Mathematical Imaging

How can we see beyond the limits of image resolution? How can we detect diseases earlier? How can we bring ancient manuscripts back to life?

In the Cambridge Image Analysis (CIA) group, researchers are doing exactly that: using mathematics and artificial intelligence to reconstruct and interpret images — from the human body, from nature, and from history. The group consists of more than 30 researchers from 15 countries and combines deep mathematical theory with modern machine learning. They develop methods that make it possible to see more — even when data are incomplete, noisy, or hidden. Their work makes advanced imaging cheaper, faster, and more accessible — and has already transformed practices in healthcare, environmental monitoring, and cultural heritage preservation.

Images take many forms — from physical and biological images to large-scale data that describe natural and engineered systems. Understanding and analysing such diverse data demands deep mathematical insight into: the mathematical models and equations that describe the physical and biological processes underlying image formation and data acquisition; the optimisation principles that enable us to extract meaningful information from incomplete or noisy measurements; and the learning algorithms that can capture and generalise complex structures in data. CIA researchers advance all these fronts — developing new mathematical models for inverse and variational problems, analysing and solving partial differential equations that describe imaging and physical processes, designing large-scale optimisation algorithms for efficient and reliable computation, and exploring the mathematical foundations of modern machine learning, from graph and hypergraph learning to generative models. In doing so, they not only deepen the mathematical foundations of imaging, but also open new horizons in the analysis of partial differential equations, large-scale optimisation, and machine learning — advances whose reach extends far beyond imaging itself. The impact of this research spans disciplines and scales — from hospitals to museums, from forests to cities.

From Hospitals to Cultural Heritage and Beyond

In healthcare, the CIA group’s work has enabled earlier disease detection and improved treatment planning through smarter analysis of medical images, blood samples, and patient records. They have developed tools that make MRI and CT scans faster and less invasive, reducing patient exposure to radiation. During the COVID-19 pandemic, the group contributed solutions that made artificial intelligence more reliable and transparent in clinical settings. Building on these advances, the group now develops mathematical techniques that support preventive medicine, enabling the identification of risks before symptoms appear and informing personalised interventions.

But their mathematics is also used far beyond hospitals. In collaboration with the Fitzwilliam Museum, the CIA group has digitally reconstructed faded manuscripts and musical recordings, allowing lost cultural treasures to be seen and heard once more. Other projects monitor forests, traffic, and urban health to help protect the environment and support sustainable city planning. Their expertise in inverse problems and large-scale data modelling reaches across physics and astronomy, driving the development of innovative and trustworthy algorithms that expand the frontiers of our understanding of nature.

The CIA group’s research is built on curiosity, collaboration, and responsibility. The group works closely with doctors, engineers, conservators, and environmental scientists. They share their data and software openly so others can build upon their work. They also place great emphasis on diversity and inclusion, creating an environment where researchers from around the world can learn and grow — from interns, PhD students and early career researchers through to experienced professors.