Abstract

Fluorescence molecular tomography suffers from being mathematically ill-conditioned resulting in non-unique solutions to the reconstruction problem. In an attempt to reduce the number of possible solutions in the underdetermined system of equations in the reconstruction, we present a method to retrieve a spatially varying regularization map outlining the feasible inclusion position. This approach can be made very simple by including a few multispectral recordings from only one source position. The results retrieved through tissue phantom experiments imply that initial reconstructions with spatially varying priors reduces artifacts and show slightly more accurate reconstruction results compared to reconstructions using no priors.

© 2007 Optical Society of America

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References

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  1. V. Ntziachristos, J. Ripoll, L. Wang, and R. Weissleder, "Looking and listening to light: the evolution of whole-body photonic imaging," Nat. Biotechnol. 23, 313-320 (2005).
    [CrossRef] [PubMed]
  2. B. Brooksby, S. Jiang, H. Dehghani, B. Pogue, K. Paulsen, J. Weaver, C. Kogel, and S. Poplack, "Combining near-infrared tomography resonance imaging to study in vivo and magnetic breast tissue: implementation of a Laplacian-type regularization to incorporate magnetic resonance structure," J. Biomed. Opt. 10 (2005).
    [CrossRef] [PubMed]
  3. M. Guven, B. Yazici, X. Intes, and B. Chance, "Diffuse optical tomography with a priori anatomical information," Phys. Med. Biol. 50, 2837-2858 (2005).
    [CrossRef] [PubMed]
  4. H. Xu, R. Springett, H. Dehghani, B. Pogue, K. Paulsen, and J. Dunn, "Magnetic-resonance-imaging-coupled broadband near-infrared tomography system for small animal brain studies," Appl. Opt. 44, 2177-2188 (2005).
    [CrossRef] [PubMed]
  5. B. Pogue, T. McBride, J. Prewitt, U. Osterberg, and K. Paulsen, "Spatially variant regularization improves diffuse optical tomography," Appl. Opt. 38, 2950-2961 (1999).
    [CrossRef]
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    [CrossRef] [PubMed]
  7. E. Graves, J. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, "Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomograpy," J. Opt. Soc. Am. 21, 231-241 (2004).
    [CrossRef]
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    [CrossRef]
  10. D. Paithankar, A. Chen, B. Pogue, M. Patterson, and E. Sevick-Muraca, "Imaging of fluorescent yield and lifetime from multiply scattered light reemitted from random media," Appl. Opt. 36, 2260-2272 (1997).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  13. A. Neumaier, "Solving ill-conditioned and singular linear systems: A tutorial on regularization," Siam Review 40, 636-666 (1998).
    [CrossRef]
  14. H. Dehghani, D. Barber, and I. Basarab-Horwath, "Incorporating a priori anatomical information into image reconstruction in electrical impedance tomography," Physiol. Measurement 20, 87-102 (1999).
    [CrossRef]
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    [CrossRef] [PubMed]
  16. J. Svensson and S. Andersson-Engels, "Modeling of spectral changes for depth localization of fluorescent inclusion," Opt. Express 13, 4263-4274 (2005).
    [CrossRef] [PubMed]
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    [CrossRef]
  18. J. Swartling, A. Pifferi, A. Enejder, and S. Andersson-Engels, "Accelerated Monte Carlo model to simulate fluorescence spectra from layered tissues," J. Opt. Soc. Am. 20, 714-727 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2005

V. Ntziachristos, J. Ripoll, L. Wang, and R. Weissleder, "Looking and listening to light: the evolution of whole-body photonic imaging," Nat. Biotechnol. 23, 313-320 (2005).
[CrossRef] [PubMed]

B. Brooksby, S. Jiang, H. Dehghani, B. Pogue, K. Paulsen, J. Weaver, C. Kogel, and S. Poplack, "Combining near-infrared tomography resonance imaging to study in vivo and magnetic breast tissue: implementation of a Laplacian-type regularization to incorporate magnetic resonance structure," J. Biomed. Opt. 10 (2005).
[CrossRef] [PubMed]

