Regularisation theory in the data driven setting
Lead Research Organisation:
University of Cambridge
Department Name: Applied Maths and Theoretical Physics
Abstract
Inverse problems deal with the reconstruction of some quantity of interest from indirectly measured data. A typical example is medical imaging, where there is no direct access to the quantity of interest (the inside of the patient's body) and imaging techniques, such X-Ray imaging and magnetic resonance imaging (MRI), are used. The classical approach to inverse problems uses models that describe the physics of the measurement. For example, in X-Ray imaging this model would describe how X-Rays pass through the body. In the era of big data, however, it becomes increasingly popular not to model the physics but to use vast amounts of data instead that relate known images with corresponding measurements.
The theory of such data driven methods, however, is not well developed yet. It is not well understood, under which conditions on the training data such methods are stable with respect to small changes in the measurement and how well they adapt to images that are different from the training images. It is important to understand this, since otherwise the reconstruction algorithm can miss important features of the image if they weren't present in the training set, such as tumours at previously unseen locations.
In this project I will extend the state-of-the-art model based theory to this data driven setting. I will study under which conditions can data driven methods achieve regularisation, i.e. when can they stably solve an otherwise unstable problem. This will make it easier to analyse stability of data driven reconstruction methods and help developing novel, stable data driven inversion methods with mathematical guarantees. I will also collaborate with the National Physical Laboratory and the Department of Chemical Engineering and Biotechnology in Cambridge on applications of my methods in imaging to reduce the time needed to acquire an image and make the reconstructions more reliable.
The theory of such data driven methods, however, is not well developed yet. It is not well understood, under which conditions on the training data such methods are stable with respect to small changes in the measurement and how well they adapt to images that are different from the training images. It is important to understand this, since otherwise the reconstruction algorithm can miss important features of the image if they weren't present in the training set, such as tumours at previously unseen locations.
In this project I will extend the state-of-the-art model based theory to this data driven setting. I will study under which conditions can data driven methods achieve regularisation, i.e. when can they stably solve an otherwise unstable problem. This will make it easier to analyse stability of data driven reconstruction methods and help developing novel, stable data driven inversion methods with mathematical guarantees. I will also collaborate with the National Physical Laboratory and the Department of Chemical Engineering and Biotechnology in Cambridge on applications of my methods in imaging to reduce the time needed to acquire an image and make the reconstructions more reliable.
Planned Impact
Inverse problems arise whenever directly accessing the quantities of interest is impossible and indirect measurements have to be used. Such problems are ubiquitous in science and technology, from microscopy to astronomy, from medical imaging to Earth exploration to non-destructive testing to airport security screening and so on.
Due to the availability of large amounts of domain-specific training data, the paradigm in many applications is shifting towards methods that rely on learning rather than careful mathematical modelling of the measurement process. However, mathematical understanding of such methods is far from being complete. The aim of this project is to reduce this gap by extending the state-of-the-art model-based inverse problems theory to the data driven setting. As a result, we will have a better understanding of data driven approaches to inverse problems and will have novel methods with improved stability properties and solid mathematical guarantees.
This will have impact on the following areas.
- Policy
Theoretical understanding of data driven methods will help shape policy on the use of such methods in sensitive applications from the societal point of view, such as medical imaging used for diagnosis or airport security screening. I will work with Cambridge based charities who help designing policy in emerging technologies in healthcare and other areas.
- Developing National Standards
In this project I will collaborate with the National Physical Laboratory, UK's National Metrology Institute responsible for developing and maintaining measurement standards. Our collaboration will help, in the long run, to design standards for the interpretation of indirect measurements using data driven approaches.
- Industry
Part of this project is applying the developed methods in image reconstruction. This type of problems occur in many areas of technology, such as material manufacturing and the energy sector. I will collaborate with the Department of Chemical Engineering and Biotechnology to speed up acquisition times in magnetic resonance imaging to enable imaging faster processes.
- Public Engagement and Outreach
An important aspect of this project is promoting mathematics to a wider audience through public engagement projects, such as the Cambridge Science Festival, and outreach, e.g., publishing articles in popular science journals.
