Abstract

A strategy is given for the design of coded apertures with respect to a given class of objects that are to be imaged. Previous knowledge of the first- and second-order statistics for the object class is assumed. The object class is characterized by its Karhunen–Loève eigenvectors and eigenvalues, whereas the imaging system is characterized by its singular-value decomposition. We introduce the concept of alignment in which the aperture parameters are adjusted until the system is tuned to measure the given object class well. A mean-square-error figure of merit that indicates degree of alignment is given, and alignment is performed by standard optimization techniques. We illustrate this technique with a simple proof-of-principle experiment. These concepts are general and may be applied to any linear imaging system.

© 1985 Optical Society of America

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References

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  1. W. E. Smith, R. G. Paxman, H. H. Barrett, “Image reconstruction from coded data: I. reconstruction algorithms and experimental results,” J. Opt. Soc. Am. A 2, 491–500 (1984).
    [CrossRef]
  2. D. Rosenfeld, A. Macovski, “Time modulated apertures for tomography in nuclear medicine,”IEEE Trans. Nucl. Sci. NS-24, 570–576 (1977).
    [CrossRef]
  3. W. L. Rogers, R. S. Adler, K. F. Koral, “A rationale for optimal coded aperture design,” Proc. Soc. Photo-Opt. Instrum. Eng. 231, 242–249 (1980).
  4. W. T. Cathey, B. R. Frieden, W. T. Rhodes, C. K. Rushforth, “Image gathering and signal processing for enhanced resolution,” J. Opt. Soc. Am. A 1, 241–249 (1984).
    [CrossRef]
  5. K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972), Chap. 8.
  6. G. Strang, Linear Algebra and its Applications (Academic, New York, 1980), Sec. 2.5.
  7. H. C. Andrews, C. L. Patterson, “Singular value decomposition and digital image processing,”IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 26–53 (1976).
    [CrossRef]
  8. T. S. Huang, P. M. Narendra, “Image restoration by singular value decomposition,” Appl. Opt. 14, 2213–2216 (1975).
    [CrossRef] [PubMed]
  9. B. P. Medoff, W. R. Brody, A. Macovski, “Iterative convolution backprojection algorithms for image reconstruction from limited data,”J. Opt. Soc. Am. 73, 1493–1500 (1983).
    [CrossRef]
  10. K. M. Hanson, G. W. Wecksung, “Bayesian approach to limited-angle reconstruction in computed tomography,”J. Opt. Soc. Am. 73, 1501–1509 (1983).
    [CrossRef]
  11. M. E. Davison, “The ill-conditioned nature of the limited angle tomography problem,” SIAM J. Appl. Math. 43, 428–448 (1983).
    [CrossRef]
  12. D. Lefkoupoulos, J. Fonroget, J. Y. Devaux, J. B. Guilhem, J. C. Roucayrol, R. Guiraud, “Quantitative 3D imaging with coded apertures by using SVD decomposition of the transmission matrix,” in Proceedings of the Third World Congress on Nuclear Medicine and Biology, C. Raynaud, ed. (Pergamon, Paris, 1982), p. 503.
  13. C. W. Helstrom, “Image restoration by the method of least squares,”J. Opt. Soc. Am. 57, 297–303 (1967).
    [CrossRef]
  14. W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), Sec. 14.6.
  15. H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977), Sec. 8.1.
  16. T. D. Milster, L. A. Selberg, H. H. Barrett, R. L. Easton, G. R. Rossi, J. Arendt, R. G. Simpson, “A modular scintillation camera for use in nuclear medicine,”IEEE Trans. Nucl. Sci. NS-31, 578–580 (1984).
    [CrossRef]
  17. L. A. Selberg, “Design studies for a modular scintillation camera,” M.S. thesis (University of Arizona, Tucson, Arizona, 1984).
  18. S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
    [CrossRef] [PubMed]
  19. N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of state calculations by fast computing machines,”J. Chem. Phys. 21, 1087–1092 (1953).
    [CrossRef]
  20. G. R. Gindi, R. G. Paxman, H. H. Barrett, “Reconstruction of an object from its coded image and object constraints,” Appl. Opt. 23, 851–856 (1984).
    [CrossRef] [PubMed]
  21. W. E. Smith, H. H. Barrett, R. G. Paxman, “Reconstruction of objects from coded images by simulated annealing,” Opt. Lett. 9, 199–201 (1983).
    [CrossRef]

1984 (4)

1983 (5)

1980 (1)

W. L. Rogers, R. S. Adler, K. F. Koral, “A rationale for optimal coded aperture design,” Proc. Soc. Photo-Opt. Instrum. Eng. 231, 242–249 (1980).

