SVD for imaging systems with discrete rotational symmetry

Eric Clarkson; Robin Palit; Mathew A. Kupinski

doi:10.1364/OE.18.025306

Abstract

The singular value decomposition (SVD) of an imaging system is a computationally intensive calculation for tomographic imaging systems due to the large dimensionality of the system matrix. The computation often involves memory and storage requirements beyond those available to most end users. We have developed a method that reduces the dimension of the SVD problem towards the goal of making the calculation tractable for a standard desktop computer. In the presence of discrete rotational symmetry we show that the dimension of the SVD computation can be reduced by a factor equal to the number of collection angles for the tomographic system. In this paper we present the mathematical theory for our method, validate that our method produces the same results as standard SVD analysis, and finally apply our technique to the sensitivity matrix for a clinical CT system. The ability to compute the full singular value spectra and singular vectors could augment future work in system characterization, image-quality assessment and reconstruction techniques for tomographic imaging systems.

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References

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Publication

H. H. Barrett, J. N. Aarsvold, and T. J. Roney, “Null functions and eigenfunctions: tools for the analysis of imaging systems,” Prog. Clin. Biol. Res. 363, 211–226 (1991).
[PubMed]
H. Barrett and K. Myers, Foundations of Image Science (John Wiley and Sons, 2004).
A. K. Jorgensen and G. L. Zeng, “SVD-Based evaluation of multiplexing in multipinhole SPECT systems,” Int. J. Biomed. Imaging 2008, 769195 (2008).
[Crossref]
Y. Hsieh, G. Zeng, and G. Gullberg, “Projection space image reconstruction using strip functions to calculate pixels more natural for modeling the geometric response of the SPECT collimator,” IEEE Trans. Med. Imaging 17(1), 24–44 (1998).
[Crossref] [PubMed]
G. Zeng and G. Gullberg, “An SVD study of truncated transmission data in SPECT,” IEEE Trans. Nucl. Sci. 44(1), 107–111 (1997).
[Crossref]
G. Gullberg and G. Zeng, “A reconstruction algorithm using singular value decomposition of a discrete representation of the exponential radon transform using natural pixels,” IEEE Trans. Nucl. Sci. 41(6), 2812–2819 (1994).
[Crossref]
S. Park and E. Clarkson, “Efficient estimation of ideal-observer performance in classification tasks involving high-dimensional complex backgrounds,” J. Opt. Soc. Am. A 26(11), 59–71 (2009).
[Crossref]
S. Park, J. Witten, and K. Myers, “Singular vectors of a linear imaging system as efficient channels for the Bayesian ideal observer,” IEEE Trans. Med. Imaging 28(5), 657–668 (2009).
[Crossref] [PubMed]
M. Hamermesh, Group Theory and its Application to Physical Problems (Dover Publications, 1989).
E. Anderson, Z. Bai, and C. Bischof, LAPACK Users’ Guide (Society for Industrial Mathematics, 1999).
[Crossref]
J. Aarsvold, “Multiple-pinhole transaxial tomography: a model and analysis,” Ph. D. Dissertation (University of Arizona, 1993).
J. Aarsvold and H. Barrett, “Symmetries of single-slice multiple-pinhole tomographs,” in Conference Record of the 1996 IEEE NSS/MIC (IEEE, 1997), vol. 3, pp. 1673–1677.
[Crossref]
P. Varatharajah, B. Tankersley, and J. Aarsvold, “Discrete models and singular-value decompositions of single-slice imagers with orthogonal detectors,” in Conference Record of the 1998 IEEE NSS/MIC (IEEE, 1999), vol. 2, pp. 1184–1188.
S. Steckmann, M. Knaup, and M. Kachelrieß, “High performance cone-beam spiral backprojection with voxel-specific weighting,” Phys. Med. Biol. 54(12), 3691–3708 (2009).
[Crossref] [PubMed]
E. Isaacson and H. Keller, Analysis of Numerical Methods (Dover Publications, 1994).

