Star profile analysis

For the following study of the star profile and the influence of deconvolution and image stretching on this profile, a medium-sized star of a stacked image has been chosen. The telescope used is a Planewave 14'' CDK, the camera a (Baader-) modified Canon 500d. The image is a stack of 38 subframes, each with an exposure time of 180 s at ISO 1600 (from the image of NGC185 - 08.10.2015). The image scale of this setup is 0.378 px/arcsec. Bias, darks and flats have been applied to the individual frames before debayering, aligning and integrating them. All performed processing steps have been done in Pixinsight. As pixel rejection algorithm, winsorized sigma-clipping has been used (sigma-low: 3.0; sigma-high: 2.0). Finally, Dynamic Background Extraction has been applied to the image.

Figure 1 Left: Image of the star used for this analysis (STF applied to this image; integration of 38 images - each 180 s at ISO 1600). It's maximum intensity is K = 0.0422.
Right: 3D-rendering of the star's profile (Pixinsight).

The star has a maximum intensity of K = 0.0422. The pixel value has been noted for a line (with a length of about 40 pixels) passing through the maximum of the profile. Two functions have been chosen for the fitting of the profile:

Circular Gauss-function: $$ G(x) = A \ e^{- \frac{1}{2} \left( \frac{x-x_C}{\sigma} \right)^2 } + B $$ The FWHM of the Gauss-function is given by the following relation: $$ FWHM_G = 2 \sigma \sqrt{2 \ln 2} $$

Circular Moffat-function: $$ M(x) = \frac{A}{ \left[ 1 + \frac{(x-x_C)^2}{\sigma^2} \right]^{\beta}} +B $$ The FWHM of the Moffat-function is given by the following relation: $$ FWHM_M = 2 \sigma \sqrt{2^{\frac{1}{\beta}} -1 } $$

Figure 2 shows the value of each pixel of the line passing through the maximum of the star. A Gauss- and a Moffat-function are fitted to the data. Although the Gauss-function is a good approximation, it doesn't reproduce the wings of the star profile correctly. The Moffat-function is a better solution for the description of the profile with it's wings. In this particular case, β = 4.195. The FWHM for this star is ... px for the Gauss-fit and ... px for the Moffat-fit. Using the image-scale of 0.378 px/arcsec, the FWHM is ... arcsec and ... arcsec for both fitting-functions.

Figure 2: Intensity profile of a star, fitted by a Gauss-function (red curve) and a Moffat-function (green curve).

Figure 3 shows the star profile after Deconvolution has been applied to the image. The following settings have been used:

Using 12 iterations, the FWHM can be reduced from 2.70 arcsec to 1.58 arcsec. Usually, I use less iterations (6 ... 10 iterations on deep-sky images). Again, the whole star profile can best be fitted with the Moffat-function.

Figure 3: Star profile after applying Deconvolution (RRL, 12 iterations). Gauss- (red) and Moffat-functions are fitted to the data.

After Deconvolution, the image has been stretched once using Histogram Transformation, once using Masked Stretch. As can be seen in figure 4 and figure 5, HT gives the best result. As expected, the FWHM increases using these non-linear transformations. The FWHM is only 2.95 arcsec for the HT but 4.61 arcsec for the Masked Stretch transformation.

Figure 4: Star profile after applying Deconvolution and HT. FWHM = 2.95 arcsec.
Figure 5: Star profile after applying Deconvolution and Masked Stretch. FWHM = 4.61 arcsec.

The FWHM is clearly larger for the MaskedStretch compared to the HT as non-linear transformation. For the MaskedStretch transformation, fainter structures (amongst others noise or deconvolution artifacts in the background) have been enhanced giving the star a larger profile.

Using HT without any deconvolution increases the FWHM to 3.69 arcsec.

Figure 6: Star profile; only HT has been applied to the original image.

All results are summarised in figure 7 showing the different transformations and the respective FWHM of the analyised star.

Figure 7: Summary of the different processing steps with the corresponding FWHM, expressed in arcsec.

Some references