Mathematical Challenges in Electron Tomography

Composite colour image showing five surface plasmon modes (labelled α - ε) on a silver nanocube (100nm across) reconstructed using 4D spectrum-tomography (see O. Nicoletti et al., Nature, 502 80-84 (2013).)

Nanotechnology lies at the forefront of advances in a wealth of important areas for modern society, such as environmentally benign chemical processes, energy harvesting and new possibilities for healthcare treatments. Optimising nanotechnology solutions requires in-depth understanding of the nanoscale materials and devices. This is creating a pressing demand for more comprehensive characterisation methods, capable of revealing nanoscale morphology,  and many other key properties (such as chemical composition) in a spatially resolved manner.

Transmission electron microscopy (TEM) is a powerful characterisation method for both the physical and biological sciences, with tomography in the TEM being particularly valuable for three-dimensional (3D) morphological analysis [1]. Owing to recent hardware advances, TEM tomography is being extended to incorporate multiple dimensions [2], enabling for example the reconstruction of 3D images with additional spectral dimensions, or even the reconstruction of dynamic 3D image sequences. However, this vast progress on the engineering side has also posed several challenging and interesting mathematical problems that can be addressed in this project. Among those are the reconstruction of 3D images from limited-angle TEM projections, joint reconstruction of multiple spectral channels and handling of the vast amount of acquired data.

In TEM tomography an angular series of projections are acquired by tilting the specimen. The series of images are then computationally ‘back-projected’ to obtain a 3D reconstruction. However, practical restrictions inherent to TEM often prevent the specimen from being tilted over the full angular range, which leads to artefacts in the reconstructions. Sophisticated reconstruction techniques using sparse and non-smooth regularisation are needed to overcome this data deficiency, e.g. [3].

Powerful reconstruction methods are also needed because it is often only possible to acquire a small number of tilt-series projections, meaning that the reconstruction processes is a highly underdetermined inverse problem [3]. On the other hand, a very recent advance has been to acquire images very rapidly while continuously tilting the specimen [4]. This provides a very high angular sampling, but at the cost of very poor signal-to-noise ratio in each image. Tomographic reconstruction from these huge data sets presents considerable new challenges – but rich rewards such as ultra-fast tomography able to capture dynamic changes in the specimen in response to applied stimuli.

The acquisition of hyperspectral data for tomographic reconstruction opens up many novel opportunities, such as the separation of different types of tissue or phases of materials within the reconstruction process. However, there are many possible ways of incorporating the additional spectral dimensions into reconstruction schemes, including non-linear approaches. It will be part of the proposed research to find the best methods for hyperspectral TEM applications.

[1] Leary, R.K., Midgley, P.A. & Thomas, J.M. Recent Advances in the Application of Electron Tomography to Materials Chemistry. Acc. Chem. Res. 45(10) (2012) 1782-1791.
[2] Thomas, J.M., Leary, R.K., Eggeman, A.S. & Midgley, P.A. The rapidly changing face of electron microscopy. Chem. Phys. Lett. 631 (2015) 103-113.
[3] Leary, R.K., Saghi, Z., Midgley, P.A, & Holland, D.J. Compressed Sensing Electron Tomography. Ultramicroscopy. 131 (2013) 70-91.
[4] Migunov, V., Ryll, H., Zhuge, X., Simson, M., Struder, L., Batenburg, K.J., Houben, L. & Dunin-Borkowski, R.E. Rapid low dose electron tomography using a direct electron detection camera. Scientific Reports. 5 (2015) 14516.

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