2009 (3)

S. Park, J. Witten, and K. Myers, “Singular vectors of a linear imaging system as efficient channels for the Bayesian ideal observer,” IEEE Trans. Med. Imaging 28(5), 657–668 (2009).
[Crossref] [PubMed]

S. Steckmann, M. Knaup, and M. Kachelrieß, “High performance cone-beam spiral backprojection with voxel-specific weighting,” Phys. Med. Biol. 54(12), 3691–3708 (2009).
[Crossref] [PubMed]

S. Park and E. Clarkson, “Efficient estimation of ideal-observer performance in classification tasks involving high-dimensional complex backgrounds,” J. Opt. Soc. Am. A 26(11), 59–71 (2009).
[Crossref]

2008 (1)

A. K. Jorgensen and G. L. Zeng, “SVD-Based evaluation of multiplexing in multipinhole SPECT systems,” Int. J. Biomed. Imaging 2008, 769195 (2008).
[Crossref]

1998 (1)

Y. Hsieh, G. Zeng, and G. Gullberg, “Projection space image reconstruction using strip functions to calculate pixels more natural for modeling the geometric response of the SPECT collimator,” IEEE Trans. Med. Imaging 17(1), 24–44 (1998).
[Crossref] [PubMed]

1997 (1)

G. Zeng and G. Gullberg, “An SVD study of truncated transmission data in SPECT,” IEEE Trans. Nucl. Sci. 44(1), 107–111 (1997).
[Crossref]

1994 (1)

G. Gullberg and G. Zeng, “A reconstruction algorithm using singular value decomposition of a discrete representation of the exponential radon transform using natural pixels,” IEEE Trans. Nucl. Sci. 41(6), 2812–2819 (1994).
[Crossref]

1991 (1)

H. H. Barrett, J. N. Aarsvold, and T. J. Roney, “Null functions and eigenfunctions: tools for the analysis of imaging systems,” Prog. Clin. Biol. Res. 363, 211–226 (1991).
[PubMed]

Aarsvold, J.

J. Aarsvold, “Multiple-pinhole transaxial tomography: a model and analysis,” Ph. D. Dissertation (University of Arizona, 1993).

P. Varatharajah, B. Tankersley, and J. Aarsvold, “Discrete models and singular-value decompositions of single-slice imagers with orthogonal detectors,” in Conference Record of the 1998 IEEE NSS/MIC (IEEE, 1999), vol. 2, pp. 1184–1188.

J. Aarsvold and H. Barrett, “Symmetries of single-slice multiple-pinhole tomographs,” in Conference Record of the 1996 IEEE NSS/MIC (IEEE, 1997), vol. 3, pp. 1673–1677.
[Crossref]

Aarsvold, J. N.

H. H. Barrett, J. N. Aarsvold, and T. J. Roney, “Null functions and eigenfunctions: tools for the analysis of imaging systems,” Prog. Clin. Biol. Res. 363, 211–226 (1991).
[PubMed]

Anderson, E.

E. Anderson, Z. Bai, and C. Bischof, LAPACK Users’ Guide (Society for Industrial Mathematics, 1999).
[Crossref]

Bai, Z.

E. Anderson, Z. Bai, and C. Bischof, LAPACK Users’ Guide (Society for Industrial Mathematics, 1999).
[Crossref]

Barrett, H.

J. Aarsvold and H. Barrett, “Symmetries of single-slice multiple-pinhole tomographs,” in Conference Record of the 1996 IEEE NSS/MIC (IEEE, 1997), vol. 3, pp. 1673–1677.
[Crossref]

H. Barrett and K. Myers, Foundations of Image Science (John Wiley and Sons, 2004).

Barrett, H. H.

H. H. Barrett, J. N. Aarsvold, and T. J. Roney, “Null functions and eigenfunctions: tools for the analysis of imaging systems,” Prog. Clin. Biol. Res. 363, 211–226 (1991).
[PubMed]

Bischof, C.

E. Anderson, Z. Bai, and C. Bischof, LAPACK Users’ Guide (Society for Industrial Mathematics, 1999).
[Crossref]

Clarkson, E.

S. Park and E. Clarkson, “Efficient estimation of ideal-observer performance in classification tasks involving high-dimensional complex backgrounds,” J. Opt. Soc. Am. A 26(11), 59–71 (2009).
[Crossref]

Gullberg, G.

Y. Hsieh, G. Zeng, and G. Gullberg, “Projection space image reconstruction using strip functions to calculate pixels more natural for modeling the geometric response of the SPECT collimator,” IEEE Trans. Med. Imaging 17(1), 24–44 (1998).
[Crossref] [PubMed]

G. Zeng and G. Gullberg, “An SVD study of truncated transmission data in SPECT,” IEEE Trans. Nucl. Sci. 44(1), 107–111 (1997).
[Crossref]

G. Gullberg and G. Zeng, “A reconstruction algorithm using singular value decomposition of a discrete representation of the exponential radon transform using natural pixels,” IEEE Trans. Nucl. Sci. 41(6), 2812–2819 (1994).
[Crossref]

Hamermesh, M.

