Abstract
Discrete tomography has proven itself as a powerful approach to image reconstruction from limited data. In recent years, algebraic reconstruction methods have been applied successfully to a range of experimental data sets. However, the computational cost of such reconstruction techniques currently prevents routine application to large data-sets. In this paper we investigate the use of adaptive refinement on QuadTree grids to reduce the number of pixels (or voxels) needed to represent an image. Such locally refined grids match well with the domain of discrete tomography as they are optimally suited for representing images containing large homogeneous regions. Reducing the number of pixels ultimately promises a reduction in both the computation time of discrete algebraic reconstruction techniques as well as reduced memory requirements. At the same time, a reduction of the number of unknowns can reduce the influence of noise on the reconstruction. The resulting refined grid can be used directly for further post-processing (such as segmentation, feature extraction or metrology). The proposed approach can also be used in a non-adaptive manner for region-of-interest tomography. We present a computational approach for automatic determination of the locations where the grid must be defined. We demonstrate how algebraic discrete tomography algorithms can be constructed based on the QuadTree data structure, resulting in reconstruction methods that are fast, accurate and memory efficient.
Chapter PDF
Similar content being viewed by others
References
Herman, G.T., Kuba, A.: Discrete Tomography: Foundations, Algorithms, and Applications. Birkhäuser (1999)
Herman, G.T., Kuba, A.: Advances in discrete tomography and its applications. Birkhäuser (2007)
Ruskó, L., Kuba, A.: Multi-resolution method for binary tomography. Electronic Notes in Discrete Mathematics 20, 299–311 (2005)
Gerard, Y.: Elementary algorithms for multiresolution geometric tomography with strip model of projections. In: 2013 8th International Symposium on Image and Signal Processing and Analysis (ISPA), pp. 600–605. University of Trieste and University of Zagreb (2013)
Schule, T., Schnorr, C., Weber, S., Hornegger, J.: Discrete tomography by convex–concave regularization and D.C. programming. Discrete Applied Mathematics 151(1-3), 229–243 (2005)
Batenburg, K.J., Sijbers, J.: DART: a practical reconstruction algorithm for discrete tomography. IEEE Transactions on Image Processing 20(9), 2542–2553 (2011)
Batenburg, K., Sijbers, J.: Dart: A Fast Heuristic Algebraic Reconstruction Algorithm for Discrete Tomography. In: IEEE International Conference on Image Processing, pp. IV-133–IV-136. IEEE (2007)
Herman, G.T., Levkowitz, H., Tuy, H.: Multilevel Image Reconstruction. In: Rosenfeld, A. (ed.) Multiresolution Image Processing and Analysis. Springer Series in Information Sciences, vol. 12, pp. 121–135. Springer, Heidelberg (1984)
Henson, V.E., Limber, M.A., McCormick, S.F., Robinson, B.T.: Multilevel Image Reconstruction with Natural Pixels. SIAM Journal on Scientific Computing 17(1), 193–216 (1996)
Kostler, H., Popa, C., Ummer, M., Rude, U.: Towards an algebraic multigrid method for tomographic image reconstruction-improving convergence of ART. In: European Conference on Computational Fluid Dynamics, pp. 1–12 (2006)
Cools, S., Ghysels, P., van Aarle, W., Vanroose, W.: A multilevel preconditioned Krylov method for algebraic tomographic reconstruction. arXiv, 26 (2013)
Bouman, C., Webb, K.: Multigrid tomographic inversion with variable resolution data and image spaces. IEEE Transactions on Image Processing 15(9), 2805–2819 (2006)
De Witte, Y., Vlassenbroeck, J., Van Hoorebeke, L.: A multiresolution approach to iterative reconstruction algorithms in X-ray computed tomography. IEEE Transactions on Image Processing: A Publication of the IEEE Signal Processing Society 19(9), 2419–2427 (2010)
Schumacher, H., Heldmann, S., Haber, E., Fischer, B.: Iterative Reconstruction of SPECT Images Using Adaptive Multi-level Refinement. In: Tolxdorff, T., Braun, J., Deserno, T.M., Horsch, A., Handels, H., Meinzer, H.P. (eds.) Bildverarbeitung für die Medizin 2008, pp. 318–322. Springer, Heidelberg (2008)
Palenstijn, W.J., Batenburg, K.J., Sijbers, J.: Performance improvements for iterative electron tomography reconstruction using graphics processing units (GPUs). Journal of Structural Biology 176(2), 250–253 (2011)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
van Leeuwen, T., Batenburg, K.J. (2014). Adaptive Grid Refinement for Discrete Tomography. In: Barcucci, E., Frosini, A., Rinaldi, S. (eds) Discrete Geometry for Computer Imagery. DGCI 2014. Lecture Notes in Computer Science, vol 8668. Springer, Cham. https://doi.org/10.1007/978-3-319-09955-2_25
Download citation
DOI: https://doi.org/10.1007/978-3-319-09955-2_25
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-09954-5
Online ISBN: 978-3-319-09955-2
eBook Packages: Computer ScienceComputer Science (R0)