American Institute of Mathematical Sciences

Advanced
February  2014, 8(1): 79-102. doi: 10.3934/ipi.2014.8.79

Solving inverse source problems by the Orthogonal Solution and Kernel Correction Algorithm (OSKCA) with applications in fluorescence tomography

Shui-Nee Chow 1, , Ke Yin 1, , Hao-Min Zhou 1, and  Ali Behrooz 2,

1. 

School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160, United States, United States, United States
2. 

School of Electrical and Computer Engineering, Georgia Institute of Technology, 777 Atlantic Drive NW Atlanta, GA 30332, United States

Received  January 2013 Revised  June 2013 Published  March 2014

We present a new approach to solve the inverse source problem arising in Fluorescence Tomography (FT). In general, the solution is non-unique and the problem is severely ill-posed. It poses tremendous challenges in image reconstructions. In practice, the most widely used methods are based on Tikhonov-type regularizations, which minimize a cost function consisting of a regularization term and a data fitting term. We propose an alternative method which overcomes the major difficulties, namely the non-uniqueness of the solution and noisy data fitting, in two separate steps. First we find a particular solution called the orthogonal solution that satisfies the data fitting term. Then we add to it a correction function in the kernel space so that the final solution fulfills other regularity and physical requirements. The key ideas are that the correction function in the kernel has no impact on the data fitting, so that there is no parameter needed to balance the data fitting and additional constraints on the solution. Moreover, we use an efficient basis to represent the source function, and introduce a hybrid strategy combining spectral methods and finite element methods in the proposed algorithm. The resulting algorithm can dramatically increase the computation speed over the existing methods. Also the numerical evidence shows that it significantly improves the image resolution and robustness against noise.
Keywords: regularization techniques, inverse source problems, augmented Lagrangian method., diffusion approximation, Image reconstruction.
Mathematics Subject Classification: 68U10, 80A2.
Citation: Shui-Nee Chow, Ke Yin, Hao-Min Zhou, Ali Behrooz. Solving inverse source problems by the Orthogonal Solution and Kernel Correction Algorithm (OSKCA) with applications in fluorescence tomography. Inverse Problems & Imaging, 2014, 8 (1) : 79-102. doi: 10.3934/ipi.2014.8.79
References:
[1]

S. R. Arridge, Optical tomography in medical imaging,, Inverse Problems, 15 (1999), 0266. doi: 10.1088/0266-5611/15/2/022. Google Scholar

[2]

S. R. Arridge and J. C. Schotland, Optical tomography: Forward and inverse problems,, Inverse Problems, 25 (2009), 0266. doi: 10.1088/0266-5611/25/12/123010. Google Scholar

[3]

G. Bao, S. Hou and P. Li, Inverse scattering by a continuation method with initial guesses from a direct imaging algorithm,, Journal of Computational Physics, 227 (2007), 755. doi: 10.1016/j.jcp.2007.08.020. Google Scholar

[4]

A. Behrooz, H. Zhou, A. Eftekhar and A. Adibi, Total variation regularization for 3d reconstruction in fluorescence tomography: experimental phantom studies,, Applied Optics, 51 (2012), 8216. doi: 10.1364/AO.51.008216. Google Scholar

[5]

C. Bremer, V. Ntziachristos and R. Weissleder, Optical-based molecular imaging: contrast agents and potential medical applications,, European Radiology, 13 (2003), 231. Google Scholar

[6]

S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods,, vol. 15, (2007). Google Scholar

[7]

J. Cai, E. Candès and Z. Shen, A singular value thresholding algorithm for matrix completion,, SIAM Journal on Optimization, 20 (2010), 1956. doi: 10.1137/080738970. Google Scholar

[8]

Y. Censor and E. Tom, Convergence of string-averaging projection schemes for inconsistent convex feasibility problems,, Optimization Methods and Software, 18 (2003), 543. doi: 10.1080/10556780310001610484. Google Scholar

[9]

