Here is an enduring example of the Donoho-Tanner phase transition, the ability to differentiate out certain reconstruction algorithms (I used to use an earlier example to talk about the peer review process).
Inverse Scattering via Transmission Matrices: Broadband Illumination and Fast Phase Retrieval Algorithms by Manoj Sharma, Christopher A. Metzler, Sudarshan Nagesh, Oliver Cossairt, Richard G. Baraniuk, Ashok Veeraraghavan
When a narrowband coherent wavefront passes through or reflects off of a scattering medium, the input and output relationship of the incident field is linear and so can be described by a transmission matrix (TM). If the TM for a given scattering medium is known, one can computationally “invert” the scattering process and image through the medium. In this work, we investigate the effect of broadband illumination, i.e., what happens when the wavefront is only partially coherent? Can one still measure a TM and “invert” the scattering? To accomplish this task, we measure TMs using the double phase retrieval technique, a method which uses phase retrieval algorithms to avoid difficult-to-capture interferometric measurements. Generally, using the double phase retrieval method requires performing massive amounts of computation. We alleviate this burden by developing a fast, GPU-accelerated algorithm, prVAMP, which lets us reconstruct 2562642 TMs in under five hours. After reconstructing several TMs using this method, we find that, as expected, reducing the coherence of the illumination significantly restricts our ability to invert the scattering process. Moreover, we find that past a certain bandwidth an incoherent, an intensity-based scattering model better describes the scattering process and is easier to invert.
Implementations:
- https://gampmatlab.fandom.com/wiki/Generalized_Approximate_Message_Passing
- Transmission matrix dataset: http://dsp.rice.edu/research/transmissionmatrices/
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