- Academic Impact
The project will have impact on several fields of research, including imaging science and other areas that use imaging, such as chemical engineering and biomedical sciences. It will also impact data science more broadly alongside with the specialist field of inverse problems. Research will be published in high quality specialist as well as interdisciplinary journals and disseminated through international conferences and workshops. Prototype software and preprints of academic papers will be made available on public repositories.
- Teaching
The results obtained in this project will be integrated into a course taught in Cambridge. Summer research projects and other research projects related to this fellowship will be offered to students, who will benefit from participating in cutting edge research.
Due to the availability of large amounts of domain-specific training data, the paradigm in many applications is shifting towards methods that rely on learning rather than careful mathematical modelling of the measurement process. However, mathematical understanding of such methods is far from being complete. The aim of this project is to reduce this gap by extending the state-of-the-art model-based inverse problems theory to the data driven setting. As a result, we will have a better understanding of data driven approaches to inverse problems and will have novel methods with improved stability properties and solid mathematical guarantees.
This will have impact on the following areas.
- Policy
Theoretical understanding of data driven methods will help shape policy on the use of such methods in sensitive applications from the societal point of view, such as medical imaging used for diagnosis or airport security screening. I will work with Cambridge based charities who help designing policy in emerging technologies in healthcare and other areas.
- Developing National Standards
In this project I will collaborate with the National Physical Laboratory, UK's National Metrology Institute responsible for developing and maintaining measurement standards. Our collaboration will help, in the long run, to design standards for the interpretation of indirect measurements using data driven approaches.
- Industry
Part of this project is applying the developed methods in image reconstruction. This type of problems occur in many areas of technology, such as material manufacturing and the energy sector. I will collaborate with the Department of Chemical Engineering and Biotechnology to speed up acquisition times in magnetic resonance imaging to enable imaging faster processes.
- Public Engagement and Outreach
An important aspect of this project is promoting mathematics to a wider audience through public engagement projects, such as the Cambridge Science Festival, and outreach, e.g., publishing articles in popular science journals.
- Academic Impact
The project will have impact on several fields of research, including imaging science and other areas that use imaging, such as chemical engineering and biomedical sciences. It will also impact data science more broadly alongside with the specialist field of inverse problems. Research will be published in high quality specialist as well as interdisciplinary journals and disseminated through international conferences and workshops. Prototype software and preprints of academic papers will be made available on public repositories.
- Teaching
The results obtained in this project will be integrated into a course taught in Cambridge. Summer research projects and other research projects related to this fellowship will be offered to students, who will benefit from participating in cutting edge research.
Organisations
- University of Cambridge (Lead Research Organisation)
- University of Manchester (Collaboration)
- University of Oulu (Collaboration)
- Medical Research Council (MRC) (Collaboration)
- DIAMOND LIGHT SOURCE (Collaboration)
- UNIVERSITY OF CAMBRIDGE (Collaboration)
- Institute of Mathematical Sciences (Collaboration)
- National Physical Laboratory (Project Partner)
- University of Bath (Fellow)
Publications
Korolev Y
(2022)
Two-Layer Neural Networks with Values in a Banach Space
in SIAM Journal on Mathematical Analysis
Toader B
(2022)
Image Reconstruction in Light-Sheet Microscopy: Spatially Varying Deconvolution and Mixed Noise.