Y. Chen, W. Hager, M. Yashtini, X. Ye and H. Zhang, Bregman operator splitting with variable stepsize for total variation image reconstruction,, Comput. Optim. Appl., 54 (2013). doi: 10.1007/s10589-012-9519-2. Google Scholar

[10]

Y. Dai and R. Fletcher, Projected barzilai-borwein methods for large-scale box-constrained quadratic programming,, Numerische Mathematik, 100 (2005), 21. doi: 10.1007/s00211-004-0569-y. Google Scholar

[11]

Y. Dai and L. Liao, R-linear convergence of the barzilai and borwein gradient method,, IMA Journal of Numerical Analysis, 22 (2002), 1. doi: 10.1093/imanum/22.1.1. Google Scholar

[12]

J. Dutta, S. Ahn, A. Joshi and R. Leahy, Optimal illumination patterns for fluorescence tomography,, in Biomedical Imaging: From Nano to Macro, (2009), 1275. doi: 10.1109/ISBI.2009.5193295. Google Scholar

[13]

J. Dutta, S. Ahn, C. Li, S. R. Cherry and R. M. Leahy, Joint l1 and total variation regularization for fluorescence molecular tomography,, Physics in Medicine and Biology, 57 (2012), 1459. Google Scholar

[14]

H. Egger, M. Freiberger and M. Schlottbom, On forward and inverse models in fluorescence diffuse optical tomography,, Inverse Problems and Imaging, 4 (2010), 411. doi: 10.3934/ipi.2010.4.411. Google Scholar

[15]

H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems,, vol. 375, (1996). doi: 10.1007/978-94-009-1740-8. Google Scholar

[16]

E. Esser, Applications of lagrangian-based alternating direction methods and connections to split bregman,, CAM Report, 9 (2009). Google Scholar

[17]

T. J. Farrell, M. S. Patterson and B. Wilson, A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,, Medical Physics, 19 (1992), 879. doi: 10.1118/1.596777. Google Scholar

[18]

H. Gao and H. Zhao, Multilevel bioluminescence tomography based on radiative transfer equation part 1: l1 regularization,, Optics Express, 18 (2010), 1854. doi: 10.1364/OE.18.001854. Google Scholar

[19]

H. Gao and H. Zhao, Multilevel bioluminescence tomography based on radiative transfer equation part 2: total variation and l1 data fidelity,, Optics Express, 18 (2010), 2894. doi: 10.1364/OE.18.002894. Google Scholar

[20]

T. Goldstein and S. Osher, The split bregman method for l1-regularized problems,, SIAM Journal on Imaging Sciences, 2 (2009), 323. doi: 10.1137/080725891. Google Scholar

[21]

G. Golub and C. Van Loan, Matrix Computations,, vol. 3, (1996). Google Scholar

[22]

P. Hansen and D. O'Leary, The use of the L-curve in the regularization of discrete ill-posed problems,, SIAM Journal on Scientific Computing, 14 (1993), 1487. doi: 10.1137/0914086. Google Scholar

[23]

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams and B. J. Tromberg, Boundary conditions for the diffusion equation in radiative transfer,, Journal of the Optical Society of America. A, 11 (1994), 2727. doi: 10.1364/JOSAA.11.002727. Google Scholar

[24]

D. Hawrysz and E. Sevick-Muraca, Developments toward diagnostic breast cancer imaging using near-infrared optical measurements and fluorescent contrast agents,, Neoplasia (New York, 2 (2000). doi: 10.1038/sj.neo.7900118. Google Scholar

[25]

J. Hebden, S. Arridge and D. Delpy, Optical imaging in medicine: I. experimental techniques,, Physics in Medicine and Biology, 42 (1999). doi: 10.1088/0031-9155/42/5/007. Google Scholar

[26]

G. Herman, A. Lent and S. Rowland, ART: Mathematics and applications: A report on the mathematical foundations and on the applicability to real data of the algebraic reconstruction techniques,, Journal of theoretical biology, 42 (1973), 1. doi: 10.1016/0022-5193(74)90135-0. Google Scholar

