Abstract

Multispectral Imaging (MSI) produces a sequence of discrete spectral slices that penetrate different light-absorbing species or chromophores and is a noninvasive technology useful for the early detection of various retinal, optic nerve and choroidal diseases. However, eye movement during the image acquisition process may introduce spatial misalignment between MSI images. This potentially causes trouble in the manual/automatic interpretation of MSI, but still remains an unresolved problem to this date. To deal with this MSI misalignment problem, we present a method on the groupwise registration of sequential images from MSI of the retina and choroid. The advantage of our algorithm is at least threefold: 1) simultaneous estimation of landmark correspondences and a parametric motion model via quadratic programming, 2) enforcement of temporal smoothness on the estimated motion, and 3) inclusion of a robust matching cost function. As validated in our experiments with a database of 22 MSI sequences, our algorithm outperforms two state-of-the-art registration techniques proposed originally in other domains. Our algorithm is potentially invaluable in ophthalmologists' clinical practice regarding various eye diseases.

© 2016 Optical Society of America

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References

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2015 (3)

J. Zhang, Z. Yu, and L. Liu, “Multimodality imaging in diagnosing polypoidal choroidal vasculopathy,” Optom. Vis. Sci. 92(1), e21–e26 (2015).
[Crossref]

J. Kutarnia and P. Pedersen, “A Markov random field approach to group-wise registration/mosaicing with application to ultrasound,” Med. Image Anal. 24(1), 106–124 (2015).
[Crossref] [PubMed]

N. T. Clancy, S. Arya, D. Stoyanov, M. Singh, G. B. Hanna, and D. S. Elson, “Intraoperative measurement of bowel oxygen saturation using a multispectral imaging laparoscope,” Biomed. Opt. Express 6(10), 4179–4190 (2015).
[Crossref] [PubMed]

2014 (4)

P. Ghassemi, T. E. Travis, L. T. Moffatt, J. W. Shupp, and J. C. Ramella-Roman, “A polarized multispectral imaging system for quantitative assessment of hypertrophic scars,” Biomed. Opt. Express 5(10), 3337–3354 (2014).
[Crossref] [PubMed]

Y. Zheng, E. Daniel, A. A. Hunter, R. Xiao, J. Gao, H. Li, M. G. Maguire, D. H. Brainard, and J. C. Gee, “Landmark matching based retinal image alignment by enforcing sparsity in correspondence matrix,” Med. Image Anal. 18(6), 903–913 (2014).
[Crossref]

C. Zimmer, D. Kahn, R. Clayton, P. Dugel, and K. Freund, “Innovation in diagnostic retinal imaging: multispectral imaging,” Retina Today 9(7), 94–99 (2014).

F. P. Oliveira and J. M. R. Tavares, “Medical image registration: a review,” Comput. Method. Biomec. 17(2), 73–93 (2014).
[Crossref]

2013 (3)

C. Wachinger and N. Navab, “Simultaneous registration of multiple images: Similarity metrics and efficient optimization,” IEEE Trans. Pattern Anal. Mach. Intell. 35(5), 1221–1233 (2013).
[Crossref] [PubMed]

P. Ghosh and B. Manjunath, “Robust simultaneous registration and segmentation with sparse error reconstruction,” IEEE Trans. Pattern Anal. Mach. Intell. 35(2), 425–436 (2013).
[Crossref]

D. Hitchmoth, “Multispectral Imaging: A revolution in retinal diagnosis and health assessment,” Adv. Ocul. Care 4(4), 76–79 (2013).

2012 (4)

D. L. Shechtman and P. M. Karpecki, “A look at MSI: multispectral imaging may help eye care providers diagnose retinal conditions earlier than conventional fundoscopy,” Rev. Opt. 149(1), 88–90 (2012).

I. Diebele, I. Kuzmina, A. Lihachev, J. Kapostinsh, A. Derjabo, L. Valeine, and J. Spigulis, “Clinical evaluation of melanomas and common nevi by spectral imaging,” Biomed. Opt. Express 3(3), 467–472 (2012).
[Crossref] [PubMed]

A. Wagner, J. Wright, A. Ganesh, Z. Zhou, H. Mobahi, and Y. Ma, “Toward a practical face recognition system: Robust alignment and illumination by sparse representation,” IEEE Trans. Pattern Anal. Mach. Intell. 34(2), 372–386 (2012).
[Crossref]

A. Rasoulian, R. Rohling, and P. Abolmaesumi, “Group-wise registration of point sets for statistical shape models,” IEEE Trans. Med. Imaging 31(11), 2025–2034 (2012).
[Crossref] [PubMed]

2011 (1)

R. Maharaj, “The clinical applications of multispectral imaging,” Rev. Opt. 148(11), SS19 (2011).

2009 (1)

M. B. Bouchard, B. R. Chen, S. A. Burgess, and E. M. Hillman, “Ultra-fast multispectral optical imaging of cortical oxygenation, blood flow, and intracellular calcium dynamics,” Opt. Express 17(18), 15,670–15,678 (2009).
[Crossref]

2007 (1)

A. Gholipour, N. Kehtarnavaz, R. Briggs, M. Devous, and K. Gopinath, “Brain functional localization: a survey of image registration techniques,” IEEE Trans. Med. Imaging 26, 427–451 (2007).
[Crossref] [PubMed]

2006 (2)

I. B. Styles, A. Calcagni, E. Claridge, F. Orihuela-Espina, and J. Gibson, “Quantitative analysis of multi-spectral fundus images,” Med. Image Anal. 10(4), 578–597 (2006).
[Crossref] [PubMed]

E. G. Learned-Miller, “Data driven image models through continuous joint alignment,” IEEE Trans. Pattern Anal. Mach. Intell. 28(2), 236–250 (2006).
[Crossref] [PubMed]

2003 (3)

J. P. Pluim, J. A. Maintz, and M. A. Viergever, “Mutual-information-based registration of medical images: a survey,” IEEE Trans. Med. Imaging 22(8), 986–1004 (2003).
[Crossref] [PubMed]

H. Chui and A. Rangarajan, “A new point matching algorithm for non-rigid registration,” Comput. Vis. Image Und. 89, 114–141 (2003).
[Crossref]

H. Chui and A. Rangarajan, “A new point matching algorithm for non-rigid registration,” Comput. Vis. Image Und. 89(2), 114–141 (2003).
[Crossref]

2002 (1)

A. Can, C. V. Stewart, B. Roysam, and H. L. Tanenbaum, “A feature-based, robust, hierarchical algorithm for registering pairs of images of the curved human retina,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 347–364 (2002).
[Crossref]

2000 (1)

C. Rorden and M. Brett, “Stereotaxic display of brain lesions,” Behav. Neurol. 12(4), 191–200 (2000).
[Crossref]

1999 (1)

C. V. Stewart, “Robust parameter estimation in computer vision,” SIAM Rev. 41(3), 513–537 (1999).
[Crossref]

1996 (1)

W. M. Wells, P. Viola, H. Atsumi, S. Nakajima, and R. Kikinis, “Multi-modal volume registration by maximization of mutual information,” Med. Image Anal. 1(1), 35–51 (1996).
[Crossref] [PubMed]

Abolmaesumi, P.

A. Rasoulian, R. Rohling, and P. Abolmaesumi, “Group-wise registration of point sets for statistical shape models,” IEEE Trans. Med. Imaging 31(11), 2025–2034 (2012).
[Crossref] [PubMed]

Anandan, P.

M. Irani and P. Anandan, “Robust multi-sensor image alignment,” in Computer Vision, 1998. Sixth International Conference on, pp. 959–966 (IEEE, 1998).

Arandjelovic, O.

O. Arandjelovic, D.-S. Pham, and S. Venkatesh, “Groupwise registration of aerial images,” Twenty-Fourth International Joint Conference on Artificial Intelligence (IJCAI 2015) (2015).

Arya, S.

Atsumi, H.

W. M. Wells, P. Viola, H. Atsumi, S. Nakajima, and R. Kikinis, “Multi-modal volume registration by maximization of mutual information,” Med. Image Anal. 1(1), 35–51 (1996).
[Crossref] [PubMed]

Balci, S. K.

S. K. Balci, P. Golland, and W. Wells, “Non-rigid groupwise registration using B-spline deformation model,” Open source and open data for MICCAI pp. 105–121 (2007).

Bouchard, M. B.

M. B. Bouchard, B. R. Chen, S. A. Burgess, and E. M. Hillman, “Ultra-fast multispectral optical imaging of cortical oxygenation, blood flow, and intracellular calcium dynamics,” Opt. Express 17(18), 15,670–15,678 (2009).
[Crossref]

Boyd, S.

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).
[Crossref]

Brainard, D. H.

Y. Zheng, E. Daniel, A. A. Hunter, R. Xiao, J. Gao, H. Li, M. G. Maguire, D. H. Brainard, and J. C. Gee, “Landmark matching based retinal image alignment by enforcing sparsity in correspondence matrix,” Med. Image Anal. 18(6), 903–913 (2014).
[Crossref]

Brett, M.

C. Rorden and M. Brett, “Stereotaxic display of brain lesions,” Behav. Neurol. 12(4), 191–200 (2000).
[Crossref]

Briggs, R.

A. Gholipour, N. Kehtarnavaz, R. Briggs, M. Devous, and K. Gopinath, “Brain functional localization: a survey of image registration techniques,” IEEE Trans. Med. Imaging 26, 427–451 (2007).
[Crossref] [PubMed]

Burgess, S. A.

M. B. Bouchard, B. R. Chen, S. A. Burgess, and E. M. Hillman, “Ultra-fast multispectral optical imaging of cortical oxygenation, blood flow, and intracellular calcium dynamics,” Opt. Express 17(18), 15,670–15,678 (2009).
[Crossref]

Calcagni, A.

I. B. Styles, A. Calcagni, E. Claridge, F. Orihuela-Espina, and J. Gibson, “Quantitative analysis of multi-spectral fundus images,” Med. Image Anal. 10(4), 578–597 (2006).
[Crossref] [PubMed]

N. Everdell, I. Styles, E. Claridge, J. Hebden, and A. Calcagni, “Multispectral imaging of the ocular fundus using LED illumination,” in European Conferences on Biomedical Optics, p. 7371C (International Society for Optics and Photonics, 2009).

Can, A.

A. Can, C. V. Stewart, B. Roysam, and H. L. Tanenbaum, “A feature-based, robust, hierarchical algorithm for registering pairs of images of the curved human retina,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 347–364 (2002).
[Crossref]

Chen, B. R.

M. B. Bouchard, B. R. Chen, S. A. Burgess, and E. M. Hillman, “Ultra-fast multispectral optical imaging of cortical oxygenation, blood flow, and intracellular calcium dynamics,” Opt. Express 17(18), 15,670–15,678 (2009).
[Crossref]

Chui, H.

H. Chui and A. Rangarajan, “A new point matching algorithm for non-rigid registration,” Comput. Vis. Image Und. 89(2), 114–141 (2003).
[Crossref]

H. Chui and A. Rangarajan, “A new point matching algorithm for non-rigid registration,” Comput. Vis. Image Und. 89, 114–141 (2003).
[Crossref]

Clancy, N. T.

Claridge, E.

I. B. Styles, A. Calcagni, E. Claridge, F. Orihuela-Espina, and J. Gibson, “Quantitative analysis of multi-spectral fundus images,” Med. Image Anal. 10(4), 578–597 (2006).
[Crossref] [PubMed]

N. Everdell, I. Styles, E. Claridge, J. Hebden, and A. Calcagni, “Multispectral imaging of the ocular fundus using LED illumination,” in European Conferences on Biomedical Optics, p. 7371C (International Society for Optics and Photonics, 2009).

Clayton, R.

C. Zimmer, D. Kahn, R. Clayton, P. Dugel, and K. Freund, “Innovation in diagnostic retinal imaging: multispectral imaging,” Retina Today 9(7), 94–99 (2014).

Daniel, E.

Y. Zheng, E. Daniel, A. A. Hunter, R. Xiao, J. Gao, H. Li, M. G. Maguire, D. H. Brainard, and J. C. Gee, “Landmark matching based retinal image alignment by enforcing sparsity in correspondence matrix,” Med. Image Anal. 18(6), 903–913 (2014).
[Crossref]

Derjabo, A.

Devous, M.

A. Gholipour, N. Kehtarnavaz, R. Briggs, M. Devous, and K. Gopinath, “Brain functional localization: a survey of image registration techniques,” IEEE Trans. Med. Imaging 26, 427–451 (2007).
[Crossref] [PubMed]

Diebele, I.

Dugel, P.

C. Zimmer, D. Kahn, R. Clayton, P. Dugel, and K. Freund, “Innovation in diagnostic retinal imaging: multispectral imaging,” Retina Today 9(7), 94–99 (2014).

Efros, A. A.

T. Zhou, Y. J. Lee, X. Y. Stella, and A. A. Efros, “Flowweb: Joint image set alignment by weaving consistent, pixel-wise correspondences,” in 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1191–1200 (IEEE, 2015).
[Crossref]

Elson, D. S.

Everdell, N.

N. Everdell, I. Styles, E. Claridge, J. Hebden, and A. Calcagni, “Multispectral imaging of the ocular fundus using LED illumination,” in European Conferences on Biomedical Optics, p. 7371C (International Society for Optics and Photonics, 2009).

Freund, K.

C. Zimmer, D. Kahn, R. Clayton, P. Dugel, and K. Freund, “Innovation in diagnostic retinal imaging: multispectral imaging,” Retina Today 9(7), 94–99 (2014).

Ganesh, A.

A. Wagner, J. Wright, A. Ganesh, Z. Zhou, H. Mobahi, and Y. Ma, “Toward a practical face recognition system: Robust alignment and illumination by sparse representation,” IEEE Trans. Pattern Anal. Mach. Intell. 34(2), 372–386 (2012).
[Crossref]

Gao, J.

Y. Zheng, E. Daniel, A. A. Hunter, R. Xiao, J. Gao, H. Li, M. G. Maguire, D. H. Brainard, and J. C. Gee, “Landmark matching based retinal image alignment by enforcing sparsity in correspondence matrix,” Med. Image Anal. 18(6), 903–913 (2014).
[Crossref]

Y. Zheng, A. A. Hunter, J. Wu, H. Wang, J. Gao, M. G. Maguire, and J. C. Gee, “Landmark matching based automatic retinal image registration with linear programming and self-similarities,” in Biennial International Conference on Information Processing in Medical Imaging, pp. 674–685 (Springer, 2011).

Gee, J. C.

Y. Zheng, E. Daniel, A. A. Hunter, R. Xiao, J. Gao, H. Li, M. G. Maguire, D. H. Brainard, and J. C. Gee, “Landmark matching based retinal image alignment by enforcing sparsity in correspondence matrix,” Med. Image Anal. 18(6), 903–913 (2014).
[Crossref]

Y. Zheng, A. A. Hunter, J. Wu, H. Wang, J. Gao, M. G. Maguire, and J. C. Gee, “Landmark matching based automatic retinal image registration with linear programming and self-similarities,” in Biennial International Conference on Information Processing in Medical Imaging, pp. 674–685 (Springer, 2011).

Ghassemi, P.

Gholipour, A.

A. Gholipour, N. Kehtarnavaz, R. Briggs, M. Devous, and K. Gopinath, “Brain functional localization: a survey of image registration techniques,” IEEE Trans. Med. Imaging 26, 427–451 (2007).
[Crossref] [PubMed]

Ghosh, P.

P. Ghosh and B. Manjunath, “Robust simultaneous registration and segmentation with sparse error reconstruction,” IEEE Trans. Pattern Anal. Mach. Intell. 35(2), 425–436 (2013).
[Crossref]

Gibson, J.

I. B. Styles, A. Calcagni, E. Claridge, F. Orihuela-Espina, and J. Gibson, “Quantitative analysis of multi-spectral fundus images,” Med. Image Anal. 10(4), 578–597 (2006).
[Crossref] [PubMed]

Golland, P.

S. K. Balci, P. Golland, and W. Wells, “Non-rigid groupwise registration using B-spline deformation model,” Open source and open data for MICCAI pp. 105–121 (2007).

Gopinath, K.

A. Gholipour, N. Kehtarnavaz, R. Briggs, M. Devous, and K. Gopinath, “Brain functional localization: a survey of image registration techniques,” IEEE Trans. Med. Imaging 26, 427–451 (2007).
[Crossref] [PubMed]

Grauman, K.

J. Kim, C. Liu, F. Sha, and K. Grauman, “Deformable spatial pyramid matching for fast dense correspondences,” Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2307–2314 (2013).

Grimson, E.

L. Zöllei, E. Learned-Miller, E. Grimson, and W. Wells, “Efficient population registration of 3D data,” in International Workshop on Computer Vision for Biomedical Image Applications, pp. 291–301 (Springer, 2005).
[Crossref]

Hanna, G. B.

Hebden, J.

N. Everdell, I. Styles, E. Claridge, J. Hebden, and A. Calcagni, “Multispectral imaging of the ocular fundus using LED illumination,” in European Conferences on Biomedical Optics, p. 7371C (International Society for Optics and Photonics, 2009).

Hillman, E. M.

M. B. Bouchard, B. R. Chen, S. A. Burgess, and E. M. Hillman, “Ultra-fast multispectral optical imaging of cortical oxygenation, blood flow, and intracellular calcium dynamics,” Opt. Express 17(18), 15,670–15,678 (2009).
[Crossref]

Hitchmoth, D.

D. Hitchmoth, “Multispectral Imaging: A revolution in retinal diagnosis and health assessment,” Adv. Ocul. Care 4(4), 76–79 (2013).

Huang, G. B.

G. B. Huang, V. Jain, and E. Learned-Miller, “Unsupervised joint alignment of complex images,” in 2007 IEEE 11th International Conference on Computer Vision, pp. 1–8 (IEEE, 2007).

Hunter, A. A.

Y. Zheng, E. Daniel, A. A. Hunter, R. Xiao, J. Gao, H. Li, M. G. Maguire, D. H. Brainard, and J. C. Gee, “Landmark matching based retinal image alignment by enforcing sparsity in correspondence matrix,” Med. Image Anal. 18(6), 903–913 (2014).
[Crossref]

Y. Zheng, A. A. Hunter, J. Wu, H. Wang, J. Gao, M. G. Maguire, and J. C. Gee, “Landmark matching based automatic retinal image registration with linear programming and self-similarities,” in Biennial International Conference on Information Processing in Medical Imaging, pp. 674–685 (Springer, 2011).

