Abstract

In the modern society we encounter digital images in many different situations: from everyday life, where analogue cameras have long been replaced by digital ones, to their professional use in medicine, earth sciences, arts, and security applications. Examples of medical imaging tools are MRI (Magnetic Resonance Imaging), PET (Positron Emission Tomography) and CT (computed tomography) for imaging the brain or other organs such as the heart. These imaging tools usually produce noisy or incomplete image data. Hence, before they can be evaluated by doctors, they have to be processed. Keywords in this context are image denoising, image deblurring, image decomposition and image inpainting.

In this talk I will present one of the most successful processing approaches: partial differential equations and variational models. Given a noisy image, its processed (denoised) version is computed as a solution of a PDE or as a minimiser of a functional (variational model). Both of these processes are regularising the given image. In favourable imaging approaches this is done by eliminating high-frequency features (noise) while preserving or even enhancing low-frequency features (object boundaries, edges). After discussing the main principles of this mathematical concept the talk is furnished with computational examples from medical imaging, forensics and the arts.

The slides of the talk are online at http://www.damtp.cam.ac.uk/user/cbs31/MI_Cambridge/MathAndInfo_Network_files/carolaschoenliebtalk.pdf