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Rich and Nonlinear Tomography - a multidisciplinary approach

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Programme
4th January 2023 to 30th June 2023
Organisers: 
Bill Lionheart University of Manchester
Marta Betcke University College London
Romina Gaburro University of Limerick, University of Limerick
Silvia Gazzola University of Bath
Sean Holman University of Manchester
Paul Ledger Keele University
Tanja Tarvainen University of Eastern Finland
Francis Watson Thales

 

Image: Ryan Warr and his paper Warr, Ryan, et al. "Enhanced hyperspectral tomography for bioimaging by spatiospectral reconstruction." arXiv preprint arXiv:2103.04796 (2021)."

 

Programme Theme

Tomographic imaging provides the ability to image properties in the interior of an object from measurements taken on its surface or in the far field. It is vital in medicine, manufacturing, geophysics,advanced material and security, to mention only few of its applications. Sensor technology is increasingly extending not only in the number of detector elements and the speed of acquisition but also by adding capability to resolve frequency, energy, polarisation state, transient response, resulting not only in higher data rates but also in richer data. Even though this holds the promise of imaging vector or tensor properties, the highly coupled and often non-linear nature of the underlying inverse problem means that
new mathematical and computational methods are needed.

Analytical and geometrical tools in inverse problems are fundamental for understanding the issues of uniqueness and stability of the solution of such ill-posed problems in terms of the data. These, in turn, are key to unearth useful subsets of possible data we can gather, transmit and store and to understand what a-priori information is needed to stabilize the problem. Where data is missing due to physical limitations of transmitter and receiver locations, microlocal analysis can provide information about the directions one can expect to be able to reconstruct singularities (abrupt changes in material properties). Solving an inverse problem also requires an efficient forward solver. Numerical analysis is
key to efficient forward solution. In particular, for non-linear problems, the forward solver is computed repeatedly with only slightly different parameters. Efficient paralleization, including both CPU and GPU cases are key to being able to solve rich tomography problems efficiently. Regularization penalty terms (e.g. total variation and its generalizations) are increasingly being applied to tomographic inverse problems. This requires the use of a new generation of primal-dual optimization algorithms, which, coupled with the non-linearity and the rich data sets, leads to challenging optimization problems. Often in inverse problems and imaging the goal is not simply the best picture. What is required is a diagnosis, location and identification of a physical property that requires a quantification of uncertainty. This might range from the quantity of oil that can be extracted from a reservoir to the detection of a land mine or
threat object in a bag. The Bayesian approach, specifically sampling schemes that are effective in high dimensional spaces, provides a tool to explore the posterior distribution in the Bayesian formulation of
an inverse problem. With fast forward solvers and efficient sampling schemes, together with dimension reduction methods, this is becoming feasible for imaging inverse problems. Calibration problems where
the geometry and characteristics of detectors for example are imprecisely known (which is typical in most real problems) can also benefit from a Machine Learning (ML) approach .The programme, organised around the four overlapping themes of

  1.  analysis and geometry;
  2. numerical analysis;
  3.  optimisation; and
  4.  Bayesian statistics,

aims at stimulating close engagement between mathematicians and statisticians working on inverse problems and instrument scientists,
engineers, physicists and image users to focus on the right problems both in the mathematical formulation and the end-user’s requirements. The main objective is to produce step changes in the capabilities
of rich and non-linear tomography through application of mathematical sciences to the system design, reconstruction and interpretation of images. By identifying rich and non-linear tomographic methods as
a large cluster of emergent techniques and by throwing a spotlight on their mathematical similarities, the programme aims to catalyse rapid development of new tomographic imaging methods. By employing
the above mathematical and statistical themes to work across diverse application areas from medical imaging to geophysics, security, defence, materials and non-destructive testing, the programme aims at
breaking down the barriers to communication across discipline areas. By bringing together end-users and instrument scientists with mathematical scientists, the programme will encourage the development of shared software platforms reflecting the underlying general structure of the mathematical problems.

By encouraging the sharing of representative data sets, it is also hoped that mathematical scientist can benefit from these when they do not have a direct collaboration with experimental groups. Finally, the programme aims at creating an open and diverse forum of researchers that includes early career researchers, women in science and, more generally, researchers coming from under-represented groups.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons