CS: Ab Initio Compressive Phase Retrieval

I was wondering when this subject would come up. Out of the many indirect sensing schemes, diffraction is a sensing scheme that most everybody has heard about. It is a lensless imaging process that requires a computational step to obtain an image. 



On top of it, one can use a high luminosity source, as high as the sample allows. Stefano Marchesini has written on the subject with a particular view on how Compressive sensing could be used to alleviate some of the very large sampling required by the current direct methods. In some of the approaches, there is some consideration for using coded apertures.

An introduction on the subject can be found in this recent presentation entitled Large scale inverse problems in ultrafast x-ray imaging. A previous abstract of the presentation reads:

The Fourier inversion of coherent diffraction patterns offers diffraction-limited three dimensional images without the aberrations, inefficiencies and depth-of field limitations of lens-based tomographic systems. We report experimental images inverted using holographic and computational phase retrieval methods. Ultimately the resolution of X-ray diffraction imaging will be limited by radiation damage. This limit can be overcome using pulses faster than the damage process. The first experimental verification of the principle of flash diffraction imaging using a soft X-ray free-electron laser will be presented. Experiments aimed at the characterization of the dynamics of organic matter under intense radiation and validation of theoretical models will be discussed.

For a more in-depth introduction on the subject, one should check A unified evaluation of iterative projection algorithms for phase retrieval. The number of measurements becomes a burden when trying to invert the computational problem as shown in the following slide:

Hence Compressive Sensing is probably becoming an obvious solution to investigate. This is done in Ab initio compressive phase retrieval by Stefano Marchesini. The abstract reads:

Any object on earth has two fundamental properties: it is finite, and it is made of atoms. Structural information about an object can be obtained from diffraction amplitude measurements that account for either one of these traits. Nyquist-sampling of the Fourier amplitudes is sufficient to image single particles of finite size at any resolution. Atomic resolution data is routinely used to image molecules replicated in a crystal structure. Here we report an algorithm that requires neither information, but uses the fact that an image of a natural object is compressible. Intended applications include tomographic diffractive imaging, crystallography, powder diffraction, small angle x-ray scattering and random Fourier amplitude measurements.

I note the use of Sparco and SPGL-1. In Stefano Marchesini's presentation page, one can find additional mateiral such as a list of most experiments that could use CS here. There are also two other talks (here and here).

I am sure I'll come back to this later.