Abstract

Research on human-observer performance for noise-limited tasks (such as those found in medical imaging) has recently progressed to investigations in which some signal or image parameters are statistically defined. In these cases the ideal-observer procedure is usually nonlinear, and analysis is mathematically intractable. Two suboptimal but linear observer models have been proposed for mathematical convenience. The Hotelling observer is the optimal linear model and has been found to give a good fit to most human results. The nonprewhitening (NPW) matched filter also has been useful for explanation of some human results. Rolland and Barrett [ J. Opt. Soc. Am. A 9, 649 ( 1992)] recently reported human results for detection of signals in white noise superimposed on statistically defined (lumpy) backgrounds in experiments that simulated nuclear medicine imaging systems. They found that the Hotelling model gave a good fit, whereas the simple NPW matched filter gave a poor fit. It is shown that the NPW model can also fit their data if a spatial frequency filter of a shape similar to the human contrast-sensitivity function is added to the NPW observer model. The best fit is achieved by use of an eye-filter model E(f) = f1.3 exp(−cf2), with c selected to yield a peak at 4 cycles/deg.

© 1994 Optical Society of America

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References

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  1. A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93 (1981).
    [CrossRef] [PubMed]
  2. P. F. Judy, R. G. Swenson, “Lesion detection and signal-to-noise ratio in CT images,” Med. Phys. 8, 13–23 (1981).
    [CrossRef] [PubMed]
  3. A. E. Burgess, R. F. Wagner, R. J. Jennings, “Human signal detection performance for noisy medical images,” in Proceedings of the IEEE Computer Society International Workshop on Medical Imaging (Institute of Electrical and Electronics Engineers, New York, 1982), pp. 99–105.
  4. R. F. Wagner, D. G. Brown, M. S. Pastel, “Application of information theory to the assessment of computed tomography,” Med. Phys. 6, 83–94 (1979).
    [CrossRef] [PubMed]
  5. K. J. Myers, H. H. Barrett, M. C. Borgstrom, D. D. Patton, G. W. Seeley, “Effect of noise correlation on detectability of disk signals in medical imaging,” J. Opt. Soc. Am. A 2, 1752–1759 (1985).
    [CrossRef] [PubMed]
  6. R. D. Fiete, H. H. Barrett, W. E. Smith, K. J. Myers, “Hotelling trace criterion and its correlation with human-observer performance,” J. Opt. Soc. Am. A 4, 945–953 (1987).
    [CrossRef] [PubMed]
  7. J. P. Rolland, H. H. Barrett, “Effect of random background inhomogeneity on observer detection performance,” J. Opt. Soc. Am. A 9, 649–658 (1992).
    [CrossRef] [PubMed]
  8. K. J. Myers, J. P. Rolland, H. H. Barrett, R. F. Wagner, “Aperture optimization for emission imaging: effect of a spatially varying background,” J. Opt. Soc. Am. A 7, 1279–1293 (1990).
    [CrossRef] [PubMed]
  9. J. Yao, H. H. Barrett, “Predicting human performance by a channelized Hotelling observer model,” in Mathematical Methods in Medical Imaging, D. C. Wilson, J. N. Wilson, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1768, 161–168 (1992).
    [CrossRef]
  10. K. J. Myers, H. H. Barrett, “Addition of a channel mechanism to the ideal-observer model,” J. Opt. Soc. Am. A 4, 2447–2457 (1987).
    [CrossRef] [PubMed]
  11. H. R. Wilson, S. C. Geise, “Threshold visibility of frequency gradient patterns,” Vision Res. 17, 1177–1190 (1977).
    [CrossRef] [PubMed]
  12. D. H. Kelly, “Spatial frequency selectivity in the retina,” Vision Res. 15, 665–672 (1975).
    [CrossRef] [PubMed]
  13. M. Hines, “Line spread function variation near the fovea,” Vision Res. 16, 567–572 (1976).
    [CrossRef] [PubMed]
  14. F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–566 (1968).
  15. F. W. Campbell, J. R. Johnstone, J. Ross, “An explanation for the visibility of low frequency gratings,” Vision Res. 21, 723–730 (1981).
    [CrossRef] [PubMed]
  16. E. R. Howell, R. F. Hess, “The functional area for summation to threshold for sinusoidal gratings,” Vision Res. 18, 369–374 (1978).
    [CrossRef] [PubMed]