[27]

A. Ishimaru, Wave Propagation and Scattering in Random Media,, vol. 2, (1997). Google Scholar

[28]

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems,, vol. 160, (2004). Google Scholar

[29]

O. A. Ladyzhenskaia and N. N. Ural'ceva, Linear and Quasilinear Elliptic Equations,, Academic Press, (1968). Google Scholar

[30]

V. A. Markel and J. C. Schotland, Inverse problem in optical diffusion tomography. II. role of boundary conditions,, Journal of the Optical Society of America. A, 19 (2002), 558. doi: 10.1364/JOSAA.19.000558. Google Scholar

[31]

A. Neumaier, Solving ill-conditioned and singular linear systems: A tutorial on regularization,, SIAM Review, 40 (1998), 636. doi: 10.1137/S0036144597321909. Google Scholar

[32]

Y. Notay, An aggregation-based algebraic multigrid method,, Electronic Transactions on Numerical Analysis, 37 (2010), 123. Google Scholar

[33]

L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D: Nonlinear Phenomena, 60 (1992), 259. doi: 10.1016/0167-2789(92)90242-F. Google Scholar

[34]

W. Rudin, Functional Analysis,, McGraw-Hill, (1991). doi: 10.1007/978-3-642-84455-3_77. Google Scholar

[35]

J. C. Schotland and V. A. Markel, Inverse scattering with diffusing waves,, Journal of the Optical Society of America. A, 18 (2001), 2767. doi: 10.1364/JOSAA.18.002767. Google Scholar

[36]

M. Schweiger, S. R. Arridge, M. Hiraoka and D. Delpy, The finite element method for the propagation of light in scattering media: Boundary and source conditions,, Medical Physics, 22 (1995). doi: 10.1118/1.597634. Google Scholar

[37]

G. Wang, Y. Li and M. Jiang, Uniqueness theorems in bioluminescence tomography,, Medical Physics, 31 (2004). doi: 10.1118/1.1766420. Google Scholar

[38]

Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction,, SIAM Journal on Imaging Sciences, 1 (2008). doi: 10.1137/080724265. Google Scholar

[39]

R. Weissleder, C. Tung, U. Mahmood and A. Bogdanov, In vivo imaging of tumors with protease-activated near-infrared fluorescent probes,, Nature Biotechnology, 17 (1999), 375. Google Scholar

[40]

C. Wu and X. Tai, Augmented lagrangian method, dual methods, and split bregman iteration for rof, vectorial tv, and high order models,, SIAM J. Imaging Sci., 3 (2010), 300. doi: 10.1137/090767558. Google Scholar

show all references

References:
[1]

S. R. Arridge, Optical tomography in medical imaging,, Inverse Problems, 15 (1999), 0266. doi: 10.1088/0266-5611/15/2/022. Google Scholar

[2]

S. R. Arridge and J. C. Schotland, Optical tomography: Forward and inverse problems,, Inverse Problems, 25 (2009), 0266. doi: 10.1088/0266-5611/25/12/123010. Google Scholar

[3]

G. Bao, S. Hou and P. Li, Inverse scattering by a continuation method with initial guesses from a direct imaging algorithm,, Journal of Computational Physics, 227 (2007), 755. doi: 10.1016/j.jcp.2007.08.020. Google Scholar

[4]

A. Behrooz, H. Zhou, A. Eftekhar and A. Adibi, Total variation regularization for 3d reconstruction in fluorescence tomography: experimental phantom studies,, Applied Optics, 51 (2012), 8216. doi: 10.1364/AO.51.008216. Google Scholar

[5]

C. Bremer, V. Ntziachristos and R. Weissleder, Optical-based molecular imaging: contrast agents and potential medical applications,, European Radiology, 13 (2003), 231. Google Scholar

[6]

S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods,, vol. 15, (2007). Google Scholar

[7]

