Multi-resolution initialisation

[herjy]

This thread is meant to brainstorm about source initialisation for multi-resolution images.

The issue being: How to initialise galaxy morphologies using blended images on different grids and with different psfs.

• The brutal deconvolution by division in the Fourier domain does not work in this limit as it does in single resolution images. The issue being that the low resolution psf vanishes at frequencies much lower that the range of frequencies present in the high-resolution frame. Therefore, interpolation and brut deconvolution leads to artefacts due to high-ish frequencies exploding. I don't think that brut deconvolution followed by interpolation has been attempted, but I'm expecting that some artefacts in troduced by the deconvolution, even if low will tend to resonate in the interpolation scheme that assumes band-limitedness.

[herjy]

Call me monomaniac, but I think wavelets might be able to get us out of there.
We can't deconvolve by fourier-space division because the band-limit of the psf is smaller than that of the whole image (yes Robert, there exist things below the psf scale).
What if, we only deconvolved what's below the band-limit of the psf. The rest we leave it, but it will be subdominant anyways. Andd to do that we have a great tool: wavelets that do exactly what we want: cut the signal into frequency chunks.

Realizing that the convolution ab = W^{-1}(Wab) (W the wavelet transform), we can easily do a deconvolution for all the Wa scales smoother than the psf band-limit through Fourier space division.

This leaves the noise (and in principle, only the noise because everything else in the convolved image is at the psf bandlimit) but for initialisation purpooses, it's more than good enough!

[pmelchior]

I was so expecting you to come up with such an idea! Give it a try!

[herjy]

The edges are blowing up and it looks like the signal at high frequencies makes quite a difference. Not sure yet, until I settle the edge issue.