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 Section for Scientific Computing in the Department of Applied Mathematics and Computer Science (DTU Compute) at the Technical University of Denmark (DTU).

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 suspicious items.

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.

Permanent Team Members

All members are from the Scientific Computing section.

Associated Members

Other Members of the Team

Visiting and Honorary Professors

Professor Bill Lionheart from Manchester University was visiting professor at DTU Compute and associated with this project. He was funded by a visiting professor scholarship from the Otto Mønsted Fonden. He visited us in October 2013 and again in June and July 2014, and traveled to Copenhagen on his boat Tui.

Professor Samuli Siltanen from University of Helsinki is adjunct professor at DTU and was also associated with this project.

Professor Todd Quinto from Tufts University was Otto Mønsted Visiting Professor at DTU Compute in the fall and winter of 2016.

Related Research Projects

Activities - Scientific Publications, Visitors, etc.

Software and Data

Accepted and Published Papers

  1. 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, 4 (2018), pp. 17-31, DOI: 10.1109/TCI.2017.2723246 (open access).
  2. H. O. Aggrawal, M. S. Andersen, and J. Modersitzki, An image registration framework for discontinuous mappings along cracks; in Ž. Špiclin, J. McClelland, J. Kybic, and O. Goksel (Eds), Biomedical Image Registration. WBIR 2020, Lecture Notes in Computer Science, vol 12120, Springer, 2020, DOI: 10.1007/978-3-030-50120-4_16 (open access). This paper received the '1st best paper award' at the WBIR 2020 conference.
  3. M. S. Andersen and P. C. Hansen, Generalized row-action methods for tomographic imaging, Numerical Algorithms, 67 (2013), pp. 121-144; DOI: 10.1007/s11075-013-9778-8.
  4. G. Bal, K. Hoffmann, and K. Knudsen, Propagation of singularities for linearized hybrid data impedance tomography, Inverse Problems, 34 (2917), 024001 (19pp); DOI: 10.1088/1361-6420/aa0d78.
  5. L. Borg, J. Frikel, J. S. Jørgensen, and E. T. Quinto, Analyzing reconstruction artifacts from arbitrary incomplete X-ray CT data, SIAM J. Imaging Sciences, 11 (2018), pp.2786-2814; DOI: 10.1137/18M1166833
  6. 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); DOI: 10.1088/1361-6501/aa8c27.
  7. D. Chen, M. E. Kilmer, and P. C. Hansen, "Plug-and-play" edge-preserving regularization, Electronic Transactions on Numerical Analysis, 41 (2014), pp. 465-477 (open access).
  8. 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. DOI: 10.1109/TAC.2014.2351851.
  9. 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), 014003 (16pp); DOI: 10.1088/1361-6501/aa950e. Software available at: github.com/vedrana/tomography-snake.
  10. F. Delbary and K. Knudsen, Numerical nonlinear complex geometrical optics for the 3D Calderón problem, Inverse Problems and Imaging, 8 (2014), pp. 991-1012; DOI: 10.3934/ipi.2014.8.991.
  11. 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).
  12. Y. Dong, T. Görner, and S. Kunis, An algorithm for total variation regularized photoacoustic imaging, Adv. Comput. Math., June 2014; DOI: 10.1007/s10444-014-9364-1
  13. Y. Dong, P. C. Hansen, M. E. Hochstenbach, and N. A. B. Riis, Fixing nonconvergence of algebraic iterative reconstruction with an unmatched backprojection, SIAM J. Sci. Comput., 40 (2019), pp. A1822-A1839. doi: 10.1137/18M1206448
  14. Y. Dong, P. C. Hansen, and H. M. Kjer, Joint CT reconstruction and segmentation with discriminative dictionary learning, IEEE Trans. Computational Imaging, 4 (2018), pp. 528-536; doi: 10.1109/TCI.2018.2858139.
  15. 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; DOI: 10.1137/120870621.
  16. 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; DOI: 10.4208/eajam.240713.120813a.
  17. T. Elfving and P. C. Hansen, Unmatched projector/backprojector pairs: perturbation and convergence analysis, SIAM J. Sci. Comp., 40 (2018), pp. A573-A591; DOI: 10.1137/17M1133828.
  18. T. Elfving, P. C. Hansen, and T. Nikazad, Convergence analysis for column-action methods in image reconstruction, Numerical Algorithms, 74 (2016), DOI: 10.1007/s11075-016-0176-x. Erratum (Fig. 3 was incorrect): DOI: 10.1007/s11075-016-0232-6.
  19. T. Elfving, P. C. Hansen, and T. Nikazad, Semi-convergence properties of Kaczmarz's method, Inverse Problems, 30 (2014), DOI: 10.1088/0266-5611/30/5/055007. This paper was selected to be part of the journal's Highlights Collection
  20. H. Garde, Comparison of linear and non-linear monotonicity-based shape reconstruction using exact matrix characterizations, Inverse Problems in Science and Engineering, 26 (2018), pp. 33-50. DOI: 10.1080/17415977.2017.1290088.
