3D Image Reconstruction from X-Ray Measurements with Overlap

  • Maria KlodtEmail author
  • Raphael Hauser
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9910)


3D image reconstruction from a set of X-ray projections is an important image reconstruction problem, with applications in medical imaging, industrial inspection and airport security. The innovation of X-ray emitter arrays allows for a novel type of X-ray scanners with multiple simultaneously emitting sources. However, two or more sources emitting at the same time can yield measurements from overlapping rays, imposing a new type of image reconstruction problem based on nonlinear constraints. Using traditional linear reconstruction methods, respective scanner geometries have to be implemented such that no rays overlap, which severely restricts the scanner design. We derive a new type of 3D image reconstruction model with nonlinear constraints, based on measurements with overlapping X-rays. Further, we show that the arising optimization problem is partially convex, and present an algorithm to solve it. Experiments show highly improved image reconstruction results from both simulated and real-world measurements.


Image reconstruction Medical imaging X-ray Overlap 



We thank Adaptix Ltd for providing the X-ray measurements used in the experiments. This work was partly supported by Adaptix Ltd and EPSRC.


  1. 1.
    Gonzales, B., Spronk, D., Cheng, Y., Tucker, A.W., Beckman, M., Zhou, O., Lu, J.: Rectangular fixed-gantry CT prototype: Combining CNT x-ray sources and accelerated compressed sensing-based reconstruction. IEEE Access 2, 971–981 (2014)CrossRefGoogle Scholar
  2. 2.
    Chen, D., Song, X., Zhang, Z., Li, Z., She, J., Deng, S., Xu, N., Chen, J.: Transmission type flat-panel x-ray source using zno nanowire field emitters. Appl. Phys. Lett. 107(24), 243105 (2015)CrossRefGoogle Scholar
  3. 3.
    Candes, E.J., Romberg, J., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math. 59, 1207–1223 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Donoho, D.L.: Compressed sensing. IEEE Trans. Inf. Theor. 52, 1289–1306 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Lustig, M., Donoho, D.L., Santos, J.M., Pauly, J.M.: Compressed sensing MRI. IEEE Sig. Process. Mag. 25(2), 72–82 (2007)Google Scholar
  6. 6.
    Ma, S., Yin, W., Zhang, Y., Chakraborty, A.: An efficient algorithm for compressed mr imaging using total variation and wavelets. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2008)Google Scholar
  7. 7.
    Yan, M., Vese, L.A.: Expectation maximization and total variation based model for computed tomography reconstruction from undersampled data. In: Proceedings of SPIE vol. 7961 Medical Imaging 2011: Physics of Medical Imaging (2011)Google Scholar
  8. 8.
    Kolev, K., Cremers, D.: Integration of multiview stereo and silhouettes via convex functionals on convex domains. In: Forsyth, D., Torr, P., Zisserman, A. (eds.) ECCV 2008. LNCS, vol. 5302, pp. 752–765. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-88682-2_57 CrossRefGoogle Scholar
  9. 9.
    Serradell, E., Romero, A., Leta, R., Gatta, C., Moreno-Noguer, F.: Simultaneous correspondence and non-rigid 3d reconstruction of the coronary tree from single x-ray images. In: International Conference on Computer Vision (ICCV), pp. 850–857. IEEE Computer Society (2011)Google Scholar
  10. 10.
    Kim, H., Thiagarajan, J.J., Bremer, P.: A randomized ensemble approach to industrial ct segmentation. In: International Conference on Computer Vision (ICCV), pp. 1707–1715. IEEE (2015)Google Scholar
  11. 11.
    Ehler, M., Fornasier, M., Sigl, J.: Quasi-linear compressed sensing. Multiscale Model. Simul. 12(2), 725–754 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Needell, D., Tropp, J.A.: Cosamp: Iterative signal recovery from incomplete and inaccurate samples. Commun. ACM 53(12), 93–100 (2010)CrossRefzbMATHGoogle Scholar
  13. 13.
    Li, X., Voroninski, V.: Sparse signal recovery from quadratic measurements via convex programming. SIAM J. Math. Anal. 45(5), 3019–3033 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Blumensath, T.: Compressed sensing with nonlinear observations and related nonlinear optimisation problems. IEEE Trans. Inf. Theor. 59(6), 3466–3474 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4(4), 1168–1200 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Donoho, D.L., Johnstone, I.M.: Minimax estimation via wavelet shrinkage. Ann. Stat. 26(3), 879–921 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Parikh, N., Boyd, S.: Proximal algorithms. Found. Trends Optim. 1(3), 127–239 (2014)CrossRefGoogle Scholar
  18. 18.
    Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Img. Sci. 2(1), 183–202 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Amanatides, J., Woo, A.: A fast voxel traversal algorithm for ray tracing. Eurographics 87, 3–10 (1987)Google Scholar
  20. 20.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK

Personalised recommendations