1977 (1)

D. Rosenfeld, A. Macovski, “Time modulated apertures for tomography in nuclear medicine,”IEEE Trans. Nucl. Sci. NS-24, 570–576 (1977).
[CrossRef]

1976 (1)

H. C. Andrews, C. L. Patterson, “Singular value decomposition and digital image processing,”IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 26–53 (1976).
[CrossRef]

1975 (1)

1967 (1)

1953 (1)

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of state calculations by fast computing machines,”J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Adler, R. S.

W. L. Rogers, R. S. Adler, K. F. Koral, “A rationale for optimal coded aperture design,” Proc. Soc. Photo-Opt. Instrum. Eng. 231, 242–249 (1980).

Andrews, H. C.

H. C. Andrews, C. L. Patterson, “Singular value decomposition and digital image processing,”IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 26–53 (1976).
[CrossRef]

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977), Sec. 8.1.

Arendt, J.

T. D. Milster, L. A. Selberg, H. H. Barrett, R. L. Easton, G. R. Rossi, J. Arendt, R. G. Simpson, “A modular scintillation camera for use in nuclear medicine,”IEEE Trans. Nucl. Sci. NS-31, 578–580 (1984).
[CrossRef]

Barrett, H. H.

Brody, W. R.

Cathey, W. T.

Davison, M. E.

M. E. Davison, “The ill-conditioned nature of the limited angle tomography problem,” SIAM J. Appl. Math. 43, 428–448 (1983).
[CrossRef]

Devaux, J. Y.

D. Lefkoupoulos, J. Fonroget, J. Y. Devaux, J. B. Guilhem, J. C. Roucayrol, R. Guiraud, “Quantitative 3D imaging with coded apertures by using SVD decomposition of the transmission matrix,” in Proceedings of the Third World Congress on Nuclear Medicine and Biology, C. Raynaud, ed. (Pergamon, Paris, 1982), p. 503.

Easton, R. L.

T. D. Milster, L. A. Selberg, H. H. Barrett, R. L. Easton, G. R. Rossi, J. Arendt, R. G. Simpson, “A modular scintillation camera for use in nuclear medicine,”IEEE Trans. Nucl. Sci. NS-31, 578–580 (1984).
[CrossRef]

Fonroget, J.

D. Lefkoupoulos, J. Fonroget, J. Y. Devaux, J. B. Guilhem, J. C. Roucayrol, R. Guiraud, “Quantitative 3D imaging with coded apertures by using SVD decomposition of the transmission matrix,” in Proceedings of the Third World Congress on Nuclear Medicine and Biology, C. Raynaud, ed. (Pergamon, Paris, 1982), p. 503.

Frieden, B. R.

Fukunaga, K.

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972), Chap. 8.

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Gindi, G. R.

Guilhem, J. B.

D. Lefkoupoulos, J. Fonroget, J. Y. Devaux, J. B. Guilhem, J. C. Roucayrol, R. Guiraud, “Quantitative 3D imaging with coded apertures by using SVD decomposition of the transmission matrix,” in Proceedings of the Third World Congress on Nuclear Medicine and Biology, C. Raynaud, ed. (Pergamon, Paris, 1982), p. 503.

Guiraud, R.

D. Lefkoupoulos, J. Fonroget, J. Y. Devaux, J. B. Guilhem, J. C. Roucayrol, R. Guiraud, “Quantitative 3D imaging with coded apertures by using SVD decomposition of the transmission matrix,” in Proceedings of the Third World Congress on Nuclear Medicine and Biology, C. Raynaud, ed. (Pergamon, Paris, 1982), p. 503.

Hanson, K. M.

Helstrom, C. W.

Huang, T. S.

Hunt, B. R.