Irani, M.

M. Irani and P. Anandan, “Robust multi-sensor image alignment,” in Computer Vision, 1998. Sixth International Conference on, pp. 959–966 (IEEE, 1998).

Jain, V.

G. B. Huang, V. Jain, and E. Learned-Miller, “Unsupervised joint alignment of complex images,” in 2007 IEEE 11th International Conference on Computer Vision, pp. 1–8 (IEEE, 2007).

Jan, J.

R. Kolar, L. Kubecka, and J. Jan, “Registration and fusion of the autofluorescent and infrared retinal images,” Int. J. Biomed. Imaging2008 (2008).
[Crossref] [PubMed]

Jia, J.

X. Shen, L. Xu, Q. Zhang, and J. Jia, “Multi-modal and multi-spectral registration for natural images,” in European Conference on Computer Vision, pp. 309–324 (Springer, 2014).

Kahn, D.

C. Zimmer, D. Kahn, R. Clayton, P. Dugel, and K. Freund, “Innovation in diagnostic retinal imaging: multispectral imaging,” Retina Today 9(7), 94–99 (2014).

Kapostinsh, J.

Karpecki, P. M.

D. L. Shechtman and P. M. Karpecki, “A look at MSI: multispectral imaging may help eye care providers diagnose retinal conditions earlier than conventional fundoscopy,” Rev. Opt. 149(1), 88–90 (2012).

Kehtarnavaz, N.

A. Gholipour, N. Kehtarnavaz, R. Briggs, M. Devous, and K. Gopinath, “Brain functional localization: a survey of image registration techniques,” IEEE Trans. Med. Imaging 26, 427–451 (2007).
[Crossref] [PubMed]

Kikinis, R.

W. M. Wells, P. Viola, H. Atsumi, S. Nakajima, and R. Kikinis, “Multi-modal volume registration by maximization of mutual information,” Med. Image Anal. 1(1), 35–51 (1996).
[Crossref] [PubMed]

Kim, J.

J. Kim, C. Liu, F. Sha, and K. Grauman, “Deformable spatial pyramid matching for fast dense correspondences,” Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2307–2314 (2013).

Kolar, R.

R. Kolar, L. Kubecka, and J. Jan, “Registration and fusion of the autofluorescent and infrared retinal images,” Int. J. Biomed. Imaging2008 (2008).
[Crossref] [PubMed]

Kubecka, L.

R. Kolar, L. Kubecka, and J. Jan, “Registration and fusion of the autofluorescent and infrared retinal images,” Int. J. Biomed. Imaging2008 (2008).
[Crossref] [PubMed]

Kutarnia, J.

J. Kutarnia and P. Pedersen, “A Markov random field approach to group-wise registration/mosaicing with application to ultrasound,” Med. Image Anal. 24(1), 106–124 (2015).
[Crossref] [PubMed]

Kuzmina, I.

Learned-Miller, E.

L. Zöllei, E. Learned-Miller, E. Grimson, and W. Wells, “Efficient population registration of 3D data,” in International Workshop on Computer Vision for Biomedical Image Applications, pp. 291–301 (Springer, 2005).
[Crossref]

G. B. Huang, V. Jain, and E. Learned-Miller, “Unsupervised joint alignment of complex images,” in 2007 IEEE 11th International Conference on Computer Vision, pp. 1–8 (IEEE, 2007).

Learned-Miller, E. G.

E. G. Learned-Miller, “Data driven image models through continuous joint alignment,” IEEE Trans. Pattern Anal. Mach. Intell. 28(2), 236–250 (2006).
[Crossref] [PubMed]

Lee, Y. J.

T. Zhou, Y. J. Lee, X. Y. Stella, and A. A. Efros, “Flowweb: Joint image set alignment by weaving consistent, pixel-wise correspondences,” in 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1191–1200 (IEEE, 2015).
[Crossref]

Lewis, J.

J. Lewis, “Fast normalized cross-correlation,” Proceedings of Vision Interface pp. 120–123 (1995).

Li, H.

Y. Zheng, E. Daniel, A. A. Hunter, R. Xiao, J. Gao, H. Li, M. G. Maguire, D. H. Brainard, and J. C. Gee, “Landmark matching based retinal image alignment by enforcing sparsity in correspondence matrix,” Med. Image Anal. 18(6), 903–913 (2014).
[Crossref]

Lihachev, A.

Liu, C.

J. Kim, C. Liu, F. Sha, and K. Grauman, “Deformable spatial pyramid matching for fast dense correspondences,” Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2307–2314 (2013).

Liu, L.

J. Zhang, Z. Yu, and L. Liu, “Multimodality imaging in diagnosing polypoidal choroidal vasculopathy,” Optom. Vis. Sci. 92(1), e21–e26 (2015).
[Crossref]

Lowe, D. G.

D. G. Lowe, “Object recognition from local scale-invariant features,” in Computer vision, 1999. The proceedings of the seventh IEEE international conference on, vol. 2, pp. 1150–1157 (IEEE, 1999).
[Crossref]

Ma, Y.

A. Wagner, J. Wright, A. Ganesh, Z. Zhou, H. Mobahi, and Y. Ma, “Toward a practical face recognition system: Robust alignment and illumination by sparse representation,” IEEE Trans. Pattern Anal. Mach. Intell. 34(2), 372–386 (2012).
[Crossref]

Maguire, M. G.

Y. Zheng, E. Daniel, A. A. Hunter, R. Xiao, J. Gao, H. Li, M. G. Maguire, D. H. Brainard, and J. C. Gee, “Landmark matching based retinal image alignment by enforcing sparsity in correspondence matrix,” Med. Image Anal. 18(6), 903–913 (2014).
[Crossref]

Y. Zheng, A. A. Hunter, J. Wu, H. Wang, J. Gao, M. G. Maguire, and J. C. Gee, “Landmark matching based automatic retinal image registration with linear programming and self-similarities,” in Biennial International Conference on Information Processing in Medical Imaging, pp. 674–685 (Springer, 2011).

Maharaj, R.

