Abstract

We derive and discuss an algorithm for separating measurement-and null-space components of an object with respect to a given imaging system. This algorithm requires a discrete-to-discrete approximation of a typically continuous-to-discrete imaging system, and problems associated with such an approximation are examined. Situations where knowledge of the measurement and null spaces is crucial for analyzing imaging systems are discussed. We conclude with two examples demonstrating the usefulness of this algorithm, even in the discrete-to-discrete approximation.

© 1998 Optical Society of America

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References

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  1. D.R. Gilland, B.M.W. Tsui, C.E. Metz, R.J. Jaszczak, and J.R. Perry, “An evaluation of maximum likelihood-EM reconstruction for SPECT by ROC analysis,” J. Nucl. Med. 33, 451–457 (1992).
    [PubMed]
  2. X-L. Xu, J-S. Liow, and S.C. Strother, “Iterative algebraic reconstruction algorithms for emission computed tomography: A unified framework and its application to positron emission tomography,” Med. Phys.,  20, 1675–1684 (1993).
    [CrossRef] [PubMed]
  3. S.C. Moore, S.P. Kijewski, S.P. Muller, and B.L. Holman, “SPECT Image Noise Power: Effects of Nonstationary Projection Noise and Attenuation Compensation,” J. Nucl. Med. 29, 1704–1709, (1988).
    [PubMed]
  4. H.H. Barrett, D.W. Wilson, and B.M.W. Tsui, “Noise properties of the EM algorithm: I. Theory,” Phys. Med. Biol. 39, 833–846 (1994).
    [CrossRef] [PubMed]
  5. O.N. Strand, “Theory and methods related to the singular-function expansion and Landweber’s iteration for integral equations of the first kind,” SIAM J. Numer. Anal.,  11, 798–825 (1974).
    [CrossRef]
  6. K.M. Hanson, “Bayesian and Related Methods in Image Reconstruction from Incomplete Data,” in Image Recovery, Theory and Application, Henry Stark, ed. (Academic, Orlando, Fla. 1987)
  7. H.H. Barrett and D.W. Wilson, “An algorithm to calculate null and measurement spaces,” http://www.radiology.arizona.edu/~fastspec/nul_spc.pdf
  8. P. Gilbert, “Iterative Methods for the Three-Dimensional Reconstruction of an Object from Projections,” J. Theor. Biol. 36, 105–117 (1972).
    [CrossRef] [PubMed]
  9. H.H. Barrett, “The FASTSPECT imaging system,” http://www.radiology.arizona.edu/~fastspec.
  10. I. Pang, “Wedge phantom for resolution measurement,” http://www.radiology.arizona.edu/~fastspec/wedge.htm.

1994 (1)

H.H. Barrett, D.W. Wilson, and B.M.W. Tsui, “Noise properties of the EM algorithm: I. Theory,” Phys. Med. Biol. 39, 833–846 (1994).
[CrossRef] [PubMed]

1993 (1)

X-L. Xu, J-S. Liow, and S.C. Strother, “Iterative algebraic reconstruction algorithms for emission computed tomography: A unified framework and its application to positron emission tomography,” Med. Phys.,  20, 1675–1684 (1993).
[CrossRef] [PubMed]

1992 (1)

D.R. Gilland, B.M.W. Tsui, C.E. Metz, R.J. Jaszczak, and J.R. Perry, “An evaluation of maximum likelihood-EM reconstruction for SPECT by ROC analysis,” J. Nucl. Med. 33, 451–457 (1992).
[PubMed]

1988 (1)

S.C. Moore, S.P. Kijewski, S.P. Muller, and B.L. Holman, “SPECT Image Noise Power: Effects of Nonstationary Projection Noise and Attenuation Compensation,” J. Nucl. Med. 29, 1704–1709, (1988).
[PubMed]

1974 (1)

O.N. Strand, “Theory and methods related to the singular-function expansion and Landweber’s iteration for integral equations of the first kind,” SIAM J. Numer. Anal.,  11, 798–825 (1974).
[CrossRef]

1972 (1)

P. Gilbert, “Iterative Methods for the Three-Dimensional Reconstruction of an Object from Projections,” J. Theor. Biol. 36, 105–117 (1972).
[CrossRef] [PubMed]

Barrett, H.H.

