Abstract

We measured human observers’ detectability of aperiodic signals in noise with two components (white and low-pass Gaussian). The white-noise component ensured that the signal detection task was always noise limited rather than contrast limited (i.e., image noise was always much larger than observer internal noise). The low-pass component can be considered to be a statistically defined background. Contrast threshold elevation was not linearly related to the rms background contrast. Our results gave power-law exponents near 0.6, similar to that found for deterministic masking. The Fisher–Hotelling linear discriminant model assessed by Rolland and Barrett [J. Opt. Soc. Am. A 9, 649 (1992)] and the modified nonprewhitening matched filter model suggested by Burgess [J. Opt. Soc. Am. A 11, 1237 (1994)] for describing signal detection in statistically defined backgrounds did not fit our more precise data. We show that it is not possible to find any nonprewhitening model that can fit our data. We investigated modified Fisher–Hotelling models by using spatial-frequency channels, as suggested by Myers and Barrett [J. Opt. Soc. Am. A 4, 2447 (1987)]. Two of these models did give good fits to our data, which suggests that we may be able to do partial prewhitening of image noise.

© 1997 Optical Society of America

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    [Crossref]
  44. J. Nachmias, “Masked detection of gratings: the standard model revisited,” Vision Res. 33, 1359–1365 (1993).
    [Crossref] [PubMed]
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  52. A. E. Burgess, X. Li, C. K. Abbey, “Nodule detection in two component noise: toward patient structure,” in Medical Imaging 1997: Image Processing, H. L. Kundel, ed., Proc. SPIE3036, 2–13 (1997).
    [Crossref]

1996 (2)

1995 (2)

A. E. Burgess, “Comparison of receiver operating characteristic and forced choice observer performance measurement methods,” Med. Phys. 22, 643–655 (1995).
[Crossref] [PubMed]

B. S. Tjan, W. L. Braje, G. E. Legge, D. Kersten, “Human efficiency for recognizing 3-D objects in luminance noise,” Vision Res. 35, 3053–3069 (1995).
[Crossref] [PubMed]

1994 (3)

1993 (2)

H. H. Barrett, J. Yao, J. P. Rolland, K. J. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. USA 90, 9758–9765 (1993).
[Crossref] [PubMed]

J. Nachmias, “Masked detection of gratings: the standard model revisited,” Vision Res. 33, 1359–1365 (1993).
[Crossref] [PubMed]

1992 (1)

1991 (1)

P. J. Bennett, A. B. Sekular, “Profile analysis in human vision?” Invest. Ophthalmol. Visual Sci. Suppl. 32, 1024 (1991).

1990 (1)

1988 (1)

1987 (4)

P. G. J. Barten, “The SQRI method: a new method for the evaluation of visible resolution on a display,” Proc. Soc. Inf. Disp. 28, 253–262 (1987).

M. Pavel, G. Sperling, T. Riedl, A. Vanderbeek, “Limits of visual communication: the effect of signal-to-noise ratio on the intelligibility of American Sign Language,” J. Opt. Soc. Am. A 41, 2355–2365 (1987).
[Crossref]

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]

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]

1985 (4)

1984 (1)

M. Ishida, K. Doi, L.-N. Loo, C. E. Metz, J. L. Lehr, “Digital image processing: effect on detectability of simulated low-contrast radiographic patterns,” Radiology 150, 569–575 (1984).
[PubMed]

1983 (2)

H. R. Wilson, D. K. MacFarlane, G. C. Phillips, “Spatial tuning of orientation selective units estimated by oblique masking,” Vision Res. 23, 873–882 (1983).
[Crossref]

D. J. Swift, R. A. Smith, “Spatial frequency masking and Weber’s law,” Vision Res. 23, 495–505 (1983).
[Crossref]

1981 (2)

G. E. Legge, “A power law for contrast discrimination,” Vision Res. 21, 457–467 (1981).
[Crossref]

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual discrimination,” Science 214, 93–94 (1981).
[Crossref] [PubMed]

1980 (1)

1978 (1)