1992 (1)

1990 (1)

1987 (2)

1985 (1)

1981 (3)

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93 (1981).
[CrossRef] [PubMed]

P. F. Judy, R. G. Swenson, “Lesion detection and signal-to-noise ratio in CT images,” Med. Phys. 8, 13–23 (1981).
[CrossRef] [PubMed]

F. W. Campbell, J. R. Johnstone, J. Ross, “An explanation for the visibility of low frequency gratings,” Vision Res. 21, 723–730 (1981).
[CrossRef] [PubMed]

1979 (1)

R. F. Wagner, D. G. Brown, M. S. Pastel, “Application of information theory to the assessment of computed tomography,” Med. Phys. 6, 83–94 (1979).
[CrossRef] [PubMed]

1978 (1)

E. R. Howell, R. F. Hess, “The functional area for summation to threshold for sinusoidal gratings,” Vision Res. 18, 369–374 (1978).
[CrossRef] [PubMed]

1977 (1)

H. R. Wilson, S. C. Geise, “Threshold visibility of frequency gradient patterns,” Vision Res. 17, 1177–1190 (1977).
[CrossRef] [PubMed]

1976 (1)

M. Hines, “Line spread function variation near the fovea,” Vision Res. 16, 567–572 (1976).
[CrossRef] [PubMed]

1975 (1)

D. H. Kelly, “Spatial frequency selectivity in the retina,” Vision Res. 15, 665–672 (1975).
[CrossRef] [PubMed]

1968 (1)

F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–566 (1968).

Barlow, H. B.

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93 (1981).
[CrossRef] [PubMed]

Barrett, H. H.

Borgstrom, M. C.

Brown, D. G.

R. F. Wagner, D. G. Brown, M. S. Pastel, “Application of information theory to the assessment of computed tomography,” Med. Phys. 6, 83–94 (1979).
[CrossRef] [PubMed]

Burgess, A. E.

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93 (1981).
[CrossRef] [PubMed]

A. E. Burgess, R. F. Wagner, R. J. Jennings, “Human signal detection performance for noisy medical images,” in Proceedings of the IEEE Computer Society International Workshop on Medical Imaging (Institute of Electrical and Electronics Engineers, New York, 1982), pp. 99–105.

Campbell, F. W.

F. W. Campbell, J. R. Johnstone, J. Ross, “An explanation for the visibility of low frequency gratings,” Vision Res. 21, 723–730 (1981).
[CrossRef] [PubMed]

F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–566 (1968).

Fiete, R. D.

Geise, S. C.

H. R. Wilson, S. C. Geise, “Threshold visibility of frequency gradient patterns,” Vision Res. 17, 1177–1190 (1977).
[CrossRef] [PubMed]

Hess, R. F.

E. R. Howell, R. F. Hess, “The functional area for summation to threshold for sinusoidal gratings,” Vision Res. 18, 369–374 (1978).
[CrossRef] [PubMed]

Hines, M.

M. Hines, “Line spread function variation near the fovea,” Vision Res. 16, 567–572 (1976).
[CrossRef] [PubMed]

Howell, E. R.

E. R. Howell, R. F. Hess, “The functional area for summation to threshold for sinusoidal gratings,” Vision Res. 18, 369–374 (1978).
[CrossRef] [PubMed]

Jennings, R. J.

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93 (1981).
[CrossRef] [PubMed]

A. E. Burgess, R. F. Wagner, R. J. Jennings, “Human signal detection performance for noisy medical images,” in Proceedings of the IEEE Computer Society International Workshop on Medical Imaging (Institute of Electrical and Electronics Engineers, New York, 1982), pp. 99–105.

Johnstone, J. R.

