AI and Inverse Problems in Medical Imaging: Carolla Schindli on ICM Insights from Cambridge

Overview

In this Voices of Mathematics episode from the University of Cambridge, Carolla Bibiana Schindli explains how AI enhances the mathematical handling of inverse problems in medical imaging, including CT and MRI. The conversation situates AI as a powerful tool that complements rigorous mathematics rather than replacing it, a theme reflected in ICM discussions and Cambridge's research culture.

• AI aids ill posed inverse problems by encoding prior information to constrain solutions
• Learned solution operators have transformed medical image reconstruction
• The interplay between mathematical guarantees and AI practicality is crucial for trustworthy results
• The ICM context highlights both optimism and concern around AI in mathematics

Introduction and Context

The podcast features Carolla Bibiana Schindli, a professor of Applied Mathematics at Cambridge, discussing the use of AI in mathematics and its application to real world problems, notably in medical imaging. The hosts reflect on the International Congress of Mathematicians (ICM) in Philadelphia, noting AI's dominance in discussions and the tension between proofs and real world utility.

Inverse Problems and the Forward Problem

Schindli explains what an inverse problem is by contrasting it with its forward counterpart. In many imaging tasks, we measure indirect data that contain information about an underlying tissue density. The forward problem describes how the measurements would look if we knew the density, while the inverse problem seeks to recover that density from the measurements. Tomography exemplifies this, with the Radon transform playing a central role in the forward model.

Ill-posedness and the Role of AI

A key mathematical challenge is that inverse problems are ill posed. They may not have a unique or stable solution without additional information. Mathematicians turn ill posed problems into well posed ones by incorporating priors about the solution. AI, Schindli argues, is especially effective at encoding these priors, narrowing the solution space and enabling accurate reconstructions from imperfect data. She discusses how this redefines the role of AI from a tool that merely automates to one that integrates with mathematical structure.

From Learning to End-to-End Inverse Problems

The conversation traces the evolution from early learned inverse problem approaches to modern deep learning, highlighting the Automap paper that demonstrated end to end learning for magnetic resonance tomography. This shift shows how parameterized models can adapt to a problem and yield high quality images, but also raises questions about guarantees and interpretability.

AI, Guarantees, and Trust

Schindli emphasizes that while AI can improve efficiency and accuracy, especially in clinical contexts, mathematics still provides guarantees such as existence and convergence proofs. In AI driven medical imaging, there is a need for reproducibility and reliability, with formal methods and verification playing a role in ensuring that AI outputs generalize correctly to new data and clinical settings.

ICM, Outreach, and Cambridge Environment

The dialogue situates AI within the broader ICM landscape, noting discussions of the value and risks of AI in proving theorems and solving real world problems. The Cambridge environment is described as offering freedom, collaboration, and inspiration that fuels research and public engagement. The episode also touches on Hannah Fry and the Lee Lavati Prize for outreach, underscoring the importance of communicating mathematics to diverse audiences.

Reflections on the Future

Overall, the talk presents a nuanced view: AI does not erase mathematics, but rather extends it. The future trajectory involves blending mathematical rigor with AI powered methods to tackle meaningful problems while maintaining guarantees and trust in results. The ICM experience reinforces that the mathematical community recognizes both opportunities and challenges as AI becomes increasingly integrated into mathematical practice and everyday life.