B. M. W. Tsui, C. E. Metz, F. B. Atkins, S. J. Starr, R. N. Beck, “A comparison of optimum spatial resolution in nuclear imaging based on statistical theory and on observer performance,” Phys. Med. Biol. 23, 654–676 (1978).
[Crossref] [PubMed]

1974 (1)

G. Revesz, H. L. Kundel, M. A. Graber, “The influence of structured noise on detection of radiologic abnormalities,” Invest. Radiol. 9, 479–486 (1974).
[Crossref] [PubMed]

1973 (1)

F. L. Engel, “Visual conspicuity and selective background interference in eccentric vision,” Vision Res. 14, 459–471 (1973).
[Crossref]

1972 (1)

1970 (1)

1967 (1)

1936 (1)

R. A. Fisher, “The use of multiple measurements in taxonic problems,” Ann. Eugenics 7, 179–188 (1936).

1931 (1)

H. Hotelling, “The generalization of student’s ratio,” Ann. Math. Stat. 2, 360–378 (1931).
[Crossref]

Abbey, C. K.

A. E. Burgess, X. Li, C. K. Abbey, “Nodule detection in two component noise: toward patient structure,” in Medical Imaging 1997: Image Processing, H. L. Kundel, ed., Proc. SPIE3036, 2–13 (1997).
[Crossref]

Ahumada, A. J.

Atkins, F. B.

B. M. W. Tsui, C. E. Metz, F. B. Atkins, S. J. Starr, R. N. Beck, “A comparison of optimum spatial resolution in nuclear imaging based on statistical theory and on observer performance,” Phys. Med. Biol. 23, 654–676 (1978).
[Crossref] [PubMed]

Barlow, H. B.

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual discrimination,” Science 214, 93–94 (1981).
[Crossref] [PubMed]

Barrett, H. H.

Barten, P. G. J.

P. G. J. Barten, “The SQRI method: a new method for the evaluation of visible resolution on a display,” Proc. Soc. Inf. Disp. 28, 253–262 (1987).

Beck, R. N.

B. M. W. Tsui, C. E. Metz, F. B. Atkins, S. J. Starr, R. N. Beck, “A comparison of optimum spatial resolution in nuclear imaging based on statistical theory and on observer performance,” Phys. Med. Biol. 23, 654–676 (1978).
[Crossref] [PubMed]

Bennett, P. J.

P. J. Bennett, A. B. Sekular, “Profile analysis in human vision?” Invest. Ophthalmol. Visual Sci. Suppl. 32, 1024 (1991).

Borgstrom, M. C.

Braje, W. L.

B. S. Tjan, W. L. Braje, G. E. Legge, D. Kersten, “Human efficiency for recognizing 3-D objects in luminance noise,” Vision Res. 35, 3053–3069 (1995).
[Crossref] [PubMed]

Brown, D. G.

R. F. Wagner, D. G. Brown, C. E. Metz, “On the multiplex advantage of coded source/aperture photon imaging,” in Digital Radiography, W. R. Brody, ed., Proc. SPIE314, 72–76 (1981).
[Crossref]

Burgess, A. E.

A. E. Burgess, “Comparison of receiver operating characteristic and forced choice observer performance measurement methods,” Med. Phys. 22, 643–655 (1995).
[Crossref] [PubMed]

A. E. Burgess, “Statistically defined backgrounds: performance of a modified nonprewhitening matched filter model,” J. Opt. Soc. Am. A 11, 1237–1242 (1994).
[Crossref]

A. E. Burgess, B. Colborne, “Visual signal detection. IV. Observer inconsistency,” J. Opt. Soc. Am. A 5, 617–627 (1988).
[Crossref] [PubMed]

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual discrimination,” Science 214, 93–94 (1981).
[Crossref] [PubMed]

A. E. Burgess, “High level visual decision efficiencies,” in Vision: Coding and Efficiency, C. Blakemore, ed. (Cambridge U. Press, Cambridge, 1990), pp. 431–440.

