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February  2015, 9(1): 79-103. doi: 10.3934/ipi.2015.9.79

Deformable multi-modal image registration by maximizing Rényi's statistical dependence measure

1. 

Department of Mathematics, 358 Little Hall, PO Box 118105, Gainesville, FL 32611, United States, United States, United States, United States

Received  January 2013 Revised  May 2014 Published  January 2015

A novel variational model for deformable multi-modal image registration is presented in this work. As an alternative to the models based on maximizing mutual information, the Rényi's statistical dependence measure of two random variables is proposed as a measure of the goodness of matching in our objective functional. The proposed model does not require an estimation of the continuous joint probability density function. Instead, it only needs observed independent instances. Moreover, the theory of reproducing kernel Hilbert space is used to simplify the computation. Experimental results and comparisons with several existing methods are provided to show the effectiveness of the model.
Citation: Yunmei Chen, Jiangli Shi, Murali Rao, Jin-Seop Lee. Deformable multi-modal image registration by maximizing Rényi's statistical dependence measure. Inverse Problems & Imaging, 2015, 9 (1) : 79-103. doi: 10.3934/ipi.2015.9.79
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A. Berlinet and C. Thomas, Reproducing kernel Hilbert spaces in Probability and Statistics,, Kluwer Academic Publishers, (2004). doi: 10.1007/978-1-4419-9096-9. Google Scholar

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N. Akhiezer and I. Glazman, Theory of Linear Operators in Hilbert Space,, Dover Publications, (1993). Google Scholar

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L. Alvarez, R. Deriche, T. Papadopoulo and J. Sanchez, Symmetrical dense optical flow estimation with occlusions detection,, International Journal of Computer Vision, 75 (2007), 371. doi: 10.1007/s11263-007-0041-4. Google Scholar

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N. Ayache, A. Guimond, A. Roche and J. Meunier, Three dimensional multimodal brain warping using the demons algorithm and adaptvie intensity correction,, IEEE Trans. Med. Imag., 20 (2001), 58. Google Scholar

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A. Bardera, M. Feixas, I. Boada and M. Sbert, High-dimensional normalized mutual information for image registration using random lines,, WBIR, 4057 (2006), 264. doi: 10.1007/11784012_32. Google Scholar

[8]

C. Broit, Optimal Registration of Deformed Images,, PhD thesis, (1981). Google Scholar

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R. Bajcsy and C. Broit, Matching of deformed images,, Proc. Int. Conf. Pattern Recognition, (1982), 351. Google Scholar

[10]

R. Bajscy and S. Kovacic, Multiresolution elastic matching,, Comput. Vision. Graph. Image Process, 46 (1989), 1. Google Scholar

[11]

R. Bajcsy, R. Lieberson and M. Reivich, A computerized system for the elastic matching of deformed radiographic images to idealized atlas images,, Journal of Computer Assisted Tomogra-phy, 7 (1983), 618. doi: 10.1097/00004728-198308000-00008. Google Scholar

[12]

A. Bardera, M. Feixas and I. Boada, Normalized similarity measures for medical image registration,, Proc. SPIE Medical Imaging SPIE, 5370 (2004). Google Scholar

[13]

P. Cachier and X. Pennec, 3d non-rigid reigistration by gradient descent on a Gaussian window similarity measure using convolutions,, IEEE workshop on mathematical methods in biomedical image analysis, (2000), 182. doi: 10.1109/MMBIA.2000.852376. Google Scholar

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A. Collignon, F. Maes, D. Delaere, D. Vandermeulen, P. Suetens and G. Marchal, Automated multi-modality image registration based on information theory,, Information Processing in Medical Imaging, (1995), 263. Google Scholar

[15]

A. Collignon, D. Vandermeulen, P. Suetens and G. Marchal, 3D multi-modality medical image registration using feature space clustering, Proceedings of the First International Conference on Computer Vision,, Virtual Reality and Robotics in Medicine, (1995), 195. Google Scholar

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H. M. Chan, A. C.S. Chung, S. C.H. Yu, A. Norbash and W. M. Wells III, Multi-modal image registration by minimizing Kullback-Leibler distance between expected and observed joint class histograms,, IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2 (2003). Google Scholar

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A. C. S. Chung, W. M. Wells III, A. Norbash and W. E. L. Grimson, Multi-modal image registration by minimizing kullback-Leibler distance., International Conference on Medical Image Computing and Computer-Assisted Intervention, 2 (2002), 525. Google Scholar

