Certain imaging and audio processing applications require the reconstruction of an image or signal from its phaseless time-frequency or time-scale representation, such as the magnitude of its Gabor or wavelet transform.

Such problems are inherently unstable, however, we formulate a relaxed notion of solution, meaningful for audio processing applications, under which stability can be restored.

The question of uniqueness becomes particularly delicate in the sampled setting. There, we show the first result evidencing the fundamental non-uniqueness property of phase retrieval from Gabor transform measurements. By restricting to appropriate function classes, positive results on the uniqueness can be obtained.

Furthermore, we present our most recent result which establishes uniqueness of phase retrieval from sampled wavelet transform measurements, without restricting the function class, when 3 wavelets are employed.

Almost all modern day imaging systems rely on digital capture of information. To this end, hardware and consumer technologies strive for high resolution quantization based acquisition. Antithetical to folk wisdom, we show that sampling quantization noise results in unconventional advantages in computational sensing and imaging. In particular, this leads to a novel, single-shot, high-dynamic-range imaging approach. Application areas include consumer and scientific imaging, computed tomography, sensor array imaging and time-resolved 3D imaging. In each case, we present a mathematically guaranteed recovery algorithm and also demonstrate a first hardware prototype for basic digital acquisition of quantization noise.

The interaction of an object with a coherent (x-ray) probe often encodes its properties in a complex-valued function, which is then detected in an intensity-only measurement. Phase retrieval methods commonly infer this complex-valued function from the intensity. However, the decoding of the object from the complex-valued function often involves some ambiguity in the phase, e.g., when the phase shift in the object exceeds $2\pi$. Here, we present a phase retrieval framework to directly recover the amplitude and phase of the object. This refractive framework is straightforward to integrate into existing algorithms. As examples, we introduce refractive algorithms for ptychography and near-field holography and demonstrate this method using measured data.

Traditional X-ray imaging achieves contrast using the attenuation of photons, making differentiation of materials of a similar density difficult. Improvements in the coherence of X-ray sources opened the way for phase changes in a material to be measured in an intensity image. In addition, the scattered component of the X-ray beam has been probed in X-ray dark-field imaging. Novel dark-field imaging techniques show promise in the detection and assessment of samples with significant micro-scale porosity, such as human lungs. Advanced dark-field imaging techniques rely on measuring sample-induced deviations on a patterned and interferometrically probed beam, requiring a highly-stable set-up and multiple exposures. Propagation-based imaging (PBI) is an experimentally-simple phase-contrast imaging technique, which relies on the downstream interference of refracted and diffracted coherent X-rays to reconstruct sample phase. Recently, PBI has been extended to dark-field reconstruction by modelling the downstream intensity using an X-ray imaging version of the Fokker-Planck diffusion equation [1, 2]. Separating the effects of refraction and diffusion on the beam requires multiple measurements, which was first achieved by imaging the sample at multiple propagation distances [3]. A multi-energy beam creates another possibility; the recent proliferation of energy-discriminating photon-counting detectors has led to an increased interest in spectral methods of coherent X-ray imaging [4]. In this talk we present the first results of inverting and solving the Fokker-Planck equation using spectral information, under assumption of a single-material sample. A linearised model is used to reconstruct sample projected thickness and dark-field in simulated and measured images. Strong attenuation energy-dependence presents challenges in reconstruction when deviating from strict single-material samples. We discuss Fokker-Planck dark-field reconstruction, and present a hybrid approach to the inverse problem, based on treating the post-sample wavefront as pseudo-patterned intensity, which improves stability in multi-material samples. Exploiting spectral dependence to reconstruct phase and dark-field would allow for imaging without having to move any part of the set-up, and would enable single-exposure imaging when combining a polychromatic source with an energy-discriminating detector. This would avoid registration issues, reduce the required dose, and open the door for time-resolved propagation-based dark-field imaging and fast CT.

[1] K. S. Morgan, D. M. Paganin. Applying the Fokker-Planck equation to grating-based x-ray phase and dark-field imaging, Scientific Reports 9(1): 17465, 2019.

[2] D. M. Paganin, K. S. Morgan. X-ray Fokker-Planck equation for paraxial imaging, Scientific Reports 9(1): 17537, 2019.

[3] T. A. Leatham, D. M. Paganin, K. S. Morgan. X-ray dark-field and phase retrieval without optics, via the Fokker-Planck equation, IEEE Transactions on Medical Imaging, 2023.

[4] F. Schaff, K. S. Morgan, J. A. Pollock, L. C. P. Croton, S. B. Hooper, & M. J. Kitchen. Material Decomposition Using Spectral Propagation-Based Phase-Contrast X-Ray Imaging, IEEE Transactions on Medical Imaging 39(12): 3891–3899, 2020.