M. Hamermesh, Group Theory and its Application to Physical Problems (Dover Publications, 1989).

Hsieh, Y.

Y. Hsieh, G. Zeng, and G. Gullberg, “Projection space image reconstruction using strip functions to calculate pixels more natural for modeling the geometric response of the SPECT collimator,” IEEE Trans. Med. Imaging 17(1), 24–44 (1998).
[Crossref] [PubMed]

Isaacson, E.

E. Isaacson and H. Keller, Analysis of Numerical Methods (Dover Publications, 1994).

Jorgensen, A. K.

A. K. Jorgensen and G. L. Zeng, “SVD-Based evaluation of multiplexing in multipinhole SPECT systems,” Int. J. Biomed. Imaging 2008, 769195 (2008).
[Crossref]

Kachelrieß, M.

S. Steckmann, M. Knaup, and M. Kachelrieß, “High performance cone-beam spiral backprojection with voxel-specific weighting,” Phys. Med. Biol. 54(12), 3691–3708 (2009).
[Crossref] [PubMed]

Keller, H.

E. Isaacson and H. Keller, Analysis of Numerical Methods (Dover Publications, 1994).

Knaup, M.

S. Steckmann, M. Knaup, and M. Kachelrieß, “High performance cone-beam spiral backprojection with voxel-specific weighting,” Phys. Med. Biol. 54(12), 3691–3708 (2009).
[Crossref] [PubMed]

Myers, K.

S. Park, J. Witten, and K. Myers, “Singular vectors of a linear imaging system as efficient channels for the Bayesian ideal observer,” IEEE Trans. Med. Imaging 28(5), 657–668 (2009).
[Crossref] [PubMed]

H. Barrett and K. Myers, Foundations of Image Science (John Wiley and Sons, 2004).

Park, S.

S. Park and E. Clarkson, “Efficient estimation of ideal-observer performance in classification tasks involving high-dimensional complex backgrounds,” J. Opt. Soc. Am. A 26(11), 59–71 (2009).
[Crossref]

S. Park, J. Witten, and K. Myers, “Singular vectors of a linear imaging system as efficient channels for the Bayesian ideal observer,” IEEE Trans. Med. Imaging 28(5), 657–668 (2009).
[Crossref] [PubMed]

Roney, T. J.

H. H. Barrett, J. N. Aarsvold, and T. J. Roney, “Null functions and eigenfunctions: tools for the analysis of imaging systems,” Prog. Clin. Biol. Res. 363, 211–226 (1991).
[PubMed]

Steckmann, S.

S. Steckmann, M. Knaup, and M. Kachelrieß, “High performance cone-beam spiral backprojection with voxel-specific weighting,” Phys. Med. Biol. 54(12), 3691–3708 (2009).
[Crossref] [PubMed]

Tankersley, B.

P. Varatharajah, B. Tankersley, and J. Aarsvold, “Discrete models and singular-value decompositions of single-slice imagers with orthogonal detectors,” in Conference Record of the 1998 IEEE NSS/MIC (IEEE, 1999), vol. 2, pp. 1184–1188.

Varatharajah, P.

P. Varatharajah, B. Tankersley, and J. Aarsvold, “Discrete models and singular-value decompositions of single-slice imagers with orthogonal detectors,” in Conference Record of the 1998 IEEE NSS/MIC (IEEE, 1999), vol. 2, pp. 1184–1188.

Witten, J.

S. Park, J. Witten, and K. Myers, “Singular vectors of a linear imaging system as efficient channels for the Bayesian ideal observer,” IEEE Trans. Med. Imaging 28(5), 657–668 (2009).
[Crossref] [PubMed]

Zeng, G.

Y. Hsieh, G. Zeng, and G. Gullberg, “Projection space image reconstruction using strip functions to calculate pixels more natural for modeling the geometric response of the SPECT collimator,” IEEE Trans. Med. Imaging 17(1), 24–44 (1998).
[Crossref] [PubMed]

G. Zeng and G. Gullberg, “An SVD study of truncated transmission data in SPECT,” IEEE Trans. Nucl. Sci. 44(1), 107–111 (1997).
[Crossref]

G. Gullberg and G. Zeng, “A reconstruction algorithm using singular value decomposition of a discrete representation of the exponential radon transform using natural pixels,” IEEE Trans. Nucl. Sci. 41(6), 2812–2819 (1994).
[Crossref]

Zeng, G. L.