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R. Gan, J. Wu, A. C. S. Chung, S. C. H. Yu and W. M. Wells III, Multiresolution image registration based on Kullback-Leibler distance,, MICCAI, 3216 (2004), 599. doi: 10.1007/978-3-540-30135-6_73. Google Scholar

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C. Guetter, C. Xu, F. Sauer and J. Hornegger, Learning based non-rigid multi-modal image registration using Kullback-Leibler divergence,, Lecture Notes in Computer Science, 3750 (2005), 255. doi: 10.1007/11566489_32. Google Scholar

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Y. He, A. B. Hamza and H. Krim, A generalized divergence measure for robust image registration,, IEEE Transactions on Signal Processing, 51 (2003), 1211. doi: 10.1109/TSP.2003.810305. Google Scholar

[25]

S. Henn and K. Witsch, Multimodal image registration using a variational approach,, SIAM J. Sci. Comput., 25 (2003), 1429. doi: 10.1137/S1064827502201424. Google Scholar

[26]

G. Hermosillo, C. C. Hotel and O. Faugeras, Variational methods for multimodal image matching,, Int. J. Computer Vision, 50 (2002), 329. Google Scholar

[27]

D. Hill, P. Batchelor, M. Holden and D. Hawkes, Topical review: medical image registration,, Physics in Medicine and Biology, 46 (2001), 1. Google Scholar

[28]

D. L. G. Hill, C. Studholme and D. J. Hawkes, Voxel similarity measures for automated image registration,, Visualization in Biomedical Computing, 2359 (1994), 205. Google Scholar

[29]

B. Jian, B. Vemuri and J. Marroquin, Robust nonrigid multimodal image registration using local frequency maps., Proc. Inf. Process. Med. Imag., 3565 (2005), 504. doi: 10.1007/11505730_42. Google Scholar

[30]

L. R. Jorge, M. S. Juan and V. M. Rafael, Generalized regularization term for non-parametric multimodal image registration,, Signal Processing, 87 (2007), 2837. Google Scholar

[31]

S. Klein, M. Staring and J. P. W. Pluim, Evaluation of optimisation methods for nonrigid medical image registration using mutual information and B-splines,, IEEE Trans. Image Process., 16 (2007), 2879. doi: 10.1109/TIP.2007.909412. Google Scholar

[32]

M. Leventon and W. E. L. Grimson, Multi-modal volume registration using joint intensity distributions,, Medical Image Computing and Computer-Assisted Interventation MICCAI98, 1496 (1998), 1057. doi: 10.1007/BFb0056295. Google Scholar

[33]

B. Likar and F. Pernus, A hierarchical approach to elastic registration based on mutual information,, Image and Vision Computing, 19 (2001), 33. doi: 10.1016/S0262-8856(00)00053-6. Google Scholar

[34]

T. Lu, P. Neittaanm and X. Tai, A parallel splitting up method and its application to Navier-Stokes equations,, Applied Mathematics Letters, 4 (1991), 25. doi: 10.1016/0893-9659(91)90161-N. Google Scholar

[35]

F. Maes, A. Collignon, D. Vandermeulen, G. Marchal and P. Suetens, Multimodality image registration by maximization of mutual information,, IEEE Trans Med Imaging, 16 (1997), 187. doi: 10.1109/42.563664. Google Scholar

[36]

F. Maes, D. Vandermeulen and P. Suetens, Comparative evaluation of multiresolution optimization strategies for multimodality image registration by maximization of mutual information,, Medical image analysis, 3 (1999), 373. doi: 10.1016/S1361-8415(99)80030-9. Google Scholar

[37]

F. Maes, D. Vandermeulen and P. Suetens, Medical image registration using mutual information,, Proc IEEE - special issue on emerging medical imaging technology, 91 (2003), 1699. doi: 10.1109/JPROC.2003.817864. Google Scholar

[38]

M. Modat, G. R. Ridgway, Z. A. Taylor, D. J. Hawkes, N. C. Fox and S. Ourselin, A parallel-friendly normalized mutual information gradient for free-form registration,, SPIE Medical Imaging: Image Processing, 7259 (2009). doi: 10.1117/12.811588. Google Scholar

[39]

J. P. W. Pluim, J. B. A. Maintz and M. A. Viergever, Mutual-information-based registration of medical images: A survey,, IEEE Trans. Med. Imaging, 22 (2003), 986. doi: 10.1109/TMI.2003.815867. Google Scholar