M. Guven, B. Yazici, X. Intes, and B. Chance, "Diffuse optical tomography with a priori anatomical information," Phys. Med. Biol. 50, 2837-2858 (2005).
[CrossRef] [PubMed]

J. Ripoll, D. Yessayan, G. Zacharakis, and V. Ntziachristos, "Experimental determination of photon propagation in highly absorbing and scattering media," J. Opt. Soc. Am. 22, 546-551 (2005).
[CrossRef]

J. Swartling, J. Svensson, D. Bengtsson, K. Terike, and S. Andersson-Engels, "Fluorescence spectra provide information on the depth of fluorescent lesions in tissue," Appl. Opt. 44, 1934-1941 (2005).
[CrossRef] [PubMed]

H. Xu, R. Springett, H. Dehghani, B. Pogue, K. Paulsen, and J. Dunn, "Magnetic-resonance-imaging-coupled broadband near-infrared tomography system for small animal brain studies," Appl. Opt. 44, 2177-2188 (2005).
[CrossRef] [PubMed]

J. Svensson and S. Andersson-Engels, "Modeling of spectral changes for depth localization of fluorescent inclusion," Opt. Express 13, 4263-4274 (2005).
[CrossRef] [PubMed]

2004

R. Schulz, J. Ripoll, and V. Ntziachristos, "Experimental Fluorescence Tomography of Tissues With Noncontact Measurements," IEEE Trans. Med. Imaging 23, 492-500 (2004).
[CrossRef] [PubMed]

E. Graves, J. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, "Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomograpy," J. Opt. Soc. Am. 21, 231-241 (2004).
[CrossRef]

2003

J. Swartling, A. Pifferi, A. Enejder, and S. Andersson-Engels, "Accelerated Monte Carlo model to simulate fluorescence spectra from layered tissues," J. Opt. Soc. Am. 20, 714-727 (2003).
[CrossRef]

2001

2000

1999

B. Pogue, T. McBride, J. Prewitt, U. Osterberg, and K. Paulsen, "Spatially variant regularization improves diffuse optical tomography," Appl. Opt. 38, 2950-2961 (1999).
[CrossRef]

H. Dehghani, D. Barber, and I. Basarab-Horwath, "Incorporating a priori anatomical information into image reconstruction in electrical impedance tomography," Physiol. Measurement 20, 87-102 (1999).
[CrossRef]

S. Arridge, "Optical tomography in medical imaging," Inverse Problems 15, R41-R93 (1999).
[CrossRef]

1998

A. Neumaier, "Solving ill-conditioned and singular linear systems: A tutorial on regularization," Siam Review 40, 636-666 (1998).
[CrossRef]

1997

1996

1982

R. Kubin and A. Fletcher, "Fluorescence Quantum Yields of Some Rhodamine Dyes," J. Lumin. 27, 455-462 (1982).
[CrossRef]

Andersson-Engels, S.

Arridge, S.

S. Arridge, "Optical tomography in medical imaging," Inverse Problems 15, R41-R93 (1999).
[CrossRef]

Barber, D.

H. Dehghani, D. Barber, and I. Basarab-Horwath, "Incorporating a priori anatomical information into image reconstruction in electrical impedance tomography," Physiol. Measurement 20, 87-102 (1999).
[CrossRef]

Basarab-Horwath, I.

H. Dehghani, D. Barber, and I. Basarab-Horwath, "Incorporating a priori anatomical information into image reconstruction in electrical impedance tomography," Physiol. Measurement 20, 87-102 (1999).
[CrossRef]

Bengtsson, D.

Boas, D.

Brooksby, B.

B. Brooksby, S. Jiang, H. Dehghani, B. Pogue, K. Paulsen, J. Weaver, C. Kogel, and S. Poplack, "Combining near-infrared tomography resonance imaging to study in vivo and magnetic breast tissue: implementation of a Laplacian-type regularization to incorporate magnetic resonance structure," J. Biomed. Opt. 10 (2005).
[CrossRef] [PubMed]

Chance, B.