F. W. Campbell, J. R. Johnstone, J. Ross, “An explanation for the visibility of low frequency gratings,” Vision Res. 21, 723–730 (1981).
[CrossRef] [PubMed]

Judy, P. F.

P. F. Judy, R. G. Swenson, “Lesion detection and signal-to-noise ratio in CT images,” Med. Phys. 8, 13–23 (1981).
[CrossRef] [PubMed]

Kelly, D. H.

D. H. Kelly, “Spatial frequency selectivity in the retina,” Vision Res. 15, 665–672 (1975).
[CrossRef] [PubMed]

Myers, K. J.

Pastel, M. S.

R. F. Wagner, D. G. Brown, M. S. Pastel, “Application of information theory to the assessment of computed tomography,” Med. Phys. 6, 83–94 (1979).
[CrossRef] [PubMed]

Patton, D. D.

Robson, J. G.

F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–566 (1968).

Rolland, J. P.

Ross, J.

F. W. Campbell, J. R. Johnstone, J. Ross, “An explanation for the visibility of low frequency gratings,” Vision Res. 21, 723–730 (1981).
[CrossRef] [PubMed]

Seeley, G. W.

Smith, W. E.

Swenson, R. G.

P. F. Judy, R. G. Swenson, “Lesion detection and signal-to-noise ratio in CT images,” Med. Phys. 8, 13–23 (1981).
[CrossRef] [PubMed]

Wagner, R. F.

K. J. Myers, J. P. Rolland, H. H. Barrett, R. F. Wagner, “Aperture optimization for emission imaging: effect of a spatially varying background,” J. Opt. Soc. Am. A 7, 1279–1293 (1990).
[CrossRef] [PubMed]

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93 (1981).
[CrossRef] [PubMed]

R. F. Wagner, D. G. Brown, M. S. Pastel, “Application of information theory to the assessment of computed tomography,” Med. Phys. 6, 83–94 (1979).
[CrossRef] [PubMed]

A. E. Burgess, R. F. Wagner, R. J. Jennings, “Human signal detection performance for noisy medical images,” in Proceedings of the IEEE Computer Society International Workshop on Medical Imaging (Institute of Electrical and Electronics Engineers, New York, 1982), pp. 99–105.

Wilson, H. R.

H. R. Wilson, S. C. Geise, “Threshold visibility of frequency gradient patterns,” Vision Res. 17, 1177–1190 (1977).
[CrossRef] [PubMed]

Yao, J.

J. Yao, H. H. Barrett, “Predicting human performance by a channelized Hotelling observer model,” in Mathematical Methods in Medical Imaging, D. C. Wilson, J. N. Wilson, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1768, 161–168 (1992).
[CrossRef]

J. Opt. Soc. Am. A (5)

J. Physiol. (London) (1)

F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–566 (1968).

Med. Phys. (2)

P. F. Judy, R. G. Swenson, “Lesion detection and signal-to-noise ratio in CT images,” Med. Phys. 8, 13–23 (1981).
[CrossRef] [PubMed]

R. F. Wagner, D. G. Brown, M. S. Pastel, “Application of information theory to the assessment of computed tomography,” Med. Phys. 6, 83–94 (1979).
[CrossRef] [PubMed]

Science (1)

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93 (1981).
[CrossRef] [PubMed]

Vision Res. (5)

F. W. Campbell, J. R. Johnstone, J. Ross, “An explanation for the visibility of low frequency gratings,” Vision Res. 21, 723–730 (1981).
[CrossRef] [PubMed]

E. R. Howell, R. F. Hess, “The functional area for summation to threshold for sinusoidal gratings,” Vision Res. 18, 369–374 (1978).
[CrossRef] [PubMed]

H. R. Wilson, S. C. Geise, “Threshold visibility of frequency gradient patterns,” Vision Res. 17, 1177–1190 (1977).
[CrossRef] [PubMed]

D. H. Kelly, “Spatial frequency selectivity in the retina,” Vision Res. 15, 665–672 (1975).
[CrossRef] [PubMed]

M. Hines, “Line spread function variation near the fovea,” Vision Res. 16, 567–572 (1976).
[CrossRef] [PubMed]

Other (2)

A. E. Burgess, R. F. Wagner, R. J. Jennings, “Human signal detection performance for noisy medical images,” in Proceedings of the IEEE Computer Society International Workshop on Medical Imaging (Institute of Electrical and Electronics Engineers, New York, 1982), pp. 99–105.

