Abstract

Diffuse optical tomography (DOT) retrieves the spatially distributed optical characteristics of a medium from external measurements. Recovering the parameters of interest involves solving a nonlinear and highly ill-posed inverse problem. This paper examines the possibility of regularizing DOT via the introduction of a priori information from alternative high-resolution anatomical modalities, using the information theory concepts of mutual information (MI) and joint entropy (JE). Such functionals evaluate the similarity between the reconstructed optical image and the prior image while bypassing the multimodality barrier manifested as the incommensurate relation between the gray value representations of corresponding anatomical features in the two modalities. By introducing structural information, we aim to improve the spatial resolution and quantitative accuracy of the solution. We provide a thorough explanation of the theory from an imaging perspective, accompanied by preliminary results using numerical simulations. In addition we compare the performance of MI and JE. Finally, we have adopted a method for fast marginal entropy evaluation and optimization by modifying the objective function and extending it to the JE case. We demonstrate its use on an image reconstruction framework and show significant computational savings.

© 2009 Optical Society of America

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2005 (5)

A. Gibson, J. Hebden, and S. R. Arridge, “Recent advances in diffuse optical tomography,” Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

Q. Zhang, T. J. Brukilacchio, A. Li, J. J. Stott, T. Chaves, E. Hillman, T. Wu, M. Chorlton, E. Rafferty, R. Moore, D. B. Kopans, and D. A. Boas, “Coregistered tomographic x-ray and optical breast imaging: initial results,” J. Biomed. Opt. 10, 024033:1-9 (2005).
[CrossRef]

Q. Zhu, S. H. Kurtzma, P. Hegde, S. Tannenbaum, M. Kane, M. Huang, N. G. Chen, B. Jagjivan, and K. Zarfos, “Utilizing optical tomography with ultrasound localization to image heterogeneous hemoglobin distribution in large breast cancers.” Neoplasia 7, 263-70 (2005).
[CrossRef] [PubMed]

S. Shwartz, M. Zibulevsky, and Y. S. Yoav, “Fast kernel entropy estimation and optimization,” SIGACT News 85, 1045-1058 (2005).

M. Schweiger, S. Arridge, and I. Nissila, “Gauss-Newton method for image reconstruction in diffuse optical tomography.” Phys. Med. Biol. 50, 2365-86 (2005).
[CrossRef] [PubMed]

2003 (3)

M. Schweiger and S. R. Arridge, “Optical tomography with local basis functions,” J. Electron. Imaging 12, 583-593 (2003).
[CrossRef]

B. Brooksby, H. Dehghani, B. W. Pogue, and K. D. Paulsen, “Near-infrared (NIR) tomography breast image reconstruction with a priori structural information from MRI: algorithm development for reconstructing heterogeneities,” IEEE J. Quantum Electron. 9, 199-209 (2003).
[CrossRef]

M. Girolami and C. He, “Probability density estimation from optimally condensed data samples,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 1253-1264 (2003).
[CrossRef]

2001 (2)

D. L. G. Hill, P. G. Batchelor, M. Holden, and D. J. Hawkes, “Medical image registration,” Phys. Med. Biol. 46, R1-R45 (2001).
[CrossRef] [PubMed]

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57-75 (2001).
[CrossRef]

2000 (2)

V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. U.S.A. 97, 2767-2772 (2000).
[CrossRef] [PubMed]

D. Rueckert, M. Clarkson, D. Hill, and D. Hawkes, “Non-rigid registration using higher order mutual information,” in Proc. SPIE 3979, 438-447 (2000).

1999 (1)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

1997 (3)

J. C. Hebden, S. R. Arridge, and D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825-840 (1997).
[CrossRef] [PubMed]

S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841-853 (1997).
[CrossRef] [PubMed]

F. Maes, A. Collignon, D. Vandermeulen, G. Marchal, and P. J. Suetens, “Multimodality image registration by maximization of mutual information,” IEEE Trans. Med. Imaging 16, 187-198 (1997).
[CrossRef] [PubMed]

1996 (1)

P. Hall and M. Wand, “On the accuracy of binned kernel density estimators,” J. Multivariate Anal. 56, 165-184 (1996).
[CrossRef]

1995 (2)

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element model for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779-1792 (1995).
[CrossRef] [PubMed]

S. R. Arridge, “Photon measurement density functions. Part 1: analytical forms,” Appl. Opt. 34, 7395-7409 (1995).
[CrossRef] [PubMed]

1994 (1)

M. Wand, “Fast computation of multivariate kernel estimators,” J. Comput. Graph. Stat. 3, 433-445 (1994).
[CrossRef]

1993 (1)

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite element method for the forward and inverse models in optical tomography,” J. Math. Imaging Vision 3, 263-283 (1993).
[CrossRef]

1984 (1)

J. Skilling and R. K. Bryan, “Maximum entropy image reconstruction--general algorithm,” Mon. Not. R. Astron. Soc. 211, 111-124 (1984).

1982 (1)

B. W. Silverman, “Algorithm AS 176: kernel density estimation using the fast Fourier transform,” Appl. Stat. 31, 93-99 (1982).
[CrossRef]

1948 (1)

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379-423 (1948).

Arridge, S.

M. Schweiger, S. Arridge, and I. Nissila, “Gauss-Newton method for image reconstruction in diffuse optical tomography.” Phys. Med. Biol. 50, 2365-86 (2005).
[CrossRef] [PubMed]

Arridge, S. R.

A. Gibson, J. Hebden, and S. R. Arridge, “Recent advances in diffuse optical tomography,” Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

M. Schweiger and S. R. Arridge, “Optical tomography with local basis functions,” J. Electron. Imaging 12, 583-593 (2003).
[CrossRef]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841-853 (1997).
[CrossRef] [PubMed]

J. C. Hebden, S. R. Arridge, and D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825-840 (1997).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element model for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779-1792 (1995).
[CrossRef] [PubMed]

S. R. Arridge, “Photon measurement density functions. Part 1: analytical forms,” Appl. Opt. 34, 7395-7409 (1995).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite element method for the forward and inverse models in optical tomography,” J. Math. Imaging Vision 3, 263-283 (1993).
[CrossRef]

S. R. Arridge, C. Panagiotou, M. Schweiger, and V. Kolehmainen, “Multimodal diffuse optical tomography: theory,” in Translational Multimodality Optical Imaging (Artech, 2008), Chap. 5, pp. 101-123.