J. Cai, E. Candès and Z. Shen, A singular value thresholding algorithm for matrix completion,, SIAM Journal on Optimization, 20 (2010), 1956. doi: 10.1137/080738970. Google Scholar

[8]

Y. Censor and E. Tom, Convergence of string-averaging projection schemes for inconsistent convex feasibility problems,, Optimization Methods and Software, 18 (2003), 543. doi: 10.1080/10556780310001610484. Google Scholar

[9]

Y. Chen, W. Hager, M. Yashtini, X. Ye and H. Zhang, Bregman operator splitting with variable stepsize for total variation image reconstruction,, Comput. Optim. Appl., 54 (2013). doi: 10.1007/s10589-012-9519-2. Google Scholar

[10]

Y. Dai and R. Fletcher, Projected barzilai-borwein methods for large-scale box-constrained quadratic programming,, Numerische Mathematik, 100 (2005), 21. doi: 10.1007/s00211-004-0569-y. Google Scholar

[11]

Y. Dai and L. Liao, R-linear convergence of the barzilai and borwein gradient method,, IMA Journal of Numerical Analysis, 22 (2002), 1. doi: 10.1093/imanum/22.1.1. Google Scholar

[12]

J. Dutta, S. Ahn, A. Joshi and R. Leahy, Optimal illumination patterns for fluorescence tomography,, in Biomedical Imaging: From Nano to Macro, (2009), 1275. doi: 10.1109/ISBI.2009.5193295. Google Scholar

[13]

J. Dutta, S. Ahn, C. Li, S. R. Cherry and R. M. Leahy, Joint l1 and total variation regularization for fluorescence molecular tomography,, Physics in Medicine and Biology, 57 (2012), 1459. Google Scholar

[14]

H. Egger, M. Freiberger and M. Schlottbom, On forward and inverse models in fluorescence diffuse optical tomography,, Inverse Problems and Imaging, 4 (2010), 411. doi: 10.3934/ipi.2010.4.411. Google Scholar

[15]

H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems,, vol. 375, (1996). doi: 10.1007/978-94-009-1740-8. Google Scholar

[16]

E. Esser, Applications of lagrangian-based alternating direction methods and connections to split bregman,, CAM Report, 9 (2009). Google Scholar

[17]

T. J. Farrell, M. S. Patterson and B. Wilson, A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,, Medical Physics, 19 (1992), 879. doi: 10.1118/1.596777. Google Scholar

[18]

H. Gao and H. Zhao, Multilevel bioluminescence tomography based on radiative transfer equation part 1: l1 regularization,, Optics Express, 18 (2010), 1854. doi: 10.1364/OE.18.001854. Google Scholar

[19]

H. Gao and H. Zhao, Multilevel bioluminescence tomography based on radiative transfer equation part 2: total variation and l1 data fidelity,, Optics Express, 18 (2010), 2894. doi: 10.1364/OE.18.002894. Google Scholar

[20]

T. Goldstein and S. Osher, The split bregman method for l1-regularized problems,, SIAM Journal on Imaging Sciences, 2 (2009), 323. doi: 10.1137/080725891. Google Scholar

[21]

G. Golub and C. Van Loan, Matrix Computations,, vol. 3, (1996). Google Scholar

[22]

P. Hansen and D. O'Leary, The use of the L-curve in the regularization of discrete ill-posed problems,, SIAM Journal on Scientific Computing, 14 (1993), 1487. doi: 10.1137/0914086. Google Scholar

[23]

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams and B. J. Tromberg, Boundary conditions for the diffusion equation in radiative transfer,, Journal of the Optical Society of America. A, 11 (1994), 2727. doi: 10.1364/JOSAA.11.002727. Google Scholar

[24]

D. Hawrysz and E. Sevick-Muraca, Developments toward diagnostic breast cancer imaging using near-infrared optical measurements and fluorescent contrast agents,, Neoplasia (New York, 2 (2000). doi: 10.1038/sj.neo.7900118. Google Scholar

[25]

