Carola-Bibiane Schönlieb

Research interests:

  1. -Nonlinear PDEs

  2. -Inverse problems in imaging

  3. -Sparse regularisation

  4. -Optimisation

  5. -Machine learning for inverse problems

  6. -Higher-order PDEs

  7. -Image analysis and processing

  8. -Large-scale and high-dimensional imaging

  9. -Biomedical imaging

  10. -Remote sensing

  11. -Arts restoration


Main research interests

  1. - 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.

  2. -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.

  3. -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.

  4. -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.

  5. -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

  6. -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.

  7. -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.

Image inpainting of a destroyed binary image with a fourth-order PDE

Domain Decomposition for TV Inpainting        Evolution of the Cahn-Hilliard equation

Cambridge Image Analysis (CIA) group

For more information on the group and our research visit CIA group webpage

Scientific activities

  1. -Editorial activities: Associate editor for SIAM Journal on Imaging Sciences, Proceedings of the Royal Society A, European Journal of Applied Mathematics, SIAM Review, IMA Journal of Numerical Analysis, Journal of Mathematical Imaging and Vision, ESAIM Proceedings and Surveys; Member of International Advisory Panel for Inverse Problems; Guest editor for Inverse Problems for Special Issue on Learning in Inverse Problems, 2015/16; Guest Editor for EJAM for Special Issue on PDE for data modelling and analysis, 2016/17.

  2. -Current projects:

  3. Leverhulme Trust project on Unveiling the invisible - mathematics for conservation in arts and humanities. January 2019 - December 2021. PI: C.-B. Schönlieb. CoI: S. Bucklow, A. Launaro, S. Panayotova.

  4. Philip Leverhulme Prize. November 2018 - October 2020. PI: C.-B. Schönlieb.

  5. Unilever & EPSRC IAA Partnership Development Award for Mathematical Image Analysis and Machine Learning for Better Food Microstructures. November 2018 - May 2020. PIs: C.-B. Schönlieb and P. Schütz.

  6. NPL postdoctoral fellowship grant for The mathematics of measurement. March 2018 - February 2021. PIs: A. Forbes and C.-B. Schönlieb.

  7. Global Alliance project on the Theory of Mathematical and Statistical Imaging. January 2017 - December 2018. PI: C.-B. Schönlieb.

  8. EPSRC Centre for Mathematical and Statistical Analysis of Multimodal Clinical Imaging. March 2016 - February 2020. PI: J. Aston. Co-Is: C.-B. Schönlieb (Co-Director), S. Bohndiek, E. Bullmore, N. Burnet, T. Fokas, F. Gilbert, A. Hansen, S. Reichelt, J. Rudd, R. Samworth, G. Treece, G. Williams

  1. Leverhulme Trust project on Breaking the non-convexity barrier. Duration: 01. November 2015 - 31. October 2018. PI: C.-B. Schönlieb. CoI: M. Benning.

- Past projects:

  1. EPSRC grant Nr. EP/M00483X/1 Efficient computational tools for inverse imaging problems. Duration: 01. December 2014 - 30. November 2017. PI: C.-B. Schönlieb. CoI: T. Valkonen.

  2. Alan Turing Institute seed funding for Personalised breast cancer screening. October 2017 - April 2018. PIs: M. van der Schaar and C.-B. Sch\"onlieb.

  3. CCI Collaborative Fund on Assessing the conservation quality of tropical forest unmanned aerial vehicles. Duration: 01. September 2014 -- 01. September 2016. PIs: D. Coomes, J. Lindsell, C.-B. Schönlieb, T. Swinfield.

  4. LMS-Scheme 3 funding for four meetings to be held in the UK on Current frontiers in inverse problems: from theory to applications. Duration: 01. Oktober 2013 - 30. September 2016. PI: C.-B. Schönlieb. CoIs: D. Lesnic, M. Soleimani.

  5. Wellcome Trust/ University of Cambridge Senior ISSF internship for the project Development of Image Analysis Algorithms for Monitoring Forest Health from Aircraft. Duration: 01. May 2014 -- 31. March 2015. PIs: X. Cai, D. Coomes, C.-B. Schönlieb.

  6. EPSRC first grant Nr. EP/J009539/1 Sparse & Higher-order Image Restoration. Duration: 03. May 2012 – 02. May 2014. PI: C.-B. Schönlieb

  7. Royal Society International Exchange Award Nr. IE110314 High-order Compressed Sensing for Medical Imaging. Duration: 01. Jan 2012 – 31. Dec 2013. PI’s: M. Burger & C.-B. Schönlieb. See online news of the University of Münster.

  8. EPSRC / Isaac Newton Trust Small Grant Non-smooth geometric reconstruction for high resolution MRI imaging of fluid transport in bed reactors. Duration: 01. July 2012 - 30. June 2013. PI: C.-B. Schönlieb

  9. Mathworks Academic Support for Development of MATLAB Tools for the Numerical Analysis Tripos. Duration: July-September 2012. P.I.’s: S. Cowley, A. Iserles, C.-B. Schönlieb and A. Shadrin.