Asma, E.

S. Somayajula, E. Asma, and R. M. Leahy, “PET image reconstruction using anatomical information through mutual information based priors,” in Proceedings of the IEEE Nuclear Science Symposium and Medical Imaging Conference (IEEE, 2005), pp. 2722-2726.

Batchelor, P. G.

D. L. G. Hill, P. G. Batchelor, M. Holden, and D. J. Hawkes, “Medical image registration,” Phys. Med. Biol. 46, R1-R45 (2001).
[CrossRef] [PubMed]

Boas, D. A.

Q. Zhang, T. J. Brukilacchio, A. Li, J. J. Stott, T. Chaves, E. Hillman, T. Wu, M. Chorlton, E. Rafferty, R. Moore, D. B. Kopans, and D. A. Boas, “Coregistered tomographic x-ray and optical breast imaging: initial results,” J. Biomed. Opt. 10, 024033:1-9 (2005).
[CrossRef]

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57-75 (2001).
[CrossRef]

Brooks, D. H.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57-75 (2001).
[CrossRef]

Brooksby, B.

B. Brooksby, H. Dehghani, B. W. Pogue, and K. D. Paulsen, “Near-infrared (NIR) tomography breast image reconstruction with a priori structural information from MRI: algorithm development for reconstructing heterogeneities,” IEEE J. Quantum Electron. 9, 199-209 (2003).
[CrossRef]

Brukilacchio, T. J.

Q. Zhang, T. J. Brukilacchio, A. Li, J. J. Stott, T. Chaves, E. Hillman, T. Wu, M. Chorlton, E. Rafferty, R. Moore, D. B. Kopans, and D. A. Boas, “Coregistered tomographic x-ray and optical breast imaging: initial results,” J. Biomed. Opt. 10, 024033:1-9 (2005).
[CrossRef]

Bryan, R. K.

J. Skilling and R. K. Bryan, “Maximum entropy image reconstruction--general algorithm,” Mon. Not. R. Astron. Soc. 211, 111-124 (1984).

Chance, B.

V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. U.S.A. 97, 2767-2772 (2000).
[CrossRef] [PubMed]

Chaves, T.

Q. Zhang, T. J. Brukilacchio, A. Li, J. J. Stott, T. Chaves, E. Hillman, T. Wu, M. Chorlton, E. Rafferty, R. Moore, D. B. Kopans, and D. A. Boas, “Coregistered tomographic x-ray and optical breast imaging: initial results,” J. Biomed. Opt. 10, 024033:1-9 (2005).
[CrossRef]

Chen, N. G.

Q. Zhu, S. H. Kurtzma, P. Hegde, S. Tannenbaum, M. Kane, M. Huang, N. G. Chen, B. Jagjivan, and K. Zarfos, “Utilizing optical tomography with ultrasound localization to image heterogeneous hemoglobin distribution in large breast cancers.” Neoplasia 7, 263-70 (2005).
[CrossRef] [PubMed]

Chorlton, M.

Q. Zhang, T. J. Brukilacchio, A. Li, J. J. Stott, T. Chaves, E. Hillman, T. Wu, M. Chorlton, E. Rafferty, R. Moore, D. B. Kopans, and D. A. Boas, “Coregistered tomographic x-ray and optical breast imaging: initial results,” J. Biomed. Opt. 10, 024033:1-9 (2005).
[CrossRef]

Clarkson, M.

D. Rueckert, M. Clarkson, D. Hill, and D. Hawkes, “Non-rigid registration using higher order mutual information,” in Proc. SPIE 3979, 438-447 (2000).

Collignon, A.

F. Maes, A. Collignon, D. Vandermeulen, G. Marchal, and P. J. Suetens, “Multimodality image registration by maximization of mutual information,” IEEE Trans. Med. Imaging 16, 187-198 (1997).
[CrossRef] [PubMed]

Cover, T. M.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley-Interscience, 1991).
[CrossRef]

Dehghani, H.

B. Brooksby, H. Dehghani, B. W. Pogue, and K. D. Paulsen, “Near-infrared (NIR) tomography breast image reconstruction with a priori structural information from MRI: algorithm development for reconstructing heterogeneities,” IEEE J. Quantum Electron. 9, 199-209 (2003).
[CrossRef]

Delpy, D. T.

J. C. Hebden, S. R. Arridge, and D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825-840 (1997).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element model for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779-1792 (1995).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite element method for the forward and inverse models in optical tomography,” J. Math. Imaging Vision 3, 263-283 (1993).
[CrossRef]

DiMarzio, C. A.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57-75 (2001).
[CrossRef]

Duda, R. O.

R. O. Duda, P. E. Hart, and D. G. Stork., Pattern Classification (Wiley, 2001).

Gaudette, R. J.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57-75 (2001).
[CrossRef]

Gibson, A.

A. Gibson, J. Hebden, and S. R. Arridge, “Recent advances in diffuse optical tomography,” Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

Gindi, G.

A. Rangarajan, I.-T. Hsiao, and G. Gindi, “Integrating anatomical priors in ECT reconstruction via joint mixtures and mutual information,” in Nuclear Science Symposium Conference Record, 1998 (IEEE, 1998), Vol. 3, pp. 1548-1588.

Girolami, M.

M. Girolami and C. He, “Probability density estimation from optimally condensed data samples,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 1253-1264 (2003).
[CrossRef]

Green, P. J.

B. W. Silverman and P. J. Green, Density Estimation for Statistics and Data Analysis (Chapman & Hall, 1986).

Hadamard, J.

J. Hadamard, “Sur les problèmes aux derivées partielles et leur signification physique,” Bulletin Princeton University (1902), Vol. 13, 49-52.

Hall, P.