R. Maharaj, “The clinical applications of multispectral imaging,” Rev. Opt. 148(11), SS19 (2011).

Maintz, J. A.

J. P. Pluim, J. A. Maintz, and M. A. Viergever, “Mutual-information-based registration of medical images: a survey,” IEEE Trans. Med. Imaging 22(8), 986–1004 (2003).
[Crossref] [PubMed]

Manjunath, B.

P. Ghosh and B. Manjunath, “Robust simultaneous registration and segmentation with sparse error reconstruction,” IEEE Trans. Pattern Anal. Mach. Intell. 35(2), 425–436 (2013).
[Crossref]

Mobahi, H.

A. Wagner, J. Wright, A. Ganesh, Z. Zhou, H. Mobahi, and Y. Ma, “Toward a practical face recognition system: Robust alignment and illumination by sparse representation,” IEEE Trans. Pattern Anal. Mach. Intell. 34(2), 372–386 (2012).
[Crossref]

Modersitzki, J.

J. Modersitzki, Numetical Methods for Image Registration (Oxford University, New York, NY, USA, 2004).

Moffatt, L. T.

Nakajima, S.

W. M. Wells, P. Viola, H. Atsumi, S. Nakajima, and R. Kikinis, “Multi-modal volume registration by maximization of mutual information,” Med. Image Anal. 1(1), 35–51 (1996).
[Crossref] [PubMed]

Navab, N.

C. Wachinger and N. Navab, “Simultaneous registration of multiple images: Similarity metrics and efficient optimization,” IEEE Trans. Pattern Anal. Mach. Intell. 35(5), 1221–1233 (2013).
[Crossref] [PubMed]

Oliveira, F. P.

F. P. Oliveira and J. M. R. Tavares, “Medical image registration: a review,” Comput. Method. Biomec. 17(2), 73–93 (2014).
[Crossref]

Orihuela-Espina, F.

I. B. Styles, A. Calcagni, E. Claridge, F. Orihuela-Espina, and J. Gibson, “Quantitative analysis of multi-spectral fundus images,” Med. Image Anal. 10(4), 578–597 (2006).
[Crossref] [PubMed]

Pedersen, P.

J. Kutarnia and P. Pedersen, “A Markov random field approach to group-wise registration/mosaicing with application to ultrasound,” Med. Image Anal. 24(1), 106–124 (2015).
[Crossref] [PubMed]

Pham, D.-S.

O. Arandjelovic, D.-S. Pham, and S. Venkatesh, “Groupwise registration of aerial images,” Twenty-Fourth International Joint Conference on Artificial Intelligence (IJCAI 2015) (2015).

Pluim, J. P.

J. P. Pluim, J. A. Maintz, and M. A. Viergever, “Mutual-information-based registration of medical images: a survey,” IEEE Trans. Med. Imaging 22(8), 986–1004 (2003).
[Crossref] [PubMed]

Ramella-Roman, J. C.

Rangarajan, A.

H. Chui and A. Rangarajan, “A new point matching algorithm for non-rigid registration,” Comput. Vis. Image Und. 89(2), 114–141 (2003).
[Crossref]

H. Chui and A. Rangarajan, “A new point matching algorithm for non-rigid registration,” Comput. Vis. Image Und. 89, 114–141 (2003).
[Crossref]

Rasoulian, A.

A. Rasoulian, R. Rohling, and P. Abolmaesumi, “Group-wise registration of point sets for statistical shape models,” IEEE Trans. Med. Imaging 31(11), 2025–2034 (2012).
[Crossref] [PubMed]

Rohling, R.

A. Rasoulian, R. Rohling, and P. Abolmaesumi, “Group-wise registration of point sets for statistical shape models,” IEEE Trans. Med. Imaging 31(11), 2025–2034 (2012).
[Crossref] [PubMed]

Rorden, C.

C. Rorden and M. Brett, “Stereotaxic display of brain lesions,” Behav. Neurol. 12(4), 191–200 (2000).
[Crossref]

Roysam, B.

A. Can, C. V. Stewart, B. Roysam, and H. L. Tanenbaum, “A feature-based, robust, hierarchical algorithm for registering pairs of images of the curved human retina,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 347–364 (2002).
[Crossref]

Sha, F.

J. Kim, C. Liu, F. Sha, and K. Grauman, “Deformable spatial pyramid matching for fast dense correspondences,” Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2307–2314 (2013).

Shechtman, D. L.

D. L. Shechtman and P. M. Karpecki, “A look at MSI: multispectral imaging may help eye care providers diagnose retinal conditions earlier than conventional fundoscopy,” Rev. Opt. 149(1), 88–90 (2012).

Shen, X.

X. Shen, L. Xu, Q. Zhang, and J. Jia, “Multi-modal and multi-spectral registration for natural images,” in European Conference on Computer Vision, pp. 309–324 (Springer, 2014).

Shupp, J. W.

Singh, M.

Spigulis, J.

Stella, X. Y.

T. Zhou, Y. J. Lee, X. Y. Stella, and A. A. Efros, “Flowweb: Joint image set alignment by weaving consistent, pixel-wise correspondences,” in 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1191–1200 (IEEE, 2015).
[Crossref]

Stewart, C. V.

A. Can, C. V. Stewart, B. Roysam, and H. L. Tanenbaum, “A feature-based, robust, hierarchical algorithm for registering pairs of images of the curved human retina,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 347–364 (2002).
[Crossref]

C. V. Stewart, “Robust parameter estimation in computer vision,” SIAM Rev. 41(3), 513–537 (1999).
[Crossref]

Stoyanov, D.

Styles, I.

N. Everdell, I. Styles, E. Claridge, J. Hebden, and A. Calcagni, “Multispectral imaging of the ocular fundus using LED illumination,” in European Conferences on Biomedical Optics, p. 7371C (International Society for Optics and Photonics, 2009).

Styles, I. B.

I. B. Styles, A. Calcagni, E. Claridge, F. Orihuela-Espina, and J. Gibson, “Quantitative analysis of multi-spectral fundus images,” Med. Image Anal. 10(4), 578–597 (2006).
[Crossref] [PubMed]

Tanenbaum, H. L.

A. Can, C. V. Stewart, B. Roysam, and H. L. Tanenbaum, “A feature-based, robust, hierarchical algorithm for registering pairs of images of the curved human retina,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 347–364 (2002).
[Crossref]

Tavares, J. M. R.