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977), Sec. 8.1.

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Koral, K. F.

W. L. Rogers, R. S. Adler, K. F. Koral, “A rationale for optimal coded aperture design,” Proc. Soc. Photo-Opt. Instrum. Eng. 231, 242–249 (1980).

Lefkoupoulos, D.

D. Lefkoupoulos, J. Fonroget, J. Y. Devaux, J. B. Guilhem, J. C. Roucayrol, R. Guiraud, “Quantitative 3D imaging with coded apertures by using SVD decomposition of the transmission matrix,” in Proceedings of the Third World Congress on Nuclear Medicine and Biology, C. Raynaud, ed. (Pergamon, Paris, 1982), p. 503.

Macovski, A.

B. P. Medoff, W. R. Brody, A. Macovski, “Iterative convolution backprojection algorithms for image reconstruction from limited data,”J. Opt. Soc. Am. 73, 1493–1500 (1983).
[CrossRef]

D. Rosenfeld, A. Macovski, “Time modulated apertures for tomography in nuclear medicine,”IEEE Trans. Nucl. Sci. NS-24, 570–576 (1977).
[CrossRef]

Medoff, B. P.

Metropolis, N.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of state calculations by fast computing machines,”J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Milster, T. D.

T. D. Milster, L. A. Selberg, H. H. Barrett, R. L. Easton, G. R. Rossi, J. Arendt, R. G. Simpson, “A modular scintillation camera for use in nuclear medicine,”IEEE Trans. Nucl. Sci. NS-31, 578–580 (1984).
[CrossRef]

Narendra, P. M.

Patterson, C. L.

H. C. Andrews, C. L. Patterson, “Singular value decomposition and digital image processing,”IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 26–53 (1976).
[CrossRef]

Paxman, R. G.

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), Sec. 14.6.

Rhodes, W. T.

Rogers, W. L.

W. L. Rogers, R. S. Adler, K. F. Koral, “A rationale for optimal coded aperture design,” Proc. Soc. Photo-Opt. Instrum. Eng. 231, 242–249 (1980).

Rosenbluth, A.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of state calculations by fast computing machines,”J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Rosenbluth, M.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of state calculations by fast computing machines,”J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Rosenfeld, D.

D. Rosenfeld, A. Macovski, “Time modulated apertures for tomography in nuclear medicine,”IEEE Trans. Nucl. Sci. NS-24, 570–576 (1977).
[CrossRef]

Rossi, G. R.

T. D. Milster, L. A. Selberg, H. H. Barrett, R. L. Easton, G. R. Rossi, J. Arendt, R. G. Simpson, “A modular scintillation camera for use in nuclear medicine,”IEEE Trans. Nucl. Sci. NS-31, 578–580 (1984).
[CrossRef]

Roucayrol, J. C.

D. Lefkoupoulos, J. Fonroget, J. Y. Devaux, J. B. Guilhem, J. C. Roucayrol, R. Guiraud, “Quantitative 3D imaging with coded apertures by using SVD decomposition of the transmission matrix,” in Proceedings of the Third World Congress on Nuclear Medicine and Biology, C. Raynaud, ed. (Pergamon, Paris, 1982), p. 503.

Rushforth, C. K.

Selberg, L. A.

T. D. Milster, L. A. Selberg, H. H. Barrett, R. L. Easton, G. R. Rossi, J. Arendt, R. G. Simpson, “A modular scintillation camera for use in nuclear medicine,”IEEE Trans. Nucl. Sci. NS-31, 578–580 (1984).
[CrossRef]

L. A. Selberg, “Design studies for a modular scintillation camera,” M.S. thesis (University of Arizona, Tucson, Arizona, 1984).

Simpson, R. G.

T. D. Milster, L. A. Selberg, H. H. Barrett, R. L. Easton, G. R. Rossi, J. Arendt, R. G. Simpson, “A modular scintillation camera for use in nuclear medicine,”IEEE Trans. Nucl. Sci. NS-31, 578–580 (1984).
[CrossRef]

Smith, W. E.

Strang, G.

G. Strang, Linear Algebra and its Applications (Academic, New York, 1980), Sec. 2.5.

Teller, A.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of state calculations by fast computing machines,”J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Teller, E.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of state calculations by fast computing machines,”J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Wecksung, G. W.