[40]

J. P. W. Pluim, J. B. A. Maintz and M. A. Viergever, f-Information measures in medical image registration,, IEEE Trans. Med. Imaging, 23 (2004), 1508. Google Scholar

[41]

A. Rényi, On measure of dependence,, Acta Mathematica Academiae Scientiarum Hungaria, 10 (1959), 441. doi: 10.1007/BF02024507. Google Scholar

[42]

A. Roche, G. Malandain and N. Ayache, Unifying Maximum Likelihood Approaches in Medical Image Registration,, International Journal of Imaging Systems and Technology, 11 (2000), 71. doi: 10.1002/(SICI)1098-1098(2000)11:1<71::AID-IMA8>3.3.CO;2-X. Google Scholar

[43]

A. Roche, G. Malandain, X. Pennec and N. Ayache, The Correlation Ratio as a New Similarity Measure for Multimodal Image Registration,, MICCAI'98, 1496 (1998), 1115. doi: 10.1007/BFb0056301. Google Scholar

[44]

M. R. Sabuncu and P. Ramadge, Using spanning graphs for efficient image registration,, IEEE Transactions on Image Processing, 17 (2008), 788. doi: 10.1109/TIP.2008.918951. Google Scholar

[45]

M. Seppa, Continuous sampling in mutual-information registration,, IEEE Trans. Med. Imaging, 17 (2008), 823. doi: 10.1109/TIP.2008.920738. Google Scholar

[46]

C. Studholme, D. L. G. Hill and D. J. Hawkes, Multiresolution voxel similarity measures for MR-PET registration,, Lecture Notes in Computer Science In Proceedings of Information Processing in Medical Imaging, 3 (1995), 287. Google Scholar

[47]

C. Studholme, D. L. G. Hill and D. J. Hawkes, An overlap invariant entropy measure of 3d medical image alignment,, Pattern Recognition, 32 (1999), 71. doi: 10.1016/S0031-3203(98)00091-0. Google Scholar

[48]

B. Schélkopf, B. K. Sriperumbudur, A. Gretton and K. Fukumizu, RKHS Representation of Measures,, In Learning Theory and Approximation Workshop, (2008). Google Scholar

[49]

P. Thevenaz, M. Bierlaire and M. Unser, Halton sampling for image registration based on mutual information,, Sampling Theory Signal Image Process, 7 (2008), 141. Google Scholar

[50]

P. A. Viola and W. M. Wells III, Alignment by maximization of mutual information,, Proceedings of International Conference on Computer Vision, (1995), 16. doi: 10.1109/ICCV.1995.466930. Google Scholar

[51]

P. Viola and W. Wells, Alignment by maximization of mutual information,, International Journal of Computer Vision, 24 (1997), 137. Google Scholar

[52]

J. Weickert, B. M. H. Romeny and M. A. Viergever, Efficient and reliable schemes for nonlinear diffusioin filtering,, IEEE Transactions on Image Processing, 7 (1998), 398. Google Scholar

[53]

Y. Weiss and D. Fleet, Velocity likelihoods in biological and machine vision,, Probabilistic Models of the brain, (2002), 81. Google Scholar

[54]

W. M. Wells III, P. Viola, H. Atsumi, S. Nakajima and R. Kikinis, Multi-modal volume registration by maximizing mutual information,, Medical Image Analysis, 1 (1996), 35. Google Scholar

[55]

Z. Zhang, Y. Jiang and H. Tsui, Consistent multi-modal non-rigid registration based on a variational approach,, Pattern Recognit. Lett., 27 (2006), 715. doi: 10.1016/j.patrec.2005.10.018. Google Scholar

[56]

A. Zaanen, Linear Analysis,, North Holland Publishing Co., (1960). Google Scholar

[57]

B. Zitova and J. Flusser, Image registration methods: A survey,, Image and Vision Computing, 21 (2003), 977. doi: 10.1016/S0262-8856(03)00137-9. Google Scholar

[58]

H. Zhang, Y. Chen and J. Shi, Nonparametric Image Segmentation Using Renyi's Statistical Dependence Measure,, Journal of Mathematical Imaging and Vision, 44 (2012), 330. doi: 10.1007/s10851-012-0329-z. Google Scholar

[59]

L. Zölei, J. Fisher and W. M. Wells III, A Unified Statistical and Information Theoretic Framework for Multi-modal Image Registration,, Information Processing in Medical Imaging, 2732 (2003), 366. Google Scholar

show all references

References:
[1]