A. E. Burgess, “Comparison of non-prewhitening and Hotelling observer models,” in Medical Imaging 1995: Image Perception, H. L. Kundel, ed., Proc. SPIE2436, 1–8 (1995).

A. E. Burgess, X. Li, “Experimental evaluation of observer models for detection of signals in statistically defined backgrounds,” in 14th Conference of Image Processing in Medical Imaging, Y. Bizais, C. Barillot, R. Di Paola, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1995), pp. 89–100.

A. E. Burgess, X. Li, C. K. Abbey, “Nodule detection in two component noise: toward patient structure,” in Medical Imaging 1997: Image Processing, H. L. Kundel, ed., Proc. SPIE3036, 2–13 (1997).
[Crossref]

Campos, J.

Colborne, B.

Daly, S.

S. Daly, “The visual differences predictor: an algorithm for the assessment of image fidelity,” in Digital Images and Human Vision, A. B. Watson., ed. (MIT Press, Cambridge, Mass., 1993), pp. 179–206.

Doi, K.

M. Ishida, K. Doi, L.-N. Loo, C. E. Metz, J. L. Lehr, “Digital image processing: effect on detectability of simulated low-contrast radiographic patterns,” Radiology 150, 569–575 (1984).
[PubMed]

Eckstein, M. P.

Engel, F. L.

F. L. Engel, “Visual conspicuity and selective background interference in eccentric vision,” Vision Res. 14, 459–471 (1973).
[Crossref]

Fiete, R. D.

Fisher, R. A.

R. A. Fisher, “The use of multiple measurements in taxonic problems,” Ann. Eugenics 7, 179–188 (1936).

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vettering, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 2, pp. 32–104.

Foley, J. M.

Fukunaga, K.

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

Graber, M. A.

G. Revesz, H. L. Kundel, M. A. Graber, “The influence of structured noise on detection of radiologic abnormalities,” Invest. Radiol. 9, 479–486 (1974).
[Crossref] [PubMed]

Graham, N.

N. Graham, “Complex channels, early nonlinearities, and normalization in texture segregation,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991), pp. 273–290.

Green, D. B.

D. B. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1968).

Heeger, D. J.

D. J. Heeger, “Nonlinear model of cat striate cortex,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991), pp. 113–135.

Hotelling, H.

H. Hotelling, “The generalization of student’s ratio,” Ann. Math. Stat. 2, 360–378 (1931).
[Crossref]

Ishida, M.

M. Ishida, K. Doi, L.-N. Loo, C. E. Metz, J. L. Lehr, “Digital image processing: effect on detectability of simulated low-contrast radiographic patterns,” Radiology 150, 569–575 (1984).
[PubMed]

Javidi, B.

Jennings, R. J.

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual discrimination,” Science 214, 93–94 (1981).
[Crossref] [PubMed]

Julesz, B.

Kersten, D.

B. S. Tjan, W. L. Braje, G. E. Legge, D. Kersten, “Human efficiency for recognizing 3-D objects in luminance noise,” Vision Res. 35, 3053–3069 (1995).
[Crossref] [PubMed]

Kober, V.

Kundel, H. L.

G. Revesz, H. L. Kundel, M. A. Graber, “The influence of structured noise on detection of radiologic abnormalities,” Invest. Radiol. 9, 479–486 (1974).
[Crossref] [PubMed]

Legge, G. E.

B. S. Tjan, W. L. Braje, G. E. Legge, D. Kersten, “Human efficiency for recognizing 3-D objects in luminance noise,” Vision Res. 35, 3053–3069 (1995).
[Crossref] [PubMed]

G. E. Legge, “A power law for contrast discrimination,” Vision Res. 21, 457–467 (1981).
[Crossref]

G. E. Legge, J. M. Foley, “Contrast masking in human vision,” J. Opt. Soc. Am. 70, 1458–1471 (1980).
[Crossref] [PubMed]

Lehr, J. L.

M. Ishida, K. Doi, L.-N. Loo, C. E. Metz, J. L. Lehr, “Digital image processing: effect on detectability of simulated low-contrast radiographic patterns,” Radiology 150, 569–575 (1984).
[PubMed]

Li, X.

