Abstract

Bisection thresholds were measured as a function of the separation of the lines. For separations of less than 1.5 min, the addition of flanking lines facilitates bisection so that thresholds of less than tmm1 sec for discriminating the direction of offset could be reliably obtained. For larger separations an interval could be bisected to an accuracy of 1 part in 60. Experiments varying the length, luminance, and overlap of the lines suggest that different cues are used in these two regimes. A dual space-size analysis is presented that can account for these bisection thresholds over a wide range of experimental conditions. This quantitative analysis produces viewprints of the stimuli (analogous to the voiceprint of audition). Each viewprint shows the output of many spatial filters of different positions and sizes. A new filter shape is introduced that has advantages for modeling the visual system. The sensitivity of each filter is fixed by the contrast-response function. The analysis further shows that the limiting factors in spatial hyperacuity are both the contrast-response function and the spatial grain.

© 1985 Optical Society of America

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1985 (2)

E. H. Adelson, J. R. Bergen, “Spatio-temporal energy models for the perception of motion,” J. Opt. Soc. Am. A. 2, 284–299 (1985).
[CrossRef] [PubMed]

K. Nakayama, G. Silverman, “Detection and discrimination of sinusoidal grating displacements,” J. Opt. Soc. Am. A. 2, 267–274 (1985).
[CrossRef] [PubMed]

1984 (5)

H. R. Wilson, D. Regan, “Spatial-frequency adaptation and grating discrimination: predictions of a line-element model,” J. Opt. Soc. Am. A. 1, 1091–1096 (1984).
[CrossRef] [PubMed]

P. H. Schiller, “The connections of the retinal on and off pathways to the lateral geniculate nucleus of the monkey,” Vision Res. 24, 923–932 (1984).
[CrossRef] [PubMed]

D. R. Badcock, “How do we discriminate relative spatial phase?” Vision Res. 24, 1847–1857 (1984).
[CrossRef] [PubMed]

R. J. Watt, “Towards a general theory of the visual acuities for shape and spatial arrangement,” Vision Res. 24, 1377–1386 (1984).
[CrossRef] [PubMed]

H. R. Wilson, D. J. Gelb, “Modified line-element theory for spatial-frequency and width discrimination,” J. Opt. Soc. Am. A. 1, 126–131 (1984).

1983 (7)

D. Regan, K. I. Beverley, “Spatial-frequency discrimination and detection: comparison of postadaptation thresholds,” J. Opt. Soc. Am. 73, 1686–1690 (1983).
[CrossRef]

R. J. Watt, M. J. Morgan, “The use of different cues in vernier acuity,” Vision Res. 23, 991–995 (1983).
[CrossRef] [PubMed]

D. M. Levi, S. A. Klein, “Spatial localization in normal and amblyopic vision,” Vision Res. 23, 1005–1017 (1983).
[CrossRef] [PubMed]

P. J. Burt, E. H. Adelson, “The Laplacian pyramid as a compact image code,” IEEE Trans. Commun. COM-31, 532–540 (1983).
[CrossRef]

R. J. Watt, M. J. Morgan, S. P. McKee, “Exposure duration affects the sensitivity of vernier acuity to target motion,” Vision Res. 23, 541–546 (1983).
[CrossRef] [PubMed]

G. E. Legge, D. Kersten, “Light and dark bars: contrast discrimination,” Vision Res. 23, 473–484 (1983).
[CrossRef]

H. Wassle, L. Peichl, B. B. Boycott, “A spatial analysis of on and off-ganglion cells in the cat retina,” Vision Res. 23, 1151–1160 (1983).
[CrossRef]

1982 (4)

J. Hirsch, R. Hylton, “Limits of spatial frequency discrimination as evidence of neural interpolation,” J. Opt. Soc. Am. 72, 1367–1374 (1982);G. Westheimer, “Line-separation discrimination curve in the human fovea: smooth or segmented?” J. Opt. Soc. Am. A. 1, 683–684 (1984);J. Hirsch, “Comment on ‘Line-separation discrimination curve in the human fovea: smooth or segmented?’ ” J. Opt. Soc. Am. A. 2, 477–478 (1985);G. Westheimer, “Reply to “Comment on ‘Line-separation discrimination curve in the human fovea: smooth or segmented?’ ” J. Opt. Soc. Am. A. 2, 479 (1985).
[CrossRef] [PubMed]

J. H. T. Jamar, J. C. Campagne, J. J. Koenderink, “Detectability of amplitude- and frequency-modulation of supra-threshold sine-wave gratings,” Vision Res. 22, 407–416 (1982).
[CrossRef]

B. Sakitt, H. B. Barlow, “A model for the economical encoding of the visual image in cerebral cortex,” Biol. Cybernet. 43, 97–108 (1982).
[CrossRef]

A. Treisman, “Perceptual grouping and attention in visual search for features and for objects,” J. Exp. Psychol. 8, 194–216 (1982).