J. Hebden, S. Arridge and D. Delpy, Optical imaging in medicine: I. experimental techniques,, Physics in Medicine and Biology, 42 (1999). doi: 10.1088/0031-9155/42/5/007. Google Scholar

[26]

G. Herman, A. Lent and S. Rowland, ART: Mathematics and applications: A report on the mathematical foundations and on the applicability to real data of the algebraic reconstruction techniques,, Journal of theoretical biology, 42 (1973), 1. doi: 10.1016/0022-5193(74)90135-0. Google Scholar

[27]

A. Ishimaru, Wave Propagation and Scattering in Random Media,, vol. 2, (1997). Google Scholar

[28]

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems,, vol. 160, (2004). Google Scholar

[29]

O. A. Ladyzhenskaia and N. N. Ural'ceva, Linear and Quasilinear Elliptic Equations,, Academic Press, (1968). Google Scholar

[30]

V. A. Markel and J. C. Schotland, Inverse problem in optical diffusion tomography. II. role of boundary conditions,, Journal of the Optical Society of America. A, 19 (2002), 558. doi: 10.1364/JOSAA.19.000558. Google Scholar

[31]

A. Neumaier, Solving ill-conditioned and singular linear systems: A tutorial on regularization,, SIAM Review, 40 (1998), 636. doi: 10.1137/S0036144597321909. Google Scholar

[32]

Y. Notay, An aggregation-based algebraic multigrid method,, Electronic Transactions on Numerical Analysis, 37 (2010), 123. Google Scholar

[33]

L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D: Nonlinear Phenomena, 60 (1992), 259. doi: 10.1016/0167-2789(92)90242-F. Google Scholar

[34]

W. Rudin, Functional Analysis,, McGraw-Hill, (1991). doi: 10.1007/978-3-642-84455-3_77. Google Scholar

[35]

J. C. Schotland and V. A. Markel, Inverse scattering with diffusing waves,, Journal of the Optical Society of America. A, 18 (2001), 2767. doi: 10.1364/JOSAA.18.002767. Google Scholar

[36]

M. Schweiger, S. R. Arridge, M. Hiraoka and D. Delpy, The finite element method for the propagation of light in scattering media: Boundary and source conditions,, Medical Physics, 22 (1995). doi: 10.1118/1.597634. Google Scholar

[37]

G. Wang, Y. Li and M. Jiang, Uniqueness theorems in bioluminescence tomography,, Medical Physics, 31 (2004). doi: 10.1118/1.1766420. Google Scholar

[38]

Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction,, SIAM Journal on Imaging Sciences, 1 (2008). doi: 10.1137/080724265. Google Scholar

[39]

R. Weissleder, C. Tung, U. Mahmood and A. Bogdanov, In vivo imaging of tumors with protease-activated near-infrared fluorescent probes,, Nature Biotechnology, 17 (1999), 375. Google Scholar

[40]

C. Wu and X. Tai, Augmented lagrangian method, dual methods, and split bregman iteration for rof, vectorial tv, and high order models,, SIAM J. Imaging Sci., 3 (2010), 300. doi: 10.1137/090767558. Google Scholar

[1]

Xiantao Xiao, Liwei Zhang, Jianzhong Zhang. On convergence of augmented Lagrangian method for inverse semi-definite quadratic programming problems. Journal of Industrial & Management Optimization, 2009, 5 (2) : 319-339. doi: 10.3934/jimo.2009.5.319
[2]

Wei Zhu, Xue-Cheng Tai, Tony Chan. Augmented Lagrangian method for a mean curvature based image denoising model. Inverse Problems & Imaging, 2013, 7 (4) : 1409-1432. doi: 10.3934/ipi.2013.7.1409
[3]

Lacramioara Grecu, Constantin Popa. Constrained SART algorithm for inverse problems in image reconstruction. Inverse Problems & Imaging, 2013, 7 (1) : 199-216. doi: 10.3934/ipi.2013.7.199
[4]

