Questions about convolving/deconvolving with a PSF

Question

A point spread function (PSF) has many different uses. Consider for instance the following quote:

To extract the maximum information out of an observation, even the smallest details of the PSF are important. Some examples include: deconvolving the PSF from an observed image to remove the blurring caused by diffraction and reveal fine structure; convolving a model image by the PSF to compare to an observed one;

My question about this is the following: If we know the PSF of a system, and use this to deconvolve the raw image (purpose 1 from the quote above), why would we convolve a model with the PSF to compare it to an image (purpose 2 from the quote)? That is, can't we just compare the original, unconvolved model, with the deconvolved image? In terms of maximizing the information that seems like the way to go.

Answer 1

Convolution is not a uniquely invertible process in the presence of random noise in your image. Deconvolving a noisy image can give misleading results, even if you have perfect knowledge of the PSF.

In general, when you are fitting models to data, it is far better to compare the models and data in the observational space of the data, where the uncertainties are reasonably well understood.

Answer 2

These two uses of PSFs are applicable to different situations. Sometimes you know your PSF well enough to deconvolve your image and get something reasonable out of it, but most of the time you have a lot of assumptions in this process and you're not going to get something perfect, so you might want to go the other way and convolve your model with the best PSF you have to get close to your observations. It also depends who you are - often observational astronomers who collect data from telescopes and theoretical astronomers who build models don't talk as much as we should, so you might not have access to all the information you need to do one of these two processes, so you work with what you've got and do the other one.