Sample PhD Projects in Mathematical Imaging

Funded PhD studentship available

Innovative mathematical approaches for the analysis of ex-vivo and in-vivo imaging - Closing date 14th January 2022

General Interests

Image restoration, segmentation and co-registration; iterative methods; inverse problems; parallel computing

Professor Ke Chen

1. Iterative Algorithms for Nonlinear Optimization Problems

Non-linear optimization problems are ubiquitous in scientific modelling and computing. While more traditional research works focus on theoretical equations for a small scale (so 20 variables), modern applications demand fats and converging algorithms that can handle millions of variables in real time. There are many challenges in the emerging field of data analysis e.g. establishing the solution existence in the correct space is a problem partly due to few analytical tools are available (for instance, theories of Banach spaces for linear functionals cannot deal with such nonlinear problems). Apart from this theoretical problem, designing fast, effective and converging algorithms is also a pressing task; issues such as convexity, non-linear optimization solvers vs PDE solvers, global vs local minimizers. The topic is rich.

2. Development of Novel Variational Models and Methods for Imaging Problems

As imaging technology is increasingly used in more and more areas, emerging demand for robust models in more challenging modalities and applications is getting high. While it is true that learning techniques can help the commonly used tasks such as segmentation and registration, they cannot work alone (due to lack of training data or unreasonable request for such) and combination with robust models will be the key.

This project will develop advanced mathematics of optimization, functions and equations (PDEs), beyond matrices and vectors, to tackle this class of problems and meet the needs of applications. A particularly interesting problem is to register images with different resolutions and overlaps of a larger scene. Others include joint models.


Mathematical Imaging

Centre for Mathematical Imaging Techniques (CMIT) website

EPSRC Liverpool Centre for Mathematics in Healthcare