A. E. Burgess, X. Li, C. K. Abbey, “Nodule detection in two component noise: toward patient structure,” in Medical Imaging 1997: Image Processing, H. L. Kundel, ed., Proc. SPIE3036, 2–13 (1997).
[Crossref]

A. E. Burgess, X. Li, “Experimental evaluation of observer models for detection of signals in statistically defined backgrounds,” in 14th Conference of Image Processing in Medical Imaging, Y. Bizais, C. Barillot, R. Di Paola, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1995), pp. 89–100.

Loo, L.-N.

M. Ishida, K. Doi, L.-N. Loo, C. E. Metz, J. L. Lehr, “Digital image processing: effect on detectability of simulated low-contrast radiographic patterns,” Radiology 150, 569–575 (1984).
[PubMed]

MacFarlane, D. K.

H. R. Wilson, D. K. MacFarlane, G. C. Phillips, “Spatial tuning of orientation selective units estimated by oblique masking,” Vision Res. 23, 873–882 (1983).
[Crossref]

Metz, C. E.

M. Ishida, K. Doi, L.-N. Loo, C. E. Metz, J. L. Lehr, “Digital image processing: effect on detectability of simulated low-contrast radiographic patterns,” Radiology 150, 569–575 (1984).
[PubMed]

B. M. W. Tsui, C. E. Metz, F. B. Atkins, S. J. Starr, R. N. Beck, “A comparison of optimum spatial resolution in nuclear imaging based on statistical theory and on observer performance,” Phys. Med. Biol. 23, 654–676 (1978).
[Crossref] [PubMed]

R. F. Wagner, D. G. Brown, C. E. Metz, “On the multiplex advantage of coded source/aperture photon imaging,” in Digital Radiography, W. R. Brody, ed., Proc. SPIE314, 72–76 (1981).
[Crossref]

Myers, K. J.

Nachmias, J.

J. Nachmias, “Masked detection of gratings: the standard model revisited,” Vision Res. 33, 1359–1365 (1993).
[Crossref] [PubMed]

Nagaraja, N. S.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1990).

Patton, D. D.

Pavel, M.

M. Pavel, G. Sperling, T. Riedl, A. Vanderbeek, “Limits of visual communication: the effect of signal-to-noise ratio on the intelligibility of American Sign Language,” J. Opt. Soc. Am. A 41, 2355–2365 (1987).
[Crossref]

Pelli, D.

D. Pelli, “Effects of visual noise,” Ph.D. dissertation (Cambridge University, Cambridge, 1981).

Pelli, D. G.

D. G. Pelli, “The quantum efficiency of vision,” in Vision: Coding and Efficiency, C. Blakemore, ed. (Cambridge U. Press, Cambridge, 1990), pp. 3–24.

Phillips, G. C.

H. R. Wilson, D. K. MacFarlane, G. C. Phillips, “Spatial tuning of orientation selective units estimated by oblique masking,” Vision Res. 23, 873–882 (1983).
[Crossref]

Pollen, H.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vettering, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 2, pp. 32–104.

Revesz, G.

G. Revesz, H. L. Kundel, M. A. Graber, “The influence of structured noise on detection of radiologic abnormalities,” Invest. Radiol. 9, 479–486 (1974).
[Crossref] [PubMed]

Riedl, T.

M. Pavel, G. Sperling, T. Riedl, A. Vanderbeek, “Limits of visual communication: the effect of signal-to-noise ratio on the intelligibility of American Sign Language,” J. Opt. Soc. Am. A 41, 2355–2365 (1987).
[Crossref]

Roehrig, H.

Rolland, J. P.

Seeley, G. W.

Sekular, A. B.

P. J. Bennett, A. B. Sekular, “Profile analysis in human vision?” Invest. Ophthalmol. Visual Sci. Suppl. 32, 1024 (1991).

Smith, R. A.