M. Guven, B. Yazici, X. Intes, and B. Chance, "Diffuse optical tomography with a priori anatomical information," Phys. Med. Biol. 50, 2837-2858 (2005).
[CrossRef] [PubMed]

M. O’Leary, D. Boas, X. Li, B. Chance, and A. Yodh, "Fluorescence lifetime imaging in turbid media," Opt. Lett. 21, 158-160 (1996).
[CrossRef] [PubMed]

Chen, A.

Contini, D.

Culver, J.

E. Graves, J. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, "Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomograpy," J. Opt. Soc. Am. 21, 231-241 (2004).
[CrossRef]

Dalgaard, T.

Dam, J.

Dehghani, H.

H. Xu, R. Springett, H. Dehghani, B. Pogue, K. Paulsen, and J. Dunn, "Magnetic-resonance-imaging-coupled broadband near-infrared tomography system for small animal brain studies," Appl. Opt. 44, 2177-2188 (2005).
[CrossRef] [PubMed]

B. Brooksby, S. Jiang, H. Dehghani, B. Pogue, K. Paulsen, J. Weaver, C. Kogel, and S. Poplack, "Combining near-infrared tomography resonance imaging to study in vivo and magnetic breast tissue: implementation of a Laplacian-type regularization to incorporate magnetic resonance structure," J. Biomed. Opt. 10 (2005).
[CrossRef] [PubMed]

H. Dehghani, D. Barber, and I. Basarab-Horwath, "Incorporating a priori anatomical information into image reconstruction in electrical impedance tomography," Physiol. Measurement 20, 87-102 (1999).
[CrossRef]

Dunn, J.

Enejder, A.

J. Swartling, A. Pifferi, A. Enejder, and S. Andersson-Engels, "Accelerated Monte Carlo model to simulate fluorescence spectra from layered tissues," J. Opt. Soc. Am. 20, 714-727 (2003).
[CrossRef]

Fabricius, P.

Fletcher, A.

R. Kubin and A. Fletcher, "Fluorescence Quantum Yields of Some Rhodamine Dyes," J. Lumin. 27, 455-462 (1982).
[CrossRef]

Graves, E.

E. Graves, J. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, "Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomograpy," J. Opt. Soc. Am. 21, 231-241 (2004).
[CrossRef]

Guven, M.

M. Guven, B. Yazici, X. Intes, and B. Chance, "Diffuse optical tomography with a priori anatomical information," Phys. Med. Biol. 50, 2837-2858 (2005).
[CrossRef] [PubMed]

Intes, X.

M. Guven, B. Yazici, X. Intes, and B. Chance, "Diffuse optical tomography with a priori anatomical information," Phys. Med. Biol. 50, 2837-2858 (2005).
[CrossRef] [PubMed]

Jiang, S.

B. Brooksby, S. Jiang, H. Dehghani, B. Pogue, K. Paulsen, J. Weaver, C. Kogel, and S. Poplack, "Combining near-infrared tomography resonance imaging to study in vivo and magnetic breast tissue: implementation of a Laplacian-type regularization to incorporate magnetic resonance structure," J. Biomed. Opt. 10 (2005).
[CrossRef] [PubMed]

Kogel, C.

B. Brooksby, S. Jiang, H. Dehghani, B. Pogue, K. Paulsen, J. Weaver, C. Kogel, and S. Poplack, "Combining near-infrared tomography resonance imaging to study in vivo and magnetic breast tissue: implementation of a Laplacian-type regularization to incorporate magnetic resonance structure," J. Biomed. Opt. 10 (2005).
[CrossRef] [PubMed]

Kubin, R.

R. Kubin and A. Fletcher, "Fluorescence Quantum Yields of Some Rhodamine Dyes," J. Lumin. 27, 455-462 (1982).
[CrossRef]

Li, X.

Martelli, F.

McBride, T.

Neumaier, A.

A. Neumaier, "Solving ill-conditioned and singular linear systems: A tutorial on regularization," Siam Review 40, 636-666 (1998).
[CrossRef]

Ntziachristos, V.