P. Hall and M. Wand, “On the accuracy of binned kernel density estimators,” J. Multivariate Anal. 56, 165-184 (1996).
[CrossRef]

Hart, P. E.

R. O. Duda, P. E. Hart, and D. G. Stork., Pattern Classification (Wiley, 2001).

Hawkes, D.

D. Rueckert, M. Clarkson, D. Hill, and D. Hawkes, “Non-rigid registration using higher order mutual information,” in Proc. SPIE 3979, 438-447 (2000).

Hawkes, D. J.

D. L. G. Hill, P. G. Batchelor, M. Holden, and D. J. Hawkes, “Medical image registration,” Phys. Med. Biol. 46, R1-R45 (2001).
[CrossRef] [PubMed]

He, C.

M. Girolami and C. He, “Probability density estimation from optimally condensed data samples,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 1253-1264 (2003).
[CrossRef]

Hebden, J.

A. Gibson, J. Hebden, and S. R. Arridge, “Recent advances in diffuse optical tomography,” Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

Hebden, J. C.

J. C. Hebden, S. R. Arridge, and D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825-840 (1997).
[CrossRef] [PubMed]

S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841-853 (1997).
[CrossRef] [PubMed]

Hegde, P.

Q. Zhu, S. H. Kurtzma, P. Hegde, S. Tannenbaum, M. Kane, M. Huang, N. G. Chen, B. Jagjivan, and K. Zarfos, “Utilizing optical tomography with ultrasound localization to image heterogeneous hemoglobin distribution in large breast cancers.” Neoplasia 7, 263-70 (2005).
[CrossRef] [PubMed]

Hill, D.

D. Rueckert, M. Clarkson, D. Hill, and D. Hawkes, “Non-rigid registration using higher order mutual information,” in Proc. SPIE 3979, 438-447 (2000).

Hill, D. L. G.

D. L. G. Hill, P. G. Batchelor, M. Holden, and D. J. Hawkes, “Medical image registration,” Phys. Med. Biol. 46, R1-R45 (2001).
[CrossRef] [PubMed]

Hillman, E.

Q. Zhang, T. J. Brukilacchio, A. Li, J. J. Stott, T. Chaves, E. Hillman, T. Wu, M. Chorlton, E. Rafferty, R. Moore, D. B. Kopans, and D. A. Boas, “Coregistered tomographic x-ray and optical breast imaging: initial results,” J. Biomed. Opt. 10, 024033:1-9 (2005).
[CrossRef]

Hiraoka, M.

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element model for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779-1792 (1995).
[CrossRef] [PubMed]

Holden, M.

D. L. G. Hill, P. G. Batchelor, M. Holden, and D. J. Hawkes, “Medical image registration,” Phys. Med. Biol. 46, R1-R45 (2001).
[CrossRef] [PubMed]

Hsiao, I.-T.

A. Rangarajan, I.-T. Hsiao, and G. Gindi, “Integrating anatomical priors in ECT reconstruction via joint mixtures and mutual information,” in Nuclear Science Symposium Conference Record, 1998 (IEEE, 1998), Vol. 3, pp. 1548-1588.

Huang, M.

Q. Zhu, S. H. Kurtzma, P. Hegde, S. Tannenbaum, M. Kane, M. Huang, N. G. Chen, B. Jagjivan, and K. Zarfos, “Utilizing optical tomography with ultrasound localization to image heterogeneous hemoglobin distribution in large breast cancers.” Neoplasia 7, 263-70 (2005).
[CrossRef] [PubMed]

Jagjivan, B.

Q. Zhu, S. H. Kurtzma, P. Hegde, S. Tannenbaum, M. Kane, M. Huang, N. G. Chen, B. Jagjivan, and K. Zarfos, “Utilizing optical tomography with ultrasound localization to image heterogeneous hemoglobin distribution in large breast cancers.” Neoplasia 7, 263-70 (2005).
[CrossRef] [PubMed]

Jones, M.

M. Wand and M. Jones, Kernel Smoothing, Vol. 60 of Monographs on Statistics and Applied Probability (Chapman & Hall, 1995).

Kane, M.

Q. Zhu, S. H. Kurtzma, P. Hegde, S. Tannenbaum, M. Kane, M. Huang, N. G. Chen, B. Jagjivan, and K. Zarfos, “Utilizing optical tomography with ultrasound localization to image heterogeneous hemoglobin distribution in large breast cancers.” Neoplasia 7, 263-70 (2005).
[CrossRef] [PubMed]

Kilmer, M.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57-75 (2001).
[CrossRef]

Kolehmainen, V.

S. R. Arridge, C. Panagiotou, M. Schweiger, and V. Kolehmainen, “Multimodal diffuse optical tomography: theory,” in Translational Multimodality Optical Imaging (Artech, 2008), Chap. 5, pp. 101-123.

Kopans, D. B.

Q. Zhang, T. J. Brukilacchio, A. Li, J. J. Stott, T. Chaves, E. Hillman, T. Wu, M. Chorlton, E. Rafferty, R. Moore, D. B. Kopans, and D. A. Boas, “Coregistered tomographic x-ray and optical breast imaging: initial results,” J. Biomed. Opt. 10, 024033:1-9 (2005).
[CrossRef]

Kurtzma, S. H.

Q. Zhu, S. H. Kurtzma, P. Hegde, S. Tannenbaum, M. Kane, M. Huang, N. G. Chen, B. Jagjivan, and K. Zarfos, “Utilizing optical tomography with ultrasound localization to image heterogeneous hemoglobin distribution in large breast cancers.” Neoplasia 7, 263-70 (2005).
[CrossRef] [PubMed]

Leahy, R. M.

S. Somayajula, E. Asma, and R. M. Leahy, “PET image reconstruction using anatomical information through mutual information based priors,” in Proceedings of the IEEE Nuclear Science Symposium and Medical Imaging Conference (IEEE, 2005), pp. 2722-2726.

S. Somayajula, A. Rangarajan, and R. M. Leahy, “PET image reconstruction using anatomical information through mutual information based priors: a scale space approach,” in Biomedical Imaging from Nano to Macro, 2007 (IEEE, 2007), pp. 165-168.
[CrossRef]

Li, A.

