tjlane/cxs-theory

Can we find the symmetries?!?

cxs-theory's Issues

Compressed Sensing

This was @VijayPande 's suggestion, which makes perfect sense. If we can phrase our problem simply as a linear measurement,

y = Ax + error

where here x are (hopefully) the g_lm terms in the expansion of the electron density, if possible, and y are the measured coefficients C_lambda of the expansion of the correlation function, then compressed sensing tells us we can get a very good answer by employing:

https://en.wikipedia.org/wiki/Basis_pursuit_denoising

which is pretty sick.

The key remaining trick is to write explicitly this linear form, i.e. write A. Note that formally A is defined in terms of inner products, so it may be OK to include a Fourier transform (which is linear) as part of that map. Need to confirm this -- just started reading.

Citations:

http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=1614066
http://onlinelibrary.wiley.com/doi/10.1002/cpa.20124/abstract

Re-think the endpoint

After a late-night chat w/Daniel Ratner, we realized that I might have been gunning for the incorrect final solution in this work. If I really computed the sqd-sum of each SH l-band (for the sample electron density), the result would be rotationally averaged in real space. This would correspond to a WAXS measurement.

The power of CXS is that it can break this symmetry, and that is what we should be leveraging!

I need to make sure this is correct, and think through it. One question: if true, why can't one extract simple symmetries from WAXS?

Previous work

I was just reading through some of the literature pile I've built up, and was thinking it would be cool to share key papers/insights about those papers here.

Some of the papers I've been leafing through are Haiguang's. Here, he uses an expansion of the electron density in Zernlike polynomials, which is similar to our expansion in spherical harmonics but slightly less elegant (it's not quite the basis he wants!).

The application is very different though -- but they are thinking along the lines of Odin.

http://scripts.iucr.org/cgi-bin/paper?S0108767313006016

also:
http://cg.cs.uni-bonn.de/aigaion2root/attachments/novotni-2004-shape.pdf