F. P. Oliveira and J. M. R. Tavares, “Medical image registration: a review,” Comput. Method. Biomec. 17(2), 73–93 (2014).
[Crossref]

Travis, T. E.

Valeine, L.

Vandenberghe, L.

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).
[Crossref]

Venkatesh, S.

O. Arandjelovic, D.-S. Pham, and S. Venkatesh, “Groupwise registration of aerial images,” Twenty-Fourth International Joint Conference on Artificial Intelligence (IJCAI 2015) (2015).

Viergever, M. A.

J. P. Pluim, J. A. Maintz, and M. A. Viergever, “Mutual-information-based registration of medical images: a survey,” IEEE Trans. Med. Imaging 22(8), 986–1004 (2003).
[Crossref] [PubMed]

Viola, P.

W. M. Wells, P. Viola, H. Atsumi, S. Nakajima, and R. Kikinis, “Multi-modal volume registration by maximization of mutual information,” Med. Image Anal. 1(1), 35–51 (1996).
[Crossref] [PubMed]

Wachinger, C.

C. Wachinger and N. Navab, “Simultaneous registration of multiple images: Similarity metrics and efficient optimization,” IEEE Trans. Pattern Anal. Mach. Intell. 35(5), 1221–1233 (2013).
[Crossref] [PubMed]

Wagner, A.

A. Wagner, J. Wright, A. Ganesh, Z. Zhou, H. Mobahi, and Y. Ma, “Toward a practical face recognition system: Robust alignment and illumination by sparse representation,” IEEE Trans. Pattern Anal. Mach. Intell. 34(2), 372–386 (2012).
[Crossref]

Wang, H.

Y. Zheng, A. A. Hunter, J. Wu, H. Wang, J. Gao, M. G. Maguire, and J. C. Gee, “Landmark matching based automatic retinal image registration with linear programming and self-similarities,” in Biennial International Conference on Information Processing in Medical Imaging, pp. 674–685 (Springer, 2011).

Wells, W.

S. K. Balci, P. Golland, and W. Wells, “Non-rigid groupwise registration using B-spline deformation model,” Open source and open data for MICCAI pp. 105–121 (2007).

L. Zöllei, E. Learned-Miller, E. Grimson, and W. Wells, “Efficient population registration of 3D data,” in International Workshop on Computer Vision for Biomedical Image Applications, pp. 291–301 (Springer, 2005).
[Crossref]

Wells, W. M.

W. M. Wells, P. Viola, H. Atsumi, S. Nakajima, and R. Kikinis, “Multi-modal volume registration by maximization of mutual information,” Med. Image Anal. 1(1), 35–51 (1996).
[Crossref] [PubMed]

Wright, J.

A. Wagner, J. Wright, A. Ganesh, Z. Zhou, H. Mobahi, and Y. Ma, “Toward a practical face recognition system: Robust alignment and illumination by sparse representation,” IEEE Trans. Pattern Anal. Mach. Intell. 34(2), 372–386 (2012).
[Crossref]

Wu, J.

Y. Zheng, A. A. Hunter, J. Wu, H. Wang, J. Gao, M. G. Maguire, and J. C. Gee, “Landmark matching based automatic retinal image registration with linear programming and self-similarities,” in Biennial International Conference on Information Processing in Medical Imaging, pp. 674–685 (Springer, 2011).

Xiao, R.

Y. Zheng, E. Daniel, A. A. Hunter, R. Xiao, J. Gao, H. Li, M. G. Maguire, D. H. Brainard, and J. C. Gee, “Landmark matching based retinal image alignment by enforcing sparsity in correspondence matrix,” Med. Image Anal. 18(6), 903–913 (2014).
[Crossref]

Xu, L.

X. Shen, L. Xu, Q. Zhang, and J. Jia, “Multi-modal and multi-spectral registration for natural images,” in European Conference on Computer Vision, pp. 309–324 (Springer, 2014).

Yu, Z.

J. Zhang, Z. Yu, and L. Liu, “Multimodality imaging in diagnosing polypoidal choroidal vasculopathy,” Optom. Vis. Sci. 92(1), e21–e26 (2015).
[Crossref]

Zhang, J.

J. Zhang, Z. Yu, and L. Liu, “Multimodality imaging in diagnosing polypoidal choroidal vasculopathy,” Optom. Vis. Sci. 92(1), e21–e26 (2015).
[Crossref]

Zhang, Q.

X. Shen, L. Xu, Q. Zhang, and J. Jia, “Multi-modal and multi-spectral registration for natural images,” in European Conference on Computer Vision, pp. 309–324 (Springer, 2014).

Zheng, Y.

Y. Zheng, E. Daniel, A. A. Hunter, R. Xiao, J. Gao, H. Li, M. G. Maguire, D. H. Brainard, and J. C. Gee, “Landmark matching based retinal image alignment by enforcing sparsity in correspondence matrix,” Med. Image Anal. 18(6), 903–913 (2014).
[Crossref]

Y. Zheng, A. A. Hunter, J. Wu, H. Wang, J. Gao, M. G. Maguire, and J. C. Gee, “Landmark matching based automatic retinal image registration with linear programming and self-similarities,” in Biennial International Conference on Information Processing in Medical Imaging, pp. 674–685 (Springer, 2011).

Zhou, T.

T. Zhou, Y. J. Lee, X. Y. Stella, and A. A. Efros, “Flowweb: Joint image set alignment by weaving consistent, pixel-wise correspondences,” in 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1191–1200 (IEEE, 2015).
[Crossref]

Zhou, Z.

A. Wagner, J. Wright, A. Ganesh, Z. Zhou, H. Mobahi, and Y. Ma, “Toward a practical face recognition system: Robust alignment and illumination by sparse representation,” IEEE Trans. Pattern Anal. Mach. Intell. 34(2), 372–386 (2012).
[Crossref]

Zimmer, C.

C. Zimmer, D. Kahn, R. Clayton, P. Dugel, and K. Freund, “Innovation in diagnostic retinal imaging: multispectral imaging,” Retina Today 9(7), 94–99 (2014).

Zöllei, L.