R. A. Smith, D. J. Swift, “Spatial-frequency masking and Birdsall’s theorem,” J. Opt. Soc. Am. A 2, 1593–1599 (1985).
[Crossref] [PubMed]

D. J. Swift, R. A. Smith, “Spatial frequency masking and Weber’s law,” Vision Res. 23, 495–505 (1983).
[Crossref]

Smith, W. E.

Sperling, G.

M. Pavel, G. Sperling, T. Riedl, A. Vanderbeek, “Limits of visual communication: the effect of signal-to-noise ratio on the intelligibility of American Sign Language,” J. Opt. Soc. Am. A 41, 2355–2365 (1987).
[Crossref]

Starr, S. J.

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Figures (10)

Fig. 1
Fig. 1

(a) Absolute detectability index, d, results for two model observers: Fisher–Hotelling (FH, which is ideal); and the modified nonprewhitening matched filter (NPWE). The Gaussian signal, g3, (σ=3 pixels) is added to white noise (spectral density, N0) and to a statistically defined background, L8, with dc spectral density, W0, and a correlation distance of 8 pixels. Note that, in the limit of no lumpy background, the value of d for the FH model equals the SNR0 value. (b) Demonstration of two methods of adjusting the models shown in (a) to fit human observer results at W0=0 (uniform background with human average d/SNR of roughly 0.7). The solid and the dashed curves show results of the preferred method of including the two components of observer internal noise in the models [FHN, with Eq. (6); and NPWE, with Eq. (3)]. The unconnected symbols (+, ) show the results of multiplicative scaling.

Fig. 2
Fig. 2

(a) Field size dependence of results for detecting a disk signal (R=4) in white noise added to a uniform background and a lumpy background (correlation distance of 8 pixels). Three observers (XL, MS, and FC) did 768 trials per datum with a standard error of roughly 5% of the mean data values. Detectability decreases as field size increases because of increased difficulty in precisely using the known signal location (dia., diameter). (b) Amplitude threshold variation as a function of the spatial standard deviation (correlation distance) of the background (bkgd corr dist). The signal is a 2D Gaussian with a spatial standard deviation of 3 pixels. The background was produced with a Gaussian low-pass filter. Observer AB was very experienced, and observer PN was both naı̈ve as to the purpose of the experiment and newly trained in signal detection tasks. Predictions of three observer models (FHN, FHCavg, and NPWE) are shown.

Fig. 3
Fig. 3

(a) Contrast threshold elevation as a function of rms background contrast (logarithmic scales, in decibels) for two signals (disk of radius 4 pixels, d4; and a 2D Gaussian with σ=3 pixels, g3) and 3 background correlation distances (sb=4, 8, and 16 pixels; labels L4, L8, L16). All the previous reports for such noise masking have shown a linear threshold increase (slope, 1; dashed line) for high rms noise contrast. The limiting slope for our results is approximately 0.6. (b) Results shown in (a) are plotted in a different way to more clearly demonstrate the anomalous slopes for threshold elevation by use of the equation Ct2=Ct02+α(W0)γ. For the identified lines (d4L4, g3L8, d4L8, and d4L16), the slopes are 0.63, 0.71, 0.55, and 0.54, respectively. The correlation (R2) values between the data and the regression fits are 0.99, 0.97, 0.82 and 0.83, respectively. Previous noise experiments gave an exponent (γ) equal to 1, corresponding to a slope of 1 in the log-log plot.

Fig. 4
Fig. 4

Human observer results for detecting a signal in white noise added to lumpy backgrounds. There are results for four human observers (AB, XL, XC, and MS) and two models (FHN and NPWE) adjusted to fit the average human observer results for a uniform background. The uniform background results (W0=0) are plotted at log(W0/N0) equal to -1.8 for convenience. The dashed curved labeled “ad hoc” is fitted with empirical Eq. (12) with parameters from Table 2. (a) Disk signal (R=4 pixels), background correlation distance sb=4 pixels; (b) disk signal (R=4 pixels), sb=8 pixels; (c) disk signal (R=4 pixels), sb=16 pixels; (d) Gaussian signal (σ=3), sb=8 pixels; (e) disk signal (R=8), sb =16 pixels.

