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An Image Quality Figure of Merit
Based on Moffat Functions?
Grant M. Hill

January 1, 2000


Moffat (1969, A& A, 3,455) first derived the function which bears his name by convolving seeing profiles (assumed to be Gaussian) with diffraction profiles and the scattering function appropriate for photographic emulsions. Despite the relation to photographic imaging, the Moffat function has been found to work very well for fitting stellar images obtained with CCD's. The function is of the form

equation (1)

where Io is the intensity at image center,is the half width at half maximum of the image in the absence of atmospheric scattering and β is the atmospheric scattering coefficient.

We have been using a so-called X-factor as our figure of merit for evaluating our efforts at image quality improvement. The X-factor is defined as:

equation (2)

where Wss is EE(50) of the stack on the sky, Wssm is the EE(50) of a single mirror on the sky and Wcs is the EE(50) at CCAS.

Important physics may be missing or wrong by defining X thus and using it as a figure of merit. An indication of this may be that it is possible for X to be imaginary as defined. Implicit in equation 2 is the assumption that all profiles are Gaussian. We know this is not true.

Although EE(50) is a very useful number, it contains little detailed information about image profile shape. In this respect it might be thought of like equivalent width. The strengths of spectral lines are useful numbers but contain no information about line shape. As an extreme example, consider an image profile in which 50 percent of the energy is confined in a narrow spike 1.0 arcsecond in diameter. The rest of the energy is spread out in extended wings 10 arcseconds across. This profile has a good EE(50) but still falls far short of being satisfactory.

Using Moffat Functions?

As we progress further in our efforts to improve the images delivered by the HET, it seems clear we need to find a way to retain information about profile shape. This could be accomplished via fitting analytic functions or splines to observed profiles. One such analytic function is the Moffat function.

It may be appropriate to derive a new figure of merit which incorporates Moffat functions. This might be accomplished by redefining X by convolutions of Moffat functions instead of using EE(50)'s and equation 2. This investigation is still at a preliminary stage but picking 5 CCAS images, 5 sky stack images and 5 sky single mirror images at random implied that a Moffat function fits either as well as, or better than a Gaussian. In all cases, the quality of the fit as determined by χ2 ranged from the same to a factor of several better for the Moffat function.

The figures below show examples of the improvement gained in fitting by using Moffat functions.

I have written some code which convolves a Moffat profile fit to a single mirror sky image with a Moffat profile fit to a CCAS image and thus predicts the expected stack image profile on the sky. The underlying assumption is that simple convolution is appropriate for determining the contributions of individual optical elements to the final image profile. If this is true then it represents a much more physical figure of merit. If it is not true, then some other formulation of a figure of merit is needed but regardless we need to retain information about profile shapes. Since IRAF already contains tasks which fit Moffat functions, they represent attractive candidates.

Figure 1: CCAS image fitted with a Moffat function (solid line) and a Gaussian (dashed line).
Figure 1

Figure 2: Sky stack image fitted with a Moffat function (solid line) and a Gaussian (dashed line).
Figure 2

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