Mathematical Approaches to Image Processing

Schönlieb began by studying partial differential equations, including the Cahn–Hilliard equation, which models how phases separate in alloys. She discovered that related mathematics could restore images by filling in damaged or occluded areas, and image restoration became the focus of her PhD.

In inverse imaging, the measurements may be projections rather than images. For CT, X-rays produce line integrals through the body, and an algorithm reconstructs the image from them. High resolution, limited measurements to reduce radiation, and noisy data make that reconstruction difficult.

Denoising methods need to remove noise while preserving edges, which carry information about boundaries between objects. Total variation is one hand-crafted approach; neural networks can perform better on familiar data but may fail on unfamiliar images or scanner data. Schönlieb describes combining learned parameters with structured models to retain guarantees and make algorithms easier to analyze.

Schönlieb describes academic collaborations with clinicians, medical physicists, and chemical engineers to reconstruct high-resolution medical images from limited data, including images of changing processes. Forest researchers use aerial, hyperspectral, and LiDAR data to study tree materials and structure. She also worked on virtual restoration of frescoes and illuminated manuscripts, allowing people to explore possible restorations without physically altering the originals.

For someone with a computer science background, Schönlieb recommends starting with foundational books on image processing, then moving on to current research. She also points to applied mathematics lectures and talks from UCLA, including work by Stan Osher and Andrea Bertozzi.