Skip to main content
SpringerLink
Log in

Halton Sampling for Image Registration Based on Mutual Information

  • Published:
Sampling Theory in Signal and Image Processing Aims and scope Submit manuscript

Abstract

Mutual information is a widely used similarity measure for aligning multimodal medical images. At its core it relies on the computation of a discrete joint histogram, which itself requires image samples for its estimation. In this paper we study the influence of the sampling process. We show that quasi-random sampling based on Halton sequences outperforms methods based on regular sampling or on random sampling. Our results suggest that sampling itself—and not interpolation, as was previously believed—is the source of two major problems associated with mutual information: the grid effect, whereby grid-aligning transformations are favored, and the overlap problem, whereby the similarity measure exhibits discontinuities. Both defects tend to impede the accuracy of registration; they also result in reduced robustness because of the presence of local optima. By estimating the joint histogram by quasi-random sampling, we solve both issues at the same time.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. K.S. Arun, T.S. Huang, and S.D. Blostein, Least-Squares Fitting of Two 3-D Point Sets, IEEE Transactions on Pattern Analysis and Machine Intelligence, 9(5), 698–700, September 1987.

    Article  Google Scholar 

  2. L.G. Brown, A Survey of Image Registration Techniques, ACM Computing Surveys, 24(4), 325–376, December 1992.

    Article  Google Scholar 

  3. J. Camp and R. Robb, A Novel Binning Method for Improved Accuracy and Speed of Volume Image Coregistration Using Normalized Mutual Information, in Proceedings of the SPIE International Conference on Image Processing: Medical Imaging, K.M. Hanson, Ed., San Diego CA, USA, February 22-25, 1999, SPIE, 3661, 24-31.

    Google Scholar 

  4. H.-M. Chen and P.K. Varshney, Registration of Multimodal Brain Images: Some Experimental Results, in Proceedings of the SPIE International Conference on Sensor Fusion, B.V. Dasarathy, Ed., Orlando FL, USA, April 3-5, 2002, SPIE, 4731, 122-133.

    Google Scholar 

  5. H.-M. Chen and P.K. Varshney, Mutual Information-Based CT-MR Brain Image Registration Using Generalized Partial Volume Joint Histogram Estimation, IEEE Transactions on Medical Imaging, 229, 1111–1119, 2003.

    Article  Google Scholar 

  6. A.A. Cole-Rhodes, K.L. Johnson, J. LeMoigne, and I. Zavorin, Multiresolution Registration of Remote Sensing Imagery by Optimization of Mutual Information Using a Stochastic Gradient, IEEE Transactions on Image Processing, 1212, 1495–1511, 2003.

    Article  MathSciNet  Google Scholar 

  7. M. Drmota and R.F. Tichy, Eds., Sequences, Discrepancies, and Applications, Lecture Notes in Mathematics, 1651, Springer-Verlag, Berlin, 1997.

  8. P.A. van den Elsen, E.-J.D. Pol, and M.A. Viergever, Medical Image Matching—A Review with Classification, IEEE Engineering and Medicine in Biology, 121, 26–39, 1993.

    Article  Google Scholar 

  9. H. Faure, Discr-pance de suites associ-es - un syst-me de num-ration (en dimension s), Acta Arithmetica, XLI, 337–351, 1982.

    Article  Google Scholar 

  10. H. Faure, Suggestions for Quasi-Monte-Carlo Users, in Proceedings of the International Conference on Finite Fields, Coding Theory and Advances in Communications and Computing, G.L. Mullen and P.J.-S. Shiue, Eds., Las Vegas NV, USA, August 7-10, 1991 (1993), Lecture Notes in Pure and Applied Mathematics, 141, 269-278, M. Dekker.

  11. J.M. Fitzpatrick, http://www.vuse.vanderbilt.edu/image/registration//, National Institute of Biomedical Imaging and Bioengineering, Project Number 8R01EB002124-03.

  12. J. Flusser and T. Suk, A Moment-Based Approach to Registration of Images with Affine Geometric Distortion, IEEE Transactions on Geoscience and Remote Sensing, 322, 382–387, 1994.

    Article  Google Scholar 

  13. K. Fukunaga, Ed., Introduction to Statistical Pattern Recognition, Electrical Science, A Series of Monographs and Texts. Academic Press, 1972.

  14. J.H. Halton, On the Efficiency of Certain Quasi-Random Sequences of Points in Evaluating Multi-Dimensional Integrals, Numerical Mathematics, 2, 84–90, 1960.

    Article  MathSciNet  Google Scholar 

  15. J.M. Hammersley, Monte-Carlo Methods for Solving Multivariate Problems, Annals of the New-York Academy of Sciences, 86, 844–874, 1960.

    Article  MathSciNet  Google Scholar 

  16. D.L.G. Hill, C. Studholme, and D.J. Hawkes, Voxel Similarity Measures for Automated Image Registration, in Proceedings of the SPIE International Symposium on Visualization in Biomedical Computing, R.A. Robb, Ed., Rochester MN, USA, October 4-7, 1994, SPIE, 2359, 205–216.