A. K. Jorgensen and G. L. Zeng, “SVD-Based evaluation of multiplexing in multipinhole SPECT systems,” Int. J. Biomed. Imaging 2008, 769195 (2008).
[Crossref]

IEEE Trans. Med. Imaging (2)

S. Park, J. Witten, and K. Myers, “Singular vectors of a linear imaging system as efficient channels for the Bayesian ideal observer,” IEEE Trans. Med. Imaging 28(5), 657–668 (2009).
[Crossref] [PubMed]

Y. Hsieh, G. Zeng, and G. Gullberg, “Projection space image reconstruction using strip functions to calculate pixels more natural for modeling the geometric response of the SPECT collimator,” IEEE Trans. Med. Imaging 17(1), 24–44 (1998).
[Crossref] [PubMed]

IEEE Trans. Nucl. Sci. (2)

G. Zeng and G. Gullberg, “An SVD study of truncated transmission data in SPECT,” IEEE Trans. Nucl. Sci. 44(1), 107–111 (1997).
[Crossref]

G. Gullberg and G. Zeng, “A reconstruction algorithm using singular value decomposition of a discrete representation of the exponential radon transform using natural pixels,” IEEE Trans. Nucl. Sci. 41(6), 2812–2819 (1994).
[Crossref]

Int. J. Biomed. Imaging (1)

A. K. Jorgensen and G. L. Zeng, “SVD-Based evaluation of multiplexing in multipinhole SPECT systems,” Int. J. Biomed. Imaging 2008, 769195 (2008).
[Crossref]

J. Opt. Soc. Am. A (1)

S. Park and E. Clarkson, “Efficient estimation of ideal-observer performance in classification tasks involving high-dimensional complex backgrounds,” J. Opt. Soc. Am. A 26(11), 59–71 (2009).
[Crossref]

Phys. Med. Biol. (1)

S. Steckmann, M. Knaup, and M. Kachelrieß, “High performance cone-beam spiral backprojection with voxel-specific weighting,” Phys. Med. Biol. 54(12), 3691–3708 (2009).
[Crossref] [PubMed]

Prog. Clin. Biol. Res. (1)

H. H. Barrett, J. N. Aarsvold, and T. J. Roney, “Null functions and eigenfunctions: tools for the analysis of imaging systems,” Prog. Clin. Biol. Res. 363, 211–226 (1991).
[PubMed]

Other (7)

H. Barrett and K. Myers, Foundations of Image Science (John Wiley and Sons, 2004).

E. Isaacson and H. Keller, Analysis of Numerical Methods (Dover Publications, 1994).

M. Hamermesh, Group Theory and its Application to Physical Problems (Dover Publications, 1989).

E. Anderson, Z. Bai, and C. Bischof, LAPACK Users’ Guide (Society for Industrial Mathematics, 1999).
[Crossref]

J. Aarsvold, “Multiple-pinhole transaxial tomography: a model and analysis,” Ph. D. Dissertation (University of Arizona, 1993).

J. Aarsvold and H. Barrett, “Symmetries of single-slice multiple-pinhole tomographs,” in Conference Record of the 1996 IEEE NSS/MIC (IEEE, 1997), vol. 3, pp. 1673–1677.
[Crossref]

P. Varatharajah, B. Tankersley, and J. Aarsvold, “Discrete models and singular-value decompositions of single-slice imagers with orthogonal detectors,” in Conference Record of the 1998 IEEE NSS/MIC (IEEE, 1999), vol. 2, pp. 1184–1188.

Supplementary Material (10)

» Media 1: MOV (40 KB)

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» Media 3: MOV (57 KB)

» Media 4: MOV (66 KB)

» Media 5: MOV (53 KB)

» Media 6: MOV (56 KB)

» Media 7: MOV (71 KB)

» Media 8: MOV (59 KB)

» Media 9: MOV (70 KB)

» Media 10: MOV (59 KB)

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Fig. 1 Plots showing the largest 16 singular values computed using the standard dimension SVD and the reduced dimension SVD. The spectra output by each SVD technique is equivalent.

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Fig. 2 Images of slices through the central plane of the object space vectors associated with the 10 largest singular values for (a) the standard dimension SVD and (b) & (c) the reduced dimension SVD.

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Fig. 3 The difference between singular vectors output by the Standard SVD and Reduced SVD algorithms.