Q. Zhang, T. J. Brukilacchio, A. Li, J. J. Stott, T. Chaves, E. Hillman, T. Wu, M. Chorlton, E. Rafferty, R. Moore, D. B. Kopans, and D. A. Boas, “Coregistered tomographic x-ray and optical breast imaging: initial results,” J. Biomed. Opt. 10, 024033:1-9 (2005).
[CrossRef]

Maes, F.

F. Maes, A. Collignon, D. Vandermeulen, G. Marchal, and P. J. Suetens, “Multimodality image registration by maximization of mutual information,” IEEE Trans. Med. Imaging 16, 187-198 (1997).
[CrossRef] [PubMed]

Marchal, G.

F. Maes, A. Collignon, D. Vandermeulen, G. Marchal, and P. J. Suetens, “Multimodality image registration by maximization of mutual information,” IEEE Trans. Med. Imaging 16, 187-198 (1997).
[CrossRef] [PubMed]

Miller, E. L.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57-75 (2001).
[CrossRef]

Moore, R.

Q. Zhang, T. J. Brukilacchio, A. Li, J. J. Stott, T. Chaves, E. Hillman, T. Wu, M. Chorlton, E. Rafferty, R. Moore, D. B. Kopans, and D. A. Boas, “Coregistered tomographic x-ray and optical breast imaging: initial results,” J. Biomed. Opt. 10, 024033:1-9 (2005).
[CrossRef]

Moulton, J. D.

J. D. Moulton, “Diffusion modelling of picosecond laser pulse propagation in turbid media,” M. Eng. thesis (McMaster University, Hamilton, Ontario, 1990).

Nissila, I.

M. Schweiger, S. Arridge, and I. Nissila, “Gauss-Newton method for image reconstruction in diffuse optical tomography.” Phys. Med. Biol. 50, 2365-86 (2005).
[CrossRef] [PubMed]

Ntziachristos, V.

V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. U.S.A. 97, 2767-2772 (2000).
[CrossRef] [PubMed]

Nuyts, J.

J. Nuyts, “The use of mutual information and joint entropy for anatomical priors in emission tomography,” in Nuclear Science Symposium Conference Record, 2007 (IEEE, 2007), Vol. 6, pp. 4149-4154.
[CrossRef]

Panagiotou, C.

S. R. Arridge, C. Panagiotou, M. Schweiger, and V. Kolehmainen, “Multimodal diffuse optical tomography: theory,” in Translational Multimodality Optical Imaging (Artech, 2008), Chap. 5, pp. 101-123.

Papoulis, A.

A. Papoulis and U. S. Pillai, Probability, Random Variables and Stochastic Processes (McGraw-Hill Science/Engineering/Math, 2001).

Paulsen, K. D.

B. Brooksby, H. Dehghani, B. W. Pogue, and K. D. Paulsen, “Near-infrared (NIR) tomography breast image reconstruction with a priori structural information from MRI: algorithm development for reconstructing heterogeneities,” IEEE J. Quantum Electron. 9, 199-209 (2003).
[CrossRef]

Pillai, U. S.

A. Papoulis and U. S. Pillai, Probability, Random Variables and Stochastic Processes (McGraw-Hill Science/Engineering/Math, 2001).

Pogue, B. W.

B. Brooksby, H. Dehghani, B. W. Pogue, and K. D. Paulsen, “Near-infrared (NIR) tomography breast image reconstruction with a priori structural information from MRI: algorithm development for reconstructing heterogeneities,” IEEE J. Quantum Electron. 9, 199-209 (2003).
[CrossRef]

Rafferty, E.

Q. Zhang, T. J. Brukilacchio, A. Li, J. J. Stott, T. Chaves, E. Hillman, T. Wu, M. Chorlton, E. Rafferty, R. Moore, D. B. Kopans, and D. A. Boas, “Coregistered tomographic x-ray and optical breast imaging: initial results,” J. Biomed. Opt. 10, 024033:1-9 (2005).
[CrossRef]

Rangarajan, A.

S. Somayajula, A. Rangarajan, and R. M. Leahy, “PET image reconstruction using anatomical information through mutual information based priors: a scale space approach,” in Biomedical Imaging from Nano to Macro, 2007 (IEEE, 2007), pp. 165-168.
[CrossRef]

A. Rangarajan, I.-T. Hsiao, and G. Gindi, “Integrating anatomical priors in ECT reconstruction via joint mixtures and mutual information,” in Nuclear Science Symposium Conference Record, 1998 (IEEE, 1998), Vol. 3, pp. 1548-1588.

Rueckert, D.

D. Rueckert, M. Clarkson, D. Hill, and D. Hawkes, “Non-rigid registration using higher order mutual information,” in Proc. SPIE 3979, 438-447 (2000).

Schnall, M.

V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. U.S.A. 97, 2767-2772 (2000).
[CrossRef] [PubMed]

Schweiger, M.

M. Schweiger, S. Arridge, and I. Nissila, “Gauss-Newton method for image reconstruction in diffuse optical tomography.” Phys. Med. Biol. 50, 2365-86 (2005).
[CrossRef] [PubMed]

M. Schweiger and S. R. Arridge, “Optical tomography with local basis functions,” J. Electron. Imaging 12, 583-593 (2003).
[CrossRef]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element model for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779-1792 (1995).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite element method for the forward and inverse models in optical tomography,” J. Math. Imaging Vision 3, 263-283 (1993).
[CrossRef]

S. R. Arridge, C. Panagiotou, M. Schweiger, and V. Kolehmainen, “Multimodal diffuse optical tomography: theory,” in Translational Multimodality Optical Imaging (Artech, 2008), Chap. 5, pp. 101-123.

Shannon, C. E.

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379-423 (1948).

Shewchuk, J. R.

J. R. Shewchuk, “An introduction to the conjugate gradient method without the agonizing pain,” Tech. Rep. CS-94-125 (Carnegie Mellon University, 1994).

Shwartz, S.