  17. B. Kim, J.L. Boes, K.A. Frey, and C.R. Meyer, Mutual Information for Automated Unwarping of Rat Brain Autoradiographs, Neuroimage, 51, 31–40, 1997.

    Article  Google Scholar 

  18. D.E. Knuth, Ed., Seminumerical Algorithms, vol.2 of The Art of Computer Programming, Addison-Wesley, third edition, 1998.

  19. L. Kocis and W.J. Whiten, Computational Investigations of Low-Discrepancy Sequences, ACM Transactions on Mathematical Software, 232, 266–294, 1997.

    Article  Google Scholar 

  20. R.K.-S. Kwan, A.C. Evans, and G.B. Pike, MRI Simulation-Based Evaluation of Image-Processing and Classification Methods, IEEE Transactions on Medical Imaging, 1811, 1085–1097, 1999.

    Article  Google Scholar 

  21. B. Likar and F. Pernus, A Hierarchical Approach to Elastic Registration Based on Mutual Information, Image and Vision Computing, 19(1-2), 33–44, 2001.

    Article  Google Scholar 

  22. B.D. Lucas and T. Kanade, An Iterative Image Registration Technique with an Application to Stereo Vision, in Proceedings of the Seventh International Joint Conference on Artificial Intelligence (IJCAI’81), A. Drinan, Ed., Vancouver BC, Canada, August 24-28, 1981, 674–679, Kaufmann, W.

    Google Scholar 

  23. F. Maes, Segmentation and Registration of Multimodal Medical Images: From Implementation and Validation to a Useful Tool in Clinical Practice, Ph.D. thesis, Faculty of Applied Sciences, Katholieke Universiteit Leuven, Leuven, Belgium, May 27, 1998, Thesis no. U.D.C. 681.3*I4 (1998).

  24. J.B.A. Maintz and M.A. Viergever, A Survey of Medical Image Registration, Medical Image Analysis, 21, 1–36, 1998.

    Article  Google Scholar 

  25. N.J.I. Mars and G.W. Arragon, Time Delay Estimation in Non-Linear Systems Using Average Amount of Mutual Information Analysis, Signal Processing, 4(2-3), 139–153, 1982.

    Article  Google Scholar 

  26. C.R. Maurer Jr. and J.M. Fitzpatrick, A Review of Medical Image Registration, in Interactive Image-Guided Neurosurgery, R.J. Maciunas, Ed., 17-44. American Associate of Neurological Surgeons, Park Ridge IL, USA, 1993.

    Google Scholar 

  27. C. Nikou, F. Heitz, J.-P. Armspach, I.-J. Namer, and D. Grucker, Registration of MR/MR and MR/SPECT Brain Images by Fast Stochastic Optimization of Robust Voxel Similarity Measures, NeuroImage, 81, 30–34, 1998.

    Article  Google Scholar 

  28. C. Nikou, F. Heitz, and J.-P. Armspach, Robust Voxel Similarity Metrics for the Registration of Dissimilar Single and Multimodal Images, Pattern Recognition, 328, 1351–1368, 1999.

    Article  Google Scholar 

  29. B.-J. Qin and T.-G. Zhuang, Similarity Measures in Voxel Intensity Based 3D Multi-Modal Medical Image Registration, Journal ofShanghai-Jiaotong University, 367, 942–945, 2002.

    Google Scholar 

  30. H.-P. Ren, H. Yang, S.-Z. Chen, and W.-K. Wu, Study of Three-Dimensional PET and MR Image Registration Based on Higher-Order Mutual Information, Nuclear Science and Techniques, 132, 65–71, 2002.

    Google Scholar 

  31. C.E. Rodr-guez-Carranza and M.H. Loew, A Weighted and Deterministic Entropy Measure for Image Registration Using Mutual Information, in Proceedings of the SPIE International Conference on Image Processing: Medical Imaging, K.M. Hanson, Ed., San Diego CA, USA, February 23-27, 1998, SPIE, 3338, 155–166.

    Google Scholar 

  32. T. Rohlfing, J.B. West, J. Beier, T. Liebig, C.A. Taschner, and U.W. Thomale, Registration of Functional and Anatomical MRI: Accuracy Assessment and Application in Navigated Neurosurgery, Computer-Aided Surgery, 56, 414–425, 2000.

    Article  Google Scholar 

  33. C.-. S-nchez Sorzano, P. Th-venaz, and M. Unser, Elastic Registration of Biological Images Using Vector-Spline Regularization, IEEE Transactions on Biomedical Engineering, 524, 652–663, 2005.

    Article  Google Scholar 

  34. I.M. Sobol, On the Distribution of Points in a Cube and the Approximate Evaluation of Integrals, USSR Computational Mathematics and Mathematical Physics, 74, 86–112, 1967.

    Article  MathSciNet  Google Scholar 

  35. H.S. Stone, M.T. Orchard, E.-C. Chang, and S.A. Martucci, A Fast Direct Fourier-Based Algorithm for Subpixel Registration of Images, IEEE Transactions on Geoscience and Remote Sensing, 3910, 2235–2243, 2001.