1981 (3)

J. G. Robson, N. Graham, “Probability summation and regional variation in contrast sensitivity across the visual field,” Vision Res. 22, 409–418 (1981).
[CrossRef]

M. Fahle, T. Poggio, “Visual hyperacuity: spatiotemporal interpolation in human vision,” Proc. R. Soc. Lond. Ser. B 213, 451–477 (1981).
[CrossRef]

G. Westheimer, “Visual hyperacuity,” Progr. Sensory Physiol. 1, 1–30 (1981).
[CrossRef]

1980 (3)

S. A. Klein, C. F. Stromeyer, “On inhibition between spatial frequency channels: adaptation to complex gratings,” Vision Res. 20, 459–466 (1980).
[CrossRef] [PubMed]

M. A. Georgeson, “Spatial frequency analysis in early visual processing,” Phil. Trans. R. Soc. London Ser. B 290, 11–22 (1980).
[CrossRef]

U. T. Keesey, “Effects of involuntary eye movements on visual acuity,” J. Opt. Soc. Am. 50, 769–774 (1980).
[CrossRef]

1979 (3)

L. E. Arend, R. V. Lange, “Phase-dependent interaction of widely separated spatial frequencies in pattern discrimination,” Vision Res. 19, 1089–1092 (1979).
[CrossRef] [PubMed]

H. B. Barlow, “Reconstructing the visual image in space and time,” Nature 279, 189–190 (1979).
[CrossRef] [PubMed]

H. R. Wilson, J. R. Bergen, “A four mechanism model for spatial vision,” Vision Res. 19, 19–32 (1979).
[CrossRef]

1978 (1)

G. Westheimer, “Spatial phase sensitivity for sinusoidal grating targets,” Vision Res. 18, 1073–1074 (1978).
[CrossRef] [PubMed]

1977 (3)

G. Westheimer, “Spatial frequency and light-spread descriptions of visual acuity and hyperacuity,” J. Opt. Soc. Am. 67, 207–212 (1977).
[CrossRef] [PubMed]

G. Westheimer, S. P. McKee, “Spatial configurations for visual hyperacuity,” Vision Res. 17, 940–947 (1977).
[CrossRef]

J. O. Limb, C. B. Rubenstein, “A model of threshold vision incorporating inhomogeneity of the visual field,” Vision Res. 17, 571–584 (1977).
[CrossRef] [PubMed]

1976 (1)

O. Estevez, C. R. Cavonious, “Low-frequency attentuation in the detection of gratings: sorting out the artifacts,” Vision Res. 16, 497–500 (1976).
[CrossRef]

1975 (5)

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

G. Westheimer, “Visual acuity and hyperacuity,” Invest. Ophthalmol. 14, 570–572 (1975).
[PubMed]

C. F. Stromeyer, S. Klein, “Evidence against narrow-band spatial frequency channels in vision: the detectability of frequency modulated gratings,” Vision Res. 15, 899–910 (1975).
[CrossRef]

G. Westheimer, S. P. McKee, “Visual acuity in the presence of retinal image motion,” J. Opt. Soc. Am. 65, 847–850 (1975).
[CrossRef] [PubMed]

D. J. Tolhurst, R. S. Dealy, “The detection and identification of lines and edges,” Vision Res. 15, 1367–1372 (1975).
[CrossRef] [PubMed]

1974 (3)

C. F. Stromeyer, S. Klein, “Spatial frequency channels in human vision as asymmetric (edge) mechanisms,” Vision Res. 14, 1409–1420 (1974).
[CrossRef] [PubMed]

J. Hoekstra, D. P. J. van der Goot, G. van den Brink, F. A. Bilsen, “The influence of the number of cycles upon the visual contrast threshold for spatial sine-wave patterns,” Vision Res. 14, 365–368 (1974).
[CrossRef] [PubMed]

S. Klein, C. F. Stromeyer, L. Ganz, “The simultaneous spatial frequency shift: a dissociation between the detection and perception of gratings,” Vision Res. 14, 1421–1432 (1974).
[CrossRef] [PubMed]

1972 (1)

1970 (1)

J. Nachmias, E. C. Kocher, “Visual detection and discrimination of luminance increments,” J. Opt. Soc. Am. 60, 381–389 (1970).
[CrossRef]

1969 (3)

C. Blakemore, P. Sutton, “Size adaptation: a new aftereffect,” Science 166, 245–247 (1969).
[CrossRef] [PubMed]

C. B. Blakemore, F. W. Campbell, “On the existence of neurones in the human visual system selectively sensitive to the orientation and size of retinal images,” J. Physiol. (London) 203, 237–260 (1969).