Fig. 5
Fig. 5

Demonstration of quantization effects as a function of the lumpy background spectral density for the FHCMB model with the discrete, nonoverlapping channels used by Myers and Barrett. Total output (d/SNR0) and individual channel outputs, (Vi)1/2, calculated with Eq. (7), are shown for one parameter selection (fc=0.87 c/deg and α=2). The signal is a 2D Gaussian (σ=3), and the lumpy background correlation distance is 8 pixels.

Fig. 6
Fig. 6

(a), (b) FHCMB model predictions compared with average human observer (obs avg) results for two signals, respectively. The parameters used for the Myers–Barrett channels are low cutoff frequencies (0.66 and 0.87 c/deg) and channel bandwidths of 0.58 octave (labeled 1.5) and 1 octave (labeled 2). Predictions for the FHN model with essentially an infinite number of channels are also shown. (a) Disk signal (radius of 4 pixels), background correlation distance of 8 pixels; (b) Gaussian signal (σ=3 pixels), background correlation distance of 8 pixels.

Fig. 7
Fig. 7

(a)–(d) Predictions of the FHCavg model with an eye filter, internal noise, and nonoverlapping channels with parameter averaging are compared with human observer results for four signals, respectively (from Fig. 4). The channels had a bandwidth of 1 octave, and averaging was done over eight values (spanning a range of 1 octave) for the frequency of the first channel boundary. The model is described by Eq. (8). (a) Disk signal (R=4 pixels), background correlation distance sb=4 pixels; (b) disk signal (R=4 pixels), sb=8 pixels; (c) disk signal (R=4 pixels), sb=16 pixels; (d) Gaussian signal (σ=3) and sb=8 pixels.

Fig. 8
Fig. 8

(a) Example amplitude spectra for a set of DOM radial channel filters with a first low-pass center frequency of 2 c/deg, a center frequency ratio of 2, and a DOM base width of 3 octaves (factor of 8). Bandwidths are described in the text. (b) Amplitude spectra for the combination of an eye filter [E(f)=f exp(-cf)] peaked at 4 c/deg and the DOM filter set shown in (a). The first three channel peaks for the product filters are 1.1, 3.1, and 6.3 c/deg. (c) Comparison of the FHCDOM model predictions, obtained by use of two different parameter sets (see text), with average human observer results. The signal is a disk (R=4 pixels) with background correlation distance of 8 pixels. (d) Same as (c), except that the signal is a 2D Gaussian (σ=3 pixels).

Fig. 9
Fig. 9

(a), (b) Human observer detection results as a function of viewing distance. Two signals were used: a disk (R =4 pixels), and a 2D Gaussian (σ=3 pixels). The predictions of the FHCDOM model for the Gaussian signal are shown with two sets of filter parameters, 1/1.4/8 and 2/2/8, for first low-pass filter center frequency, center frequency ratio, and DOM filter base width. (a) Results for uniform background. (b) Results for lumpy background (sb=8 pixels, and W0/N0 =515). (c) Examples of viewing distance dependence for four models (FHN, FHCavg, FHCMB, and NPWE) for the Gaussian signal and lumpy background case. The FHCMB model shows oscillation of total output with distance because of the tuning of individual channels. The Myers–Barrett channels parameters for this example are 0.88-c/deg cutoff frequency and bandwidth of 1 octave.

Fig. 10
Fig. 10

Same background experimental results. Identical background samples were used in both the 2AFC noise fields in a decision trial. There are results for three human observers (AB, XL, and MS). The dashed curves labeled “diff - fit” are the ad hoc Eq. (12) fits to the observer data shown in Fig. 4 (where different background samples were used in the two alternative fields). The solid curves labeled We=W0/10 were obtained with the ad hoc equation but with an effective background spectral density, We, equal to one tenth the actual background spectral density (see text). Exponents for the ad hoc equation are listed in Table 2. (a) Disk signal (R=4 pixels), and background correlation distance sb=4 pixels; (b) disk signal (R=4 pixels), and sb=8 pixels; (c) disk signal (R=4 pixels), and sb=16 pixels; (d) Gaussian signal (σ=3), and sb=8 pixels.

