HD-Tomo: High-Definition Tomography
This project was funded by an Advanced Grant from the
European Research Council
The project started August 1, 2012 and it expired July 31, 2017.
It was located at the
Scientific Computing in the
Applied Mathematics and Computer Science (DTU Compute) at
the Technical University of
Follow this link
to learn more about the project and its objectives.
Additional information is available in the
Midway Report, and a summary of the
project and the scientific results is available in the
Final Activity Report.
Summary of the Achievements of the Project
With computed tomography (CT) we can see inside objects - we send signals
through an object and measure the response, from which we compute an image
of the object's interior. Medical doctors can look for cancer, physicists
can study microscopic details of new materials, engineers can identify
internal defects in pipes, and security personnel can inspect luggage for
It is of vital importance that the images are as sharp, detailed and reliable
as possible, so scientists, engineers, doctors etc. can make the correct
decisions. To achieve high-definition tomography - sharper images with more
reliable details - we must use prior information consisting of accumulated
knowledge about the object.
Overall Outcome: Insight and Framework
Previous efforts were often based on ad-hoc techniques and naive algorithms with limited applications and ill-defined results. This project focused on obtaining deeper insight and developing a rigorous framework. We carefully analyzed the underlying mathematical problems and algorithms, and we developed new theory that provides better understanding of their challenges and possibilities. This insight allowed us to develop a solid framework for precisely formulated CT algorithms that compute much more well-defined results. We laid the groundwork for the next generation of rigorously defined algorithms that will further optimize the use of prior information.
The road to this insight involved specific case studies related to the formulation and use of prior information, involving such applications as X-ray phase-contrast tomography, fusion plasma physics, and underwater pipeline inspection. Below we list the highlights of these cases.
Understanding of Sparsity for Low-Dose CT
We characterize how the prior information that an object is "simple" - in
mathematical terms, sparse - allows us to compute reliable images from very
limited data, and we show that the sufficient amount of CT data depends in a simple way on the sparsity. This is essential in medical and engineering CT where one must minimize the X-ray dose and shorten measurement time.
Superior Localization in Electrical Impedance Tomography
By incorporation the prior information that the details stand out from the background, we can now compute images with superior localization and contrast. Moreover we developed new theory that, for the first time, precisely describes the obtainable resolution and the optimal measurement configuration. This is essential in industrial process monitoring where measurement constraints often limit the amount of data.
Superior Use of Textural Training Images
For textural images, we developed a new mathematical and computational framework that is superior to other methods for limited-data. It is particularly suited for computing reliable segmentations of these images. To do this we use prior information in the form of training images that the computed image must resemble.
Novel Convergence Analysis of Iterative Methods
We developed novel theoretical insight into the advantages and limitations of the iterative methods that are required for 3D tomography computations. This insight guided the development of new software suited for many-core and GPU computers, as well as public-domain software with model implementations of these algorithms.
Correct Handling of Noise
We formulate correct mathematical models for the measurement noise and we develop new computational algorithms especially suited for using prior information about non-Gaussian noise. We show that these noise priors improve both the algorithms and the images, compared to the standard algorithms that are based on cruder models.
Novel Use of Prior Information about Structure
Structural prior information states that the image contains visual
structures, e.g., texture along certain directions.
Incorporation of this kind of information prompted us to develop
new anisotropic higher-order techniques that avoid the unwanted artifacts
of traditional methods (such as total variation).
Motion Modeling for Dynamic Tomography
The precised motion behavior of an object is unknown, and therefore
we jointly perform motion estimation and image reconstruction.
We derive a motion model to handle the challenging task of
representing formation and closing af cracks in the object.
- Professor Anders Bjorholm Dahl, DTU Compute.
- Kristoffer Hoffmann. PhD project: Reconstruction Methods for Inverse
Problems with Partial Data (finished 2014).
- Associate Professor Mirza Karamehmedovic, DTU Compute.
- Federica Sciacchitano,
now with University of Genoa.
PhD project at DTU Compute:
Image Reconstruction Under Non-Gaussian Noise (September 2013 - October 2016),
- Angeliki Xenaki, now with GN ReSound.
PhD project: High-Resoluiton Imaging Methods in
Arrays Signal Processing (finished 2015).
- Jürgen Frikel,
now with University of Regensburg.