Appl. Opt. (2)

IEEE Trans. Acoust. Speech Signal Process. (1)

H. C. Andrews, C. L. Patterson, “Singular value decomposition and digital image processing,”IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 26–53 (1976).
[CrossRef]

IEEE Trans. Nucl. Sci. (2)

T. D. Milster, L. A. Selberg, H. H. Barrett, R. L. Easton, G. R. Rossi, J. Arendt, R. G. Simpson, “A modular scintillation camera for use in nuclear medicine,”IEEE Trans. Nucl. Sci. NS-31, 578–580 (1984).
[CrossRef]

D. Rosenfeld, A. Macovski, “Time modulated apertures for tomography in nuclear medicine,”IEEE Trans. Nucl. Sci. NS-24, 570–576 (1977).
[CrossRef]

J. Chem. Phys. (1)

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equation of state calculations by fast computing machines,”J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

Opt. Lett. (1)

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

W. L. Rogers, R. S. Adler, K. F. Koral, “A rationale for optimal coded aperture design,” Proc. Soc. Photo-Opt. Instrum. Eng. 231, 242–249 (1980).

Science (1)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

SIAM J. Appl. Math. (1)

M. E. Davison, “The ill-conditioned nature of the limited angle tomography problem,” SIAM J. Appl. Math. 43, 428–448 (1983).
[CrossRef]

Other (6)

D. Lefkoupoulos, J. Fonroget, J. Y. Devaux, J. B. Guilhem, J. C. Roucayrol, R. Guiraud, “Quantitative 3D imaging with coded apertures by using SVD decomposition of the transmission matrix,” in Proceedings of the Third World Congress on Nuclear Medicine and Biology, C. Raynaud, ed. (Pergamon, Paris, 1982), p. 503.

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972), Chap. 8.

G. Strang, Linear Algebra and its Applications (Academic, New York, 1980), Sec. 2.5.

L. A. Selberg, “Design studies for a modular scintillation camera,” M.S. thesis (University of Arizona, Tucson, Arizona, 1984).

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), Sec. 14.6.

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977), Sec. 8.1.

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Figures (7)

Fig. 1
Fig. 1

Orthogonal-view coded-aperture system for two-dimensional object.

Fig. 2
Fig. 2

Vector representation of three-pixel objects. The K-L eigenvectors for a given object class, ϕ1, ϕ2, and ϕ3, are oriented along the Z, X, and Y axes, respectively. The SVD singular vectors, f1, f2, and f3, also form an orthonormal basis set. a, Unaligned system: Interest space (the Z axis) does not lie in measurement space; b, aligned system: The imaging system has been adjusted so that interest space falls within measurement space.

Fig. 3
Fig. 3

Simulation geometry used in alignment. The design parameters included the location and number of pinholes in the aperture.

Fig. 4
Fig. 4

Optimization for a single object. a, Original object; b, reconstruction from unoptimized-aperture data; c, reconstruction from optimized-aperture data.

Fig. 5
Fig. 5

Singular vectors for a particular aperture. Each of the 18 cells contains an eigenimage of H*H. These singular vectors are arranged according to magnitude of the associated singular values with the largest corresponding to the upper left cell and the least corresponding to the lower right cell. The eigenimages are bipolar, and the border between cells is set at zero for reference.

Fig. 6
Fig. 6

Optimization for a single object with positivity constraint. a, Original object; b, reconstruction from unoptimized-aperture data with positivity constraint applied; c, reconstruction from optimized-aperture data with positivity constraint applied.

Fig. 7
Fig. 7

System alignment for a three-object class. a, Three equally likely objects which constitute a zero-mean object class; b, reconstruction of each object from unoptimized aperture data; c, reconstruction of each object from optimized aperture data.