F. W. Campbell, R. H. S. Carpenter, J. Z. Levinson, “Visibility of aperiodic patterns compared with that of sinusoidal gratings,” J. Physiol. (London) 204, 283–298 (1969).

1968 (2)

D. H. Hubel, T. N. Wiesel, “Receptive fields and functional architecture of monkey striate cortex,” J. Physiol. (London) 195, 215–243 (1968).

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

1967 (1)

G. Westheimer, “Spatial interaction in human cone vision,” J. Physiol. 190, 139–154 (1967).
[PubMed]

1965 (1)

F. W. Campbell, D. G. Green, “Optical and retinal factors affecting visual resolution,” J. Physiol. (London) 181, 576–593 (1965).

1958 (1)

G. T. Fechner, “Uber ein psychophysisches Grundgesetz,” Abh. Leipziger Ges. VI, 457–532 (1958).

1946 (1)

D. Gabor, “Theory of communication,” Proc. Inst. Electr. Eng. Part 3, 93, 439–457 (1946).

1939 (1)

S. Hecht, E. U. Mintz, “The visibility of single lines of various illuminations and the retinal basis of visual resolution,” J. Gen. Physiol. 22, 593–612 (1939).
[CrossRef] [PubMed]

1858 (1)

A. W. Volkmann, “Uber den Einfluss der Ilbung auf das Erkennen raumlicher Distanzen,” Leipsziger Ber. X, 38–69 (1858).

Adelson, E. H.

E. H. Adelson, J. R. Bergen, “Spatio-temporal energy models for the perception of motion,” J. Opt. Soc. Am. A. 2, 284–299 (1985).
[CrossRef] [PubMed]

P. J. Burt, E. H. Adelson, “The Laplacian pyramid as a compact image code,” IEEE Trans. Commun. COM-31, 532–540 (1983).
[CrossRef]

Arend, L. E.

L. E. Arend, R. V. Lange, “Phase-dependent interaction of widely separated spatial frequencies in pattern discrimination,” Vision Res. 19, 1089–1092 (1979).
[CrossRef] [PubMed]

Badcock, D. R.

D. R. Badcock, “How do we discriminate relative spatial phase?” Vision Res. 24, 1847–1857 (1984).
[CrossRef] [PubMed]

Barlow, H. B.

B. Sakitt, H. B. Barlow, “A model for the economical encoding of the visual image in cerebral cortex,” Biol. Cybernet. 43, 97–108 (1982).
[CrossRef]

H. B. Barlow, “Reconstructing the visual image in space and time,” Nature 279, 189–190 (1979).
[CrossRef] [PubMed]

Bedell, H. E.

H. E. Bedell, University of Houston, Houston, Tex. 77004 (personal communication).

Bergen, J. R.

E. H. Adelson, J. R. Bergen, “Spatio-temporal energy models for the perception of motion,” J. Opt. Soc. Am. A. 2, 284–299 (1985).
[CrossRef] [PubMed]

H. R. Wilson, J. R. Bergen, “A four mechanism model for spatial vision,” Vision Res. 19, 19–32 (1979).
[CrossRef]

Beverley, K. I.

D. Regan, K. I. Beverley, “Spatial-frequency discrimination and detection: comparison of postadaptation thresholds,” J. Opt. Soc. Am. 73, 1686–1690 (1983).
[CrossRef]

Bilsen, F. A.

J. Hoekstra, D. P. J. van der Goot, G. van den Brink, F. A. Bilsen, “The influence of the number of cycles upon the visual contrast threshold for spatial sine-wave patterns,” Vision Res. 14, 365–368 (1974).
[CrossRef] [PubMed]

Blakemore, C.

C. Blakemore, P. Sutton, “Size adaptation: a new aftereffect,” Science 166, 245–247 (1969).
[CrossRef] [PubMed]

Blakemore, C. B.

C. B. Blakemore, F. W. Campbell, “On the existence of neurones in the human visual system selectively sensitive to the orientation and size of retinal images,” J. Physiol. (London) 203, 237–260 (1969).

Boycott, B. B.

H. Wassle, L. Peichl, B. B. Boycott, “A spatial analysis of on and off-ganglion cells in the cat retina,” Vision Res. 23, 1151–1160 (1983).
[CrossRef]

Burt, P. J.

P. J. Burt, E. H. Adelson, “The Laplacian pyramid as a compact image code,” IEEE Trans. Commun. COM-31, 532–540 (1983).
[CrossRef]

Campagne, J. C.

J. H. T. Jamar, J. C. Campagne, J. J. Koenderink, “Detectability of amplitude- and frequency-modulation of supra-threshold sine-wave gratings,” Vision Res. 22, 407–416 (1982).
[CrossRef]

Campbell, F. W.

F. W. Campbell, R. H. S. Carpenter, J. Z. Levinson, “Visibility of aperiodic patterns compared with that of sinusoidal gratings,” J. Physiol. (London) 204, 283–298 (1969).