Martin Hanke, William Rundell. On rational approximation methods for inverse source problems. Inverse Problems & Imaging, 2011, 5 (1) : 185-202. doi: 10.3934/ipi.2011.5.185
[5]

Qingsong Duan, Mengwei Xu, Yue Lu, Liwei Zhang. A smoothing augmented Lagrangian method for nonconvex, nonsmooth constrained programs and its applications to bilevel problems. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1241-1261. doi: 10.3934/jimo.2018094
[6]

Guanghui Hu, Peijun Li, Xiaodong Liu, Yue Zhao. Inverse source problems in electrodynamics. Inverse Problems & Imaging, 2018, 12 (6) : 1411-1428. doi: 10.3934/ipi.2018059
[7]

Jianjun Zhang, Yunyi Hu, James G. Nagy. A scaled gradient method for digital tomographic image reconstruction. Inverse Problems & Imaging, 2018, 12 (1) : 239-259. doi: 10.3934/ipi.2018010
[8]

Chunlin Wu, Juyong Zhang, Xue-Cheng Tai. Augmented Lagrangian method for total variation restoration with non-quadratic fidelity. Inverse Problems & Imaging, 2011, 5 (1) : 237-261. doi: 10.3934/ipi.2011.5.237
[9]

Xueyong Wang, Yiju Wang, Gang Wang. An accelerated augmented Lagrangian method for multi-criteria optimization problem. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-9. doi: 10.3934/jimo.2018136
[10]

Li Jin, Hongying Huang. Differential equation method based on approximate augmented Lagrangian for nonlinear programming. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-15. doi: 10.3934/jimo.2019053
[11]

Zhousheng Ruan, Sen Zhang, Sican Xiong. Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method. Evolution Equations & Control Theory, 2018, 7 (4) : 669-682. doi: 10.3934/eect.2018032
[12]

Razvan C. Fetecau, Mitchell Kovacic, Ihsan Topaloglu. Swarming in domains with boundaries: Approximation and regularization by nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1815-1842. doi: 10.3934/dcdsb.2018238
[13]

Gabriel Peyré, Sébastien Bougleux, Laurent Cohen. Non-local regularization of inverse problems. Inverse Problems & Imaging, 2011, 5 (2) : 511-530. doi: 10.3934/ipi.2011.5.511
[14]

Janne M.J. Huttunen, J. P. Kaipio. Approximation errors in nonstationary inverse problems. Inverse Problems & Imaging, 2007, 1 (1) : 77-93. doi: 10.3934/ipi.2007.1.77
[15]

Victor Isakov, Shuai Lu. Inverse source problems without (pseudo) convexity assumptions. Inverse Problems & Imaging, 2018, 12 (4) : 955-970. doi: 10.3934/ipi.2018040
[16]

Xi-Hong Yan. A new convergence proof of augmented Lagrangian-based method with full Jacobian decomposition for structured variational inequalities. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 45-54. doi: 10.3934/naco.2016.6.45
[17]

Wei Zhu. A numerical study of a mean curvature denoising model using a novel augmented Lagrangian method. Inverse Problems & Imaging, 2017, 11 (6) : 975-996. doi: 10.3934/ipi.2017045
[18]

Egil Bae, Xue-Cheng Tai, Wei Zhu. Augmented Lagrangian method for an Euler's elastica based segmentation model that promotes convex contours. Inverse Problems & Imaging, 2017, 11 (1) : 1-23. doi: 10.3934/ipi.2017001
[19]

Xihong Yan. An augmented Lagrangian-based parallel splitting method for a one-leader-two-follower game. Journal of Industrial & Management Optimization, 2016, 12 (3) : 879-890. doi: 10.3934/jimo.2016.12.879
[20]

Anis Theljani, Ke Chen. An augmented lagrangian method for solving a new variational model based on gradients similarity measures and high order regulariation for multimodality registration. Inverse Problems & Imaging, 2019, 13 (2) : 309-335. doi: 10.3934/ipi.2019016

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences