Skip to main content
Springer Nature Link
Log in

Regularized graph cuts based discrete tomography reconstruction methods

  • Published:
Journal of Combinatorial Optimization Aims and scope Submit manuscript

Abstract

The topic of this paper includes graph cuts based computed tomography reconstruction methods in binary and multi-gray level cases. This approach combines the graph cuts and a gradient based method. The present paper introduces and analyses the shape circularity as a new regularization and incorporates it in a graph cuts based computed tomography reconstruction approach, thus introducing a new energy-minimization based reconstruction algorithm for binary tomography. Proposed method is capable for reconstructions in cases of limited projection view availability. Results of experimental evaluation of the considered graph cuts type reconstruction methods for both binary and multi-level tomography are presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Data availability

The data (digital test images) that support the findings of this study are available from the corresponding author, upon reasonable request.

References

  • Barzilai J, Borwein JM (1988) Two point step size gradient methods. IMA J Num Anal 8:141–148

    Article  MathSciNet  Google Scholar 

  • Batenburg KJ, Sijbers J (2007) DART: A Fast heuristic algebraic reconstruction algorithm for discrete tomography. In Proceedings of International conference on image processing (ICIP), pages 133–136

  • Birchfield S, Tomasi C (1999) Multiway cut for stereo and motion with slanted surfaces. Proc. International Conf. Computer Vision, pages 489–495

  • Birgin E, Martínez J (2001) Spectral conjugate gradient method for unconstrained optimization. Appl Math Optim 43:117–128

    Article  MathSciNet  Google Scholar 

  • Birgin E, Martínez J, Raydan M (2000) Nonmonotone spectral projected gradient methods on convex sets. SIAM J Optim 10:1196–1211

    Article  MathSciNet  Google Scholar 

  • Birgin EG, Martínez JM, Raydan M (2001) Algorithm: 813: SPG - software for convex-constrained optimization. ACM Trans Math Softw 27:340–349

    Article  Google Scholar 

  • Boykov Y, Jolly MP (2001) Interactive graph cuts for optimal boundary and region segmentation of objects in N-D images. In Proc. International Conf. Computer Vision, pages 105–112

  • Boykov Y, Kolmogorov V (2003) Computing geodesics and minimal surfaces via graph cuts. In Proc. International Conf. Computer Vision, pages 26–33

  • Boykov Y, Kolmogorov V (2004) An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision. IEEE Trans. PAMI 26(9):1124–1137

    Article  Google Scholar 

  • Boykov Y, Veksler O, Zabih R (1998) Markov random fields with efficient approximations. In Proceedings of IEEE Conf. Computer Vision and Pattern Recognition, pages 648–655

  • Boykov Y, Veksler O, Zabih R (2001) Fast approximate energy minimization via graph cuts. IEEE Trans PAMI 23(11):1222–1239

    Article  Google Scholar 

  • Delong A, Osokin A, Isack H. N, Boykov Y (2010) Fast approximate energy minimization with label costs. Proc. IEEE Conf. computer vision and pattern recognition 96(1):1–27

  • Grippo L, Lampariello F, Lucidi S (1986) A nonmonotone line search technique for Newton’s method. SIAM J Numer Anal 23:707–716

    Article  MathSciNet  Google Scholar 

  • Herman GT, Kuba A (1999) Discrete tomography: foundations, algorithms and applications. Birkhäuser

  • Herman GT, Kuba A (2006) Advances in discrete tomography and its applications. Birkhäuser

  • Kim J, Zabih R (2003) Automatic segmentation of contrast- enhanced image sequences. In Proc. International Conf. Computer Vision, pages 502–509

  • Kolmogorov V, Zabih R (2001) Visual correspondence with occlusions using graph cuts. In Proc. International Conf. Computer Vision, pages 508–515

  • Kolmogorov V, Zabih R (2004) What energy functions can be minimized via graph cuts? IEEE Trans PAMI 26(2):147–159

    Article  Google Scholar 

  • Kwatra V, Schoedl A, Essa I, Turk G, Bobick A (2003) Graphcut textures: image and video synthesis using graph cuts. In Proc. SIGGRAPH 2003, pages 277–286. ACM Trans. Graphics

  • Lukić T (2011) Discrete tomography reconstruction based on the multi-well potential. In Proceedings of combinatorial image analysis - 14th international workshop (IWCIA), volume 6636 of LNCS, pages 335–345, Madrid, Spain. Springer-Verlag

  • Lukić T, Balázs P (2016) Binary tomography reconstruction based on shape orientation. Pattern Recognit Lett 79:18–24

    Article  Google Scholar 

  • Lukić T, Balázs P (2020) Shape circularity assisted tomography reconstruction. Physica Scripta 95(10):105211

    Article  Google Scholar 

  • Lukić T, Marčeta M (2017) Gradient and graph cuts based method for multi-level discrete tomography. In Proceedings of combinatorial image analysis - 18th International workshop (IWCIA), LNCS, pages 322–333, Plovdiv, Bulgaria, Springer-Verlag

  • Lukić T, Nagy B (2014) Deterministic discrete tomography reconstruction method for images on triangular grid. Pattern Recognit Lett 49:11–16

    Article  Google Scholar 

  • Marčeta M, Lukić T (2020) Graph cuts based tomography enhanced by shape orientation. In T. Lukic, R. P. Barneva, V. E. Brimkov, L. Comic, and N. Sladoje, editors, Combinatorial image analysis - 20th International workshop, IWCIA 2020, Novi Sad, Serbia, July 16-18, 2020, Proceedings, volume 12148 of lecture notes in computer science, pages 219–235. Springer. https://doi.org/10.1007/978-3-030-51002-2_16

  • Nagy B, Lukić T (2016) Dense projection tomography on the triangular tiling. Fundamenta Informaticae 145:125–141

    Article  MathSciNet  Google Scholar 

  • Raydan M (1997) The Barzilai and Browein gradient method for the large scale unconstrained minimization problem. SIAM J Optim 7:26–33

    Article  MathSciNet  Google Scholar 

  • Schüle T, Schnörr C, Weber S, Hornegger J (2005) Discrete tomography by convex-concave regularization and D.C. programming. Discrete Appl Math 151:229–243

    Article  MathSciNet  Google Scholar 

  • Sonka M, Hlavac V, Boyle R (2007) Image processing, analysis, and machine vision. Thomosn-Engineering

  • Žunić J, Rosin PL, Kopanja L (2006) On the orientability of shapes. IEEE Trans Image Process 15:3478–3487

    Article  Google Scholar 

  • Žunić J, Hirota K, Rosin PL (2010) A hu invariant as a shape circularity measure. Pattern Recognit 43:47–57

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia

    Marina Marčeta & Tibor Lukić

Authors
  1. Marina Marčeta
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Tibor Lukić
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Marina Marčeta.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest.

Funding

The research is not financially supported.

Code availability

The Matlab code used in this paper is available on request per email from the corresponding author.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Authors acknowledge the project ”Innovative scientific and artistic research from the domain of the Faculty of Technical Sciences”, grant no. \(451-03-68/2020-14/200156\).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Marčeta, M., Lukić, T. Regularized graph cuts based discrete tomography reconstruction methods. J Comb Optim 44, 2324–2346 (2022). https://doi.org/10.1007/s10878-021-00730-4

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10878-021-00730-4

Keywords