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Fig. 4 An illustration of the geometry of a simulated clinical x-ray CT system for which we computed the SVD using the reduced dimension SVD algorithm. This system specifications were chosen to roughly approximate the Siemens Sensation 64 scanner ( Media 1, Media 2, Media 3, Media 4, Media 5, Media 6, Media 7, Media 8, Media 9, and Media 10).

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Fig. 5 A plot showing the singular value spectra for the simulated Siemens x-ray CT system. These data were computed using the reduced dimension SVD analysis.

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Fig. 6 Images of slices through the central plane of the object space vectors for the modeled Siemens x-ray CT systems associated with singular value indices 1–10.

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Fig. 7 Images of slices through the central plane of the object space vectors for the modeled Siemens x-ray CT systems associated with singular value indices 88–92.

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Fig. 8 Images of slices through the central plane of the object space vectors for the modeled Siemens x-ray CT systems associated with singular value indices 175–179.

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Tables (2)

Table 1 The magnitudes of the inner products of singular vectors that were computed using the reduced dimension SVD. Shown in this table are $| u_{m}^{†} u_{n} |$ for m = {1, 2,...,8} and n = {1, 2,...,16}

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Table 2 The magnitudes of the inner products of singular vectors that were computed using the reduced dimension SVD. Shown in this table are $| u_{m}^{†} u_{n} |$ for m = {9, 10,...,16} and n = {1, 2,...,16}

View Table | View all tables in this article

Equations (27)

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{(\bar{g})}_{m} = {\bar{g}}_{m} = {(ℋ f)}_{m} = \int_{S} h_{m}^{*} (r) f (r) d^{3} r,

\bar{g} = [\begin{matrix} {\bar{g}}_{0} \\ {\bar{g}}_{1} \\ ⋮ \\ {\bar{g}}_{J - 1} \end{matrix}] .

V = \oplus_{j = 0}^{J - 1} V_{j} .

{({\bar{g}}_{j})}_{p} = {\bar{g}}_{j p} = {(ℋ_{j} f)}_{p} = \int_{S} 𝒯_{j} h_{p}^{*} (r) f (r) d^{3} r .

ℋ_{j} f = [\begin{matrix} 0 \\ ⋮ \\ 0 \\ {\bar{g}}_{j} \\ 0 \\ ⋮ \\ 0 \end{matrix}] .

ℋ_{f} = \sum_{j = 0}^{J - 1} ℋ_{j} f,

S_{1} \bar{g} = [\begin{matrix} {\bar{g}}_{J - 1} \\ {\bar{g}}_{0} \\ ⋮ \\ {\bar{g}}_{J - 2} \end{matrix}],

{(f, f^{'})}_{U} = \int_{S} f^{*} (r) f^{'} (r) d^{3} r,

{(g, g^{'})}_{V} = \sum_{m} g_{m}^{*} {g^{'}}_{m} .

(ℋ^{†} g) (r) = \sum_{m} g_{m} h_{m} (r) .

ℋ^{†} g = \sum_{j = 0}^{J - 1} ℋ_{j}^{†} g = \sum_{j = 0}^{J - 1} 𝒯_{j} ℋ_{0}^{†} S_{- j} g .

𝒬_{k} f = \frac{1}{J} \sum_{j = 0}^{J - 1} exp (\frac{i 2 π kj}{J}) 𝒯_{j} f

P_{k} g = \frac{1}{J} \sum_{j = 0}^{J - 1} exp (\frac{i 2 π kj}{J}) S_{j} g .

𝒯_{j} 𝒬_{k} f = exp (- \frac{i 2 π kj}{J}) 𝒬_{k} f

S_{j} P_{k} g = exp (- \frac{i 2 π kj}{J}) P_{k} g .

f = \sum_{k = 0}^{J - 1} 𝒬_{k} f

g = \sum_{k = 0}^{J - 1} P_{k} g .

U = \oplus_{j = 0}^{J - 1} {\tilde{U}}_{j}

V = \oplus_{j = 0}^{J - 1} {\tilde{V}}_{j} .

ℋ = \sum_{j = 0}^{J - 1} {\tilde{ℋ}}_{j} .

ℋ^{†} = \sum_{j = 0}^{J - 1} {\tilde{ℋ}}_{j}^{†} .

ℋ ℋ^{†} = \sum_{j = 0}^{J - 1} {\tilde{ℋ}}_{j} {\tilde{ℋ}}_{j}^{†} .