R. B. Ash, Information Theory,, Dover Publications, (1990). Google Scholar

[2]

A. Berlinet and C. Thomas, Reproducing kernel Hilbert spaces in Probability and Statistics,, Kluwer Academic Publishers, (2004). doi: 10.1007/978-1-4419-9096-9. Google Scholar

[3]

N. Akhiezer and I. Glazman, Theory of Linear Operators in Hilbert Space,, Dover Publications, (1993). Google Scholar

[4]

L. Alvarez, R. Deriche, T. Papadopoulo and J. Sanchez, Symmetrical dense optical flow estimation with occlusions detection,, International Journal of Computer Vision, 75 (2007), 371. doi: 10.1007/s11263-007-0041-4. Google Scholar

[5]

N. Aronszajn, Theory of reproducing kernels,, Transactions of the American mathematical society, 68 (1950), 337. doi: 10.1090/S0002-9947-1950-0051437-7. Google Scholar

[6]

N. Ayache, A. Guimond, A. Roche and J. Meunier, Three dimensional multimodal brain warping using the demons algorithm and adaptvie intensity correction,, IEEE Trans. Med. Imag., 20 (2001), 58. Google Scholar

[7]

A. Bardera, M. Feixas, I. Boada and M. Sbert, High-dimensional normalized mutual information for image registration using random lines,, WBIR, 4057 (2006), 264. doi: 10.1007/11784012_32. Google Scholar

[8]

C. Broit, Optimal Registration of Deformed Images,, PhD thesis, (1981). Google Scholar

[9]

R. Bajcsy and C. Broit, Matching of deformed images,, Proc. Int. Conf. Pattern Recognition, (1982), 351. Google Scholar

[10]

R. Bajscy and S. Kovacic, Multiresolution elastic matching,, Comput. Vision. Graph. Image Process, 46 (1989), 1. Google Scholar

[11]

R. Bajcsy, R. Lieberson and M. Reivich, A computerized system for the elastic matching of deformed radiographic images to idealized atlas images,, Journal of Computer Assisted Tomogra-phy, 7 (1983), 618. doi: 10.1097/00004728-198308000-00008. Google Scholar

[12]

A. Bardera, M. Feixas and I. Boada, Normalized similarity measures for medical image registration,, Proc. SPIE Medical Imaging SPIE, 5370 (2004). Google Scholar

[13]

P. Cachier and X. Pennec, 3d non-rigid reigistration by gradient descent on a Gaussian window similarity measure using convolutions,, IEEE workshop on mathematical methods in biomedical image analysis, (2000), 182. doi: 10.1109/MMBIA.2000.852376. Google Scholar

[14]

A. Collignon, F. Maes, D. Delaere, D. Vandermeulen, P. Suetens and G. Marchal, Automated multi-modality image registration based on information theory,, Information Processing in Medical Imaging, (1995), 263. Google Scholar

[15]

A. Collignon, D. Vandermeulen, P. Suetens and G. Marchal, 3D multi-modality medical image registration using feature space clustering, Proceedings of the First International Conference on Computer Vision,, Virtual Reality and Robotics in Medicine, (1995), 195. Google Scholar

[16]

H. M. Chan, A. C.S. Chung, S. C.H. Yu, A. Norbash and W. M. Wells III, Multi-modal image registration by minimizing Kullback-Leibler distance between expected and observed joint class histograms,, IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2 (2003). Google Scholar

[17]

A. C. S. Chung, W. M. Wells III, A. Norbash and W. E. L. Grimson, Multi-modal image registration by minimizing kullback-Leibler distance., International Conference on Medical Image Computing and Computer-Assisted Intervention, 2 (2002), 525. Google Scholar

[18]

T. M. Cover and J. A. Thomas, Elements of Information Theory,, Wiley-Interscience, (2006). Google Scholar

[19]

P. T. Evenaz , M. Bierlaire and M. Unser, Halton sampling for image registration based on mutual information,, Sampling Theory Signal Image Process, 7 (2008), 141. Google Scholar

[20]

G. H. Golub and C. F. Van Loan, Matrix Computations,, 3rd Edition, (1996). Google Scholar

[21]

Y. Guo and C. Lu, Multi-modality image registration using mutual information based on gradient vector flow,, 18th International Conference on Pattern Recognition (ICPR'06), 3 (2006), 697. Google Scholar

[22]