C. B. Blakemore, F. W. Campbell, “On the existence of neurones in the human visual system selectively sensitive to the orientation and size of retinal images,” J. Physiol. (London) 203, 237–260 (1969).

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

F. W. Campbell, D. G. Green, “Optical and retinal factors affecting visual resolution,” J. Physiol. (London) 181, 576–593 (1965).

Carpenter, R. H. S.

F. W. Campbell, R. H. S. Carpenter, J. Z. Levinson, “Visibility of aperiodic patterns compared with that of sinusoidal gratings,” J. Physiol. (London) 204, 283–298 (1969).

Cavonious, C. R.

O. Estevez, C. R. Cavonious, “Low-frequency attentuation in the detection of gratings: sorting out the artifacts,” Vision Res. 16, 497–500 (1976).
[CrossRef]

Crick, F. H. C.

F. H. C. Crick, D. C. Marr, T. Poggio, “An information-processing approach to understanding the visual cortex,” in The Organization of the Cerebral Cortex, F. O. Schmidt, ed. (MIT, Boston, Mass., 1981), pp. 505–533.

Dealy, R. S.

D. J. Tolhurst, R. S. Dealy, “The detection and identification of lines and edges,” Vision Res. 15, 1367–1372 (1975).
[CrossRef] [PubMed]

Estevez, O.

O. Estevez, C. R. Cavonious, “Low-frequency attentuation in the detection of gratings: sorting out the artifacts,” Vision Res. 16, 497–500 (1976).
[CrossRef]

Fahle, M.

M. Fahle, T. Poggio, “Visual hyperacuity: spatiotemporal interpolation in human vision,” Proc. R. Soc. Lond. Ser. B 213, 451–477 (1981).
[CrossRef]

Fechner, G. T.

G. T. Fechner, “Uber ein psychophysisches Grundgesetz,” Abh. Leipziger Ges. VI, 457–532 (1958).

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pp. 367–375.

Gabor, D.

D. Gabor, “Theory of communication,” Proc. Inst. Electr. Eng. Part 3, 93, 439–457 (1946).

Ganz, L.

S. Klein, C. F. Stromeyer, L. Ganz, “The simultaneous spatial frequency shift: a dissociation between the detection and perception of gratings,” Vision Res. 14, 1421–1432 (1974).
[CrossRef] [PubMed]

Gelb, D. J.

H. R. Wilson, D. J. Gelb, “Modified line-element theory for spatial-frequency and width discrimination,” J. Opt. Soc. Am. A. 1, 126–131 (1984).

Georgeson, M. A.

M. A. Georgeson, “Spatial frequency analysis in early visual processing,” Phil. Trans. R. Soc. London Ser. B 290, 11–22 (1980).
[CrossRef]

Graham, N.

J. G. Robson, N. Graham, “Probability summation and regional variation in contrast sensitivity across the visual field,” Vision Res. 22, 409–418 (1981).
[CrossRef]

Green, D. G.

F. W. Campbell, D. G. Green, “Optical and retinal factors affecting visual resolution,” J. Physiol. (London) 181, 576–593 (1965).

Hecht, S.

S. Hecht, E. U. Mintz, “The visibility of single lines of various illuminations and the retinal basis of visual resolution,” J. Gen. Physiol. 22, 593–612 (1939).
[CrossRef] [PubMed]

Hirsch, J.

Hoekstra, J.

J. Hoekstra, D. P. J. van der Goot, G. van den Brink, F. A. Bilsen, “The influence of the number of cycles upon the visual contrast threshold for spatial sine-wave patterns,” Vision Res. 14, 365–368 (1974).
[CrossRef] [PubMed]

Hubel, D. H.

D. H. Hubel, T. N. Wiesel, “Receptive fields and functional architecture of monkey striate cortex,” J. Physiol. (London) 195, 215–243 (1968).

Hylton, R.

Jamar, J. H. T.

J. H. T. Jamar, J. C. Campagne, J. J. Koenderink, “Detectability of amplitude- and frequency-modulation of supra-threshold sine-wave gratings,” Vision Res. 22, 407–416 (1982).
[CrossRef]

Julesz, B.

Keesey, U. T.

Kelly, D. H.

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

Kersten, D.

G. E. Legge, D. Kersten, “Light and dark bars: contrast discrimination,” Vision Res. 23, 473–484 (1983).
[CrossRef]

Klein, S.

C. F. Stromeyer, S. Klein, “Evidence against narrow-band spatial frequency channels in vision: the detectability of frequency modulated gratings,” Vision Res. 15, 899–910 (1975).
[CrossRef]

S. Klein, C. F. Stromeyer, L. Ganz, “The simultaneous spatial frequency shift: a dissociation between the detection and perception of gratings,” Vision Res. 14, 1421–1432 (1974).
[CrossRef] [PubMed]

C. F. Stromeyer, S. Klein, “Spatial frequency channels in human vision as asymmetric (edge) mechanisms,” Vision Res. 14, 1409–1420 (1974).
[CrossRef] [PubMed]

Klein, S. A.