S. Shwartz, M. Zibulevsky, and Y. S. Yoav, “Fast kernel entropy estimation and optimization,” SIGACT News 85, 1045-1058 (2005).

Silverman, B. W.

B. W. Silverman, “Algorithm AS 176: kernel density estimation using the fast Fourier transform,” Appl. Stat. 31, 93-99 (1982).
[CrossRef]

B. W. Silverman and P. J. Green, Density Estimation for Statistics and Data Analysis (Chapman & Hall, 1986).

Simonoff, J.

J. Simonoff, Smoothing Methods in Statistics (Springer, 1996).
[CrossRef]

Skilling, J.

J. Skilling and R. K. Bryan, “Maximum entropy image reconstruction--general algorithm,” Mon. Not. R. Astron. Soc. 211, 111-124 (1984).

Somayajula, S.

S. Somayajula, A. Rangarajan, and R. M. Leahy, “PET image reconstruction using anatomical information through mutual information based priors: a scale space approach,” in Biomedical Imaging from Nano to Macro, 2007 (IEEE, 2007), pp. 165-168.
[CrossRef]

S. Somayajula, E. Asma, and R. M. Leahy, “PET image reconstruction using anatomical information through mutual information based priors,” in Proceedings of the IEEE Nuclear Science Symposium and Medical Imaging Conference (IEEE, 2005), pp. 2722-2726.

Stork., D. G.

R. O. Duda, P. E. Hart, and D. G. Stork., Pattern Classification (Wiley, 2001).

Stott, J. J.

Q. Zhang, T. J. Brukilacchio, A. Li, J. J. Stott, T. Chaves, E. Hillman, T. Wu, M. Chorlton, E. Rafferty, R. Moore, D. B. Kopans, and D. A. Boas, “Coregistered tomographic x-ray and optical breast imaging: initial results,” J. Biomed. Opt. 10, 024033:1-9 (2005).
[CrossRef]

Suetens, P. J.

F. Maes, A. Collignon, D. Vandermeulen, G. Marchal, and P. J. Suetens, “Multimodality image registration by maximization of mutual information,” IEEE Trans. Med. Imaging 16, 187-198 (1997).
[CrossRef] [PubMed]

Tannenbaum, S.

Q. Zhu, S. H. Kurtzma, P. Hegde, S. Tannenbaum, M. Kane, M. Huang, N. G. Chen, B. Jagjivan, and K. Zarfos, “Utilizing optical tomography with ultrasound localization to image heterogeneous hemoglobin distribution in large breast cancers.” Neoplasia 7, 263-70 (2005).
[CrossRef] [PubMed]

Thomas, J. A.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley-Interscience, 1991).
[CrossRef]

Vandermeulen, D.

F. Maes, A. Collignon, D. Vandermeulen, G. Marchal, and P. J. Suetens, “Multimodality image registration by maximization of mutual information,” IEEE Trans. Med. Imaging 16, 187-198 (1997).
[CrossRef] [PubMed]

Viola, P. A.

P. A. Viola, “Alignment by maximization of mutual information,” Tech. Rep. AITR-1548 (MIT, 1995).

Vogel, C. R.

C. R. Vogel, Computational Methods for Inverse Problems (SIAM, 2002).
[CrossRef]

Wand, M.

P. Hall and M. Wand, “On the accuracy of binned kernel density estimators,” J. Multivariate Anal. 56, 165-184 (1996).
[CrossRef]

M. Wand, “Fast computation of multivariate kernel estimators,” J. Comput. Graph. Stat. 3, 433-445 (1994).
[CrossRef]

M. Wand and M. Jones, Kernel Smoothing, Vol. 60 of Monographs on Statistics and Applied Probability (Chapman & Hall, 1995).

Wu, T.

Q. Zhang, T. J. Brukilacchio, A. Li, J. J. Stott, T. Chaves, E. Hillman, T. Wu, M. Chorlton, E. Rafferty, R. Moore, D. B. Kopans, and D. A. Boas, “Coregistered tomographic x-ray and optical breast imaging: initial results,” J. Biomed. Opt. 10, 024033:1-9 (2005).
[CrossRef]

Yoav, Y. S.

S. Shwartz, M. Zibulevsky, and Y. S. Yoav, “Fast kernel entropy estimation and optimization,” SIGACT News 85, 1045-1058 (2005).

Yodh, A. G.

V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. U.S.A. 97, 2767-2772 (2000).
[CrossRef] [PubMed]

Zarfos, K.

Q. Zhu, S. H. Kurtzma, P. Hegde, S. Tannenbaum, M. Kane, M. Huang, N. G. Chen, B. Jagjivan, and K. Zarfos, “Utilizing optical tomography with ultrasound localization to image heterogeneous hemoglobin distribution in large breast cancers.” Neoplasia 7, 263-70 (2005).
[CrossRef] [PubMed]

Zhang, Q.

Q. Zhang, T. J. Brukilacchio, A. Li, J. J. Stott, T. Chaves, E. Hillman, T. Wu, M. Chorlton, E. Rafferty, R. Moore, D. B. Kopans, and D. A. Boas, “Coregistered tomographic x-ray and optical breast imaging: initial results,” J. Biomed. Opt. 10, 024033:1-9 (2005).
[CrossRef]

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57-75 (2001).
[CrossRef]

Zhu, Q.

Q. Zhu, S. H. Kurtzma, P. Hegde, S. Tannenbaum, M. Kane, M. Huang, N. G. Chen, B. Jagjivan, and K. Zarfos, “Utilizing optical tomography with ultrasound localization to image heterogeneous hemoglobin distribution in large breast cancers.” Neoplasia 7, 263-70 (2005).
[CrossRef] [PubMed]

Zibulevsky, M.

S. Shwartz, M. Zibulevsky, and Y. S. Yoav, “Fast kernel entropy estimation and optimization,” SIGACT News 85, 1045-1058 (2005).

Appl. Opt. (1)

Appl. Stat. (1)

B. W. Silverman, “Algorithm AS 176: kernel density estimation using the fast Fourier transform,” Appl. Stat. 31, 93-99 (1982).
[CrossRef]

Bell Syst. Tech. J. (1)

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379-423 (1948).

