I spent the day writing code to create mixtures of Gaussians and (importantly) their Fourier transforms. I can't count the number of times I have written mixtures-of-Gaussians code! But each use case is at least slightly different. Today the application is diffraction microscopy. I want to explore bases other than the standard grid-of-pixels basis.
The funny thing about the diffraction-microscopy problem is that it is simultaneously trivial and impossible: It is to infer all the phases of the Fourier transform given only a noisy, censored measurement of its square modulus. All the approaches that work apply very informative priors or regularization. My biggest concern with them is that they often put the most informative part of the prior on the space outside the object. Hence my goal of using a basis that is compact to begin with.
As a teaser and demo, here is an unlabeled set of figures that “test” my code:
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