R. Gan, J. Wu, A. C. S. Chung, S. C. H. Yu and W. M. Wells III, Multiresolution image registration based on Kullback-Leibler distance,, MICCAI, 3216 (2004), 599. doi: 10.1007/978-3-540-30135-6_73. Google Scholar

[23]

C. Guetter, C. Xu, F. Sauer and J. Hornegger, Learning based non-rigid multi-modal image registration using Kullback-Leibler divergence,, Lecture Notes in Computer Science, 3750 (2005), 255. doi: 10.1007/11566489_32. Google Scholar

[24]

Y. He, A. B. Hamza and H. Krim, A generalized divergence measure for robust image registration,, IEEE Transactions on Signal Processing, 51 (2003), 1211. doi: 10.1109/TSP.2003.810305. Google Scholar

[25]

S. Henn and K. Witsch, Multimodal image registration using a variational approach,, SIAM J. Sci. Comput., 25 (2003), 1429. doi: 10.1137/S1064827502201424. Google Scholar

[26]

G. Hermosillo, C. C. Hotel and O. Faugeras, Variational methods for multimodal image matching,, Int. J. Computer Vision, 50 (2002), 329. Google Scholar

[27]

D. Hill, P. Batchelor, M. Holden and D. Hawkes, Topical review: medical image registration,, Physics in Medicine and Biology, 46 (2001), 1. Google Scholar

[28]

D. L. G. Hill, C. Studholme and D. J. Hawkes, Voxel similarity measures for automated image registration,, Visualization in Biomedical Computing, 2359 (1994), 205. Google Scholar

[29]

B. Jian, B. Vemuri and J. Marroquin, Robust nonrigid multimodal image registration using local frequency maps., Proc. Inf. Process. Med. Imag., 3565 (2005), 504. doi: 10.1007/11505730_42. Google Scholar

[30]

L. R. Jorge, M. S. Juan and V. M. Rafael, Generalized regularization term for non-parametric multimodal image registration,, Signal Processing, 87 (2007), 2837. Google Scholar

[31]

S. Klein, M. Staring and J. P. W. Pluim, Evaluation of optimisation methods for nonrigid medical image registration using mutual information and B-splines,, IEEE Trans. Image Process., 16 (2007), 2879. doi: 10.1109/TIP.2007.909412. Google Scholar

[32]

M. Leventon and W. E. L. Grimson, Multi-modal volume registration using joint intensity distributions,, Medical Image Computing and Computer-Assisted Interventation MICCAI98, 1496 (1998), 1057. doi: 10.1007/BFb0056295. Google Scholar

[33]

B. Likar and F. Pernus, A hierarchical approach to elastic registration based on mutual information,, Image and Vision Computing, 19 (2001), 33. doi: 10.1016/S0262-8856(00)00053-6. Google Scholar

[34]

T. Lu, P. Neittaanm and X. Tai, A parallel splitting up method and its application to Navier-Stokes equations,, Applied Mathematics Letters, 4 (1991), 25. doi: 10.1016/0893-9659(91)90161-N. Google Scholar

[35]

F. Maes, A. Collignon, D. Vandermeulen, G. Marchal and P. Suetens, Multimodality image registration by maximization of mutual information,, IEEE Trans Med Imaging, 16 (1997), 187. doi: 10.1109/42.563664. Google Scholar

[36]

F. Maes, D. Vandermeulen and P. Suetens, Comparative evaluation of multiresolution optimization strategies for multimodality image registration by maximization of mutual information,, Medical image analysis, 3 (1999), 373. doi: 10.1016/S1361-8415(99)80030-9. Google Scholar

[37]

F. Maes, D. Vandermeulen and P. Suetens, Medical image registration using mutual information,, Proc IEEE - special issue on emerging medical imaging technology, 91 (2003), 1699. doi: 10.1109/JPROC.2003.817864. Google Scholar

[38]

M. Modat, G. R. Ridgway, Z. A. Taylor, D. J. Hawkes, N. C. Fox and S. Ourselin, A parallel-friendly normalized mutual information gradient for free-form registration,, SPIE Medical Imaging: Image Processing, 7259 (2009). doi: 10.1117/12.811588. Google Scholar

[39]

J. P. W. Pluim, J. B. A. Maintz and M. A. Viergever, Mutual-information-based registration of medical images: A survey,, IEEE Trans. Med. Imaging, 22 (2003), 986. doi: 10.1109/TMI.2003.815867. Google Scholar

[40]