D. M. Levi, S. A. Klein, “Spatial localization in normal and amblyopic vision,” Vision Res. 23, 1005–1017 (1983).
[CrossRef] [PubMed]

S. A. Klein, C. F. Stromeyer, “On inhibition between spatial frequency channels: adaptation to complex gratings,” Vision Res. 20, 459–466 (1980).
[CrossRef] [PubMed]

Kocher, E. C.

J. Nachmias, E. C. Kocher, “Visual detection and discrimination of luminance increments,” J. Opt. Soc. Am. 60, 381–389 (1970).
[CrossRef]

Koenderink, J. J.

J. H. T. Jamar, J. C. Campagne, J. J. Koenderink, “Detectability of amplitude- and frequency-modulation of supra-threshold sine-wave gratings,” Vision Res. 22, 407–416 (1982).
[CrossRef]

Lange, R. V.

L. E. Arend, R. V. Lange, “Phase-dependent interaction of widely separated spatial frequencies in pattern discrimination,” Vision Res. 19, 1089–1092 (1979).
[CrossRef] [PubMed]

Legge, G. E.

G. E. Legge, D. Kersten, “Light and dark bars: contrast discrimination,” Vision Res. 23, 473–484 (1983).
[CrossRef]

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D. M. Levi, S. A. Klein, “Spatial localization in normal and amblyopic vision,” Vision Res. 23, 1005–1017 (1983).
[CrossRef] [PubMed]

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F. W. Campbell, R. H. S. Carpenter, J. Z. Levinson, “Visibility of aperiodic patterns compared with that of sinusoidal gratings,” J. Physiol. (London) 204, 283–298 (1969).

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J. O. Limb, C. B. Rubenstein, “A model of threshold vision incorporating inhomogeneity of the visual field,” Vision Res. 17, 571–584 (1977).
[CrossRef] [PubMed]

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D. Marr, Vision (Freeman, San Francisco, Calif., 1982).We have recently become aware of some interesting theorems for the zero crossings of space–size plots by A. L. Yuille, T. Poggio, in “Scaling theorems for zero-crossings,” A. I. Memo 722,and “Fingerprint theorems for zero-crossings,” A. I. Memo 730 (Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, Mass., 1983).

Marr, D. C.

F. H. C. Crick, D. C. Marr, T. Poggio, “An information-processing approach to understanding the visual cortex,” in The Organization of the Cerebral Cortex, F. O. Schmidt, ed. (MIT, Boston, Mass., 1981), pp. 505–533.

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R. J. Watt, M. J. Morgan, S. P. McKee, “Exposure duration affects the sensitivity of vernier acuity to target motion,” Vision Res. 23, 541–546 (1983).
[CrossRef] [PubMed]

G. Westheimer, S. P. McKee, “Spatial configurations for visual hyperacuity,” Vision Res. 17, 940–947 (1977).
[CrossRef]

G. Westheimer, S. P. McKee, “Visual acuity in the presence of retinal image motion,” J. Opt. Soc. Am. 65, 847–850 (1975).
[CrossRef] [PubMed]

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N. McWhirter, Guiness Book of World Records (Bantam, New York, 1984), p. 33.

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S. Hecht, E. U. Mintz, “The visibility of single lines of various illuminations and the retinal basis of visual resolution,” J. Gen. Physiol. 22, 593–612 (1939).
[CrossRef] [PubMed]

Morgan, M. J.

R. J. Watt, M. J. Morgan, S. P. McKee, “Exposure duration affects the sensitivity of vernier acuity to target motion,” Vision Res. 23, 541–546 (1983).
[CrossRef] [PubMed]

R. J. Watt, M. J. Morgan, “The use of different cues in vernier acuity,” Vision Res. 23, 991–995 (1983).
[CrossRef] [PubMed]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pp. 367–375.

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J. Nachmias, E. C. Kocher, “Visual detection and discrimination of luminance increments,” J. Opt. Soc. Am. 60, 381–389 (1970).
[CrossRef]

Nakayama, K.

K. Nakayama, G. Silverman, “Detection and discrimination of sinusoidal grating displacements,” J. Opt. Soc. Am. A. 2, 267–274 (1985).
[CrossRef] [PubMed]

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H. Wassle, L. Peichl, B. B. Boycott, “A spatial analysis of on and off-ganglion cells in the cat retina,” Vision Res. 23, 1151–1160 (1983).
[CrossRef]

Poggio, T.