H.C. Ørsted Post Doc with DTU Compute (April 2015 - August 2016).
- Ivan G. Kazantsev, DTU Compute and DTU Physics (project ended June 2015).
Other Members of the Team
- Post Doc Jakob Sauer Jørgensen, now with Manchester University.
Project: Computations with Sparse Representations (September 1, 2013 -
July 31, 2017)
Sara Soltani, now with
Fingerprint Cards ApS.
PhD project: Training Sets in Large-Scale Reconstruction Methods
(September 1, 2012 - August 31, 2015),
Mikhail (Mike) Romanov,
PhD project: Statistical Priors in Variational Reconstruction Methods
(November 1, 2012 - October 31, 2015),
Henrik Garde, now with Aalborg University.
PhD project: Prior Information in Inverse Boundary Problems
(March 1, 2013 - February 29, 2016),
Rasmus Dalgas Kongskov, now with 3Shape A/S.
PhD project: Segmentation-Driven Tomographic Reconstruction
(started September 1, 2014),
Om Aggrawal, now with University of L\uuml;beck.
PhD project: Priors for Temporal Tomographic Image Reconstruction
(April 1, 2015 - May 31, 2018).
Post Doc Lauri Harhanen, now with KaVo Kerr, Finland.
Project: Formulation and Application of Priors in Spectral CT
(May 1, 2015 - July 31, 2016).
Hans Martin Kjer, now with DTU Compute and the Danish Research Centre
for Magentic Resonance.
Project: Joint CT Reconstruction and Segmentation (January 1, 2016 -
July 31, 2017).
Related Research Projects
Activities - Scientific Publications, Visitors, etc.
Software and Data
AIR Tools II,
a Matlab package of algebraic iterative reconstruction methods - improved
a Matlab pacage of iterative regularization methods and test problems
for large-scale linear inverse problems.
- DLCT-Toolbox, a Matlab package for the dictionary learning approach to
tomograhic image reconstruction, written by Sara Soltani:
zip file and
- DTGV-Reg, a Matlab package for Directional Total Variation (DTV) and
Directional Total Generalized Variation (DTGV) regularization, written
by Rasmus Dalgas Kongskov:
zip file and
- Matlab software for handling non-Gaussian noise: available from the
- Software for kernel regularization and ring reduction:
available from the homepage of
- SparseBeads Dataset
for benchmarking of sparsity-regularized
reconstruction methods (Jakob Sauer Jørgensen et al.).
Accepted and Published Papers
- H. O. Aggrawal, M. S. Andersen, S. Rose, and E. Y. Sidky,
A convex reconstruction model for X-ray tomographic imaging with
uncertaint flat-fields, IEEE Trans. Comput. Imaging,
10.1109/TCI.2017.2723246 (open access).
- M. S. Andersen and P. C. Hansen, Generalized row-action methods
for tomographic imaging, Numerical Algorithms, 67 (2013), pp. 121-144;
- G. Bal, K. Hoffmann, and K. Knudsen,
Propagation of singularities for linearized hybrid data impedance
tomography, Inverse Problems, 34 (2917), 024001 (19pp);
- L. Borg, J. S. Jørgensen, J. Frikel, and J. Sporring,
Reduction of variable-truncation artifacts from beam occlusion
during in situ X-ray tomography, Meas. Sci. Technol., 28
(2017), 124004 (19pp);
- D. Chen, M. E. Kilmer, and P. C. Hansen, "Plug-and-play"
Electronic Transactions on
Numerical Analysis, 41 (2014), pp. 465-477 (open access).
- T. Chen, M. S. Andersen, L. Ljung, A. Chiuso, and G. Pillonetto,
System identification via sparse multiple kernel-based regularization
using sequential convex optimization techniques,
IEEE Trans. on Automatic Control, 59 (2014), pp. 2933-2945.
- V. A. Dahl, A. B. Dahl, and P. C. Hansen,
Computing segmentations directly from X-ray projection data via
parametric deformable curves, Meas. Sci. Technol., 29 (2018),
- F. Delbary and K. Knudsen, Numerical nonlinear complex geometrical
optics for the 3D Calderón problem, Inverse Problems and
Imaging, 8 (2014), pp. 991-1012;
- Y. Dong, H. Garde, and P. C. Hansen, R3GMRES: including
prior information in GMRES-type methods for discrete inverse problems,
Electronic Transactions on Numerical Analysis,
42 (2014), pp. 136-146 (open access).