{\tilde{ℋ}}_{k} {\tilde{ℋ}}_{k}^{†} = \frac{1}{J} \sum_{j = 0}^{J - 1} \sum_{l - 0}^{J - 1} \sum_{p = 0}^{J - 1} exp (\frac{i 2 π kj}{J}) S_{j} S_{l} ℋ_{0} 𝒯_{- l} 𝒯_{p} H_{0}^{†} S_{- p} .

{\tilde{ℋ}}_{k} {\tilde{ℋ}}_{k}^{†} = \frac{1}{J} \sum_{j = 0}^{J - 1} \sum_{q = 0}^{J - 1} \sum_{p = 0}^{J - 1} exp (\frac{i 2 π kj}{J}) S_{j} S_{p - q} ℋ_{0} 𝒯_{q} ℋ_{0}^{†} S_{- p} .

{\tilde{ℋ}}_{k} {\tilde{ℋ}}_{k}^{†} = \frac{1}{J} \sum_{r = 0}^{J - 1} \sum_{q = 0}^{J - 1} \sum_{p = 0}^{J - 1} exp (\frac{i 2 π k (r - p + q)}{J}) S_{r} ℋ_{0} 𝒯_{q} ℋ_{0}^{†} S_{- p} .

g = [\begin{matrix} g_{0} \\ 0 \\ ⋮ \\ 0 \end{matrix}] .

u_{m}^{†} u_{n} = δ_{nm}, n, m = 1, 2, \dots N .

	$u_{1}^{†}$	$u_{2}^{†}$	$u_{3}^{†}$	$u_{4}^{†}$	$u_{5}^{†}$	$u_{6}^{†}$	$u_{7}^{†}$	$u_{8}^{†}$

u₁	1.0E+0	2.7E-3	2.7E-3	3.9E-4	3.9E-4	1.2E-15	5.3E-5	5.3E-5
u₂	2.7E-3	1.0E+0	1.6E-3	1.8E-5	2.8E-3	2.8E-3	4.3E-4	6.3E-4
u₃	2.7E-3	1.6E-3	1.0E+0	2.8E-3	1.8E-5	2.8E-3	6.3E-4	4.3E-4
u₄	3.9E-4	1.8E-5	2.8E-3	1.0E+0	2.8E-4	1.5E-3	2.9E-3	2.9E-5
u₅	3.9E-4	2.8E-3	1.8E-5	2.8E-4	1.0E+0	1.5E-3	2.9E-5	2.9E-3
u₆	1.2E-15	2.8E-3	2.8E-3	1.5E-3	1.5E-3	1.0E+0	8.5E-6	8.5E-6
u₇	5.3E-5	4.3E-4	6.3E-4	2.9E-3	2.9E-5	8.5E-6	1.0E+0	1.5E-3
u₈	5.3E-5	6.3E-4	4.3E-4	2.9E-5	2.9E-3	8.5E-6	1.5E-3	1.0E+0
u₉	2.8E-3	4.2E-15	8.1E-5	4.7E-5	2.9E-3	2.9E-3	2.3E-4	1.4E-3
u₁₀	2.8E-3	8.1E-5	2.4E-15	2.9E-3	4.7E-5	2.9E-3	1.4E-3	2.3E-4
u₁₁	1.5E-4	1.7E-5	3.2E-5	6.6E-4	7.2E-4	4.7E-4	2.9E-3	6.0E-5
u₁₂	1.5E-4	3.2E-5	1.7E-5	7.2E-4	6.6E-4	4.7E-4	6.0E-5	2.9E-3
u₁₃	2.9E-4	2.9E-3	1.5E-5	2.0E-4	4.7E-15	1.6E-4	2.9E-5	3.0E-3
u₁₄	2.9E-4	1.5E-5	2.9E-3	1.7E-15	2.0E-4	1.6E-4	3.0E-3	2.9E-5
u₁₅	4.1E-16	2.9E-3	2.9E-3	1.6E-4	1.6E-4	1.2E-15	5.4E-5	5.4E-5
u₁₆	1.7E-5	3.0E-5	1.8E-4	2.6E-5	5.8E-5	2.1E-5	7.3E-4	1.2E-3

