Abstract
We study a problem involving the reconstruction of an image from its horizontal and vertical projections in the case where some parts of these projections are unavailable. The desired goal is to model applications where part of the x-rays used for the analysis of an object are blocked by particularly dense components that do not allow the rays to pass through the material. This is a common issue in many tomographic scans, and while there are several heuristics to handle quite efficiently the problem in applications, the underlying theory has not been extensively developed. In this paper, we study the properties of consistency and uniqueness of this problem, and we propose an efficient reconstruction algorithm. We also show how this task can be reduced to a network flow problem, similarly to the standard reconstruction algorithm, allowing the determination of a solution even in the case where some pixels of the output image must have some prescribed values.
Chapter PDF
Similar content being viewed by others
References
Barcucci, E., Brlek, S., Brocchi, S.: PCIF: an algorithm for lossless true color image compression. In: Wiederhold, P., Barneva, R.P. (eds.) IWCIA 2009. LNCS, vol. 5852, pp. 224–237. Springer, Heidelberg (2009)
Barcucci, E., Brocchi, S.: Solving multicolor discrete tomography problems by using prior knowledge. Fundamenta Informaticae 125, 313–328 (2013)
Barcucci, E., Brocchi, S., Frosini, A.: Solving the two color problem - An heuristic algorithm. In: Aggarwal, J.K., Barneva, R.P., Brimkov, V.E., Koroutchev, K.N., Korutcheva, E.R. (eds.) IWCIA 2011. LNCS, vol. 6636, pp. 298–310. Springer, Heidelberg (2011)
Barcucci, E., Del Lungo, A., Nivat, M., Pinzani, R.: Reconstructing convex polyominoes from horizontal and vertical projections. Theoretical Computer Science 155, 321–347 (1996)
Batenburg, K.J.: A network flow algorithm for reconstructing binary images from discrete X-rays. Journal of Mathematical Imaging and Vision 27(2), 175–191 (2013)
Batenburg, K.J., Sijbers, J.: DART: a practical reconstruction algorithm for discrete tomography. IEEE Transactions on Image Processing 20(9), 2542–2553 (2011)
Costa, M.-C., de Werra, D., Picouleau, C., Schindl, D.: A solvable case of image reconstruction in discrete tomography. Discrete Applied Mathematics 148(3), 240–245 (2005)
De Man, B., Nuyts, J., Dupont, P., Marchal, G., Suetens, P.: Reduction of metal streak artifacts in x-ray computed tomography using a transmission maximum a posteriori algorithm. In: Nuclear Science Symposium, pp. 850–854 (1999)
Duan, X., Zhang, L., Xiao, Y., Cheng, J., Chen, Z., Xing, Y.: Metal artifact reduction in CT images by sinogram TV inpainting. In: Nuclear Science Symposium Conference, pp. 4175–4177 (2008)
Durr, C., Guinez, F., Matamala, M.: Reconstructing 3-Colored Grids from Horizontal and Vertical Projections is NP-Hard, A Solution to the 2-Atom Problem in Discrete Tomography. SIAM J. Discrete Math. 26(1), 330–352 (2012)
Faggiano, E., Lorenzi, T., Quarteroni, A.: Metal Artifact Reduction in Computed Tomography Images by Variational Inpainting Methods, MOX-report No. 16/2013 (2013)
Gardner, R.J., Gritzmann, P., Pranenberg, D.: On the computational complexity of reconstructing lattice sets from their X-rays. Discrete Mathematics 202(1-3), 45–71 (1999)
Guinez, F.: A formulation of the wide partition conjecture using the atom problem in discrete tomography. Discrete Applied Mathematics (2013) (in press)
Hantos, N., Balazs, P.: A uniqueness result for reconstructing hv-convex polyominoes from horizontal and vertical projections and morphological skeleton. In: 8th International Symposium on Image and Signal Processing and Analysis (ISPA). IEEE (2013)
Hantos, N., Balazs, P.: The Reconstruction of Polyominoes from Horizontal and Vertical Projections and Morphological Skeleton is NP-complete. Fundamenta Informaticae 125(3), 343–359 (2013)
Herman, G., Kuba, A.: Advances in discrete tomography and its applications. Birkhauser, Boston (2007)
Kalender, W.A., Hebel, R., Ebersberger, J.: Reduction of CT artifacts caused by metallic implants. Radiology 164(2), 576–577 (1987)
Picouleau, C., Brunetti, S., Frosini, A.: Reconstructing a binary matrix under timetabling constraints. Electronic Notes in Discrete Mathematics 20, 99–112 (2005)
Ryser, H.J.: Combinatorial properties of matrices of zeros and ones. Canadian Journal of Mathematics 9, 371–377 (1957)
Wang, G., Snyder, D.L., O’Sullivan, J., Vannier, M.: Iterative deblurring for ct metal artifact reduction. IEEE Transactions on Medical Imaging 15(5), 657–664 (1996)
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
Bilotta, S., Brocchi, S. (2014). Discrete Tomography Reconstruction Algorithms for Images with a Blocking Component. 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_21
Download citation
DOI: https://doi.org/10.1007/978-3-319-09955-2_21
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-09954-5
Online ISBN: 978-3-319-09955-2
eBook Packages: Computer ScienceComputer Science (R0)