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).

- SpeX Iterated PSFs: http://www.skycoyote.com/skycoyote/swri/psfiter.html.
- Iterated PSF Proof of Concept: http://www.skycoyote.com/skycoyote/swri/proof.html.
- Hilo talk 10/05: http://www.skycoyote.com/skycoyote/swri/hilo0105/.
- Hilo talk partial summary: http://www.skycoyote.com/skycoyote/swri/hilo0105/summary.txt.
- Sky Coyote: skycoyote@comcast.net.

©Copyright Sky Coyote and Eliot Young 2005.