J. Yao, H. H. Barrett, “Predicting human performance by a channelized Hotelling observer model,” in Mathematical Methods in Medical Imaging, D. C. Wilson, J. N. Wilson, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1768, 161–168 (1992).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Relative amplitude spectra as a function of radial spatial frequency for three model detection filters. All spectra were calculated with the pinhole–signal ratio equal to 1 (ap1) and are normalized to have a peak value of unity. The spectra include the NPW model with eye response function, f1.3 exp(−cf2), as well as the filter functions for two Hotelling observer cases [for lumpy-background spectral densities of (L1) 1.2 × 108 and (L2) 2.4 × 108 counts2/pixel, respectively].

Fig. 2
Fig. 2

Observer performance (detectability index d′) as a function of the ratio of imaging-system pinhole-aperture size to Gaussian signal size: (A) Rolland and Barrett observer data [d′ from receiver operating characteristic (ROC) studies] fitted with the Hotelling and the basic (no eye filter) NPW observer models for a Gaussian signal, four imaging-system pinhole diameters (Gaussian profile), and three different lumpy-background spectral densities (in counts2/pixel; L0 = 0, L1 = 1.2 × 108, L2 = 2.4 × 108). (B) Human-observer data (solid symbols). Theoretical curves are based on Eq. (2) for the NPW observer with the f1.3 exp(–cf2) eye-filter model.

Fig. 3
Fig. 3

Same as Fig. 2 except that the independent variable is exposure time, which gives different ratios of Poisson noise and lumpy-background spectral density in the images. The ratio is proportional to exposure time. For this experiment lumpy-background spectral densities were L1 = 1.3 × 105 counts/(s2 pixel) and L2 = 8.2 × 105 counts/(s2 pixel) referred to the object.

Fig. 4
Fig. 4

Time-dependence results for the human, the Hotelling, and the two modified NPW observers, shown again in a transformed plot. The use of the independent-variable log(W0T) causes results for different lumpy-background intensities to fall on a common curve for a given model, whereas the use of the dependent-variable log(d′/T1/2) removes the expected T1/2 dependence from the uniform-background data. Note that the uniform-background results (L0) have been placed at log(W0T) equal to 3 rather than −∞ for display convenience. One of the NPW models is E(f) = f1.3 exp(−cf2), and the other is E(f) = f exp(−cf2).

Equations (6)

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( d H ) 2 = d u d υ | H ( u , υ ) | 2 | S ( u , υ ) | 2 [ N 0 + | H ( u , υ ) | 2 W ( u , υ ) ] .
( d npwe ) 2 = [ d u d υ | S ( u , υ ) | 2 | E ( u , υ ) | 2 | H ( u , υ ) | 2 ] 2 d u d υ | S ( u , υ ) | 2 | E ( u , υ ) | 4 | H ( u , υ ) | 2 G ,
T H ( u , υ ) = H ( u , υ ) S ( u , υ ) N q + H 2 ( u , υ ) W ( u , υ ) ,
( d ) 2 = π 2 κ T r p 2 a s 2 n ! / h 2 ( n + 1 ) [ B m / k ( 2 n + 1 ) ] + [ κ T r p 2 W 0 / m ( 2 n + 1 ) ] ,
h = 2 π 2 ( r s 2 + r e 2 + r p 2 ) , k = h + 2 π 2 r e 2 , m = h + 2 π 2 ( r e 2 + r p 2 + r b 2 ) ,
( d ) 2 π a s 2 n ! / h ( B m / κ T r p 2 ) + [ W 0 ( h / m ) ( 2 n + 1 ) ] .

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