- Y. Dong, T. Görner, and S. Kunis, An algorithm for total
variation regularized photoacoustic imaging,
Adv. Comput. Math., June 2014;
- Y. Dong, P. C. Hansen, and H. M. Kjer, Joint CT reconstruction
and segmentation with discriminative dictionary learning,
IEEE Trans. Computational Imaging, online 2018;
- Y. Dong and T. Zeng, A convex variational model for restoring
blurred images with multiplicative noise, SIAM J. Imaging Sci.,
6 (2013), pp. 1598-1625;
- Y. Dong and T. Zeng, New hybrid variational recovery model for
blurred images with multiplicative noise, East Asian Journal on
Appl. Math., 3 (2013), pp. 263-282;
- T. Elfving and P. C. Hansen,
Unmatched projector/backprojector pairs: perturbation and convergence
analysis, SIAM J. Sci. Comp., 40 (2018), pp. A573-A591;
- T. Elfving, P. C. Hansen, and T. Nikazad,
Convergence analysis for column-action methods in image reconstruction,
Numerical Algorithms, 74 (2016),
10.1007/s11075-016-0176-x. Erratum (Fig. 3 was incorrect):
- T. Elfving, P. C. Hansen, and T. Nikazad, Semi-convergence
properties of Kaczmarz's method, Inverse Problems, 30 (2014),
This paper was selected to be part of the journal's
- H. Garde, Comparison of linear and non-linear
monotonicity-based shape reconstruction using exact matrix
characterizations, Inverse Problems in Science and Engineering,
published online 2017,
- H. Garde and K. Knudsen, Distinguishability revisited: depth dependent
bounds on reconstruction quality in electrical impedance tompography
SIAM J. Appl. Math., 77 (2017);
- H. Garde and K. Knudsen, Sparsity prior for electrical impedance
tomography with partial data, Inverse Problems in Science and
Engineering, 24 (2016), pp. 524-541;
- H. Garde and S. Staboulis, Convergence and regularization for
monotonicity-based shape reconstruction in electrical impedance
tomography, Numer. Math., 135 (2017), pp. 1221-1251;
- S. Gazzola, P. C. Hansen, and J. G. Nagy,
IR Tools - A MATLAB package of iterative regularization methods
and large-scale test problems, Numerical Algorithms, online 2018;
- P. C. Hansen and J. S. Jørgensen,
AIR Tools II: algebraic iterative reconstruction methods, improved
implementation, Numerical Algorithms, 79 (2018), pp. 107-137;
- P. C. Hansen, J. G. Nagy, and K. Tigkos, Rotational image
deblurring with sparse matrices, BIT Numerial Mathematics, 54 (2014),
pp. 649-671, DOI:
- K. Hoffmann and K. Knudsen, Iterative reconstruction methods
for hybrid inverse problems in impedance tomography, Sensing and
Imaging, 15 (2014), pp. 1-27;
- J. S. Jørgensen, S. B. Coban, W. R. B Lionheart, S. A. McDonald, and
P. J. Withers,
SparseBeads data: benchmarking sparsity-regularized computed
tomography, Meas. Sci. Technol., 28 (2017), 124005 (18pp);
- J. S. Jørgensen, C. Kruschel, and D. Lorenz,
Testable uniqueness conditions for empirical assessment of
undersampling levels in total variation-regularized x-ray CT,
Inverse Problems in Science and Engineering, 23 (2014), pp. 1283-1305;
- J. S. Jørgensen and E. Y. Sidky, How little data is enough?
Phase-diagram analysis of sparsity-regularized X-ray CT, Phil.
Trans. Royal Soc. A, 373 (2015), 20140387 (special issue "X-ray tomographic
reconstruction for materials science");
Data and code to reproduce the results are available from
- J. S. Jørgensen, E. Y. Sidky, P. C. Hansen, and X. Pan,
Empirical average-case relation between undersampling and sparsity
in X-ray CT, Inverse Problems and Imaging, 9 (2015), pp. 431-446;
10.3934/ipi.2015.9.431 (open access).