M. Fahle, T. Poggio, “Visual hyperacuity: spatiotemporal interpolation in human vision,” Proc. R. Soc. Lond. Ser. B 213, 451–477 (1981).
[CrossRef]

F. H. C. Crick, D. C. Marr, T. Poggio, “An information-processing approach to understanding the visual cortex,” in The Organization of the Cerebral Cortex, F. O. Schmidt, ed. (MIT, Boston, Mass., 1981), pp. 505–533.

Regan, D.

H. R. Wilson, D. Regan, “Spatial-frequency adaptation and grating discrimination: predictions of a line-element model,” J. Opt. Soc. Am. A. 1, 1091–1096 (1984).
[CrossRef] [PubMed]

D. Regan, K. I. Beverley, “Spatial-frequency discrimination and detection: comparison of postadaptation thresholds,” J. Opt. Soc. Am. 73, 1686–1690 (1983).
[CrossRef]

Robson, J. G.

J. G. Robson, N. Graham, “Probability summation and regional variation in contrast sensitivity across the visual field,” Vision Res. 22, 409–418 (1981).
[CrossRef]

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

Rubenstein, C. B.

J. O. Limb, C. B. Rubenstein, “A model of threshold vision incorporating inhomogeneity of the visual field,” Vision Res. 17, 571–584 (1977).
[CrossRef] [PubMed]

Sakitt, B.

B. Sakitt, H. B. Barlow, “A model for the economical encoding of the visual image in cerebral cortex,” Biol. Cybernet. 43, 97–108 (1982).
[CrossRef]

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P. H. Schiller, “The connections of the retinal on and off pathways to the lateral geniculate nucleus of the monkey,” Vision Res. 24, 923–932 (1984).
[CrossRef] [PubMed]

Silverman, G.

K. Nakayama, G. Silverman, “Detection and discrimination of sinusoidal grating displacements,” J. Opt. Soc. Am. A. 2, 267–274 (1985).
[CrossRef] [PubMed]

Stromeyer, C. F.

S. A. Klein, C. F. Stromeyer, “On inhibition between spatial frequency channels: adaptation to complex gratings,” Vision Res. 20, 459–466 (1980).
[CrossRef] [PubMed]

C. F. Stromeyer, S. Klein, “Evidence against narrow-band spatial frequency channels in vision: the detectability of frequency modulated gratings,” Vision Res. 15, 899–910 (1975).
[CrossRef]

S. Klein, C. F. Stromeyer, L. Ganz, “The simultaneous spatial frequency shift: a dissociation between the detection and perception of gratings,” Vision Res. 14, 1421–1432 (1974).
[CrossRef] [PubMed]

C. F. Stromeyer, S. Klein, “Spatial frequency channels in human vision as asymmetric (edge) mechanisms,” Vision Res. 14, 1409–1420 (1974).
[CrossRef] [PubMed]

C. F. Stromeyer, B. Julesz, “Spatial-frequency masking in vision: critical bands and spread of masking,” J. Opt. Soc. Am. 62, 1221–1232 (1972).
[CrossRef] [PubMed]

Sutton, P.

C. Blakemore, P. Sutton, “Size adaptation: a new aftereffect,” Science 166, 245–247 (1969).
[CrossRef] [PubMed]

Tolhurst, D. J.

D. J. Tolhurst, R. S. Dealy, “The detection and identification of lines and edges,” Vision Res. 15, 1367–1372 (1975).
[CrossRef] [PubMed]

Treisman, A.

A. Treisman, “Perceptual grouping and attention in visual search for features and for objects,” J. Exp. Psychol. 8, 194–216 (1982).

van den Brink, G.

J. Hoekstra, D. P. J. van der Goot, G. van den Brink, F. A. Bilsen, “The influence of the number of cycles upon the visual contrast threshold for spatial sine-wave patterns,” Vision Res. 14, 365–368 (1974).
[CrossRef] [PubMed]

van der Goot, D. P. J.

J. Hoekstra, D. P. J. van der Goot, G. van den Brink, F. A. Bilsen, “The influence of the number of cycles upon the visual contrast threshold for spatial sine-wave patterns,” Vision Res. 14, 365–368 (1974).
[CrossRef] [PubMed]

Volkmann, A. W.

A. W. Volkmann, “Uber den Einfluss der Ilbung auf das Erkennen raumlicher Distanzen,” Leipsziger Ber. X, 38–69 (1858).

Wassle, H.

H. Wassle, L. Peichl, B. B. Boycott, “A spatial analysis of on and off-ganglion cells in the cat retina,” Vision Res. 23, 1151–1160 (1983).
[CrossRef]

Watt, R. J.

