Example projects by Marta M. Betcke

My work in mainly in the are of inverse problems, numerical analysis and scientific computing. I am also teaching Numerical Optimization (COMPGV19) . Below are some example projects I am offering. If you are intersted please contact me to dicuss the details. Note this are examples but I am happy to supervise other projects related to my research.

Vessel promoting regularisation via Hessian Schatten norms

Vessel image (maximum projection)

Co-supervised by Felix Lucka

Pre-requisites: Matlab and some knowledge of Numerical Optimisation and Inverse Problems

Work summary:
The student will develop and implement a vessel promoting regularisation method based on vessel filter which utilises image curvature (Hessian) information. The regularisation method will be tested on photoacoustic tomography problem on simulated and in-vivo data.

Scientific aims:
Vascular structure present a challenge to popular regularisers like e.g. total variation or sparsity in most of known frames. On the other hand vessel filtering has proven to be successful post processing technique for such images. The aim is to develop a convex regulariser which mimics the action of a vessel filter for use in variational reconstruction.

Overview:
The Hessian based vessel filter [1] is a popular tool in post processing photoacoustic vasculature images [2]. The basic idea involves pre-smoothing of the images to extract appropriate scale and subsequently using curvature information to characterise the structures in the image as e.g. vessel or plate based on the singular values of the Hessian. The aim of this project is to develop this post processing tool into a regularisation functional. To this end a family of Hessian Schatten norms will be investigated to provide models for different structures. The resulting problem can then be solved for instance by a primal dual method [3].

References:
[1] http://link.springer.com/chapter/10.1007%2FBFb0056195
[2] http://spie.org/Publications/Proceedings/Paper/10.1117/12.2005988
[3] http://arxiv.org/pdf/1209.3318.pdf

Transmission and reflection mode for Ultrasound Computed Tomography (USCT) with incident planar waves

Co-supervised by Ben Cox

Pre-requisites: Matlab and some knowledge of Numerical Optimization and Inverse Problems

Work summary:
The student will implement the transmission and reflection problem for Ultrasound Computed Tomography (USCT) using k-wave Toolbox [1]. They will perform numerical experiments to ascertain the limitations of such system in different scenarios including simultaneous recovery of sound speed and density, and limited number of measurements.

Scientific aims:
The majority of the USCT systems use small, point-like ultrasound sources and measure in transmission mode. The goal of this project is to simulate a USCT system which uses plane wave as sources in both transmission and reflection mode and investigate the discriminative properties of such system for sound speed and density recovery.

Overview:
Preclinical imaging of small animals is crucial for the study of disease and in the development of new drugs. Photoacoustic tomography is an emerging imaging modality that shows great promise for preclinical imaging, but the resolution of the images is limited by the heterogeneities in the acoustic properties of the tissue, such as the sound speed. One way to overcome this would be to measure (quantitatively image) the properties and incorporate them into the photoacoustic reconstruction algorithm. Ultrasound computed tomography (USCT) is one way in which the spatially-varying acoustic properties of a medium may be imaged quantitatively. At UCL there is an existing pre-clinical photoacoustic imaging system based on a planar scanner that could potentially be adapted to make USCT measurements. It is common, in USCT, to use small, point-like, sources of ultrasound but with this scanner it may be easier to use plane ultrasonic waves as sources. This project will explore how USCT images can be recovered from measurements of the scattered, or transmitted, waves when the incident waves are planar

Directional total variation image reconstruction models for limited-view X-ray and photoacoustic tomography

Co-supervised by Felix Lucka

Pre-requisites: Matlab and some knowledge of Numerical Optimisation and Inverse Problems

Work summary:
The student will develop, implement and evaluate total-variation based imaging models that take into account that in limited-view tomography applications - we will consider X-ray computed tomography (CT) and photoacoustic tomography (PAT) - the imaging modality has a non-uniform sensitivity to feature edges. First, a variational image reconstruction algorithm using the conventional, isotopic TV energy will be implemented with the primal-dual hybrid gradient algorithm. Then, the TV energy will be locally modified to account for the different sensitivity of the measurement device towards edges at this location and potential benefits of this modification will be evaluated on simulated data scenarios. Finally, results for experimental data from sub-sampled, limited-view PAT will be computed.

Scientific aims:
A qualitative and quantitative improvement of the reconstructed images in limited-view CT and PAT.

Overview:
Limited view tomography applications like CT or PAT typically suffer from a locally-varying reduced sensitivity towards certain image features, most notably edges, while simultaneously introducing unwanted artefacts into the reconstruction. While it is possible to precisely characterize this sensitivity and even understand the nature and appearance of the artefacts through micro-local analysis (e.g., [1]), few image reconstruction approaches make use of this information. Total variation (TV) based regularization [2] is a popular image reconstruction technique to reliably detect and recover edges in challenging tomographic applications, but in its general form, it is ignorant towards the edges direction, which leads to sub-optimal results in the limited-view settings described above (see, e.g., [3,4] ). We want to investigate how to optimally modify TV-based methods locally to account for the different directional sensitivity of the measurement device (see, e.g., [5] for one approach how to locally adapt TV methods). The resulting optimization problems will be solved with the primal dual hybrid gradient algorithm [6].

References:
[1] http://epubs.siam.org/doi/abs/10.1137/140977709
[2] http://www.springer.com/cda/content/document/cda_downloaddocument/9783319017112-c1.pdf
[3] http://arxiv.org/abs/1602.02027
[4] http://arxiv.org/abs/1605.00133
[5] http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6527975
[3] http://link.springer.com/article/10.1007/s10851-010-0251-1