J. Ripoll, D. Yessayan, G. Zacharakis, and V. Ntziachristos, "Experimental determination of photon propagation in highly absorbing and scattering media," J. Opt. Soc. Am. 22, 546-551 (2005).
[CrossRef]

V. Ntziachristos, J. Ripoll, L. Wang, and R. Weissleder, "Looking and listening to light: the evolution of whole-body photonic imaging," Nat. Biotechnol. 23, 313-320 (2005).
[CrossRef] [PubMed]

R. Schulz, J. Ripoll, and V. Ntziachristos, "Experimental Fluorescence Tomography of Tissues With Noncontact Measurements," IEEE Trans. Med. Imaging 23, 492-500 (2004).
[CrossRef] [PubMed]

E. Graves, J. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, "Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomograpy," J. Opt. Soc. Am. 21, 231-241 (2004).
[CrossRef]

V. Ntziachristos and R. Weissleder, "Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation," Opt. Lett. 26, 893-895 (2001).
[CrossRef]

O’Leary, M.

Osterberg, U.

Paithankar, D.

Patterson, M.

Paulsen, K.

B. Brooksby, S. Jiang, H. Dehghani, B. Pogue, K. Paulsen, J. Weaver, C. Kogel, and S. Poplack, "Combining near-infrared tomography resonance imaging to study in vivo and magnetic breast tissue: implementation of a Laplacian-type regularization to incorporate magnetic resonance structure," J. Biomed. Opt. 10 (2005).
[CrossRef] [PubMed]

H. Xu, R. Springett, H. Dehghani, B. Pogue, K. Paulsen, and J. Dunn, "Magnetic-resonance-imaging-coupled broadband near-infrared tomography system for small animal brain studies," Appl. Opt. 44, 2177-2188 (2005).
[CrossRef] [PubMed]

B. Pogue, T. McBride, J. Prewitt, U. Osterberg, and K. Paulsen, "Spatially variant regularization improves diffuse optical tomography," Appl. Opt. 38, 2950-2961 (1999).
[CrossRef]

Pifferi, A.

J. Swartling, A. Pifferi, A. Enejder, and S. Andersson-Engels, "Accelerated Monte Carlo model to simulate fluorescence spectra from layered tissues," J. Opt. Soc. Am. 20, 714-727 (2003).
[CrossRef]

Pogue, B.

Poplack, S.

B. Brooksby, S. Jiang, H. Dehghani, B. Pogue, K. Paulsen, J. Weaver, C. Kogel, and S. Poplack, "Combining near-infrared tomography resonance imaging to study in vivo and magnetic breast tissue: implementation of a Laplacian-type regularization to incorporate magnetic resonance structure," J. Biomed. Opt. 10 (2005).
[CrossRef] [PubMed]

Prewitt, J.

Ripoll, J.

J. Ripoll, D. Yessayan, G. Zacharakis, and V. Ntziachristos, "Experimental determination of photon propagation in highly absorbing and scattering media," J. Opt. Soc. Am. 22, 546-551 (2005).
[CrossRef]

V. Ntziachristos, J. Ripoll, L. Wang, and R. Weissleder, "Looking and listening to light: the evolution of whole-body photonic imaging," Nat. Biotechnol. 23, 313-320 (2005).
[CrossRef] [PubMed]

R. Schulz, J. Ripoll, and V. Ntziachristos, "Experimental Fluorescence Tomography of Tissues With Noncontact Measurements," IEEE Trans. Med. Imaging 23, 492-500 (2004).
[CrossRef] [PubMed]

E. Graves, J. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, "Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomograpy," J. Opt. Soc. Am. 21, 231-241 (2004).
[CrossRef]

Schulz, R.

R. Schulz, J. Ripoll, and V. Ntziachristos, "Experimental Fluorescence Tomography of Tissues With Noncontact Measurements," IEEE Trans. Med. Imaging 23, 492-500 (2004).
[CrossRef] [PubMed]

Sevick-Muraca, E.

Springett, R.

Svensson, J.

Swartling, J.

J. Swartling, J. Svensson, D. Bengtsson, K. Terike, and S. Andersson-Engels, "Fluorescence spectra provide information on the depth of fluorescent lesions in tissue," Appl. Opt. 44, 1934-1941 (2005).
[CrossRef] [PubMed]

J. Swartling, A. Pifferi, A. Enejder, and S. Andersson-Engels, "Accelerated Monte Carlo model to simulate fluorescence spectra from layered tissues," J. Opt. Soc. Am. 20, 714-727 (2003).
[CrossRef]

Terike, K.

