# Combining Knowledge And Data Driven Approaches to Inverse Imaging Problems

Lead Research Organisation:
University of Cambridge

Department Name: Applied Maths and Theoretical Physics

### Abstract

Imaging plays an important role in many applications in the natural sciences, medicine and the life sciences, as well as in engineering and industrial applications. An example is an MRI image of a brain used by a physician to detect a brain tumour such as glioblastoma. At the core of many imaging applications is an inverse problem, i.e. the mathematical problem of reconstructing the image from data produced by the imaging machine, for example the MRI machine. Such inverse imaging problems have been approached for many years in a "knowledge-driven" way, using information about the device and the imaging procedure. However, the knowledge-driven models cannot always be solved, are computationally very expensive, or deliver suboptimal images.

In recent years, new "data-driven" methods, which use past examples of successfully reconstructed images together with the data that produced them, have been shown to produce some impressive results in image reconstruction. The problem with such data-driven methods, however, is that currently they do not have "mathematical guarantees", in other words one cannot state the degree to which the results are reliable. They also have the property that even small deviations in the data could result in large differences in the results. This clearly could have devastating implications for many applications.

In this proposal, we will develop a new hybrid approach that combines the best of knowledge-driven and data-driven methods for inverse imaging problems, crucially providing the mathematical guarantees essential to being able to use the methods in real-world applications. Once the challenging task of developing these mathematical methods is achieved, we will apply this learning to produce an imaging pipeline that draws into a single step the stages of the imaging process, thus optimising the process further. We will apply the new methods to real-world applications. For example, using the data driven mathematical methods developed in the project and working closely with the Radiology Department, we will create an end-to-end workflow where multi-modal image acquisition, reconstruction, segmentation and image analyses are performed jointly and optimised for the end task of real time treatment response assessment in patients with metastatic cancer.

In recent years, new "data-driven" methods, which use past examples of successfully reconstructed images together with the data that produced them, have been shown to produce some impressive results in image reconstruction. The problem with such data-driven methods, however, is that currently they do not have "mathematical guarantees", in other words one cannot state the degree to which the results are reliable. They also have the property that even small deviations in the data could result in large differences in the results. This clearly could have devastating implications for many applications.

In this proposal, we will develop a new hybrid approach that combines the best of knowledge-driven and data-driven methods for inverse imaging problems, crucially providing the mathematical guarantees essential to being able to use the methods in real-world applications. Once the challenging task of developing these mathematical methods is achieved, we will apply this learning to produce an imaging pipeline that draws into a single step the stages of the imaging process, thus optimising the process further. We will apply the new methods to real-world applications. For example, using the data driven mathematical methods developed in the project and working closely with the Radiology Department, we will create an end-to-end workflow where multi-modal image acquisition, reconstruction, segmentation and image analyses are performed jointly and optimised for the end task of real time treatment response assessment in patients with metastatic cancer.

## People |
## ORCID iD |

Carola-Bibiane Schönlieb (Principal Investigator / Fellow) |

### Publications

Budd J
(2023)

*Joint Reconstruction-Segmentation on Graphs*in SIAM Journal on Imaging Sciences
Chen D
(2023)

*Imaging With Equivariant Deep Learning: From unrolled network design to fully unsupervised learning*in IEEE Signal Processing Magazine
Esteve-Yagüe C
(2023)

*Spectral decomposition of atomic structures in heterogeneous cryo-EM*in Inverse Problems
Grossmann T
(2023)

*Extracting chain lines and laid lines from digital images of medieval paper using spectral total variation decomposition*in Heritage Science
Tan H
(2023)

*Data-Driven Mirror Descent with Input-Convex Neural Networks*in SIAM Journal on Mathematics of Data Science