Wednesday, October 28, 2020
12:00 PM -
1:00 PM
Online Event
CMX Lunch Seminar
Series: CMX Lunch Series
Combining knowledge and data driven methods for solving inverse imaging problems - getting the best from both worlds
Carola-Bibiane Schonlieb,
Professor of Applied Mathematics and head of the Cambridge Image Analysis (CIA) group,
Department of Applied Mathematics and Theoretical Physics (DAMTP),
University of Cambridge,
Speaker's Bio:
Main research interests - Noise estimation, model selection & bilevel optimisation: A key issue in image denoising, and in inverse problems as a whole, is the correct choice of data priors and fidelity terms. Depending on this choice, different results are obtained. Several strategies, both physical (dictated by the physics behind the acquisition process) and statistically grounded (e.g. by estimating or learning noise and structure in the data), have been considered in the literature. Recent approaches in the community also propose to learn the model and the parameter choice by bilevel optimisation techniques. -Sparse and higher-order variational & PDE regularization: One of the most successful image processing approaches is PDEs 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). This gives rise to non-smooth, nonlinear terms in the model of possibly high differential order. The total variation regularizer is a typical example in this class. Beyond image denoising, such regularization procedures are successfully applied to image deblurring, inpainting and inverse problems in imaging in general. -High-resolution magnetic resonance imaging & emission tomography: The quality of images reconstructed from measurements acquired with medical imaging tools such as magnetic resonance imaging (MRI) and emission tomography (PET and SPECT) usually suffers from acquisition noise and undersampling. For still being able to reconstruct high-resolution images the solution of the respective inverse problem is equipped with non-smooth regularizers - as outlined above. In this context I am interested in PET and dynamic MRI. -Image Inpainting: Additionally I consider higher-order PDEs in image inpainting. This important task in image processing is the process of filling in missing parts of damaged images based on the information gleaned from the surrounding areas. It is essentially a type of interpolation and is called inpainting. Second order variational inpainting methods (where the order of the method is determined by the derivatives of highest order in the corresponding Euler-Lagrange equation), like total variation (TV) inpainting, have drawbacks as in the connection of edges over large distances or the continuous propagation of level lines into the damaged domain. In an attempt to solve both the connectivity principle and the so called staircasing effect resulting from second order image diffusions, a number of third and fourth order diffusions have been suggested for image inpainting. Among them is Eulers elastica inpainting, inpainting with a modified Cahn-Hilliard equation and so called TV-H^{-1} inpainting to just name a few. -Restoration of artwork: A specific application of inpainting is the restoration of digital photographs from historic artworks. See the following online article for more information on a project which deals with the restoration of medieval frescoes: Restoring profanity -Higher-order PDEs: The study of higher order PDEs is still very young and therefore both their analytical analysis and suitable numerical solutions are challenging problems. A famous higher-order PDE in material sciences is the so called Cahn-Hilliard equation. This equation models phase separation and subsequent coarsening in binary alloys. I studied instabilities of solutions of the Cahn-Hilliard equation and their connection to the Willmore functional. Further I am interested in nonlocal higher order PDEs, like the higher order nonlocal Laplace equation. -Domain decomposition methods: I am interested in domain decomposition methods used in image processing. The following link gives more information on my research in this area: Domain decomposition methods for TV-minimisation.
Main research interests - Noise estimation, model selection & bilevel optimisation: A key issue in image denoising, and in inverse problems as a whole, is the correct choice of data priors and fidelity terms. Depending on this choice, different results are obtained. Several strategies, both physical (dictated by the physics behind the acquisition process) and statistically grounded (e.g. by estimating or learning noise and structure in the data), have been considered in the literature. Recent approaches in the community also propose to learn the model and the parameter choice by bilevel optimisation techniques. -Sparse and higher-order variational & PDE regularization: One of the most successful image processing approaches is PDEs 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). This gives rise to non-smooth, nonlinear terms in the model of possibly high differential order. The total variation regularizer is a typical example in this class. Beyond image denoising, such regularization procedures are successfully applied to image deblurring, inpainting and inverse problems in imaging in general. -High-resolution magnetic resonance imaging & emission tomography: The quality of images reconstructed from measurements acquired with medical imaging tools such as magnetic resonance imaging (MRI) and emission tomography (PET and SPECT) usually suffers from acquisition noise and undersampling. For still being able to reconstruct high-resolution images the solution of the respective inverse problem is equipped with non-smooth regularizers - as outlined above. In this context I am interested in PET and dynamic MRI. -Image Inpainting: Additionally I consider higher-order PDEs in image inpainting. This important task in image processing is the process of filling in missing parts of damaged images based on the information gleaned from the surrounding areas. It is essentially a type of interpolation and is called inpainting. Second order variational inpainting methods (where the order of the method is determined by the derivatives of highest order in the corresponding Euler-Lagrange equation), like total variation (TV) inpainting, have drawbacks as in the connection of edges over large distances or the continuous propagation of level lines into the damaged domain. In an attempt to solve both the connectivity principle and the so called staircasing effect resulting from second order image diffusions, a number of third and fourth order diffusions have been suggested for image inpainting. Among them is Eulers elastica inpainting, inpainting with a modified Cahn-Hilliard equation and so called TV-H^{-1} inpainting to just name a few. -Restoration of artwork: A specific application of inpainting is the restoration of digital photographs from historic artworks. See the following online article for more information on a project which deals with the restoration of medieval frescoes: Restoring profanity -Higher-order PDEs: The study of higher order PDEs is still very young and therefore both their analytical analysis and suitable numerical solutions are challenging problems. A famous higher-order PDE in material sciences is the so called Cahn-Hilliard equation. This equation models phase separation and subsequent coarsening in binary alloys. I studied instabilities of solutions of the Cahn-Hilliard equation and their connection to the Willmore functional. Further I am interested in nonlocal higher order PDEs, like the higher order nonlocal Laplace equation. -Domain decomposition methods: I am interested in domain decomposition methods used in image processing. The following link gives more information on my research in this area: Domain decomposition methods for TV-minimisation.
Inverse problems in imaging range from tomographic reconstruction (CT, MRI, etc) to image deconvolution, segmentation, and classification, just to name a few. In this talk I will discuss approaches to inverse imaging problems which have both a mathematical modelling (knowledge driven) and a machine learning (data driven) component. Mathematical modelling is crucial in the presence of ill-posedness, making use of information about the imaging data, for narrowing down the search space. Such an approach results in highly generalizable reconstruction and analysis methods which come with desirable solutions guarantees. Machine learning on the other hand is a powerful tool for customising methods to individual data sets. Highly parametrised models such as deep neural networks in particular, are powerful tools for accurately modelling prior information about solutions. The combination of these two paradigms, getting the best from both of these worlds, is the topic of this talk, furnished with examples for image classification under minimal supervision and for tomographic image reconstruction.
Event Sponsors:
For more information, please contact Jolene Brink by phone at 6263952813 or by email at [email protected] or visit CMX Website.