R. J. Watt, “Towards a general theory of the visual acuities for shape and spatial arrangement,” Vision Res. 24, 1377–1386 (1984).
[CrossRef] [PubMed]

R. J. Watt, M. J. Morgan, “The use of different cues in vernier acuity,” Vision Res. 23, 991–995 (1983).
[CrossRef] [PubMed]

R. J. Watt, M. J. Morgan, S. P. McKee, “Exposure duration affects the sensitivity of vernier acuity to target motion,” Vision Res. 23, 541–546 (1983).
[CrossRef] [PubMed]

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G. Westheimer, “Visual hyperacuity,” Progr. Sensory Physiol. 1, 1–30 (1981).
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G. Westheimer, “Spatial phase sensitivity for sinusoidal grating targets,” Vision Res. 18, 1073–1074 (1978).
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G. Westheimer, S. P. McKee, “Visual acuity in the presence of retinal image motion,” J. Opt. Soc. Am. 65, 847–850 (1975).
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D. H. Hubel, T. N. Wiesel, “Receptive fields and functional architecture of monkey striate cortex,” J. Physiol. (London) 195, 215–243 (1968).

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H. R. Wilson, D. Regan, “Spatial-frequency adaptation and grating discrimination: predictions of a line-element model,” J. Opt. Soc. Am. A. 1, 1091–1096 (1984).
[CrossRef] [PubMed]

H. R. Wilson, D. J. Gelb, “Modified line-element theory for spatial-frequency and width discrimination,” J. Opt. Soc. Am. A. 1, 126–131 (1984).

H. R. Wilson, J. R. Bergen, “A four mechanism model for spatial vision,” Vision Res. 19, 19–32 (1979).
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A. Treisman, “Perceptual grouping and attention in visual search for features and for objects,” J. Exp. Psychol. 8, 194–216 (1982).

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S. Hecht, E. U. Mintz, “The visibility of single lines of various illuminations and the retinal basis of visual resolution,” J. Gen. Physiol. 22, 593–612 (1939).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (7)

G. Westheimer, “Spatial frequency and light-spread descriptions of visual acuity and hyperacuity,” J. Opt. Soc. Am. 67, 207–212 (1977).
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J. Nachmias, E. C. Kocher, “Visual detection and discrimination of luminance increments,” J. Opt. Soc. Am. 60, 381–389 (1970).
[CrossRef]

C. F. Stromeyer, B. Julesz, “Spatial-frequency masking in vision: critical bands and spread of masking,” J. Opt. Soc. Am. 62, 1221–1232 (1972).
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J. Opt. Soc. Am. A. (4)

H. R. Wilson, D. J. Gelb, “Modified line-element theory for spatial-frequency and width discrimination,” J. Opt. Soc. Am. A. 1, 126–131 (1984).

H. R. Wilson, D. Regan, “Spatial-frequency adaptation and grating discrimination: predictions of a line-element model,” J. Opt. Soc. Am. A. 1, 1091–1096 (1984).
[CrossRef] [PubMed]

K. Nakayama, G. Silverman, “Detection and discrimination of sinusoidal grating displacements,” J. Opt. Soc. Am. A. 2, 267–274 (1985).
[CrossRef] [PubMed]

E. H. Adelson, J. R. Bergen, “Spatio-temporal energy models for the perception of motion,” J. Opt. Soc. Am. A. 2, 284–299 (1985).
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J. Physiol. (1)

G. Westheimer, “Spatial interaction in human cone vision,” J. Physiol. 190, 139–154 (1967).
[PubMed]

J. Physiol. (London) (5)

F. W. Campbell, D. G. Green, “Optical and retinal factors affecting visual resolution,” J. Physiol. (London) 181, 576–593 (1965).

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

F. W. Campbell, R. H. S. Carpenter, J. Z. Levinson, “Visibility of aperiodic patterns compared with that of sinusoidal gratings,” J. Physiol. (London) 204, 283–298 (1969).

D. H. Hubel, T. N. Wiesel, “Receptive fields and functional architecture of monkey striate cortex,” J. Physiol. (London) 195, 215–243 (1968).

C. B. Blakemore, F. W. Campbell, “On the existence of neurones in the human visual system selectively sensitive to the orientation and size of retinal images,” J. Physiol. (London) 203, 237–260 (1969).

Leipsziger Ber. (1)

A. W. Volkmann, “Uber den Einfluss der Ilbung auf das Erkennen raumlicher Distanzen,” Leipsziger Ber. X, 38–69 (1858).

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M. A. Georgeson, “Spatial frequency analysis in early visual processing,” Phil. Trans. R. Soc. London Ser. B 290, 11–22 (1980).
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D. Gabor, “Theory of communication,” Proc. Inst. Electr. Eng. Part 3, 93, 439–457 (1946).

