FWF Project P29192

Our perception of the world is, to a great extent, visual.  Its manifestation, images, are an integral part of human culture.  With today's technology, images can digitally be acquired, processed and stored, enabling imaging as a scientific discipline.  Imaging sciences increasingly affect our everyday lives, playing a role for digital photo and video technology, for instance capturing our holiday memories, and being a crucial part of modern diagnostic medical technologies such as computed tomography (CT) and magnetic resonance imaging (MRI).

Mathematical imaging as scientific discipline comprises studying the translation of acquired data to a meaningful visual representation.  This task is far from being trivial, in particular, in situations where there is no direct relation between visual representation and measured data.  For tomography applications such as CT or MRI, where one aims at creating an image of the inside of a body, this is indeed the case.  Additionally, difficulties arise from the measurements being corrupted by noise and potentially highly incomplete.  The translation to a clean, accurate image, say, of the heart of a patient, requires solid mathematics and constitutes a challenging research problem.

Variational methods contribute significantly to the progress towards the solution of such problems.  They base on efficient mathematical models that can be transferred into computer programs performing the actual calculations.  These models incorporate the relation between the image one aims to recover and the measurements, but also provide an abstract description of the qualitative properties the image satisfies.  The latter, which is called regularization, is the key ingredient for solving the difficulties associated with perturbed and incomplete data.  Its choice is challenging as it is the decisive factor for the performance of the method. Nonetheless, once a good regularization approach has been found, it can broadly be applied.  In MRI, for instance, this allows to obtain reconstructions from only 10% of the data that is normally required and, consequently, for shorter scan time.  Without appropriate regularization, such results would not be possible, making respective research an important topic within variational imaging.

In the past years, great progress was achieved with seemingly very different regularization approaches.  The effectiveness of those, however, is in fact driven by a similar underlying structure.  This structure bears great potential in unifying and extending the state of the art.  The project's goal is to provide, in theory and application, such a unification and extension by virtue of regularization graphs.  It is designed to be easily transferable into application, helping practitioners to push today's limits for reconstruction in diverse fields of imaging sciences.  Such advances could lay the foundations for concrete benefits, for instance, a scan-time reduction in MRI that enables new real-time applications.

• K. Bredies, M. Carioni, M. Holler.
  Regularization Graphs — A unified framework for variational regularization of inverse problems.
  Inverse Problems, 2022.
  (Open access, preprint arXiv:2111.03509)
   
• K. Bredies, M. Carioni, S. Fanzon, F. Romero.
  A generalized conditional gradient method for dynamic inverse problems with optimal transport regularization.
  Foundations of Computational Mathematics, 2022.
  (Open access, preprint arXiv:2012.11706)
   
• K. Bredies, M. Carioni, S. Fanzon, D. Walter.
  Linear convergence of accelerated generalized conditional gradient methods.
  Submitted for publication, 2021.
  (Preprint arXiv:2110.06756)
   
• K. Bredies, M. Carioni, S. Fanzon and F. Romero.
  On the extremal points of the ball of the Benamou-Brenier energy.
  Bulletin of the London Mathematical Society, 53(5):1436-1452, 2021.
  (Open access, preprint arXiv:1907.11589)
   
• K. Bredies, M. Carioni, S. Fanzon.
  A superposition principle for the inhomogeneous continuity equation with Hellinger-Kantorovich-regular coefficients.
  Zur Veröffentlichung eingereicht, 2020.
  (Preprint arXiv:2007.06964)
   
• K. Bredies, M. Holler.
  Higher-order total variation approaches and generalisations.
  Inverse Problems, 36(12):123001, 2020.
  (Open access, preprint arXiv:1912.01587)
   
• K. Bredies, R. Nuster, R. Watschinger.
  TGV-regularized inversion of the Radon transform for photoacoustic tomography.
  Biomedical Optics Express, 11(2):994-1019, 2020.
  (Open access)
   
• K. Bredies, M. Carioni.
  Sparsity of solutions for variational inverse problems with finite-dimensional data.
  Calculus of Variations and Partial Differential Equations, 59:14, 2020.
  (Open access, preprint arXiv:1809.05045)
   
• R. Huber, G. Haberfehlner, M. Holler, G. Kothleitner, K. Bredies.
  Total Generalized Variation regularization for multi-modal electron tomography.
  Nanoscale, 11:5617-5632, 2019.
  (Open access)
   
• Y. Gao, K. Bredies.
  Infimal Convolution of Oscillation Total Generalized Variation for the Recovery of Images with Structured Texture.
  SIAM Journal on Imaging Sciences, 11(3):2021-2063, 2018.
  (Open access, preprint arXiv:1710.11591)
   
• K. Bredies, M. Holler, M. Storath, A. Weinmann.
  Total Generalized Variation for Manifold-valued Data.
  SIAM Journal on Imaging Sciences, 11(3):1785-1848, 2018.
  (Open access, preprint arXiv:1709.01616)
   

• M. Holler, R. Huber, F. Knoll.
  Coupled regularization with multiple data discrepancies.
  Inverse Problems, 34(8):084003, 2018.
  (Open access, preprint arXiv:1711.11512)

• M. Hintermüller, M. Holler, K. Papafitsoros.
  A function space framework for structural total variation regularization with application in inverse problems.
  Inverse Problems, 34(6):064002, 2018.
  (Open access, preprint arXiv:1710.01527)

• Organization of the minisymposium: Variational Methods for Inverse Problems in Imaging
  (10th International Conference on Inverse Problems: Modeling and Simulation)
  22-28 May 2022 in Cirkewwa/Mellieha (Malta)
   
• 29th IFIP TC7 Conference on System Modelling and Optimization
  30 August-3 September 2021 in Quito (Ecuador)
   
• SIAM Conference on Imaging Science (IS20)
  6-17 July 2020 (virtual)
   
• 1st Austrian Calculus of Variations Day
  17-18 October 2019 in Vienna (Austria)
   
• Signal Processing with Adaptive Sparse Structured Representations (SPARS)
  1-4 July 2019 in Toulouse (France)
   
• 9th ASEM Workshop for Advanced Electron Microscopy
  25-26 April 2019 in Graz (Austria)
   
• 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM)
  18-22 February 2019 in Vienna (Austria)
   
• Variational methods and optimization in imaging
  4-8 February 2019 in Paris (France)
   
• 5th European Conference on Computational Optimization (EUCCO)
  10-12 September 2018 in Trier (Germany)
   
• Organization of the minisymposium: Variational Approaches for Regularizing Nonlinear Geometric Data
  (SIAM Conference on Imaging Science (IS18))
  5-8 June 2018 in Bologna (Italy)
   
• SIAM Conference on Imaging Science
  5-8 June 2018 in Bologna (Italy)
   
• 9th International Conference "Inverse Problems: Modeling & Simulation"
  21-25 May 2018 in Cirkewwa/Mellieha (Malta)
   
• Parisian Seminar on the Mathematics of Imaging
  8 Mai 2018 in Paris (France)
   
• 8th ASEM Workshop for Advanced Electron Microscopy
  26-27 April 2018 in Vienna (Austria)
   
• Mathematics and Image Analysis (MIA)
  15-17 January 2018 in Berlin (Germany)
   
• Isaac Newton Institute Workshop "Generative models, parameter learning and sparsity"
  30 October-3 November 2017 in Cambridge (United Kingdom)
   
• Isaac Newton Institute Workshop "Variational methods, new optimisation techniques and new fast numerical algorithms"
  4-8 September 2017 in Cambridge (United Kingdom)