- M. Karamehmedovic and K. Knudsen, Inclusion estimation from a
single electrostatic boundary measurement, Inverse Problems, 29 (2013);
- R. D. Kongskov and Y. Dong,
Tomographic reconstruction methods for decomposing directional
To appear in Inverse Problems and Imaging.
- R. D. Kongskov, J. S. Jørgensen, H. F. Poulsen, and P. C. Hansen,
Noise robustness of a combined phase retrieval and reconstruction method
for phase-contrast tomography, J. Optical Society of America A,
33 (2016), pp. 447-454;
- J.-J. Mei, Y. Dong, T.-Z. Huang, and W. Yin,
Cauchy noise removal by nonconvex ADMM with convergence guarantees,
J. Scientific Computing, to appear.
- S. K. Pakazad, M. S. Andersen, and A. Hansson, Distributed
solutions for loosely coupled feasibility problems using proximal
splitting methods, Optimization Methods and Software, 30 (2015),
- V. Paoletti, P. C. Hansen, M. F. Hansen, and M. Fedi,
A computationally efficient tool for assessing the depth
resolution in large-scale potential-field inversion,
Geophysics, 79 (2014), pp. A33-A38;
- T. Ramos, J. S. Jørgensen, and J. W. Andreasen,
Automated angular and translational tomographic alignment and
application to phase-contrast imaging,
J. Optical Society of America A, 34 (2017), pp. 1830-1843;
- N. A. B. Riis, J. Frøsig, Y. Dong, and P. C. Hansen,
Limited-data X-ray CT for underwater pipeline inspection,
Inverse Problems, 34 (2018), 034002 /16pp);
- M. Romanov, A. B. Dahl, Y. Dong, and P. C. Hansen,
Simultaneous tomographic reconstruction and segmentation with
class priors; Inverse Problems in Science and Engineering, 24 (2015),
- S. Rose, E. Y. Sidky, X. Pan, and M. S. Andersen,
Noise properties of CT images reconstructed by use of constrained
total-variation, data-discrepancy minimization, Medical Physics,
42 (2015), pp. 2690-2698;
- M. Salewski, B. Geiger, A. Jacobsen, P. C. Hansen + 12,
High-definition velocity-space tomography of fast-ion dynamics,
Nuclear Fusion, 56 (2016), DOI:
- M. F. Schmidt, M. Benning, and C.-B. Schönlieb,
Inverse scale space decomposition, Inverse Problems, 34 (2018),
- F. Sciacchitano, Y. Dong, and M. S. Andersen,
Total variation based parameter-free model for Impulse noise removal,
Numerical Mathematics: Theory, Methods and Applications, 10 (2017),
pp. 186-204, DOI:
- F. Sciacchitano, Y. Dong, and T. Zeng,
Variational approach for restoring blurred images with Cauchy noise,
SIAM J. Imaging Sc., 8 (2015), pp. 1896-1922,
- S. Soltani, M. S. Andersen, and P. C. Hansen,
Tomographic image reconstruction using training images,
Journal of Computational and Applied Mathematics, 313 (2017), pp. 243-258;
- S. Soltani, M. E. Kilmer, and P. C. Hansen,
A tensor-based dictionary learning approach to tomographic
BIT Numerical Mathematics, 56 (2016), pp. 1425-1454;
This paper is mentioned on the front page of SIAM News Vol. 50 Issue 6, 2017:
A Computationally Efficient
Solution of Large-Scale Image Reconstruction Problems.
- Y. Sun, M. S. Andersen, and L. Vandenberghe, Decomposition in
conic optimization with partially separable structure,
SIAM J. Optimiz., 24 (2014), pp. 873-897;
- H. H. B. Sørensen and P. C. Hansen, Multicore performance of block
algebraic iterative reconstruction methods, SIAM J. Sci. Comp,
36 (2014), pp. C524-C546.
- P. Weiss, P. Escande, and Y. Dong,
Contrast Invariant SNR, DTU Compute
Technical Report 2016-09.
To appear in International Journal of Computer Vision.
Refereed Conference Proceedings
- H. Garde and K. Knudsen, 3D reconstruction for partial data
electrical impedance tomography using a sparsity prior,
Dynamical Systems, Differential Equations and Applications - AIMS
Proceedings (2015), pp. 495-504; open access
- J. S. Jørgensen, S. B. Coban, W. R. B. Lionheart, and P. J. Withers,
Effect of sparsity and exposure on total variation
regularized X-ray tomography from few projections,
Proc. 4th International Conference on
Image Formation in X-Ray Computed Tomography, Bamberg, Germany,
2016, pp. 279-282.
