Astronomy 302

Lecture 9

Image formation, Deconvolution & PSF fitting Photometry




1.0 How An Image Is formed

Images of an Object appear in the detector as the convolution of the true object with the system PSF.
The PSF is the Point Spread Function, and is shape of a star (or point source) as imaged by the telescope.


Here we see how the true object is transformed (by convolution by the PSF)
(this is from the HygensDeconvolution site )


Mathematically it is simply that an

image (g) =  convolution of the object (f)  with the PSF (h)    (1)

g = f * h   (where * is the convolution operator)

we can rewrite equ (1) as

in other words an image is simply the intensity of the Object weighted by the PSF

But one must integrate over all x, y and z. This is a lot of calculation space.

The convolution theorem simplifies this. If the Fourier Transform of g is G, and f -> F and h -> H then:

G = F times H

just a simple multiplication of the transforms of f and h

So any image can be calculated by:
  1. Compute the Fourier transforms F and H of f and h
  2. Multiply F times H to obtain G
  3. Transform G back to g, the convolved image.

See here for some examples:


Here are the intensity images (top; black & white) then
on the bottom the Fourier Transforms in color


 
Now we can estimate the image by multiplying the transforms of the Object and the PSF
and then by taking the inverse Fourier transform we have the actual image of the object
produced by the PSF.



3.0 Image Deconvolution

Now it should obvious that since we observe g and we can estimate h (like from the image of a star)
it should be possible to deconvolve the image g to estimate the true image f !

F = G/H         (2)

this is the deconvolution equation. It is a very powerful concept.

But H is bandlimited (the telescope's entrance pupil (its primary mirror) is limited to D in size)
hence there is only an approximate knowledge of F in G.

For example
the diffraction limit of a 6.5 meter at 2.2 microns is lambda/D ~ 69 mas (milliarcseconds)
so if there is a binary that is 10 mas in separation there will be no "support" from H (the PSF) at the frequencies where the object is clearly resolved into a binary.

In other words deconvolution can only restore an image to the lambda/D resolution limit of the telescope.
Also photon noise limits our knowledge of the PSF, as well the PSF can change with time... Often it is
really hard to get a good PSF image. All this limits the power of deconvolution.

However, in cases of very high S/N PSFs and very stable PSFs it is possible to restore the object past the
lambda/D limit. This is called super resolution deconvolution.

4.0 Super Resolution at the MMT


VERY HIGH STREHL MID-IR AO IMAGING with the MMT Adaptive Optics System

With a low emissivity adaptive secondary (~8% emissive) adaptive optics can finally reach Mid-IR wavelengths.

Even with 53 corrected Ao modes Strehl ratios of 98% are predicted at 10 microns. The Strehl ratios reached are 98+/-2% with MMT AO in 1" seeing at 9.8 microns. Similarly high strehls were reached at 11.7 and 18 microns.


Above we see the first Mid-IR AO images made of Post-AGB stars (Close et al. 2003b). AC Her is a post-AGB star that is transiting from the AGB to the planetary nebula phase (an RV Tauri star). Due to the very high Strehls achieved the PSF standards are an excellent match independent of seeing, airmass, or time. This creates a whole new world for AO science when PSF calibration can be done on different stars at different times. Note how similar the post-AGB star AC Her appears to the other 2 PSF stars. Also not how the morphology of AC Her is very different from that of the Keck 18 um image (upper right - false color).


Graphical proof of how similar AC Her is to the other PSF stars (Close et al. 2003b). Note how incompatible it is with the previous keck image (Jura et al. 2000).



Above we see how similar the PSF really are. If we simply subtract a scaled version of the alpha Her PSF from the AC Her 11.7 um image there is hardly any residual remaining. This is a remarkable degree of PSF subtraction considering that these 2 stars were observed 2 hours apart and at different airmasses.


DIRECT DETECTION OF A DISK AROUND RV BOO
(Biller & Close et al. 2003)


We further exploited the excellent PSF stability to detect the thermal disk around the AGB star RV Boo (which is known to have keplarian CO disk).


We noticed that RV Boo was slightly more elliptical than the other PSFs at 9.8 microns.


RV Boo was uniquely wide and elliptical.



RV Boo's "disk" rotated on the sky with the parallactic angle



After Lucy deconvolution a small (R~50 AU) disk was revealed around RV Boo (At a PA=120 degrees, similar to the CO disk). Such a disk size at 10 microns is expected from the IRAS fluxes using a simple dust emission model (Biller et al. 2003).

5.0 Some notes about getting a good PSF

It critical to estimate a good PSF for photometry in crowded regions (aperture phot will not work).

So we use PSF fitting photometry packages like DAOPHOT's ALLSTAR to fit individual stars that may have overlapping PSFs

see here for more details.

The PSF is the Point Spread Function, and is shape of a star (or point source) as imaged by the telescope.
Note that the PSF changes due to seeing from one exposure to the next. It will also change if the filter is changed.
(usually redder filters have smaller FWHM PSFs since ro varies as lambda6/5 since FWHM = lambda/ro; hence FWHM varies lambda-1/5)
Due to field dependent optical aberrations, like coma, the PSF of a star will change as one moves off the optical axis of the detector.

For example the images of the globular cluster in Project 2 are more elongated as one approaches the edges of the 4k CCD. This is mainly coma at the Cassegrain focus from the 61inch telescope.

Hence a PSF standard that is good in the center of the field will be a poor match at the edges.

One solution is to cut out a smaller section of the image and use that for PSF fitting photometry (like we do in
project 2). Or you can pick an array of PSFs across a large field and allow the theoretical PSF to change as function of X and Y of the star. You can do this in DAOPHOT's PSF task by setting the PSF order  from 0 to 1.