Abstract
Combinations of spectroscopic analysis and microscopic techniques are used across many disciplines of scientific research, including material science, chemistry, and biology. X-ray spectromicroscopy, in particular, is a powerful tool used for studying chemical state distributions at the micro- and nanoscales. With the beam fixed, a specimen is typically rastered through the probe with continuous motion, and a range of multimodal data is collected at fixed time intervals.The recorded absorption coefficients are naturally stored in a third-order tensor, with spatial horizontal and vertical axes, and an energy axis; however, current analytical approaches require the tensors to be flattened into a matrix prior to performing the analysis. The spectral responses of the different materials contained within the specimen sum linearly, leading to datasets that allow for accurate reduced-order approximations. In particular, the flattened datasets have low numerical ranks.
There is a need to perform more in-situ experiments, in which chemical changes within specimen can be observed over a period of time. However, factors such as scanning times, radiation damage, and thermal drift during measurements prohibit wide-scale adoption of spectromicroscopic techniques in this setting. In particular, biological and environmental samples are vulnerable to high doses and long exposure times, and applications in these areas are limited.
The proposed solution to this problem is to undersample the data by taking only a small subset of the total measurements and recovering the missing entries using numerical techniques. Indeed, by scanning just a few randomly selected rasters for each X-ray energy, empty rows can be skipped over quickly, effectively reducing experiment times and the total X-ray dose.
Broadly, proposed methods of data recovery from sparse measurements follow two approaches: low-rank matrix completion on flattened data, and low-rank tensor completion of the data in its native space.
Low-rank matrix completion is a well-studied field in which, under suitable low-rank assumptions, sampled matrices can be recovered from only a small number of its entries. Due to their low numerical rank, spectromicroscopy datasets are suitable candidates for these techniques. In this thesis, we develop robust sampling methods and a novel completion algorithm, LoopedASD, to improve the accuracy of recovery from smaller sets of measurements. LoopedASD is a variation of the Alternating Steepest Descent (ASD) method for matrix completion, first proposed in [J. Tanner and K. Wei, Appl. Comput. Harmon. Anal., 40 (2016), pp. 417-429].
To avoid the loss of information that occurs when flattening a tensor, we develop two new Tensor Alternating Steepest Descent algorithms for tensor completion in the low-rank *M-product format. In this setting, we aim to reconstruct the entire low-rank tensor directly from the sparse set of measured rasters. Both methods are related to the ASD family of data completion algorithms, and extends the previously established matrix completion approaches to 3D tensors, and further to higher order multi-way data representations. In this framework, we can apply any unitary transform (such as the discrete Fourier transform) to the tensor tube by tube, providing a natural way to work with the *M-tensor algebra. Tensor algorithms can be easily adapted to fit particular datasets, using prior information from the experiment or other datasets to determine/learn a suitable transform.
The proposed sampling and completion steps are designed to be easily integrated into the existing experimental pipeline, so that sparse spectromicroscopy can be widely adopted by researchers from the scientific community. The low-rank completion approach can be adapted more broadly to any spectral or spectro-microscopy measurement where a low-rank approximation can be made.
All methods derived from the two approaches (matrix & tensor completion) are extensively tested on both synthetic low-rank data and real X-ray spectromicroscopy data. In practice, sparse spectromicroscopy has allowed for a 5 to 6-fold reduction in sampling. Experimental results obtained with real data are illustrated.
| Date of Award | 14 Jan 2026 |
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| Original language | English |
| Awarding Institution |
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| Sponsors | Diamond Light Source Ltd |
| Supervisor | Silvia Gazzola (Supervisor), Sergey Dolgov (Supervisor), Paul D. Quinn (Supervisor) & Julia Parker (Supervisor) |