Wang, L.

V. Ntziachristos, J. Ripoll, L. Wang, and R. Weissleder, "Looking and listening to light: the evolution of whole-body photonic imaging," Nat. Biotechnol. 23, 313-320 (2005).
[CrossRef] [PubMed]

Weaver, J.

B. Brooksby, S. Jiang, H. Dehghani, B. Pogue, K. Paulsen, J. Weaver, C. Kogel, and S. Poplack, "Combining near-infrared tomography resonance imaging to study in vivo and magnetic breast tissue: implementation of a Laplacian-type regularization to incorporate magnetic resonance structure," J. Biomed. Opt. 10 (2005).
[CrossRef] [PubMed]

Weissleder, R.

V. Ntziachristos, J. Ripoll, L. Wang, and R. Weissleder, "Looking and listening to light: the evolution of whole-body photonic imaging," Nat. Biotechnol. 23, 313-320 (2005).
[CrossRef] [PubMed]

E. Graves, J. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, "Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomograpy," J. Opt. Soc. Am. 21, 231-241 (2004).
[CrossRef]

V. Ntziachristos and R. Weissleder, "Experimental three-dimensional fluorescence reconstruction of diffuse media by use of a normalized Born approximation," Opt. Lett. 26, 893-895 (2001).
[CrossRef]

Xu, H.

Yazici, B.

M. Guven, B. Yazici, X. Intes, and B. Chance, "Diffuse optical tomography with a priori anatomical information," Phys. Med. Biol. 50, 2837-2858 (2005).
[CrossRef] [PubMed]

Yessayan, D.

J. Ripoll, D. Yessayan, G. Zacharakis, and V. Ntziachristos, "Experimental determination of photon propagation in highly absorbing and scattering media," J. Opt. Soc. Am. 22, 546-551 (2005).
[CrossRef]

Yodh, A.

Zaccanti, G.

Zacharakis, G.

J. Ripoll, D. Yessayan, G. Zacharakis, and V. Ntziachristos, "Experimental determination of photon propagation in highly absorbing and scattering media," J. Opt. Soc. Am. 22, 546-551 (2005).
[CrossRef]

Appl. Opt.

IEEE Trans. Med. Imaging

R. Schulz, J. Ripoll, and V. Ntziachristos, "Experimental Fluorescence Tomography of Tissues With Noncontact Measurements," IEEE Trans. Med. Imaging 23, 492-500 (2004).
[CrossRef] [PubMed]

Inverse Problems

S. Arridge, "Optical tomography in medical imaging," Inverse Problems 15, R41-R93 (1999).
[CrossRef]

J. Biomed. Opt.

B. Brooksby, S. Jiang, H. Dehghani, B. Pogue, K. Paulsen, J. Weaver, C. Kogel, and S. Poplack, "Combining near-infrared tomography resonance imaging to study in vivo and magnetic breast tissue: implementation of a Laplacian-type regularization to incorporate magnetic resonance structure," J. Biomed. Opt. 10 (2005).
[CrossRef] [PubMed]

J. Lumin.

R. Kubin and A. Fletcher, "Fluorescence Quantum Yields of Some Rhodamine Dyes," J. Lumin. 27, 455-462 (1982).
[CrossRef]

J. Opt. Soc. Am.

E. Graves, J. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, "Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomograpy," J. Opt. Soc. Am. 21, 231-241 (2004).
[CrossRef]

J. Ripoll, D. Yessayan, G. Zacharakis, and V. Ntziachristos, "Experimental determination of photon propagation in highly absorbing and scattering media," J. Opt. Soc. Am. 22, 546-551 (2005).
[CrossRef]

J. Swartling, A. Pifferi, A. Enejder, and S. Andersson-Engels, "Accelerated Monte Carlo model to simulate fluorescence spectra from layered tissues," J. Opt. Soc. Am. 20, 714-727 (2003).
[CrossRef]

Nat. Biotechnol.