J. P. W. Pluim, J. B. A. Maintz and M. A. Viergever, f-Information measures in medical image registration,, IEEE Trans. Med. Imaging, 23 (2004), 1508. Google Scholar

[41]

A. Rényi, On measure of dependence,, Acta Mathematica Academiae Scientiarum Hungaria, 10 (1959), 441. doi: 10.1007/BF02024507. Google Scholar

[42]

A. Roche, G. Malandain and N. Ayache, Unifying Maximum Likelihood Approaches in Medical Image Registration,, International Journal of Imaging Systems and Technology, 11 (2000), 71. doi: 10.1002/(SICI)1098-1098(2000)11:1<71::AID-IMA8>3.3.CO;2-X. Google Scholar

[43]

A. Roche, G. Malandain, X. Pennec and N. Ayache, The Correlation Ratio as a New Similarity Measure for Multimodal Image Registration,, MICCAI'98, 1496 (1998), 1115. doi: 10.1007/BFb0056301. Google Scholar

[44]

M. R. Sabuncu and P. Ramadge, Using spanning graphs for efficient image registration,, IEEE Transactions on Image Processing, 17 (2008), 788. doi: 10.1109/TIP.2008.918951. Google Scholar

[45]

M. Seppa, Continuous sampling in mutual-information registration,, IEEE Trans. Med. Imaging, 17 (2008), 823. doi: 10.1109/TIP.2008.920738. Google Scholar

[46]

C. Studholme, D. L. G. Hill and D. J. Hawkes, Multiresolution voxel similarity measures for MR-PET registration,, Lecture Notes in Computer Science In Proceedings of Information Processing in Medical Imaging, 3 (1995), 287. Google Scholar

[47]

C. Studholme, D. L. G. Hill and D. J. Hawkes, An overlap invariant entropy measure of 3d medical image alignment,, Pattern Recognition, 32 (1999), 71. doi: 10.1016/S0031-3203(98)00091-0. Google Scholar

[48]

B. Schélkopf, B. K. Sriperumbudur, A. Gretton and K. Fukumizu, RKHS Representation of Measures,, In Learning Theory and Approximation Workshop, (2008). Google Scholar

[49]

P. Thevenaz, M. Bierlaire and M. Unser, Halton sampling for image registration based on mutual information,, Sampling Theory Signal Image Process, 7 (2008), 141. Google Scholar

[50]

P. A. Viola and W. M. Wells III, Alignment by maximization of mutual information,, Proceedings of International Conference on Computer Vision, (1995), 16. doi: 10.1109/ICCV.1995.466930. Google Scholar

[51]

P. Viola and W. Wells, Alignment by maximization of mutual information,, International Journal of Computer Vision, 24 (1997), 137. Google Scholar

[52]

J. Weickert, B. M. H. Romeny and M. A. Viergever, Efficient and reliable schemes for nonlinear diffusioin filtering,, IEEE Transactions on Image Processing, 7 (1998), 398. Google Scholar

[53]

Y. Weiss and D. Fleet, Velocity likelihoods in biological and machine vision,, Probabilistic Models of the brain, (2002), 81. Google Scholar

[54]

W. M. Wells III, P. Viola, H. Atsumi, S. Nakajima and R. Kikinis, Multi-modal volume registration by maximizing mutual information,, Medical Image Analysis, 1 (1996), 35. Google Scholar

[55]

Z. Zhang, Y. Jiang and H. Tsui, Consistent multi-modal non-rigid registration based on a variational approach,, Pattern Recognit. Lett., 27 (2006), 715. doi: 10.1016/j.patrec.2005.10.018. Google Scholar

[56]

A. Zaanen, Linear Analysis,, North Holland Publishing Co., (1960). Google Scholar

[57]

B. Zitova and J. Flusser, Image registration methods: A survey,, Image and Vision Computing, 21 (2003), 977. doi: 10.1016/S0262-8856(03)00137-9. Google Scholar

[58]

H. Zhang, Y. Chen and J. Shi, Nonparametric Image Segmentation Using Renyi's Statistical Dependence Measure,, Journal of Mathematical Imaging and Vision, 44 (2012), 330. doi: 10.1007/s10851-012-0329-z. Google Scholar

[59]

L. Zölei, J. Fisher and W. M. Wells III, A Unified Statistical and Information Theoretic Framework for Multi-modal Image Registration,, Information Processing in Medical Imaging, 2732 (2003), 366. Google Scholar

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