	$u_{9}^{†}$	$u_{10}^{†}$	$u_{11}^{†}$	$u_{12}^{†}$	$u_{13}^{†}$	$u_{14}^{†}$	$u_{15}^{†}$	$u_{16}^{†}$

u₁	2.8E-3	2.8E-3	1.5E-4	1.5E-4	2.9E-4	2.9E-4	4.1E-16	1.7E-5
u₂	4.2E-15	8.1E-5	1.7E-5	3.2E-5	2.9E-3	1.5E-5	2.9E-3	3.0E-5
u₃	8.1E-5	2.4E-15	3.2E-5	1.7E-5	1.5E-5	2.9E-3	2.9E-3	1.8E-4
u₄	4.7E-5	2.9E-3	6.6E-4	7.2E-4	2.0E-4	1.7E-15	1.6E-4	2.6E-5
u₅	2.9E-3	4.7E-5	7.2E-4	6.6E-4	4.7E-15	2.0E-4	1.6E-4	5.8E-5
u₆	2.9E-3	2.9E-3	4.7E-4	4.7E-4	1.6E-4	1.6E-4	1.2E-15	2.1E-5
u₇	2.3E-4	1.4E-3	2.9E-3	6.0E-5	2.9E-5	3.0E-3	5.4E-5	7.3E-4
u₈	1.4E-3	2.3E-4	6.0E-5	2.9E-3	3.0E-3	2.9E-5	5.4E-5	1.2E-3
u₉	1.0E+0	1.6E-3	5.8E-6	1.4E-5	3.1E-3	3.1E-5	3.0E-3	9.6E-4
u₁₀	1.6E-3	1.0E+0	1.4E-5	5.8E-6	3.1E-5	3.1E-3	3.0E-3	5.1E-4
u₁₁	5.8E-6	1.4E-5	1.0E+0	9.1E-4	1.3E-3	1.4E-3	2.2E-4	3.0E-3
u₁₂	1.4E-5	5.8E-6	9.1E-4	1.0E+0	1.4E-3	1.3E-3	2.2E-4	2.6E-5
u₁₃	3.1E-3	3.1E-5	1.3E-3	1.4E-3	1.0E+0	2.3E-4	1.6E-3	1.1E-4
u₁₄	3.1E-5	3.1E-3	1.4E-3	1.3E-3	2.3E-4	1.0E+0	1.6E-3	3.3E-5
u₁₅	3.0E-3	3.0E-3	2.2E-4	2.2E-4	1.6E-3	1.6E-3	1.0E+0	9.5E-5
u₁₆	9.6E-4	5.1E-4	3.0E-3	2.6E-5	1.1E-4	3.3E-5	9.5E-5	1.0E+0

	$u_{1}^{†}$	$u_{2}^{†}$	$u_{3}^{†}$	$u_{4}^{†}$	$u_{5}^{†}$	$u_{6}^{†}$	$u_{7}^{†}$	$u_{8}^{†}$

u₁	1.0E+0	2.7E-3	2.7E-3	3.9E-4	3.9E-4	1.2E-15	5.3E-5	5.3E-5
u₂	2.7E-3	1.0E+0	1.6E-3	1.8E-5	2.8E-3	2.8E-3	4.3E-4	6.3E-4
u₃	2.7E-3	1.6E-3	1.0E+0	2.8E-3	1.8E-5	2.8E-3	6.3E-4	4.3E-4
u₄	3.9E-4	1.8E-5	2.8E-3	1.0E+0	2.8E-4	1.5E-3	2.9E-3	2.9E-5
u₅	3.9E-4	2.8E-3	1.8E-5	2.8E-4	1.0E+0	1.5E-3	2.9E-5	2.9E-3
u₆	1.2E-15	2.8E-3	2.8E-3	1.5E-3	1.5E-3	1.0E+0	8.5E-6	8.5E-6
u₇	5.3E-5	4.3E-4	6.3E-4	2.9E-3	2.9E-5	8.5E-6	1.0E+0	1.5E-3
u₈	5.3E-5	6.3E-4	4.3E-4	2.9E-5	2.9E-3	8.5E-6	1.5E-3	1.0E+0
u₉	2.8E-3	4.2E-15	8.1E-5	4.7E-5	2.9E-3	2.9E-3	2.3E-4	1.4E-3
u₁₀	2.8E-3	8.1E-5	2.4E-15	2.9E-3	4.7E-5	2.9E-3	1.4E-3	2.3E-4
u₁₁	1.5E-4	1.7E-5	3.2E-5	6.6E-4	7.2E-4	4.7E-4	2.9E-3	6.0E-5
u₁₂	1.5E-4	3.2E-5	1.7E-5	7.2E-4	6.6E-4	4.7E-4	6.0E-5	2.9E-3
u₁₃	2.9E-4	2.9E-3	1.5E-5	2.0E-4	4.7E-15	1.6E-4	2.9E-5	3.0E-3
u₁₄	2.9E-4	1.5E-5	2.9E-3	1.7E-15	2.0E-4	1.6E-4	3.0E-3	2.9E-5
u₁₅	4.1E-16	2.9E-3	2.9E-3	1.6E-4	1.6E-4	1.2E-15	5.4E-5	5.4E-5
u₁₆	1.7E-5	3.0E-5	1.8E-4	2.6E-5	5.8E-5	2.1E-5	7.3E-4	1.2E-3