Group seminars

  1. Applied and Computational Analysis (ACA) seminar

  2. Cambridge Image Analysis Seminars

  3. Cambridge Image Analysis (CIA) meeting, Every Friday during term time
    from 4-5 pm (please contact for details)

Selected Talks (recent)

  1. Bilevel learning of variational models, ETH Colloqium, Zürich, Switzerland, 8 March 2017.

  2. Seeing More in Pictures - Customised Image Analysis, CCIMI meeting, Cambridge, UK, 21 November 2016.

  3. Fellows short talk, Alan Turing Institute, London, UK, 11 November 2016.

  4. SIAM PDE 2015 Plenary talk Customizing Image Analysis Using Nonlinear Partial Differential Equations

  5. Optimising the optimisers - what is the right image and data model?, Isaac Newton Institute, UK, February 2014.

Organised meetings (recent and upcoming)

  1. Isaac Newton Institute Programme on Variational methods and effective algorithms for imaging and vision, August - December 2017, Isaac Newton Institute, Cambridge, UK. Co-organisers: K. Chen, A. Fitzgibbon, M. Hintermüller, X.-C. Tai.

  2. IMA Conference on Inverse Problems, 19-21 September 2017, Isaac Newton Institute, Cambridge, UK. Co-organisers: P. Ledger, B. Lionheart, C. Sebu.

  3. BIRS Workshop on Optimal Transport meets Probability, Statistics and Machine Learning, 30 April - 5 May 5, 2017, Oaxaca, Mexico. Co-organiser: G. Carlier, M. Cuturi, B. Pass.

  4. Launch of the Cantab Capital Institute for the Mathematics of Information,  Isaac Newton Institute, Cambridge, 9 May 2016 with an inaugural lecture by Ron DeVore.

  5. Launch of EPSRC Centre for Mathematical and Statistical Analysis of Multimodal Clinical Imaging, Isaac Newton Institute, Cambridge, 8 March 2016. Co-organiser: J. Aston.

  1. POEMS workshop on Big Data, Multimodality & Dynamic Models in Biomedical Imaging, Isaac Newton Institute, Cambridge, 9 March 2016. Co-organisers: J. Aston, C. Dyer-Smith, S. Panayotova, S. Reichelt, TGM.

  2. TGM Workshop on Challenges in Dynamic Imaging Data, Isaac Newton Institute, 9-11 June 2015. Co-organisers: J. Aston, TGM.

  1. ICMS Workshop on Gradient flows: from theory to application, 20-24 April 2015, International Centre for Mathematical Sciences (ICMS), Edinburgh, UK. Co-organisers: B. Düring, M.T. Wolfram.

  2. LMS meetings on Current frontiers in inverse problems: from theory to applications. Co-organisers: D. Lesnic, M. Soleimani.
    - LMS inverse day on statistical inverse problems in Bath, 4th November 2016.
    - BIG inverse problems, University of Nottingham, 29 February 2016.
    - Large-scale and nonlinear inverse problems, U. of Edinburgh, 21 September 2015.
    Inverse problems in wave propagation, U. Cardiff, 12 June 2015.
    - Learning in inverse problems, UCL, 9. January 2015.
    - Tomographic reconstructions from boundary data, University of Leeds, 22. September 2014
    - Hybrid and multi-modal imaging, University of Manchester, 4. July 2014.
    - Data assimilation and numerical weather prediction, University of Bath, 7. May 2014.
    - Sparse Regularisation for Inverse Problems, Isaac Newton Institute, Cambridge, 7. February 2014.

Outreach activities

  1. -London Mathematical Society / Gresham College lecture Mathematics can make you fly?, 23 May 2017, Gresham College, London, UK.

  2. -Talks on Seeing More in Images: A Mathematical Perspective, Maths Open Days 2016, Cambridge, UK.

  3. -Talk with Stella Panayotova (Fitzwilliam Museum, Cambridge) on Arts & Science, CRASSH, 11 May 2016. Video.

  4. -Talk on Seeing More in Images: A Mathematical Perspective, University of Cambridge Mathematics and Big Data Showcase, Centre for Mathematical Sciences, Cambridge, UK, 20 April 2016. Video.

  5. -Talk on Seeing More in Images: A Mathematical Perspective, Step Up! Mathematical Problem-Solving Day for Y12 Girls, 18 April 2016, Centre for Mathematical Sciences, Cambridge, UK. Video.

  6. -Mathematical moments: Interview with the Plus Magazine, April 2016. Video. Also featured on

  7. -Plus article on What the eye can't see, April 2016.

  8. -100 seconds science video on What is a Fourier transform, Physicsworld, October 2014.

  9. -Plus article on Restoring profanity, March 2009.