  21. H. Garde and K. Knudsen, Distinguishability revisited: depth dependent bounds on reconstruction quality in electrical impedance tompography SIAM J. Appl. Math., 77 (2017); DOI: 10.1137/16M1072991.
  22. 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; DOI: 10.1080/17415977.2015.1047365.
  23. H. Garde and S. Staboulis, The regularized monotonicity method: detecting irregular indefinite inclusions, Inverse Problems and Imaging, 13 (2019), pp. 93-116. DOI: 10.3934/ipi.2019006.
  24. H. Garde and S. Staboulis, Convergence and regularization for monotonicity-based shape reconstruction in electrical impedance tomography, Numer. Math., 135 (2017), pp. 1221-1251; DOI: 10.1007/s00211-016-0830-1.
  25. 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, 81 (2019), pp. 773-811. doi: 10.1007/s11075-018-0570-7.
  26. P. C. Hansen, Y. Dong, and K. Abe, Hybrid enriched bidiagonalization for discrete ill-posed problems, Numer. Linear Algebra Appl., 26 (2019), e2230; DOI: 10.1002/nla.2230.
  27. P. C. Hansen and J. S. Jørgensen, AIR Tools II: algebraic iterative reconstruction methods, improved implementation, Numerical Algorithms, 79 (2018), pp. 107-137; DOI: 10.1007/s11075-017-0430-x.
  28. P. C. Hansen, J. G. Nagy, and K. Tigkos, Rotational image deblurring with sparse matrices, BIT Numerial Mathematics, 54 (2014), pp. 649-671, DOI: 10.1007/s10543-013-0464-y
  29. K. Hoffmann and K. Knudsen, Iterative reconstruction methods for hybrid inverse problems in impedance tomography, Sensing and Imaging, 15 (2014), pp. 1-27; DOI: 10.1007/s11220-014-0096-6
  30. 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); DOI: 10.1088/1361-6501/aa8c29
  31. 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; DOI: 10.1080/17415977.2014.986724
  32. 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"); DOI: 10.1098/rsta.2014.0387. Data and code to reproduce the results are available from DOI: 10.5061/dryad.3jg57.
  33. 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; DOI: 10.3934/ipi.2015.9.431 (open access).
  34. M. Karamehmedovic and K. Knudsen, Inclusion estimation from a single electrostatic boundary measurement, Inverse Problems, 29 (2013); DOI: 10.1088/0266-5611/29/2/025005.
  35. R. D. Kongskov and Y. Dong, Tomographic reconstruction methods for decomposing directional components, Inverse Problems and Imaging, 12 (2018), pp. 1429-1442. DOI: 10.3934/ipi.2018060.
  36. R. D. Kongskov, Y. Dong, and K. Knudsen, Directional total generalized variation regularization, BIT Numerical Mathematics, 59 (2019), pp. 903-928. DOI: 10.1007/s10543-019-00755-6
  37. 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; DOI: 10.1364/JOSAA.33.000447.
  38. J.-J. Mei, Y. Dong, T.-Z. Huang, and W. Yin, Cauchy noise removal by nonconvex ADMM with convergence guarantees, J. Sci. Comput., 74 (2018), pp. 743-766. DOI: 10.1007/s10915-017-0460-5.
  39. 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), pp. 128-161; DOI: 10.1080/10556788.2014.902056.
  40. 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; DOI: 10.1190/geo2014-0017.1.
  41. 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; DOI: 10.1364/JOSAA.34.001830.
  42. 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); DOI: 10.1088/1361-6420/aaa49c.
  43. 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), pp. 1432-1453; DOI: 10.1080/17415977.2015.1124428.
  44. 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; DOI: 10.1118/1.4914148
  45. M. Salewski, B. Geiger, A. Jacobsen, P. C. Hansen + 12, High-definition velocity-space tomography of fast-ion dynamics, Nuclear Fusion, 56 (2016), DOI: 10.1088/0029-5515/56/10/106024.
  46. M. F. Schmidt, M. Benning, and C.-B. Schönlieb, Inverse scale space decomposition, Inverse Problems, 34 (2018), 045008 (34pp); DOI: 10.1088/1361-6420/aab0ae.
  47. 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: 10.4208/nmtma.2017.m1613
  48. 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, DOI: 10.1137/140997816
  49. 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; DOI: 10.1016/j.cam.2016.09.019.
  50. S. Soltani, M. E. Kilmer, and P. C. Hansen, A tensor-based dictionary learning approach to tomographic image reconstruction, BIT Numerical Mathematics, 56 (2016), pp. 1425-1454; DOI: 10.1007/s10543-016-0607-z. 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.
  51. Y. Sun, M. S. Andersen, and L. Vandenberghe, Decomposition in conic optimization with partially separable structure, SIAM J. Optimiz., 24 (2014), pp. 873-897; DOI: 10.1137/130926924.
  52. 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. DOI: 10.1137/130920642.
  53. P. Weiss, P. Escande, G. Bathie, and Y. Dong, Contrast invariant SNR and isotonic regressions, International Journal of Computer Vision, published online 2019. DOI: 10.1007/s11263-019-01161-9

Refereed Conference Proceedings

Technical Reports


Seminars, Workshops, PhD Courses, etc.

Bachelor and Master Projects