in Journal of mathematical imaging and vision
Description | New course developed |
Geographic Reach | Local/Municipal/Regional |
Policy Influence Type | Contribution to new or Improved professional practice |
Title | A new method for deconvolution of light-sheet microscopy images |
Description | Together with my collaborators from the Cambridge Advanced Imaging Centre and the MRC Laboratory of Molecular Biology, I proposed a new method for improving the quality of light-sheet microscopy images. We study the problem of deconvolution for light-sheet microscopy, where the data is corrupted by spatially varying blur and a combination of Poisson and Gaussian noise. The spatial variation of the point spread function (PSF) of a light-sheet microscope is determined by the interaction between the excitation sheet and the detection objective PSF. First, we introduce a model of the image formation process that incorporates this interaction, therefore capturing the main characteristics of this imaging modality. Then, we formulate a variational model that accounts for the combination of Poisson and Gaussian noise. |
Type Of Material | Data analysis technique |
Year Produced | 2021 |
Provided To Others? | Yes |
Impact | no impacts yet |
URL | https://arxiv.org/abs/2108.03642 |
Description | Data driven operator correction |
Organisation | University of Manchester |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | I contribute my expertise in inverse problems and machine learning. |
Collaborator Contribution | My collaborators contribute their expertise in electrical impedance tomography, experimental data, and computing resources. |
Impact | This collaboration involves experimental data in electrical impedance tomography available from my collaborator at the University of Oulu. |
Start Year | 2021 |
Description | Data driven operator correction |
Organisation | University of Oulu |
Country | Finland |
Sector | Academic/University |
PI Contribution | I contribute my expertise in inverse problems and machine learning. |
Collaborator Contribution | My collaborators contribute their expertise in electrical impedance tomography, experimental data, and computing resources. |
Impact | This collaboration involves experimental data in electrical impedance tomography available from my collaborator at the University of Oulu. |
Start Year | 2021 |
Description | Image reconstruction in X-ray microscopy |
Organisation | Diamond Light Source |
Country | United Kingdom |
Sector | Private |
PI Contribution | I contribute my expertise in inverse problems and imaging. |
Collaborator Contribution | My collaborator contributes experimental data and expertise in X-ray microscopy. |
Impact | This collaboration is interdisciplinary (experimental physics + mathematics). |
Start Year | 2023 |
Description | Image reconstruction in light microscopy |
Organisation | Medical Research Council (MRC) |
Department | MRC Laboratory of Molecular Biology (LMB) |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | I contribute my expertise in inverse problems and imaging. We jointly supervised a student research project in summer 2022 funded jointly by the Department of Applied Mathematics and Theoretical Physics and the Cambridge Advanced Imaging Centre. |
Collaborator Contribution | My collaborators contribute their expertise in microscopy. They also provide experimental data and computing facilities. We jointly supervised a student research project in summer 2022 funded jointly by the Department of Applied Mathematics and Theoretical Physics and the Cambridge Advanced Imaging Centre. |
Impact | We proposed a new method of improving light-sheet microscopy images described in the Research Datasets, Databases & Models section. We are applying for further funding to work on cell tracking. It is a multidisciplilnary collaboration with biologists (Laboratory of Molecular Biology) and microscopists (Cambridge Advanced Imaging Centre). This collaboration started in 2018 but has been growing in depth, especially since the start of this project. |
Start Year | 2018 |
Description | Image reconstruction in light microscopy |
Organisation | University of Cambridge |
Department | Cambridge Advanced Imaging Centre |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | I contribute my expertise in inverse problems and imaging. We jointly supervised a student research project in summer 2022 funded jointly by the Department of Applied Mathematics and Theoretical Physics and the Cambridge Advanced Imaging Centre. |
Collaborator Contribution | My collaborators contribute their expertise in microscopy. They also provide experimental data and computing facilities. We jointly supervised a student research project in summer 2022 funded jointly by the Department of Applied Mathematics and Theoretical Physics and the Cambridge Advanced Imaging Centre. |
Impact | We proposed a new method of improving light-sheet microscopy images described in the Research Datasets, Databases & Models section. We are applying for further funding to work on cell tracking. It is a multidisciplilnary collaboration with biologists (Laboratory of Molecular Biology) and microscopists (Cambridge Advanced Imaging Centre). This collaboration started in 2018 but has been growing in depth, especially since the start of this project. |
Start Year | 2018 |
Description | Machine learning and partially ordered spaces |
Organisation | Institute of Mathematical Sciences |
Country | Spain |
Sector | Charity/Non Profit |
PI Contribution | I contribute my expertise in applied mathematics, in particular inverse problems and mathematics of machine learning. |
Collaborator Contribution | My collaborator contributes his expertise in functional analysis, in particular partially ordered spaces and Banach lattices. |
Impact | This collaboration is intradisciplinary, i.e. it involves several areas of mathematics/several research communities. My collaborator in Spain, Pedro Tradacete, is one of leading figures in partially ordered spaces, in particular Banach lattices. We are working on establishing connections between partially ordered spaces and machine learning. |
Start Year | 2023 |