IEEE J. Quantum Electron. (1)

B. Brooksby, H. Dehghani, B. W. Pogue, and K. D. Paulsen, “Near-infrared (NIR) tomography breast image reconstruction with a priori structural information from MRI: algorithm development for reconstructing heterogeneities,” IEEE J. Quantum Electron. 9, 199-209 (2003).
[CrossRef]

IEEE Signal Process. Mag. (1)

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57-75 (2001).
[CrossRef]

IEEE Trans. Med. Imaging (1)

F. Maes, A. Collignon, D. Vandermeulen, G. Marchal, and P. J. Suetens, “Multimodality image registration by maximization of mutual information,” IEEE Trans. Med. Imaging 16, 187-198 (1997).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

M. Girolami and C. He, “Probability density estimation from optimally condensed data samples,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 1253-1264 (2003).
[CrossRef]

Inverse Probl. (1)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

J. Biomed. Opt. (1)

Q. Zhang, T. J. Brukilacchio, A. Li, J. J. Stott, T. Chaves, E. Hillman, T. Wu, M. Chorlton, E. Rafferty, R. Moore, D. B. Kopans, and D. A. Boas, “Coregistered tomographic x-ray and optical breast imaging: initial results,” J. Biomed. Opt. 10, 024033:1-9 (2005).
[CrossRef]

J. Comput. Graph. Stat. (1)

M. Wand, “Fast computation of multivariate kernel estimators,” J. Comput. Graph. Stat. 3, 433-445 (1994).
[CrossRef]

J. Electron. Imaging (1)

M. Schweiger and S. R. Arridge, “Optical tomography with local basis functions,” J. Electron. Imaging 12, 583-593 (2003).
[CrossRef]

J. Math. Imaging Vision (1)

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite element method for the forward and inverse models in optical tomography,” J. Math. Imaging Vision 3, 263-283 (1993).
[CrossRef]

J. Multivariate Anal. (1)

P. Hall and M. Wand, “On the accuracy of binned kernel density estimators,” J. Multivariate Anal. 56, 165-184 (1996).
[CrossRef]

Med. Phys. (1)

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element model for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779-1792 (1995).
[CrossRef] [PubMed]

Mon. Not. R. Astron. Soc. (1)

J. Skilling and R. K. Bryan, “Maximum entropy image reconstruction--general algorithm,” Mon. Not. R. Astron. Soc. 211, 111-124 (1984).

Neoplasia (1)

Q. Zhu, S. H. Kurtzma, P. Hegde, S. Tannenbaum, M. Kane, M. Huang, N. G. Chen, B. Jagjivan, and K. Zarfos, “Utilizing optical tomography with ultrasound localization to image heterogeneous hemoglobin distribution in large breast cancers.” Neoplasia 7, 263-70 (2005).
[CrossRef] [PubMed]

Phys. Med. Biol. (5)

A. Gibson, J. Hebden, and S. R. Arridge, “Recent advances in diffuse optical tomography,” Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

D. L. G. Hill, P. G. Batchelor, M. Holden, and D. J. Hawkes, “Medical image registration,” Phys. Med. Biol. 46, R1-R45 (2001).
[CrossRef] [PubMed]

M. Schweiger, S. Arridge, and I. Nissila, “Gauss-Newton method for image reconstruction in diffuse optical tomography.” Phys. Med. Biol. 50, 2365-86 (2005).
[CrossRef] [PubMed]

J. C. Hebden, S. R. Arridge, and D. T. Delpy, “Optical imaging in medicine: I. Experimental techniques,” Phys. Med. Biol. 42, 825-840 (1997).
[CrossRef] [PubMed]

S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841-853 (1997).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (1)

V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. U.S.A. 97, 2767-2772 (2000).
[CrossRef] [PubMed]

Proc. SPIE (1)

D. Rueckert, M. Clarkson, D. Hill, and D. Hawkes, “Non-rigid registration using higher order mutual information,” in Proc. SPIE 3979, 438-447 (2000).

SIGACT News (1)

S. Shwartz, M. Zibulevsky, and Y. S. Yoav, “Fast kernel entropy estimation and optimization,” SIGACT News 85, 1045-1058 (2005).

Other (16)

P. A. Viola, “Alignment by maximization of mutual information,” Tech. Rep. AITR-1548 (MIT, 1995).

A. Rangarajan, I.-T. Hsiao, and G. Gindi, “Integrating anatomical priors in ECT reconstruction via joint mixtures and mutual information,” in Nuclear Science Symposium Conference Record, 1998 (IEEE, 1998), Vol. 3, pp. 1548-1588.

S. Somayajula, E. Asma, and R. M. Leahy, “PET image reconstruction using anatomical information through mutual information based priors,” in Proceedings of the IEEE Nuclear Science Symposium and Medical Imaging Conference (IEEE, 2005), pp. 2722-2726.

S. Somayajula, A. Rangarajan, and R. M. Leahy, “PET image reconstruction using anatomical information through mutual information based priors: a scale space approach,” in Biomedical Imaging from Nano to Macro, 2007 (IEEE, 2007), pp. 165-168.
[CrossRef]

J. Nuyts, “The use of mutual information and joint entropy for anatomical priors in emission tomography,” in Nuclear Science Symposium Conference Record, 2007 (IEEE, 2007), Vol. 6, pp. 4149-4154.
[CrossRef]

S. R. Arridge, C. Panagiotou, M. Schweiger, and V. Kolehmainen, “Multimodal diffuse optical tomography: theory,” in Translational Multimodality Optical Imaging (Artech, 2008), Chap. 5, pp. 101-123.

J. Hadamard, “Sur les problèmes aux derivées partielles et leur signification physique,” Bulletin Princeton University (1902), Vol. 13, 49-52.

A. Papoulis and U. S. Pillai, Probability, Random Variables and Stochastic Processes (McGraw-Hill Science/Engineering/Math, 2001).