    Article  Google Scholar 

  36. A. Tan-cs, K. Pal-gyi, and A. Kuba, A Fully Automatic Medical Image Registration Algorithm Based on Mutual Information, in Proceedings of the Third Conference of PhD Students in Computer Science, Szeged, Hungary, July 1-4, 2002.

    Google Scholar 

  37. P. Thevenaz, U.E. Ruttimann, and M. Unser, A Pyramid Approach to Subpixel Registration Based on Intensity, IEEE Transactions on Image Processing, 71, 27–41, 1998.

    Article  Google Scholar 

  38. P. Thevenaz, T. Blu, and M. Unser, Interpolation Revisited, IEEE Transactions on Medical Imaging, 197, 739–758, 2000.

    Article  Google Scholar 

  39. P. Thevenaz and M. Unser, Optimization of Mutual Information for Multiresolution Image Registration, IEEE Transactions on Image Processing, 912, 2083–2099, 2000.

    Article  Google Scholar 

  40. M. Unser, Splines: A Perfect Fit for Signal and Image Processing, IEEE Signal Processing Magazine, 166, 22–38, 1999.

    Article  Google Scholar 

  41. J. West, J.M. Fitzpatrick, M.Y. Wang, B.M. Dawant, C.R. Maurer Jr., R.M. Kessler, R.J. Maciunas, C. Barillot, D. Lemoine, A. Collignon, F. Maes, P. Suetens, D. Vandermeulen, P.A. van den Elsen, P.F. Hemler, S. Napel, T.S. Sumanaweera, B. Harkness, D.L.G. Hill, C. Studholme, G. Malandain, X. Pennec, M.E. Noz, G.Q. Maguire Jr., M. Pollack, C.A. Pelizzari, R.A. Robb, D. Hanson, and R.P. Woods, Comparison and Evaluation of Retrospective Intermodality Image Registration Techniques, in Proceedings of the SPIE International Conference on Image Processing: Medical Imaging, K.M. Hanson, Ed., Newport Beach CA, USA, February 12-15, 1996, SPIE, 2710, 332–347.

  42. J. West, J.M. Fitzpatrick, M.Y. Wang, B.M. Dawant, C.R. Maurer Jr., R.M. Kessler, R.J. Maciunas, C. Barillot, D. Lemoine, A. Collignon, F. Maes, P. Suetens, D. Vandermeulen, P.A. van den Elsen, S. Napel, T.S. Sumanaweera, B. Harkness, P.F. Hemler, D.L.G. Hill, D.J. Hawkes, C. Studholme, J.B.A. Maintz, M.A. Viergever, G. Malandin, X. Pennec, M.E. Noz, G.Q. Maguire Jr., M. Pollack, C.A. Pelizzari, R.A. Robb, D. Hanson, and R.P. Woods, Comparison and Evaluation of Retrospective Intermodality Brain Image Registration Techniques, Journal of Computer Assisted Tomography, 214, 554–566, 1997.

    Article  Google Scholar 

  43. J. West, J.M. Fitzpatrick, M.Y. Wang, B.M. Dawant, C.R. Maurer Jr., R.M. Kessler, and R.J. Maciunas, Retrospective Intermodality Registration Techniques for Images of the Head: Surface-Based Versus Volume-Based, IEEE Transactions on Medical Imaging, 182, 144–150, 1999.

    Article  Google Scholar 

  44. H. Weyl, -ber die Gleichverteilung von Zahlen mod Eins, Annals of Mathematics, 77, 313–352, 1916.

    Article  Google Scholar 

  45. R.P. Woods, J.C. Mazziotta, and S.R. Cherry, MRI-PET Registration with Automated Algorithm, Journal ofComputer Assisted Tomography, 174, 536–546, 1993.

    Article  Google Scholar 

Download references

Acknowledgments

This work was supported by the Center for Biomedical Imaging of the Geneva-Lausanne Universities and the EPFL, as well as by the Hasler, Leenaards, and Louis-Jeantet foundations. The images and the gold-standard transformations of Section 5 were provided as part of the project, “Evaluation of Retrospective Image Registration,” National Institutes of Health, 1 R01 NS33926-02, Principal Investigator, J.M. Fitzpatrick, Vanderbilt University, Nashville TN, USA.

Author information

Authors and Affiliations

  1. Biomedical Imaging Group, École polytechnique fédérale de Lausanne (EPFL), Station 17, CH-1015, Lausanne VD, Switzerland

    Philippe Thévenaz & Michael Unser

  2. Transport and Mobility Laboratory, École polytechnique fédérale de Lausanne (EPFL), Station 18, CH-1015, Lausanne VD, Switzerland

    Michel Bierlaire

Authors
  1. Philippe Thévenaz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michel Bierlaire
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michael Unser
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Philippe Thévenaz.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Thévenaz, P., Bierlaire, M. & Unser, M. Halton Sampling for Image Registration Based on Mutual Information. STSIP 7, 141–171 (2008). https://doi.org/10.1007/BF03549492

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03549492

Key words and phrases

2000 AMS Mathematics Subject Classification

Search

Navigation