Equations (47)

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g = Hf ,
c l m = [ f ( l ) - f ( l ) ] [ f ( m ) - f ( m ) ] .
f = j = 1 N a j ϕ j ,
a j = ( f · ϕ j ) .
a j a k = λ j δ j k .
MSE = j = k + 1 N λ j .
H * H f i = μ i 2 f i .
g i = ( 1 / μ i ) Hf i .
g = H { f } = H { i = 1 N ( f · f i ) f i } = i = 1 N ( f · f i ) H f i . = i = 1 N ( f · f i ) μ i g i .
null space = { f Hf = 0 } .
f null = i = R + 1 N ( f · f i ) f i ,
MSE = f null 2 .
MSE = i = R + 1 N ( f · f i ) 2 .
σ i 2 ( f · f i ) 2 .
σ i 2 = [ ( j = 1 N a j ϕ j ) · f i ] 2 = [ j = 1 N a j ( ϕ j · f i ) ] 2 = ( j = 1 N a j ϕ j · f i ) ( k = 1 N a k ϕ k · f i ) = j = 1 N k = 1 N a j a k ϕ j · f i ( ϕ k · f i ) .
a j a k = λ j δ j k ,
σ i 2 = j = 1 N λ j ( ϕ j · f i ) 2 .
MSE = i = R + 1 N σ i 2 .
f ^ = i = 1 R ( 1 / μ i ) [ ( g + n ) · g i ] f i ,
f = i = 1 R ( 1 / μ i ) ( g · g i ) f i .
f - f ^ = i = 1 R - ( 1 - / μ i ) ( n · g i ) f i ,
f - f ^ 2 = i = 1 R ( 1 / μ i 2 ) ( n · g i ) 2 .
n = i = 1 M b i g i .
f - f ^ 2 = i = 1 R ( b i 2 / μ i 2 ) .
MSE = i = 1 R b i 2 μ i 2 = i = 1 R b i 2 μ i 2 .
σ n 2 b i 2 .
MSE = σ n 2 i = 1 R ( 1 / μ i 2 ) .
f ^ = i = 1 R ( 1 / μ i ) [ ( g + n ) · g i ] f i ,
f - f ^ = i = 1 R { [ ( 1 / μ i ) - ( 1 / μ i ) ] ( g · g i ) - ( 1 / μ i ) ( n · g i ) } f i .
f - f ^ 2 = i = 1 R [ μ i - μ i μ i μ i ( g · g i ) - 1 μ i ( n · g i ) ] 2 = i = 1 R [ ( μ i - μ i ) 2 ( μ i μ i ) 2 ( g · g i ) 2 - 2 ( μ i - μ i ) μ i μ i 2 × ( g · g i ) ( n · g i ) + 1 μ i 2 ( n · g i ) 2 ] .
( g · g i ) 2 = μ i 2 ( f · f i ) 2 = μ i 2 σ i 2 ,
( g · g i ) ( n · g i ) = ( g · g i ) ( n · g i ) = ( g · g i ) ( n · g i ) .
f - f ^ 2 = i = 1 R [ ( μ i - μ i ) 2 μ i 2 σ i 2 + ( 1 / μ i 2 ) σ n 2 ] .
μ i [ ( μ i - μ i ) 2 μ i 2 σ i 2 + ( 1 / μ i 2 ) σ n 2 ] = 0.
μ i = σ n 2 μ i σ i 2 + μ i ,
1 μ i = μ i μ i 2 + σ n 2 σ i 2 .
σ n 2 σ i 2 μ i 2 ,
MMSE = i = 1 R [ 1 ( 1 / σ i 2 ) + ( μ i 2 / σ n 2 ) ] .
MSE = i = R + 1 N j = 1 N λ j ( ϕ j · f i ) 2 .
MSE = j = 1 N λ j [ 1 - i = 1 R ( ϕ j - f i ) 2 ] .
MSE = i = R + 1 N j = 1 N λ j ( ϕ j · f j ) 2 .
MSE = j = 1 N λ j i = R + 1 N ( ϕ j · f i ) 2 .
ϕ j = ϕ j N + ϕ j N .
ϕ j 2 = ϕ j M 2 + ϕ j N 2 .
ϕ j 2 = 1.
ϕ j N 2 = 1 - ϕ j M 2 .
MSE = j = 1 N λ j ϕ j N 2 = j = 1 N λ j [ 1 - ϕ j M 2 ] = j = 1 N λ j [ 1 - i = 1 R ( ϕ j · f i ) 2 ] ,

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