Assume we have a ‘point’ source, or a source that is far smaller than the maximum resolution (a pixel). When we take an image of it, it will ‘spread’ over an area. To quantify that spread, we can define a ‘function’. This is how the “point spread function” or the PSF of an image is defined.

This ‘spread’ can have various causes, for example, in ground-based astronomy, due to the atmosphere. In practice we can never surpass the ‘spread’ due to the diffraction of the telescope aperture (even in Space!). Various other effects can also be quantified through a PSF. For example, the simple fact that we are sampling in a discrete space, namely the pixels, also produces a very small ‘spread’ in the image.

Convolution is the mathematical process by which we can apply a ‘spread’ to an image, or in other words blur the image, see Convolution process. The sum of pixels of an image should remain unchanged after convolution. Therefore, it is important that the sum of all the pixels of the PSF be unity. The PSF image also has to have an odd number of pixels on its sides so one pixel can be defined as the center.

In MakeProfiles, the PSF can be set by the two methods explained below:

- Parametric functions ¶
A known mathematical function is used to make the PSF. In this case, only the parameters to define the functions are necessary and MakeProfiles will make a PSF based on the given parameters for each function. In both cases, the center of the profile has to be exactly in the middle of the central pixel of the PSF (which is automatically done by MakeProfiles). When talking about the PSF, usually, the full width at half maximum or FWHM is used as a scale of the width of the PSF.

- Gaussian ¶
In the older papers, and to a lesser extent even today, some researchers use the 2D Gaussian function to approximate the PSF of ground based images. In its most general form, a Gaussian function can be written as:

$$f(r)=a \exp \left( -(x-\mu)^2 \over 2\sigma^2 \right)+d$$

Since the center of the profile is pre-defined, \(\mu\) and \(d\) are constrained. \(a\) can also be found because the function has to be normalized. So the only important parameter for MakeProfiles is the \(\sigma\). In the Gaussian function we have this relation between the FWHM and \(\sigma\):

$$\rm{FWHM}_g=2\sqrt{2\ln{2}}\sigma \approx 2.35482\sigma$$

- Moffat ¶
The Gaussian profile is much sharper than the images taken from stars on photographic plates or CCDs. Therefore in 1969, Moffat proposed this functional form for the image of stars:

$$f(r)=a \left[ 1+\left( r\over \alpha \right)^2 \right]^{-\beta}$$

Again, \(a\) is constrained by the normalization, therefore two parameters define the shape of the Moffat function: \(\alpha\) and \(\beta\). The radial parameter is \(\alpha\) which is related to the FWHM by

$$\rm{FWHM}_m=2\alpha\sqrt{2^{1/\beta}-1}$$

Comparing with the PSF predicted from atmospheric turbulence theory with a Moffat function, Trujillo et al.

^{228}claim that \(\beta\) should be 4.765. They also show how the Moffat PSF contains the Gaussian PSF as a limiting case when \(\beta\to\infty\).

- An input FITS image
An input image file can also be specified to be used as a PSF. If the sum of its pixels are not equal to 1, the pixels will be multiplied by a fraction so the sum does become 1.

Gnuastro has tools to extract the non-parametric (extended) PSF of any image as a FITS file (assuming there are a sufficient number of stars in it), see Building the extended PSF. This method is not perfect (will have noise if you do not have many stars), but it is the actual PSF of the data that is not forced into any parametric form.

While the Gaussian is only dependent on the FWHM, the Moffat function is also dependent on \(\beta\). Comparing these two functions with a fixed FWHM gives the following results:

- Within the FWHM, the functions do not have significant differences.
- For a fixed FWHM, as \(\beta\) increases, the Moffat function becomes sharper.
- The Gaussian function is much sharper than the Moffat functions, even when \(\beta\) is large.

Trujillo, I., J. A. L. Aguerri, J. Cepa, and C. M. Gutierrez (2001). “The effects of seeing on Sérsic profiles - II. The Moffat PSF”. In: MNRAS 328, pp. 977—985.

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