Deconvolution and Dawes limit

[Melty]

Hi,

Consider the following scenario.

I have a 60mm F6 telescope with a calculated Dawes limit of 1.9 "/pixel, and an astrocam with 2.3 micron pixels. This gives me an image scale of:

2.3x206.265/360 = 1.32 "/pixel.

Now this clearly outresolves the optics, but along with Decon module's intended purpose of improving diffraction limited images in bad seeing, can it also remedy the optics limitations of the above example?

Thanks

Melty

[admin]

Hi Melty,

Indeed, as long as you can 1. record sufficient samples, 2. your samples are sufficiently clean, 3. you can accurately model the point spread function 4. you have sufficient dynamic range, then deconvolution can absolutely still help.

The image on the Wikipedia page on the Dawes limit shows two overlapping Airy discs at the Dawes limit. Deconvolution, given that precise Airy Disc, will happily resolve these discs into their 2 constituent point lights.

See also the page about the related Rayleigh Criterion, which indeed states that;

A calculation using Airy discs as point spread function shows that at Dawes' limit there is a 5% dip between the two maxima, whereas at Rayleigh's criterion there is a 26.3% dip.[3] Modern image processing techniques including deconvolution of the point spread function allow resolution of binaries with even less angular separation.

Deconvolution is an extremely powerful tool, but...
The problem in practice is that the 4 aforementioned variables (I will have left some out) are all hard to achieve to perfection. Due to the imperfections involved and it being an inverse problem, deconvolution is an ill-posed problem with no one "true" solution. Once more from Wikipedia;

The usual method is to assume that the optical path through the instrument is optically perfect, convolved with a point spread function (PSF), that is, a mathematical function that describes the distortion in terms of the pathway a theoretical point source of light (or other waves) takes through the instrument.[3] Usually, such a point source contributes a small area of fuzziness to the final image. If this function can be determined, it is then a matter of computing its inverse or complementary function, and convolving the acquired image with that. The result is the original, undistorted image.

In practice, finding the true PSF is impossible, and usually an approximation of it is used, theoretically calculated[4] or based on some experimental estimation by using known probes. Real optics may also have different PSFs at different focal and spatial locations, and the PSF may be non-linear. The accuracy of the approximation of the PSF will dictate the final result. Different algorithms can be employed to give better results, at the price of being more computationally intensive. Since the original convolution discards data, some algorithms use additional data acquired at nearby focal points to make up some of the lost information. Regularization in iterative algorithms (as in expectation-maximization algorithms) can be applied to avoid unrealistic solutions.

When the PSF is unknown, it may be possible to deduce it by systematically trying different possible PSFs and assessing whether the image has improved. This procedure is called blind deconvolution.[3] Blind deconvolution is a well-established image restoration technique in astronomy, where the point nature of the objects photographed exposes the PSF thus making it more feasible. It is also used in fluorescence microscopy for image restoration, and in fluorescence spectral imaging for spectral separation of multiple unknown fluorophores. The most common iterative algorithm for the purpose is the Richardson–Lucy deconvolution algorithm; the Wiener deconvolution (and approximations) are the most common non-iterative algorithms.

All this then limits how much you can wring out of deconvolving images. Being close to the Dawes limit (or even exceeding it) will make it harder, but certainly not impossible to achieve better resolution.

Hope this helps (or at least helps you on your way finding an answer!)

[Melty]

Wow, that's a lot to digest. I'll take a deeper look in due course.

Excellent answers as always.

Thanks Ivo!