Mathematical methods for Ptychography Imaging

Ptychography is a method of recovering the phase and amplitude of an image via an inverse Fourier transform
problem. The method can be used to reconstruct images that are noisy and have low resolution. The ptychographical
iterative engine (PIE) is a phase retrieval algorithm that can perform this method. There are several different
iterations of this algorithm such as parallel PIE (pPIE), extended PIE (ePIE), regularized PIE (rPIE) and momentumaccelerated
PIE (mPIE). My aim is to develop, test and analyse algorithms for recovering full ptychography images
from undersampled data, particularly in tomography, which consists of many similar images at slightly different
angles, and/or spectral ptychography at multiple beam energies. Initial objectives will be:
- To study the error between fully sampled and undersampled images under random data collection.
- Design a more efficient data collection pattern, especially for big data scenarios like spectral tomography.
Other methods to look at are regularisation and relaxation techniques when applied to the optimisation problems
underlying ptychography and tomography. In particular, similar images in (spectral) tomography can be approximated
by a low-rank matrix or tensor. In turn, the nonlinearity in the ptychography model can be lifted by considering a
product of image with its copy. Time permitting, the project will consider uncertainty quantification (such as Bayesian
inference) if the noise in the image is large, for example, in the case of low photon counts.
Ptychography is used in the reconstruction of images from x-ray spectrometers and images from electron
microscopes. If under-sampling images has a comparable performance to an existing reconstruction algorithm,
reconstruction times might be quicker as fewer diffraction patterns will need to be recorded. This in turn will reduce
the x-ray dosage required for the experiments. For synchrotron facilities that use this technique such as Diamond
Light Source, UK, poss ible faster reconstruction times will mean less time running the experiment at the end of a
beamline so the cost will be cheaper. It will also mean that more experiments can be run since each individual
experiment will take less time.