H.H. Barrett, D.W. Wilson, and B.M.W. Tsui, “Noise properties of the EM algorithm: I. Theory,” Phys. Med. Biol. 39, 833–846 (1994).
[CrossRef] [PubMed]

H.H. Barrett and D.W. Wilson, “An algorithm to calculate null and measurement spaces,” http://www.radiology.arizona.edu/~fastspec/nul_spc.pdf

H.H. Barrett, “The FASTSPECT imaging system,” http://www.radiology.arizona.edu/~fastspec.

Gilbert, P.

P. Gilbert, “Iterative Methods for the Three-Dimensional Reconstruction of an Object from Projections,” J. Theor. Biol. 36, 105–117 (1972).
[CrossRef] [PubMed]

Gilland, D.R.

D.R. Gilland, B.M.W. Tsui, C.E. Metz, R.J. Jaszczak, and J.R. Perry, “An evaluation of maximum likelihood-EM reconstruction for SPECT by ROC analysis,” J. Nucl. Med. 33, 451–457 (1992).
[PubMed]

Hanson, K.M.

K.M. Hanson, “Bayesian and Related Methods in Image Reconstruction from Incomplete Data,” in Image Recovery, Theory and Application, Henry Stark, ed. (Academic, Orlando, Fla. 1987)

Holman, B.L.

S.C. Moore, S.P. Kijewski, S.P. Muller, and B.L. Holman, “SPECT Image Noise Power: Effects of Nonstationary Projection Noise and Attenuation Compensation,” J. Nucl. Med. 29, 1704–1709, (1988).
[PubMed]

Jaszczak, R.J.

D.R. Gilland, B.M.W. Tsui, C.E. Metz, R.J. Jaszczak, and J.R. Perry, “An evaluation of maximum likelihood-EM reconstruction for SPECT by ROC analysis,” J. Nucl. Med. 33, 451–457 (1992).
[PubMed]

Kijewski, S.P.

S.C. Moore, S.P. Kijewski, S.P. Muller, and B.L. Holman, “SPECT Image Noise Power: Effects of Nonstationary Projection Noise and Attenuation Compensation,” J. Nucl. Med. 29, 1704–1709, (1988).
[PubMed]

Liow, J-S.

X-L. Xu, J-S. Liow, and S.C. Strother, “Iterative algebraic reconstruction algorithms for emission computed tomography: A unified framework and its application to positron emission tomography,” Med. Phys.,  20, 1675–1684 (1993).
[CrossRef] [PubMed]

Metz, C.E.

D.R. Gilland, B.M.W. Tsui, C.E. Metz, R.J. Jaszczak, and J.R. Perry, “An evaluation of maximum likelihood-EM reconstruction for SPECT by ROC analysis,” J. Nucl. Med. 33, 451–457 (1992).
[PubMed]

Moore, S.C.

S.C. Moore, S.P. Kijewski, S.P. Muller, and B.L. Holman, “SPECT Image Noise Power: Effects of Nonstationary Projection Noise and Attenuation Compensation,” J. Nucl. Med. 29, 1704–1709, (1988).
[PubMed]

Muller, S.P.

S.C. Moore, S.P. Kijewski, S.P. Muller, and B.L. Holman, “SPECT Image Noise Power: Effects of Nonstationary Projection Noise and Attenuation Compensation,” J. Nucl. Med. 29, 1704–1709, (1988).
[PubMed]

Pang, I.

I. Pang, “Wedge phantom for resolution measurement,” http://www.radiology.arizona.edu/~fastspec/wedge.htm.

Perry, J.R.

D.R. Gilland, B.M.W. Tsui, C.E. Metz, R.J. Jaszczak, and J.R. Perry, “An evaluation of maximum likelihood-EM reconstruction for SPECT by ROC analysis,” J. Nucl. Med. 33, 451–457 (1992).
[PubMed]

Strand, O.N.

O.N. Strand, “Theory and methods related to the singular-function expansion and Landweber’s iteration for integral equations of the first kind,” SIAM J. Numer. Anal.,  11, 798–825 (1974).
[CrossRef]

Strother, S.C.

X-L. Xu, J-S. Liow, and S.C. Strother, “Iterative algebraic reconstruction algorithms for emission computed tomography: A unified framework and its application to positron emission tomography,” Med. Phys.,  20, 1675–1684 (1993).
[CrossRef] [PubMed]

Tsui, B.M.W.