V. Ntziachristos, J. Ripoll, L. Wang, and R. Weissleder, "Looking and listening to light: the evolution of whole-body photonic imaging," Nat. Biotechnol. 23, 313-320 (2005).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Med. Biol.

M. Guven, B. Yazici, X. Intes, and B. Chance, "Diffuse optical tomography with a priori anatomical information," Phys. Med. Biol. 50, 2837-2858 (2005).
[CrossRef] [PubMed]

Physiol. Measurement

H. Dehghani, D. Barber, and I. Basarab-Horwath, "Incorporating a priori anatomical information into image reconstruction in electrical impedance tomography," Physiol. Measurement 20, 87-102 (1999).
[CrossRef]

Siam Review

A. Neumaier, "Solving ill-conditioned and singular linear systems: A tutorial on regularization," Siam Review 40, 636-666 (1998).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

The experimental setup

Fig. 2.
Fig. 2.

Optical properties for the experimental tissue phantom obtained from integrating sphere measurements. In (a) absorption coefficient and (b) the reduced scattering coefficient. In (c) the normalized fluorescence spectrum for the fluorophore is shown.

Fig. 3.
Fig. 3.

UmR for detector (a) 5, (b) 15 and (c) 27 extracted for the same excitation source position. The source position (•) and detector position (x) is indicated. The true position of the cylinder is marked by the circular ring for reference. In (d) the summation of the minima arcs in (a)-(c) are shown schematically. The colouring indicate the detector each arc is originating from.

Fig. 4.
Fig. 4.

Regularization maps retrieved using the ratio of two spectral bands within the fluorescence emission profile. The fluorophore concentration (c) and position (z) is (a) c = 0.5 μM and z= 11 mm, (b) c =0.5 μM and z = 7 mm, (c) c = 1 μM and z=11 mm, (d) c=1 μM and z = 7 mm. A circular ring indicates the true inclusion position.

Fig. 5.
Fig. 5.

Reconstructions of a cylinder with a concentration of 1 μM positioned at z = 11 mm using (a) no prior and (b) spatially varying regularization map. In (c) and (d) the cross-sections indicated by dotted lines in (a) and (b) are shown. The full line (–) represents the reconstruction with prior information while the dashed line (––) represent reconstruction without prior. A circular ring indicates the true position of the inclusion.

Fig. 6.
Fig. 6.

Reconstructions of two cylinder with concentrations 1 μM (lower) and 0.5 μM (upper) positioned at z = 9 mm using (a) no prior and (c) spatially varying regularization map. In b) the regularization map is shown. The circular rings indicate the true position of the inclusions.

Equations (16)

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

u x r s r = P 0 D x G x r s r
G x r s r = exp ( μ effx r r s ) 4 π r r s .
D x , m = 1 3 ( μ s x , m ' + α μ a x , m )
μ eff x , m = μ a x , m D x , m
u m ( r s , r , r d ) = dV u x r s r × μ af ( r ) γ m D m G m ( r , r d ) .
U m r s r d = V u m ( r s , r , r d ) dV .
U nb r s r d = U m r s r d u x r s r d = 1 u x r s r d × V u x r s r G m r r d D m μ af ( r ) γ m dV .
U nb i r s i r d i j = 1 N voxels w i , j μ af j γ m
w i , j = Δ V j u x r s i r j u x r s i r d i G m r j r d i D m .
x j = μ af j γ m
X = ( W T W + β I ) 1 W T U nb .
u mR r r d = u m 1 ( r s , r , r d ) u m 2 ( r s , r , r d ) = D m 2 γ m 1 D m 1 γ m 2 G m 1 r r d G m 2 r r d .
Δ U mR d ( r j ) = In ( u mR r d r j ) In ( U ˜ mR ( r d ) ) .
Δ U mR ¯ ( r j ) = 1 N detectors d = 1 N det ectors Δ U mR d ( r j ) .
p j , j = In ( Δ U mR ¯ ( r j ) min ( Δ U mR ¯ ( r j ) ) ) + 1 .
γ m = Q e λ m 10 λ m + 10 γ ( λ ) 0 γ ( λ )

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