L. Zöllei, E. Learned-Miller, E. Grimson, and W. Wells, “Efficient population registration of 3D data,” in International Workshop on Computer Vision for Biomedical Image Applications, pp. 291–301 (Springer, 2005).
[Crossref]

Adv. Ocul. Care (1)

D. Hitchmoth, “Multispectral Imaging: A revolution in retinal diagnosis and health assessment,” Adv. Ocul. Care 4(4), 76–79 (2013).

Behav. Neurol. (1)

C. Rorden and M. Brett, “Stereotaxic display of brain lesions,” Behav. Neurol. 12(4), 191–200 (2000).
[Crossref]

Biomed. Opt. Express (3)

Comput. Method. Biomec. (1)

F. P. Oliveira and J. M. R. Tavares, “Medical image registration: a review,” Comput. Method. Biomec. 17(2), 73–93 (2014).
[Crossref]

Comput. Vis. Image Und. (2)

H. Chui and A. Rangarajan, “A new point matching algorithm for non-rigid registration,” Comput. Vis. Image Und. 89(2), 114–141 (2003).
[Crossref]

H. Chui and A. Rangarajan, “A new point matching algorithm for non-rigid registration,” Comput. Vis. Image Und. 89, 114–141 (2003).
[Crossref]

IEEE Trans. Med. Imaging (3)

A. Gholipour, N. Kehtarnavaz, R. Briggs, M. Devous, and K. Gopinath, “Brain functional localization: a survey of image registration techniques,” IEEE Trans. Med. Imaging 26, 427–451 (2007).
[Crossref] [PubMed]

A. Rasoulian, R. Rohling, and P. Abolmaesumi, “Group-wise registration of point sets for statistical shape models,” IEEE Trans. Med. Imaging 31(11), 2025–2034 (2012).
[Crossref] [PubMed]

J. P. Pluim, J. A. Maintz, and M. A. Viergever, “Mutual-information-based registration of medical images: a survey,” IEEE Trans. Med. Imaging 22(8), 986–1004 (2003).
[Crossref] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell. (5)

A. Can, C. V. Stewart, B. Roysam, and H. L. Tanenbaum, “A feature-based, robust, hierarchical algorithm for registering pairs of images of the curved human retina,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 347–364 (2002).
[Crossref]

P. Ghosh and B. Manjunath, “Robust simultaneous registration and segmentation with sparse error reconstruction,” IEEE Trans. Pattern Anal. Mach. Intell. 35(2), 425–436 (2013).
[Crossref]

A. Wagner, J. Wright, A. Ganesh, Z. Zhou, H. Mobahi, and Y. Ma, “Toward a practical face recognition system: Robust alignment and illumination by sparse representation,” IEEE Trans. Pattern Anal. Mach. Intell. 34(2), 372–386 (2012).
[Crossref]

C. Wachinger and N. Navab, “Simultaneous registration of multiple images: Similarity metrics and efficient optimization,” IEEE Trans. Pattern Anal. Mach. Intell. 35(5), 1221–1233 (2013).
[Crossref] [PubMed]

E. G. Learned-Miller, “Data driven image models through continuous joint alignment,” IEEE Trans. Pattern Anal. Mach. Intell. 28(2), 236–250 (2006).
[Crossref] [PubMed]

Med. Image Anal. (4)

J. Kutarnia and P. Pedersen, “A Markov random field approach to group-wise registration/mosaicing with application to ultrasound,” Med. Image Anal. 24(1), 106–124 (2015).
[Crossref] [PubMed]

Y. Zheng, E. Daniel, A. A. Hunter, R. Xiao, J. Gao, H. Li, M. G. Maguire, D. H. Brainard, and J. C. Gee, “Landmark matching based retinal image alignment by enforcing sparsity in correspondence matrix,” Med. Image Anal. 18(6), 903–913 (2014).
[Crossref]

I. B. Styles, A. Calcagni, E. Claridge, F. Orihuela-Espina, and J. Gibson, “Quantitative analysis of multi-spectral fundus images,” Med. Image Anal. 10(4), 578–597 (2006).
[Crossref] [PubMed]

W. M. Wells, P. Viola, H. Atsumi, S. Nakajima, and R. Kikinis, “Multi-modal volume registration by maximization of mutual information,” Med. Image Anal. 1(1), 35–51 (1996).
[Crossref] [PubMed]

Opt. Express (1)

M. B. Bouchard, B. R. Chen, S. A. Burgess, and E. M. Hillman, “Ultra-fast multispectral optical imaging of cortical oxygenation, blood flow, and intracellular calcium dynamics,” Opt. Express 17(18), 15,670–15,678 (2009).
[Crossref]

Optom. Vis. Sci. (1)

J. Zhang, Z. Yu, and L. Liu, “Multimodality imaging in diagnosing polypoidal choroidal vasculopathy,” Optom. Vis. Sci. 92(1), e21–e26 (2015).
[Crossref]

Retina Today (1)

C. Zimmer, D. Kahn, R. Clayton, P. Dugel, and K. Freund, “Innovation in diagnostic retinal imaging: multispectral imaging,” Retina Today 9(7), 94–99 (2014).

Rev. Opt. (2)

D. L. Shechtman and P. M. Karpecki, “A look at MSI: multispectral imaging may help eye care providers diagnose retinal conditions earlier than conventional fundoscopy,” Rev. Opt. 149(1), 88–90 (2012).

R. Maharaj, “The clinical applications of multispectral imaging,” Rev. Opt. 148(11), SS19 (2011).

SIAM Rev. (1)

C. V. Stewart, “Robust parameter estimation in computer vision,” SIAM Rev. 41(3), 513–537 (1999).
[Crossref]

Other (15)

Y. Zheng, A. A. Hunter, J. Wu, H. Wang, J. Gao, M. G. Maguire, and J. C. Gee, “Landmark matching based automatic retinal image registration with linear programming and self-similarities,” in Biennial International Conference on Information Processing in Medical Imaging, pp. 674–685 (Springer, 2011).

R. Kolar, L. Kubecka, and J. Jan, “Registration and fusion of the autofluorescent and infrared retinal images,” Int. J. Biomed. Imaging2008 (2008).
[Crossref] [PubMed]

M. Irani and P. Anandan, “Robust multi-sensor image alignment,” in Computer Vision, 1998. Sixth International Conference on, pp. 959–966 (IEEE, 1998).

