSF is better than the shift-and-add PSF, especially in the "skirt". The average squared magnitude PSF  is slightly better than the average magnitude PSF, but not by much. Movies made with different filters produce different PSFs for each wavelength . A simulated telescope aperture has a PSF very similar to those computed from movies. An Airy pattern is the PSF for an unobstructed pupil. An annular mirror produces the most exaggerated PSF. The double-correlation of the Fourier transform  produces an "unrolled" phase which reconstructs the data. The average "unrolled" phase does not produce a good PSF, although the average phase does. Decoupling the correlation calculation from the phase reconstruction produces a good PSF which is slightly shifted due to the phase gradient. An iterated multiplicative procedure turns a flat image into a deconvolved image.

The iterated procedure is:
```   A(t) = I(t) *' P
P(t) = I(t) *' A(t)
P' = f({P(t)})

where P(t) = 'partial' telescope PSF of each frame
P' = new PSF
f = some function of all P(t)
*' = deconvolution
``` The atmosphere PSF can then be used to iterate a new "partial" telescope PSF which has detail in the "skirt". The 3 PSFS (average magnitude, average correlation, and iterated) can be compared on several frames of data. In all cases, the average iterated PSF reduces the sum-of-squares error with fewer iterations.

```Error1 = 0.000215016   Error1 = 0.000162754
Error2 = 0.000204401   Error2 = 0.000152583
Error3 = 0.000126753   Error3 = 0.000104833

Error1 = 0.000271488   Error1 = 0.000192831
Error2 = 0.000244765   Error2 = 0.000191867
Error3 = 0.000160357   Error3 = 0.000123493

Error1 = 0.000255344   Error1 = 0.00018255
Error2 = 0.000243201   Error2 = 0.000173685
Error3 = 0.000165362   Error3 = 0.000119835

Error1 = 0.000152502   Error1 = 0.000325866
Error2 = 0.000141432   Error2 = 0.000328347
Error3 = 9.84253e-05   Error3 = 0.000198643

Error1 = 0.000185886   Error1 = 0.000190741
Error2 = 0.000183091   Error2 = 0.000180752
Error3 = 0.000116223   Error3 = 0.00011305
```

An additive (and subtractive) iteration can be used to create images and PSFs with negative values. A simulated band-limited compact object can be used to try to find an "inverse" to a PSF which might turn (slow) deconvolution into (fast) convolution. What's next?
• Finding inverse PSFs.
• Using the triple-correlation .
• Using IDAC and blind-deconvolution.
• Processing Venus cloud data.
• Automated PSF movie acquisition and extraction for IRTF.

References:
1. "Attainment of Diffraction Limited Resolution in Large Telescopes by Fourier Analysis Speckle Patterns in Star Images", Labeyrie, Astron. & Astrophys. 6, 85-87 (1970).
2. SpeX movie log: http://www.skycoyote.com/skycoyote/swri/hilo0105/log.txt.
3. "Recovery of Images From Atmospherically Degraded Short-Exposure Photographs", Knox and Thompson, The Astrophysical Journal, 193: L45-L48 (1974).
4. "Speckle masking in astronomy: triple correlation theory and applications", Lohmann et. al., Applied Optics, vol. 22, no. 24 (1983).
5. "An iterative technique for the rectification of observed distributions", Lucy, The Astronomical Journal, vol. 79, no. 6 (1974).

©Copyright Sky Coyote and Eliot Young 2005.