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 [1] is slightly better than
the average magnitude PSF, but not by much.
Movies made with different filters produce different PSFs
for each wavelength [2].
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 [3] 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 [4].
- Using IDAC and blind-deconvolution.
- Processing Venus cloud data.
- Automated PSF movie acquisition and extraction for IRTF.
References:
- "Attainment of Diffraction Limited Resolution in Large
Telescopes by Fourier Analysis Speckle Patterns in Star
Images", Labeyrie, Astron. & Astrophys. 6, 85-87 (1970).
- SpeX movie log:
http://www.skycoyote.com/skycoyote/swri/hilo0105/log.txt.
- "Recovery of Images From Atmospherically Degraded
Short-Exposure Photographs", Knox and Thompson, The
Astrophysical Journal, 193: L45-L48 (1974).
- "Speckle masking in astronomy: triple correlation
theory and applications", Lohmann et. al., Applied Optics,
vol. 22, no. 24 (1983).
- "An iterative technique for the rectification of
observed distributions", Lucy, The Astronomical Journal,
vol. 79, no. 6 (1974).