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

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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

Abstract
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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.
[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.
[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.
[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.
[5]	C. Bremer, V. Ntziachristos and R. Weissleder, Optical-based molecular imaging: contrast agents and potential medical applications,, European Radiology, 13 (2003), 231.
[6]	S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods,, vol. 15, (2007).
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems,, vol. 375, (1996). doi: 10.1007/978-94-009-1740-8.
[16]	E. Esser, Applications of lagrangian-based alternating direction methods and connections to split bregman,, CAM Report, 9 (2009).
[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.
[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.
[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.
[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.
[21]	G. Golub and C. Van Loan, Matrix Computations,, vol. 3, (1996).
[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.
[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.
[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.
[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.
[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.
[27]	A. Ishimaru, Wave Propagation and Scattering in Random Media,, vol. 2, (1997).
[28]	J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems,, vol. 160, (2004).
[29]	O. A. Ladyzhenskaia and N. N. Ural'ceva, Linear and Quasilinear Elliptic Equations,, Academic Press, (1968).
[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.
[31]	A. Neumaier, Solving ill-conditioned and singular linear systems: A tutorial on regularization,, SIAM Review, 40 (1998), 636. doi: 10.1137/S0036144597321909.
[32]	Y. Notay, An aggregation-based algebraic multigrid method,, Electronic Transactions on Numerical Analysis, 37 (2010), 123.
[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.
[34]	W. Rudin, Functional Analysis,, McGraw-Hill, (1991). doi: 10.1007/978-3-642-84455-3_77.
[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.
[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.
[37]	G. Wang, Y. Li and M. Jiang, Uniqueness theorems in bioluminescence tomography,, Medical Physics, 31 (2004). doi: 10.1118/1.1766420.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[5]	C. Bremer, V. Ntziachristos and R. Weissleder, Optical-based molecular imaging: contrast agents and potential medical applications,, European Radiology, 13 (2003), 231.
[6]	S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods,, vol. 15, (2007).
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems,, vol. 375, (1996). doi: 10.1007/978-94-009-1740-8.
[16]	E. Esser, Applications of lagrangian-based alternating direction methods and connections to split bregman,, CAM Report, 9 (2009).
[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.
[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.
[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.
[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.
[21]	G. Golub and C. Van Loan, Matrix Computations,, vol. 3, (1996).
[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.
[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.
[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.
[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.
[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.
[27]	A. Ishimaru, Wave Propagation and Scattering in Random Media,, vol. 2, (1997).
[28]	J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems,, vol. 160, (2004).
[29]	O. A. Ladyzhenskaia and N. N. Ural'ceva, Linear and Quasilinear Elliptic Equations,, Academic Press, (1968).
[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.
[31]	A. Neumaier, Solving ill-conditioned and singular linear systems: A tutorial on regularization,, SIAM Review, 40 (1998), 636. doi: 10.1137/S0036144597321909.
[32]	Y. Notay, An aggregation-based algebraic multigrid method,, Electronic Transactions on Numerical Analysis, 37 (2010), 123.
[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.
[34]	W. Rudin, Functional Analysis,, McGraw-Hill, (1991). doi: 10.1007/978-3-642-84455-3_77.
[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.
[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.
[37]	G. Wang, Y. Li and M. Jiang, Uniqueness theorems in bioluminescence tomography,, Medical Physics, 31 (2004). doi: 10.1118/1.1766420.
[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.
[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.
[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.

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