The Point Spread Function (PSF)

A 3-panel image show the same spiral galaxy core with the left image not deconvolved, the middle deconvolved with more detail visible, and the right deconvolved with ringing artifacts visible.
Left: original, middle: deconvolved image with appropriate settings, right: deconvolved image with ringing artifacts due to an inappropriate (too high) choice for the Radius parameter.

The Deconvolution algorithm's task, is to reverse the blur caused by the atmosphere and optics. Stars, for example, are so far away that they should really render as single-pixel point lights. However in most images, stellar profiles of non-overexposing stars show the point light "smeared" out, yielding a core surrounded by light tapering off. Further diffraction may be caused by spider vanes and/or other obstructions in the Optical Tube Array, for example yielding diffraction spikes.

The point light's energy is scattered around its actual location, yielding the blur. The way a point light is blurred like this, is also called a Point Spread Function (PSF). Deconvolution is all about modelling this PSF, then finding and applying its reverse to the best of our abilities.

Atmospheric or lens-related blur is more easily modelled, as its behaviour and effects on long exposure photography has been well studied over the decades. 5 subtly different models are available for selection via the 'Primary Point Spread Function' parameter;

  • 'Gaussian' uses a Gaussian distribution to model atmospheric blurring.
  • 'Circle of Confusion' models the way light rays from a lens are unable to come to a perfect focus when imaging a point source (aka the 'Circle of Confusion'). This distribution is suitable for images taken outside of Earth's atmosphere or images where Earth's atmosphere did otherwise not distort the image. It may also be used succesfully on marginally oversampled datasets.
  • 'Moffat Beta=4.765 (Trujillo)' uses a Moffat distribution with a Beta factor of 4.765. Trujillo et al (2001) propose in their paper that this value (and its resulting PSF) is the best fit for prevailing Atmospheric turbulence theory.
  • 'Moffat Beta=3.0 (Saglia, FALT)' uses Moffat distribution with a Beta factor of 3.0, which is a rough average of the values tested by Saglia et al (1993). The value of ~3.0 also corresponds with the findings Bendinelli et al (1988) and was implemented as the default in the FALT software at ESO, as a result of studying the Mayall II cluster.
  • 'Moffat Beta=2.5 (IRAF)' uses a Moffat distribution with a Beta factor of 2.5, as implemented in the IRAF software suite by the United States National Optical Astronomy Observatory.

A three-panel image showing an excerpt of a Hubble Space Telescope dataset.
Even this noisy and heavily drizzled Hubble dataset can be corrected by the StarTools' Decon module at its native, drizzled resolution. Left: not deconvolved, middle: deconvolved, right: deconvolved and noise grain equalized.

The size (aka 'kernel size') of the chosen 'Primary Point Spread Function' is controlled by the 'Primary Radius' parameter. A good rule of thumb is to increase this value until ringing artefacts become noticeable, and then back off a little until it disappears again. An 'Enhanced Deringing' parameter is available than can further ameliorate ringing artefacts.

Deconvolution module interface with a guide star selected.
When choosing a star as PSF guide star and wishing to use a Dynamic Star Sample setting, make sure that the star is not masked out; masked out pixels are displayed in red.

Converging on an optimal solution is an iterative process in the Deconvolution module. In general, more iterations, controlled by the 'Iterations' parameter, will yield a better result but will take longer to compute. More iterations tend to yield diminishing returns. Different datasets may benefit from more or fewer iterations. You may wish to experiment on a smaller preview section to evaluate improvements before computing deconvolution of the entire image. Deconvolution in StarTools always converges on an optimal solution and does not destabilise as seen in other software, except when 'Error Diffusion' is set to a non-zero value.

A 'Secondary Point Spread Function' may be specified by clicking on a guide star. The Deconvolution module will then use the star as a guide to construct a suitable total PSF. Good star samples are stars that do not overexpose, but are not too dim, are closer to the center of the image and have a flat background. When a 'Secondary Point Spread Function' is provided, the total/final PSF used is a combination of that PSF modulated by the 'Primary Point Spread Function'. This allows you to create a final PSF that is tightly controlled by the ideal atmospheric profile (and its radius) as specified by the 'Primary Point Spread Function', while exhibiting a custom measure of deformity as seen in the selected star's PSF.

For example, to make Decon use the 'Secondary Point Spread Function' only, set the 'Primary Point Spread Function' to 'Circle of Confusion (No Atmosphere)' and specify a very large 'Primary PSF Radius'. As expected, smaller radii will start cutting off the 'Secondary Point Spread Function' in a circular fashion. For a gentler tapering off of the 'Secondary Point Spread Function', you can use, for example,a 'Gaussian (Fast)' profile for the 'Primary Point Spread Function'.

Uniquely, any star chosen as a 'Secondary Point Spread Function' can be made to iteratively deconvolve along with the image (by choosing one of the 'Dynamic' Star Sample settings). This effectively means that the deconvolution process deconvolves with an ever-changing total PSF. This mode can yield very good, even superior results, depending on the fidelity of the initial star sample. If this mode is selected and you are using a preview, make sure that the chosen star is included in the preview and falls well into the preview area - a message will be shown if this is not the case.

When choosing a star as PSF guide star and wishing to use a Dynamic Star Sample feature, make sure that the star is not masked out; masked out pixels are displayed in red if a guide star is set. Masked out stars (and thus their derivative PSF) will not iteratively deconvolve along with the image and are hence not suitable for this mode.