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.

© 2010 Optical Society of America

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References

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  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]
  2. H. Barrett, and K. Myers, Foundations of Image Science (John Wiley and Sons, 2004).
  3. A. K. Jorgensen, and G. L. Zeng, "SVD-based evaluation of multiplexing in multipinhole SPECT systems," Int. J. Biomed. Imaging 2008, 769195 (2008).
    [CrossRef]
  4. Y. L. Hsieh, G. L. Zeng, and G. T. 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]
  5. G. Zeng, and G. Gullberg, "An SVD study of truncated transmission data in SPECT," IEEE Trans. Nucl. Sci. 44(1), 107-111 (1997).
    [CrossRef]
  6. 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]
  7. 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]
  8. S. Park, J. M. Witten, and K. J. 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]
  9. M. Hamermesh, Group Theory and its Application to Physical Problems (Dover Publications, 1989).
  10. E. Anderson, Z. Bai, and C. Bischof, LAPACK Users’ Guide (Society for Industrial Mathematics, 1999).
    [CrossRef]
  11. J. Aarsvold, "Multiple-pinhole transaxial tomography: a model and analysis," Ph. D. Dissertation (University of Arizona, 1993).
  12. 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]
  13. 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.
  14. 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]
  15. E. Isaacson, and H. Keller, Analysis of Numerical Methods (Dover Publications, 1994).

2009 (3)

S. Park, J. M. Witten, and K. J. 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. L. Hsieh, G. L. Zeng, and G. T. 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. 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]

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]

Clarkson, E.

Gullberg, G.

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]

Gullberg, G. T.

Y. L. Hsieh, G. L. Zeng, and G. T. 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]

Hsieh, Y. L.

Y. L. Hsieh, G. L. Zeng, and G. T. 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]

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]

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. J.

S. Park, J. M. Witten, and K. J. 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]

Park, S.

S. Park, J. M. Witten, and K. J. 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. 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]

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]

Witten, J. M.

S. Park, J. M. Witten, and K. J. 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.

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]

Y. L. Hsieh, G. L. Zeng, and G. T. 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. Med. Imaging (2)

Y. L. Hsieh, G. L. Zeng, and G. T. 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]

S. Park, J. M. Witten, and K. J. 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]

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)

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)

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Figures (8)

Fig. 1
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.

Fig. 2
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.

Fig. 3
Fig. 3

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

Fig. 4
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).

Fig. 5
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.

Fig. 6
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.

Fig. 7
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.

Fig. 8
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.

Tables (2)

Tables Icon

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}

Tables Icon

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}

Equations (27)

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( g ¯ ) m = g ¯ m = ( f ) m = S h m * ( r ) f ( r ) d 3 r ,
g ¯ = [ g ¯ 0 g ¯ 1 g ¯ J 1 ] .
V = j = 0 J 1 V j .
( g ¯ j ) p = g ¯ j p = ( j f ) p = S 𝒯 j h p * ( r ) f ( r ) d 3 r .
j f = [ 0 0 g ¯ j 0 0 ] .
f = j = 0 J 1 j f ,
S 1 g ¯ = [ g ¯ J 1 g ¯ 0 g ¯ J 2 ] ,
( f , f ) U = S f * ( r ) f ( r ) d 3 r ,
( g , g ) V = m g m * g m .
( g ) ( r ) = m g m h m ( r ) .
g = j = 0 J 1 j g = j = 0 J 1 𝒯 j 0 S j g .
𝒬 k f = 1 J j = 0 J 1 exp ( i 2 π kj J ) 𝒯 j f
P k g = 1 J j = 0 J 1 exp ( i 2 π kj J ) S j g .
𝒯 j 𝒬 k f = exp ( i 2 π kj J ) 𝒬 k f
S j P k g = exp ( i 2 π kj J ) P k g .
f = k = 0 J 1 𝒬 k f
g = k = 0 J 1 P k g .
U = j = 0 J 1 U ˜ j
V = j = 0 J 1 V ˜ j .
= j = 0 J 1 ˜ j .
= j = 0 J 1 ˜ j .
= j = 0 J 1 ˜ j ˜ j .
˜ k ˜ k = 1 J j = 0 J 1 l 0 J 1 p = 0 J 1 exp ( i 2 π kj J ) S j S l 0 𝒯 l 𝒯 p H 0 S p .
˜ k ˜ k = 1 J j = 0 J 1 q = 0 J 1 p = 0 J 1 exp ( i 2 π kj J ) S j S p q 0 𝒯 q 0 S p .
˜ k ˜ k = 1 J r = 0 J 1 q = 0 J 1 p = 0 J 1 exp ( i 2 π k ( r p + q ) J ) S r 0 𝒯 q 0 S p .
g = [ g 0 0 0 ] .
u m u n = δ nm , n , m = 1 , 2 , N .

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