- I. G. Kazantsev, U. L. Olsen, H. F. Poulsen, and P. C. Hansen,
A spectral geometrical model for Compton scatter tomography
based on the SSS approximation,
of the 4th International Conference
on Image Formation in X-Ray Computed Tomography, July 18-22, Bamberg,
Germany, pp. 577-580.
- H. M. Kjer, Y. Dong, and P. C. Hansen,
User-friendly simultaneous tomographic reconstruction and segmentation
with class priors, Proc. SSVM 2017, pp. 260-270, Springer, 2017;
- R. D. Kongskov and Y. Dong,
Directional total generalized variation regularization for impulse
noise removal, Proc. SSVM 2017, pp. 221-231, Springer, 2017;
- S. Rose, M. S. Andersen, E. Y. Sidky, and X. Pan, Application of
incremental algorithms to CT reconstruction for sparse-view, noisy data,
Proc. 3rd International Conference on Image Formation in X-Ray Computed
Tomography, 2014, pp. 351-354.
- L. Borg, J. Frikel, J. S. Jørgensen, and E. T. Quinto,
Theorems that characterize artifacts for arbitrary limited
X-ray CT data,
- Y. Dong, P. C. Hansen, M. E. Hochstenbach, and N. A. B. Riis,
Fixing nonconvergence of algebraic iterative reconstruction with
an unmatched backprojection, submitted to SISC.
- H. Garde and S. Staboulis,
The regularized monotonicity method: detecting irregular indefinite
inclusions, 2017; submitted to Inverse Problems.
- P. C. Hansen, Y. Dong, and K. Abe,
Hybrid enriched bidiagonalization for discrete ill-posed problems,
submitted to J. Numer. Lin. Alg. Appl.
- L. Borg, J. S. Jørgensen, and J. Sporring,
Towards characterizing and reducing artifacts caused by varying
Technical Report 2017/1,
Department of Computer Science, University of Copenhagen, 2017.
- M. Burger, Y. Dong, and F. Sciacchitano,
Bregman cost for non-Gaussian noise, DTU Compute
Technical Report 2016-8.
- P. C. Hansen and K. Abe,
LBAS: Lanczos bidiagonalization with subspace augmentation for
discrete inverse problems,
Technical Report 2017-03.
- R. D. Kongskov, Y. Dong, and K. Knudsen,
Directional total generalized variation regularization,
- M. Romanov, A. B. Dahl, Y. Dong, and P. C. Hansen,
Relaxed Simultaneous Tomographic Reconstruction and Segmentation with
Class Priors for Poisson Noise, DTU Compute
Technical Report 2015-6.
- M. Romanov, P. C. Hansen, and A. B. Dahl,
A Parameter Choice Method for Simultaneous Reconstruction and
Segmentation, DTU Compute
Technical Report 2015-5.
- S. Soltani, Studies of Sensitivity in the Dictionary Learning Approach
to Computed Tomography: Simplifying the Reconstruction Problem, Rotation,
DTU Compute Technical Report 2015-4, July 2, 2015.
- Kuniyoshi Abe, Gifu Shotoku Gakuen University.
- Joost Batenburg, CWI, Amsterdam.
- Martin Benning, Univ. of Cambridge
- Marta Betcke, UCL.
- Raymond Chan, Chinese Univ. of Hong Kong.
- Julianne Chung, Virginia Tech.
- Fabrice Delbary, Univ. of Genova.
- Tommy Elfving, Linkoping Univ.
- Oliver G. Ernst, TU Chemnitz.
- David Franck, TU Munich.
- Jürgen Frikel, TU Munich.
- Silvia Gazzola, Univ. of Bath.
- Sarah Hamilton, Helsinki University.
- Andreas Hauptmann, Helsinki University.
- Michiel Hochstenbach, Technische Universiteit Eindhoven.
- Yinyi (Larry) Hu, Emory Univ.
- Simon Hubmer, Johannes Kepler University, Linz.
- Daniil Kazantsev, Manchester Univ.