H.H. Barrett, D.W. Wilson, and B.M.W. Tsui, “Noise properties of the EM algorithm: I. Theory,” Phys. Med. Biol. 39, 833–846 (1994).
[CrossRef] [PubMed]

D.R. Gilland, B.M.W. Tsui, C.E. Metz, R.J. Jaszczak, and J.R. Perry, “An evaluation of maximum likelihood-EM reconstruction for SPECT by ROC analysis,” J. Nucl. Med. 33, 451–457 (1992).
[PubMed]

Wilson, D.W.

H.H. Barrett, D.W. Wilson, and B.M.W. Tsui, “Noise properties of the EM algorithm: I. Theory,” Phys. Med. Biol. 39, 833–846 (1994).
[CrossRef] [PubMed]

H.H. Barrett and D.W. Wilson, “An algorithm to calculate null and measurement spaces,” http://www.radiology.arizona.edu/~fastspec/nul_spc.pdf

Xu, X-L.

X-L. Xu, J-S. Liow, and S.C. Strother, “Iterative algebraic reconstruction algorithms for emission computed tomography: A unified framework and its application to positron emission tomography,” Med. Phys.,  20, 1675–1684 (1993).
[CrossRef] [PubMed]

J. Nucl. Med. (2)

S.C. Moore, S.P. Kijewski, S.P. Muller, and B.L. Holman, “SPECT Image Noise Power: Effects of Nonstationary Projection Noise and Attenuation Compensation,” J. Nucl. Med. 29, 1704–1709, (1988).
[PubMed]

D.R. Gilland, B.M.W. Tsui, C.E. Metz, R.J. Jaszczak, and J.R. Perry, “An evaluation of maximum likelihood-EM reconstruction for SPECT by ROC analysis,” J. Nucl. Med. 33, 451–457 (1992).
[PubMed]

J. Theor. Biol. (1)

P. Gilbert, “Iterative Methods for the Three-Dimensional Reconstruction of an Object from Projections,” J. Theor. Biol. 36, 105–117 (1972).
[CrossRef] [PubMed]

Med. Phys. (1)

X-L. Xu, J-S. Liow, and S.C. Strother, “Iterative algebraic reconstruction algorithms for emission computed tomography: A unified framework and its application to positron emission tomography,” Med. Phys.,  20, 1675–1684 (1993).
[CrossRef] [PubMed]

Phys. Med. Biol. (1)

H.H. Barrett, D.W. Wilson, and B.M.W. Tsui, “Noise properties of the EM algorithm: I. Theory,” Phys. Med. Biol. 39, 833–846 (1994).
[CrossRef] [PubMed]

SIAM J. Numer. Anal. (1)

O.N. Strand, “Theory and methods related to the singular-function expansion and Landweber’s iteration for integral equations of the first kind,” SIAM J. Numer. Anal.,  11, 798–825 (1974).
[CrossRef]

Other (4)

K.M. Hanson, “Bayesian and Related Methods in Image Reconstruction from Incomplete Data,” in Image Recovery, Theory and Application, Henry Stark, ed. (Academic, Orlando, Fla. 1987)

H.H. Barrett and D.W. Wilson, “An algorithm to calculate null and measurement spaces,” http://www.radiology.arizona.edu/~fastspec/nul_spc.pdf

H.H. Barrett, “The FASTSPECT imaging system,” http://www.radiology.arizona.edu/~fastspec.

I. Pang, “Wedge phantom for resolution measurement,” http://www.radiology.arizona.edu/~fastspec/wedge.htm.

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Figures (5)

Fig. 1.
Fig. 1.

A slice from the reconstructed image of the wedge phantom.

Fig. 2.
Fig. 2.

A slice from (a) the digital wedge phantom, (b) the measurement space of the phantom, and (c) the null space of the phantom.

Fig. 3.
Fig. 3.

The wedge phantom for the redesigned system, with (a) measurement space for the digitized phantom and (b) reconstruction of the actual phantom.

Fig. 4.
Fig. 4.

A slice from the with best agreement with a g collected by our FASTSPECT imaging system.

Fig. 5.
Fig. 5.

The (a) null space and (b) measurement space of the reconstruction image shown in Fig. 6.

Equations (11)

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

g = 𝛨 f ,
g m = h m ( r ) f ( r ) d 3 r
g a = H f ,
f n = v n f ( r ) d 3 r ,
h mn = v n h m ( r ) d 3 r .
H = UD V t
g a = i = 1 p d ii ( v i f ) u i ,
H + VD 1 U t ,
H + = lim η 0 [ H t H + η I ] 1 H t .
[ I + X ] 1 = n = 0 X n .
Nf = n = 0 ( I H t H ) n f .

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