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley-Interscience, 1991).
[CrossRef]

B. W. Silverman and P. J. Green, Density Estimation for Statistics and Data Analysis (Chapman & Hall, 1986).

J. Simonoff, Smoothing Methods in Statistics (Springer, 1996).
[CrossRef]

R. O. Duda, P. E. Hart, and D. G. Stork., Pattern Classification (Wiley, 2001).

M. Wand and M. Jones, Kernel Smoothing, Vol. 60 of Monographs on Statistics and Applied Probability (Chapman & Hall, 1995).

J. D. Moulton, “Diffusion modelling of picosecond laser pulse propagation in turbid media,” M. Eng. thesis (McMaster University, Hamilton, Ontario, 1990).

C. R. Vogel, Computational Methods for Inverse Problems (SIAM, 2002).
[CrossRef]

J. R. Shewchuk, “An introduction to the conjugate gradient method without the agonizing pain,” Tech. Rep. CS-94-125 (Carnegie Mellon University, 1994).

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Figures (12)

Fig. 1
Fig. 1

Top row, three images all of the same size. Image 1 has three distinct gray values. Image 2 is a random permutation of the pixel locations of Image 1; thus it shares the color bar with Image 1. Image 3 was created by applying an arbitrary nonlinear injective transformation, in this case I 3 = 2 ( I 1 ) 2 + 3 ( I 1 ) ln ( ( I 1 ) + 1 ) and rescaled in [0,255]. Image 3 has the same features (or structure) as image 1 but has completely different gray values. However, the entropies are almost identical, with H ( 1 ) = H ( 2 ) H ( 3 ) , as their pdfs (bottom row) are very close in entropic terms. Each pdf has three modes of almost equal amplitude. The centering of the modes reflects the different gray values, but entropy is invariant to their location. Observe that in the pdf of Image 3, two modes partially overlap. This is due to the resolution of the pdf estimation method. The case is deliberately included in order to note that the true entropy of the images is identical but that the numerical estimate might be subject to inaccuracies propagating as a result of the pdf estimation method.

Fig. 2
Fig. 2

Top row, test images; middle row; marginal pdf of image 1 and joint pdf between image pairs [1, Z] where Z = [ 1 2 3 4 5 ] ; bottom row; marginal pdf for each case. All joint pdfs but P ( 1 , 5 ) , have the same number of modes (or clusters) with the same amplitude, but these are centered in different locations across the horizontal axis. Hence, the first four cases share the same JE. The highest JE is found in image pair (1, 5), where the corresponding joint pdf is more spread out. The marginal entropy of the test images Z varies, with H ( 5 ) > H ( 1 ) H ( 2 ) > H ( 3 ) > H ( 4 ) . It is interesting to note that JE is more invariant to the mode overlapping problem of P ( 2 ) because, due to the extra dimension, entries in the joint pdf are more distant except that the least distance occurs when one image is homogeneous, i.e., P ( 1 , 4 ) , which actually has slightly lower JE than all the other cases.

Fig. 3
Fig. 3

Nonparametric kernel density estimation. The black vertical lines under the horizontal axes denote the N continuous pixel gray values x i of an image, forming the finite sample A. p x ( x ; A ) denotes the continuous probability density estimates of the gray value r.v. x, at regularly spaced locations g. K σ x ( g x i ) denotes a Gaussian kernel centered at each sample point x i . Effectively, the pdf estimate p ̂ x ( g j ; A ) at a point g j equals the sum of the support from all kernels K σ x ( g j x i ) , i .

Fig. 4
Fig. 4

The continuous gray values x A are interpolated to a regularly spaced grid g. Since g is fixed, the original density of x is now reflected by a weight w ( a i # ) . The collection of w ( a # ) forms the reduced equispaced sample A # .

Fig. 5
Fig. 5

The pdf estimate is a product of the convolution of A # with the sampled version of K σ x , denoted by K σ x # . The convolution is performed in the frequency domain as an elementwise multiplication. The pdf estimates p ̂ x ( g ; A ) and the fast binned analog p ̂ x ( g ; A # ) are superimposed, because the assessment of the match acts as an indicator regarding the accuracy of the fast estimate.

Fig. 6
Fig. 6

Left graph, normal distribution with a set of 100 samples. On the right we superimposed the analytic derivatives and the derivatives computed from the FD scheme. We note that in this instance the normalized norm distance between the quantities is A D F D 2 F D 2 = 0.0038 .

Fig. 7
Fig. 7

Computation time for JE and derivative evaluation for multiple image pairs of variable image size among pairs (image size is equal within a pair). The images consist of a collection of pixels where the value of each pixel is randomly drawn from a 1D normal distribution. The image size in the graph is provided in its 2D format and its 3D counterpart with the same number of pixels, as the computational time is the same for both. Standard entropy was not estimated for the last three cases due to the increased time requirements. The number of bins was set to 400 for all image pairs, and both methods utilized kernels of equal finite support.

Fig. 8
Fig. 8

Target distributions and the five reference image pairs, incommensurately related to the target gray values. “Ref. [1]” displays full correspondence between its features and the ones in the target distributions. “Ref. [2]” contains features not existing in the target space. “Ref. [3]” is missing features. The gradient in “Ref. [4]” rises by centering a 2D Gaussian (σ: 50   pixels ) on top of Ref. [1] and multiplying the pixel values underneath. We also add 5% Gaussian multiplicative noise.

Fig. 9
Fig. 9

μ a reconstructions produced by introducing the available reference image pairs with JE or MI. The converged TK1 reconstructions are provided for comparison along with the initial guess.

Fig. 10
Fig. 10

μ s reconstructions produced by introducing the available reference image pairs with JE and MI.

Fig. 11
Fig. 11

Profiles of μ a at y = 40 or y = 100 for reconstructions using TK1, JE, and MI. We also provide the profiles for the Target and Reference image. The choice of y targets the profiling of different features and was made taking into account the feature correspondence between target and prior space for each case.

Fig. 12
Fig. 12

Profiles for μ s at y = 40 or y = 100 .