Tables (3)

Tables Icon

Table 1 Summary of Descriptions and Properties of Models Used in This Paper a

Tables Icon

Table 2 Ad Hoc Model Parameter Values for the Different Signals and Lumpy Background Correlation Distances Used in Our Experiments a

Tables Icon

Table 3 Root-Mean-Square Error of Model Fits to Experiment 1 Data a

Equations (42)

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i(x, y)=s(x-x0, y-y0)+n(x, y)+b(x, y)+C,
(dFH)2=|S(u, v)|2N(u, v)+W(u, v)dudv.
(dNPWE)2=|S(u, v)|2|E(u, v)|2dudv2[N(u, v)+W(u, v)]|S(u, v)|2|E(u, v)|4dudv+Ni(u, v)dudv.
Ct2-Ct02=αW0,(d0/d)2-1=βW0/N0,
(dNPWE)2=ACt02/(N0B+C),
A=|F(u, v)|2|E(u, v)|2dudv2,
B=H2(u, v)|F(u, v)|2|E(u, v)|4dudv,
C=Ni(u, v)dudv.
(dFHN)2=2π0E(f)2S(f)2E(f)2[N0+W(f)]+Ni fdf.
(dFHCMB)2=i=1N Vi,
Vi=2π RiS(f)fdf2/2π Ri[N0+W(f)]fdf.
Vi=2π Ri S(f)E(f)fdf2/2π×Ri{E2(f)[N0+W(f)]+Ni(f)}fdf.
H(f, f1/2)=12 1+cosπ(f-f1/2+w/2)w.
An(θ, θw)=12 1+cosπ(θ-θc)θw.
Vm=Vint+002πCm(f, θ)I(f, θ)E(f)fdfdθ=Vint+0Dm(f)I(f)E(f)fdf02πAm(θ)dθ.
(d/d0)2-1=α(W0/N0)γ.
E=--S(u, v)I02dudv,
Ct2=aN0,wherea-1=P(u, v)2H(u, v)2dudv.
Ct2=aN0,where
a-1=E(u, v)2P(u, v)2E(f)2H(f)2+Ni(u, v)/N0dudv.
Ct2=aN0,where
a-1=i=0MRiP(f)2fdf2/RiH(f)2fdf.
Ct2=aN0,where
a=|H(u, v)|2||P(u, v)|2|E(u, v)|4dudv|P(u, v)|2|E(u, v)|2dudv2.
Ct2=bN2,where
b-1=|P(u, v)|2(N1/N2)|H1(u, v)|2+|H2(u, v)|2dudv.
Ct2=b1N1+b2N2,where
bk=|Hk(u, v)|2||P(u, v)|2|E(u, v)|4dudv|P(u, v)|2|E(u, v)|2dudv2.
Ct2=Ct12+b2N2.
d0dNPW2-1=βW0N0,
(dNPWE)2=I12(1+Bn)N0I2+W0I3+Ni0I4,
I1=|S(u, v)|2|E(u, v)|2dudv,
I2=|S(u, v)|2|E(u, v)|4dudv,
I3=|S(u, v)|2|H(u, v)|2|E(u, v)|4dudv,
I4=|Hi(u, v)|2dudv.
(dFHN)2=|S(u, v)|2|E(u, v)|2|E(u, v)|2[(1+Bn)N0+W(u, v)]+Ni0×dudv.
Vm=002πCm(f, θ)I(f, θ)E(f)fdfdθ=0Dm(f)I(f)E(f)fdf02πAm(θ)dθ.
Vm=0Dm(f)S(f)E(f)fdf02πAm(θ)dθ=ρmαm.
Kmn=VmVn=Um(f, θ)fdfdθUn(f1, θ1)f1df1dθ1.
Kmn=Um(f, θ)Un(f, θ)fdfdθ.
Kmn=0R2(f)Dm(f)Dn(f)E2(f)fdf×02πAm(θ)An(θ)dθ.
R2(f)=W0|H(f)|2+N0,

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