J. Lewis, “Fast normalized cross-correlation,” Proceedings of Vision Interface pp. 120–123 (1995).

X. Shen, L. Xu, Q. Zhang, and J. Jia, “Multi-modal and multi-spectral registration for natural images,” in European Conference on Computer Vision, pp. 309–324 (Springer, 2014).

D. G. Lowe, “Object recognition from local scale-invariant features,” in Computer vision, 1999. The proceedings of the seventh IEEE international conference on, vol. 2, pp. 1150–1157 (IEEE, 1999).
[Crossref]

G. B. Huang, V. Jain, and E. Learned-Miller, “Unsupervised joint alignment of complex images,” in 2007 IEEE 11th International Conference on Computer Vision, pp. 1–8 (IEEE, 2007).

J. Kim, C. Liu, F. Sha, and K. Grauman, “Deformable spatial pyramid matching for fast dense correspondences,” Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2307–2314 (2013).

T. Zhou, Y. J. Lee, X. Y. Stella, and A. A. Efros, “Flowweb: Joint image set alignment by weaving consistent, pixel-wise correspondences,” in 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1191–1200 (IEEE, 2015).
[Crossref]

N. Everdell, I. Styles, E. Claridge, J. Hebden, and A. Calcagni, “Multispectral imaging of the ocular fundus using LED illumination,” in European Conferences on Biomedical Optics, p. 7371C (International Society for Optics and Photonics, 2009).

S. K. Balci, P. Golland, and W. Wells, “Non-rigid groupwise registration using B-spline deformation model,” Open source and open data for MICCAI pp. 105–121 (2007).

O. Arandjelovic, D.-S. Pham, and S. Venkatesh, “Groupwise registration of aerial images,” Twenty-Fourth International Joint Conference on Artificial Intelligence (IJCAI 2015) (2015).

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University, 2004).
[Crossref]

J. Modersitzki, Numetical Methods for Image Registration (Oxford University, New York, NY, USA, 2004).

L. Zöllei, E. Learned-Miller, E. Grimson, and W. Wells, “Efficient population registration of 3D data,” in International Workshop on Computer Vision for Biomedical Image Applications, pp. 291–301 (Springer, 2005).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 A collection of MSI images from Annidis RHA, in which from left to right and from top to bottom, the first 11 sequential images are captured with short wavelengths of MSI-550, MSI-580, MSI-590, MSI-620, MSI-660, MSI-690, MSI-740, MSI-760, MSI-780, MSI-810 and MSI-850, respectively.
Fig. 2
Fig. 2 Significant motion exists between sequential MSI images and our groupwise registration algorithm can handle it effectively. Top row, left to right: MSI images of MSI-550 and MSI-850 in Fig. 1, respectively. Bottom row, left to right: patches in the images created by overlaying MSI-550 and MSI-850 before and after applying our algorithm, respectively. Misalignment is eliminated in the right image in contrast to the left one.
Fig. 3
Fig. 3 Errors (in pixels) of our algorithm when different matching cost functions (SSD, MI and RCC) and different τ values in Eq. (20) are used.
Fig. 4
Fig. 4 Registration results from a pair of MSI images of a healthy subject. Left to right and top to down : MSI-550, MSI-850, overlapping MSI-850 to MSI-550, overlapping MSI-850 to MSI-550 transformed by the method of Chui & Rangarajan [40], Wells et al. [12] and ours, respectively.
Fig. 5
Fig. 5 Registration results from a pair of MSI images of a patient diagnosed with hypertensive retinopathy. Left to right and top to down: MSI-550, MSI-850, overlapping MSI-850 to MSI-550, overlapping MSI-850 to MSI-550 transformed by the method of Chui & Rangarajan [40], Wells et al. [12] and ours, respectively.

Tables (1)

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Table 1 Mean (standard deviation) of distance between manually marked points and the transformed ones based on the estimated motion models.

Equations (22)

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min i = 1 N j = 1 N tr ( C i , j T E i , j )
X j = X i Θ i , j .
E i , j { 0 , 1 } N i × N j ,
E i , j 1 = 1 ,
C i , j ( k , l ) = w ρ ( 1 γ ( I i ( k ) , I j ( l ) ) ) + ( 1 w ) ρ ( 1 γ ( I i ( k ) , I j ( l ) ) )
ρ ( t ) = t 2 1 + t 2 / a 2
X i ( j , : ) = [ x i k 2 x i k y i k y i k 2 x i k y i k 1 ]
E i , j 0 .
X ¯ i , j = E i , j X j
D i , j = ( X ¯ i , j ( : , 1 ) 1 T 1 ( X j ( : , 1 ) ) T ) 2 + ( X ¯ i , j ( : , 2 ) 1 T 1 ( X j ( : , 2 ) ) T ) 2
min tr ( D i , j T E i , j ) .
min i = 1 N j = 1 N tr ( D i , j T E i , j ) .
min i = 1 N j = 1 N ( 1 λ ) tr ( C i , j T E i , j ) + λ tr ( D i , j T E i , j ) = min i = 1 N j = 1 N tr ( ( 1 λ ) C i , j T E i , j + λ D i , j T E i , j )
min ( i = 1 N j = 1 N tr ( 1 λ ) ( C i , j T E i , j + λ D i , j T E i , j ) + τ k = 2 N 1 tr ( ( Θ k 1 , k Θ k , k + 1 ) T ( Θ k 1 , k Θ k , k + 1 ) ) )
P e i , j e i , j = 1
E i , j X j ( : , 1 ) = Q i , j 1 e i , j .
E i , j X j ( : , 2 ) = Q i , j 2 e i , j .
χ i Θ i , j = E i , j X j .
[ χ i 0 0 χ i ] [ θ i , j ] = [ Q i , j 1 Q i , j 2 ] e i , j
min i = 1 N j = 1 N [ ( 1 λ ) c i , j + λ d i , j 0 ] T [ e i , j θ i , j ] + τ min k = 2 N 1 θ k 1 , k θ k , k + 1 2 2
[ P e i , j 0 0 Q i , j 1 χ i 0 Q i , j 2 0 χ i ] [ e i , j θ i , j ] = [ 1 0 0 ]
e i , j 0 .

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