- Ivan Kazantsev, Novisibirsk Scientific Centre.
- Ville Kolehmainen, Univ. of Eastern Finland.
- Tobias Lasser, Technische Universität München.
- Julia Mrongowius, Univ. of Lübeck.
- James G. Nagy, Emory University.
- Touraj Nikazad, Iran Univ. of Science and Technology.
- Mila Nikolova, Universite Paris-Saclay.
- Willem Jan Palenstijn, CWI, Amsterdam.
- Valeria Paoletti, Univ, of Naples Federico II.
- Stefania Petra, Univ. of Heidelberg.
- Martin Plesinger, Technical Univ. of Liberec, Czech Republic.
- David Rigie, University of Chicago.
- Carola-Bibiane Schönlieb, Univ. of Cambridge.
- Ekatarina Sherina, Tomsk State Univ.
- Gabriele Steidl, Univ. of Kaiserslautern.
- Gabriele Stocchino, Univ. of Padua.
- Tristan van Leeuwen, Universiteit Utrecht.
- Pierre Weiss, Institut des Technologies Avancées du Vivant.
- Matthias Wieczovek, Technische Universität München.
- Tieyong Zeng, Hong Kong Baptist University.
- Xiaoqun Zhang, Shanghai Jiao Tong University.
Seminars, Workshops, PhD Courses, etc.
- Training School:
Algebraic Reconstruction Methods in Tomography, April 4-6, 2016,
as part of the EXTREMA COST
- Training School
Scientific Computing for X-Ray Computed Tomography,
January 2017, as part of the EXTREMA COST
Insights and Algorithms for Incomplete Data Tomography,
September 14, 2016:
HD-Tomo Days, April 6-8,
2016. The program with abstracts is available
- Workshop: Sparse
Tomo Days, March 26-28, 2014. The abstracts are available
- Three seminars on Microlocal characterization in limited-angle
tomography Jürgen Frikel:
- Five seminars by Prof. Bill Lionheart, October 2013 and July 2014:
- Online short-course: Algebraic Iterative Reconstruction Methods.
- MSc course
Introduction to Inverse Problems (at DTU).
- PhD course
Scientific Computing for X-Ray Computed Tomography (at DTU).
- PhD course Measurement Techniques and Mathematical Modeling
in Tomography in the spring of 2014.
- Weekly seminars by project members, visitors, and invited speakers.
Bachelor and Master Projects
- Sparsity Regularization for Electrical Impedance Tomography (MSc), 2013;
this thesis by H. Garde won the
award by the Danish Mathematical Society.
- Sparsity Regularization for Inverse Problems using Curvelets (MSc), 2013.
- Magnetic Resonance Electrical Impedance Tomography for Anisotropic
Conductivity Distribution (MSc), 2013.
- Characterization and Modeling of Structured Noise in Seismic Reflection
Data (MSc), 2013.
- Shape Optimization for Electrical Impedance Tomography (MSc), 2013.
- Computational Methods for Multi-Spectral Tomography (BSc),
- Multi-Spectral Tomography: Models and Computational Methods (BSc),
- Frequency Dependent Stability for Inverse Boundary Value Problems (MSc),
- Image Denoising and Tomography for Non-Linear Diffusion (MSc), 2014.
- Simulated Phase Contrast Tomography Experiments Using Total Variation
Regularization (MSc), June 2014; joint with DTU Physics.
Block Algebraic Methods for 3D Image Reconstructions on GPUs (MSc),
Robust Computed Tomography with Incomplete Data (BSc),
2015; joint with School of Mathematics, Manchester University.
- GPU Implementation of a Toolbox for Tomographic Reconstruction (BSc),
- Linearized Acousto-Electric Impedance Tomography (MSc), 2015.
- Material Decomposition Using Spectral Tomography (BSc), 2016.
- Hybrid Data Impedance Tomogarphy with the Complete Electrode Model (MSc),
- Preconditioning for PDE-Constrained Optimization Problems (MSc), 2016.
- Computational Methods for Hybrid Impedance Tomography (MSc), 2016.
for Block-Iterative Tomography Reconstructions (MSc), July 2016.
Iterative Tomographic Reconstruction with Priorconditioning (BSc),
- Computed Tomography for Region-of-Interest Problems with Limited Data (MSc),