Tables (4)

Tables Icon

Table 1 Computational Complexity for Marginal Entropy and Its Derivative

Tables Icon

Table 2 Computational Complexity for Joint Entropy and Its Derivative

Tables Icon

Table 3 Sobolev Norm Distance between Target and Reconstructed Absorption Distributions

Tables Icon

Table 4 Sobolev Norm Distance between Target and Reconstructed Scattering Distributions

Equations (48)

Equations on this page are rendered with MathJax. Learn more.

κ ( r ) Φ s ( r ; ω ) + ( μ a ( r ) + i ω c ) Φ s ( r ; ω ) = 0 ,
Φ s ( m ; ω ) + 2 ζ κ ( m ) Φ s ( m ; ω ) ν = f s ( m ) ,
κ ( r ) = 1 [ 3 ( μ a ( r ) + μ s ( r ) ) ]
z s ( m ; ω ) = κ ( m ) Φ s ( m ; ω ) ν , m Ω .
y ( m ) = ( log z s arg ( z s ) ) = ( Re Im ) log ( z s ) .
F : X Y ,
x ̃ ( μ a c κ c ) , x ̃ R 2 N .
L ( x ̃ ) = exp ( s , i y s , i F s , i ( x ̃ ) 2 2 ) .
x ̃ opt = arg min x ̃ [ E ( x ̃ ) = l ( x ̃ ) + τ Ψ ( x ̃ , x ref ) ] ,
x opt = arg min x [ E ( x ) = l ( exp ( x ) ) + τ Ψ ( x , x ref ) ] .
E ( x ) x = l ( exp ( x ) ) x + τ Ψ ( x , x ref ) x .
x k + 1 = x k + λ M ( x cg ) ,
Ψ ( x , x ref ) = H ( x , x ref ) for JE ,
Ψ ( x , x ref ) = MI ( x , x ref ) for MI ,
H ( x ) = p x ( x ) log ( p x ( x ) ) d x ,
H ( x , x ref ) = [ p x , x ref ( x , x ref ) log ( p x , x ref ( x , x ref ) ) d x d x ref ] .
MI ( x , x ref ) = H ( x ) + H ( x ref ) H ( x , x ref ) .
p ̂ x ( x k ; A ) = 1 N i = 1 N K σ x ( x k α i ) ,
K σ x ( t ) = 1 σ x 2 π e ( t 2 2 σ x 2 ) ,
w ( a i # ) = 1 N j = 1 N tri ( a i # α j ) ,
tri ( u ) = { 1 u , if u < d g 0 , otherwise } .
p ̂ x b ( g i ; A # ) = 1 M j = 1 M K σ x ( g i a j # ) w ( a j # ) .
p ̂ x b ( g ; A # ) p ̂ x ( g ; A # ) as M 0 .
p ̂ x b ( g ; A # ) = ( K σ x # w ) ( g )
= F ( K σ x # ( t ) ) × F ( w ( g ) ) ,
H ̂ x b ( A # ) = j = 1 M p ̂ x b ( g j ; A # ) log ( p ̂ x b ( g j ; A # ) ) d g ,
H ̂ x b ( A # ) H ̂ x ( A # ) , as M 0 ,
H ̂ x ( A # ) = j = 1 M p ̂ x ( g j ; A # ) log ( p ̂ x ( g j ; A # ) ) d g .
H ̂ x ( A # ) a i # = H ̂ x ( A # ) p ̂ x ( g ; A # ) p ̂ x ( g ; A # ) a i #
= j = 1 M log ( p ̂ x ( g j ; A # ) + 1 ) p ̂ x ( g j ; A # ) a i # d g
= 1 M j = 1 M log ( p ̂ x ( g j ; A # ) + 1 ) K σ x ( ( a i # g j ) ) a i # d g ,
K σ x ( g j a i # ) a i # = K σ x ( g j a i # ) ( g j a i # ) σ 2 .
H ̂ x b ( A # ) a i # = 1 M j = 1 M log ( p ̂ x b ( g j ; A # ) + 1 ) K σ x ( ( a i # g j ) ) a i # d g .
H ̂ x b ( A # ) a # = F ( log ( p ̂ x b ( g ; A # ) + 1 ) ) × F ( K σ x # ( t ) t ) .
H ̂ x b ( A # ) α j = k = i i + 1 [ tri ( α j a k # ) × H ̂ x b ( A # ) a k # ] ,
K Σ ( ζ i , j ) = 1 2 π Σ 1 2 exp ( 1 2 ζ i , j T Σ 1 ζ i , j ) ,
Σ = [ σ x 2 0 0 σ x ref 2 ]
p ̂ z b ( ζ i , j ; Γ # ) = 1 M k = 1 M K Σ ( [ g i a k # , f j β k # ] ) w ( a k # , β k # ) .
p ̂ z b ( ζ ; Γ # ) = ( K Σ # w ) ( ζ )
= F ( K Σ # ( t ) ) × F ( w ( ζ ) ) ,
H ̂ z b ( Γ # ) = i , j = 1 M p ̂ z b ( ζ i , j ; Γ # ) log ( p ̂ z b ( ζ i , j ; Γ # ) ) d g d f
H ̂ z b ( Γ # ) a k # = H ̂ z b ( Γ # ) p ̂ z b ( ζ i , j ; Γ # ) p ̂ z b ( ζ i , j ; Γ # ) a k #
= 1 M i , j = 1 M log ( p ̂ z b ( ζ i , j ; Γ # ) + 1 ) K Σ ( ζ i , j γ k , l # ) a k # d g d f ,
K Σ ( ζ i , j ) a k # = K Σ ( ζ i , j γ k # ) ( g i a k # ) σ x 2 .
H ̂ z b ( Γ # ) a # = F ( log ( p ̂ z b ( ζ ; Γ # ) + 1 ) ) × F ( K Σ # ( t ) a # ) .
p ̂ x b ( g i ; A # ) = j p ̂ x , x ref b ( ζ i , j ; Γ # ) d f ,
p ̂ x ref b ( f j ; B # ) = i p ̂ x , x ref b ( ζ i , j ; Γ # ) d g .
x x S , 2 2 = 0.5 x x 2 2 x 2 2 + 0.5 ( x x ) 2 2 x 2 2 .

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