[School of Natural Sciences PhD Scholarships] Mathematical Foundations of Learned Priors for Coherent Imaging

About the project

About the Project How can machine learning be used to aide image formation when these images are complex-valued, resolution varies between acquisitions, and reliable ground truth is unavailable? Modern imaging systems often recover images from incomplete or indirect measurements, making reconstruction the image an inverse problem. To achieve stable and accurate reconstructions, prior information about the underlying image is often incorporated through regularisation. In recent years, learned image priors have become one of the most successful ways to combine machine learning with inverse problems methods. These approaches can capture rich image structure directly from real data, avoiding the need to specify explicit mathematical models of image content. This PhD project will develop new mathematical theory and computational methods for learning image priors for coherent imaging systems such as synthetic aperture radar (SAR). SAR is a widely used remote sensing modality with applications ranging from environmental monitoring and Earth observation to security and defence. The combination of complex-valued measurements, limited ground-truth data and varying image discretisations presents fundamental challenges for current machine learning approaches and motivates the development of new mathematical foundations. The research lies at the interface of inverse problems, optimisation and machine learning, with a particular focus on developing methods that remain effective across different image resolutions and discretisations. The student will join the inverse problems group in the Department of Mathematics, as well as a growing radar research focus. There will be further opportunity and encouragement to engage with radar researchers and practitioners outside of mathematics to help guide the mathematical development towards practical application and impact, and support interdisciplinary and impactful mathematics skills development. This project is expected to start in September 2027. Before you apply: We strongly recommend that you contact the supervisors for this project before you apply. How to apply: To be considered for this project you must complete a formal application through our online application portal. If you already have an applicant account this link will directly open an application for PhD School of Natural Sciences Scholarships . If you don’t already have an applicant account, please follow the instructions here . When applying, please specify the full title and supervisor/s of the project, details of your previous study, and names and contact details of two referees. You must also upload a Supporting Statement describing your motivation to apply to the project, your CV and transcripts of awarded and in-progress university qualifications . Please note late or incomplete applications will not be considered. Equality, diversity and inclusion are fundamental to the success of The University of Manchester and central to all our activities. A diverse research community strengthens creativity, productivity and quality, while increasing the societal and economic impact of our work. We welcome applicants from all career paths, backgrounds and sections of the community, regardless of age, disability, ethnicity, gender, gender expression, sexual orientation or transgender status. We welcome applications from candidates returning to study after a career break or experience in other roles. Flexible study arrangements may be available, including part-time study at 50%, 60% or 80%, subject to the requirements of the project and funder. Eligibility : The standard academic entry requirement for this PhD is an upper second-class (2:1) honours degree (or international equivalent) in Mathematics or a closely related discipline OR any upper-second class (2:1) honours degree and a Master’s degree at merit (or international equivalent) in Mathematics or a closely related discipline. An interest in applied mathematics and the mathematical foundations of machine learning is essential. Prior experience in areas such as inverse problems, optimisation, numerical analysis, signal processing, or scientific computing would be beneficial. The project will involve both theoretical mathematical development and scientific computing in Python or other suitable languages. This project will remain open until filled. If your application is submitted by 1st November 2026, you can expect a decision by 18th December 2026. If your application is submitted by 15th January 2027, you can expect a decision by 30th March 2027. Self or externally funded students can also be considered for this project. FSESoNS

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