	$u_{9}^{†}$	$u_{10}^{†}$	$u_{11}^{†}$	$u_{12}^{†}$	$u_{13}^{†}$	$u_{14}^{†}$	$u_{15}^{†}$	$u_{16}^{†}$

u₁	2.8E-3	2.8E-3	1.5E-4	1.5E-4	2.9E-4	2.9E-4	4.1E-16	1.7E-5
u₂	4.2E-15	8.1E-5	1.7E-5	3.2E-5	2.9E-3	1.5E-5	2.9E-3	3.0E-5
u₃	8.1E-5	2.4E-15	3.2E-5	1.7E-5	1.5E-5	2.9E-3	2.9E-3	1.8E-4
u₄	4.7E-5	2.9E-3	6.6E-4	7.2E-4	2.0E-4	1.7E-15	1.6E-4	2.6E-5
u₅	2.9E-3	4.7E-5	7.2E-4	6.6E-4	4.7E-15	2.0E-4	1.6E-4	5.8E-5
u₆	2.9E-3	2.9E-3	4.7E-4	4.7E-4	1.6E-4	1.6E-4	1.2E-15	2.1E-5
u₇	2.3E-4	1.4E-3	2.9E-3	6.0E-5	2.9E-5	3.0E-3	5.4E-5	7.3E-4
u₈	1.4E-3	2.3E-4	6.0E-5	2.9E-3	3.0E-3	2.9E-5	5.4E-5	1.2E-3
u₉	1.0E+0	1.6E-3	5.8E-6	1.4E-5	3.1E-3	3.1E-5	3.0E-3	9.6E-4
u₁₀	1.6E-3	1.0E+0	1.4E-5	5.8E-6	3.1E-5	3.1E-3	3.0E-3	5.1E-4
u₁₁	5.8E-6	1.4E-5	1.0E+0	9.1E-4	1.3E-3	1.4E-3	2.2E-4	3.0E-3
u₁₂	1.4E-5	5.8E-6	9.1E-4	1.0E+0	1.4E-3	1.3E-3	2.2E-4	2.6E-5
u₁₃	3.1E-3	3.1E-5	1.3E-3	1.4E-3	1.0E+0	2.3E-4	1.6E-3	1.1E-4
u₁₄	3.1E-5	3.1E-3	1.4E-3	1.3E-3	2.3E-4	1.0E+0	1.6E-3	3.3E-5
u₁₅	3.0E-3	3.0E-3	2.2E-4	2.2E-4	1.6E-3	1.6E-3	1.0E+0	9.5E-5
u₁₆	9.6E-4	5.1E-4	3.0E-3	2.6E-5	1.1E-4	3.3E-5	9.5E-5	1.0E+0

Abstract

References

2009 (3)

2008 (1)

1998 (1)

1997 (1)

1994 (1)

1991 (1)

Aarsvold, J.

Aarsvold, J. N.

Anderson, E.

Bai, Z.

Barrett, H.

Barrett, H. H.

Bischof, C.

Clarkson, E.

Gullberg, G.

Hamermesh, M.

Hsieh, Y.

Isaacson, E.

Jorgensen, A. K.

Kachelrieß, M.

Keller, H.

Knaup, M.

Myers, K.

Park, S.

Roney, T. J.

Steckmann, S.

Tankersley, B.

Varatharajah, P.

Witten, J.

Zeng, G.

Zeng, G. L.

IEEE Trans. Med. Imaging (2)

IEEE Trans. Nucl. Sci. (2)

Int. J. Biomed. Imaging (1)

J. Opt. Soc. Am. A (1)

Phys. Med. Biol. (1)

Prog. Clin. Biol. Res. (1)

Other (7)

Supplementary Material (10)

Cited By

Figures (8)

Tables (2)

Equations (27)

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