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M. Fahle, T. Poggio, “Visual hyperacuity: spatiotemporal interpolation in human vision,” Proc. R. Soc. Lond. Ser. B 213, 451–477 (1981).
[CrossRef]

Progr. Sensory Physiol. (1)

G. Westheimer, “Visual hyperacuity,” Progr. Sensory Physiol. 1, 1–30 (1981).
[CrossRef]

Science (1)

C. Blakemore, P. Sutton, “Size adaptation: a new aftereffect,” Science 166, 245–247 (1969).
[CrossRef] [PubMed]

Vision Res. (23)

R. J. Watt, “Towards a general theory of the visual acuities for shape and spatial arrangement,” Vision Res. 24, 1377–1386 (1984).
[CrossRef] [PubMed]

S. Klein, C. F. Stromeyer, L. Ganz, “The simultaneous spatial frequency shift: a dissociation between the detection and perception of gratings,” Vision Res. 14, 1421–1432 (1974).
[CrossRef] [PubMed]

D. J. Tolhurst, R. S. Dealy, “The detection and identification of lines and edges,” Vision Res. 15, 1367–1372 (1975).
[CrossRef] [PubMed]

H. Wassle, L. Peichl, B. B. Boycott, “A spatial analysis of on and off-ganglion cells in the cat retina,” Vision Res. 23, 1151–1160 (1983).
[CrossRef]

P. H. Schiller, “The connections of the retinal on and off pathways to the lateral geniculate nucleus of the monkey,” Vision Res. 24, 923–932 (1984).
[CrossRef] [PubMed]

J. H. T. Jamar, J. C. Campagne, J. J. Koenderink, “Detectability of amplitude- and frequency-modulation of supra-threshold sine-wave gratings,” Vision Res. 22, 407–416 (1982).
[CrossRef]

L. E. Arend, R. V. Lange, “Phase-dependent interaction of widely separated spatial frequencies in pattern discrimination,” Vision Res. 19, 1089–1092 (1979).
[CrossRef] [PubMed]

G. Westheimer, “Spatial phase sensitivity for sinusoidal grating targets,” Vision Res. 18, 1073–1074 (1978).
[CrossRef] [PubMed]

R. J. Watt, M. J. Morgan, “The use of different cues in vernier acuity,” Vision Res. 23, 991–995 (1983).
[CrossRef] [PubMed]

D. M. Levi, S. A. Klein, “Spatial localization in normal and amblyopic vision,” Vision Res. 23, 1005–1017 (1983).
[CrossRef] [PubMed]

G. Westheimer, S. P. McKee, “Spatial configurations for visual hyperacuity,” Vision Res. 17, 940–947 (1977).
[CrossRef]

R. J. Watt, M. J. Morgan, S. P. McKee, “Exposure duration affects the sensitivity of vernier acuity to target motion,” Vision Res. 23, 541–546 (1983).
[CrossRef] [PubMed]

C. F. Stromeyer, S. Klein, “Evidence against narrow-band spatial frequency channels in vision: the detectability of frequency modulated gratings,” Vision Res. 15, 899–910 (1975).
[CrossRef]

H. R. Wilson, J. R. Bergen, “A four mechanism model for spatial vision,” Vision Res. 19, 19–32 (1979).
[CrossRef]

C. F. Stromeyer, S. Klein, “Spatial frequency channels in human vision as asymmetric (edge) mechanisms,” Vision Res. 14, 1409–1420 (1974).
[CrossRef] [PubMed]

J. O. Limb, C. B. Rubenstein, “A model of threshold vision incorporating inhomogeneity of the visual field,” Vision Res. 17, 571–584 (1977).
[CrossRef] [PubMed]

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

J. Hoekstra, D. P. J. van der Goot, G. van den Brink, F. A. Bilsen, “The influence of the number of cycles upon the visual contrast threshold for spatial sine-wave patterns,” Vision Res. 14, 365–368 (1974).
[CrossRef] [PubMed]

O. Estevez, C. R. Cavonious, “Low-frequency attentuation in the detection of gratings: sorting out the artifacts,” Vision Res. 16, 497–500 (1976).
[CrossRef]

J. G. Robson, N. Graham, “Probability summation and regional variation in contrast sensitivity across the visual field,” Vision Res. 22, 409–418 (1981).
[CrossRef]

S. A. Klein, C. F. Stromeyer, “On inhibition between spatial frequency channels: adaptation to complex gratings,” Vision Res. 20, 459–466 (1980).
[CrossRef] [PubMed]

G. E. Legge, D. Kersten, “Light and dark bars: contrast discrimination,” Vision Res. 23, 473–484 (1983).
[CrossRef]

D. R. Badcock, “How do we discriminate relative spatial phase?” Vision Res. 24, 1847–1857 (1984).
[CrossRef] [PubMed]

Other (6)

The rounding up and truncation of the differential response was done by the following formula: differential response = ±INT[(ABS(d′1− d′2) + 0.175)/1.675], where INT and ABS stand for the integer part and the absolute value. The 0.175 term produces a slight rounding up, so that the integer value of the differential response displayed in Figs. 12–14 is nonvanishing when d′ ≥ 0.5.

D. Marr, Vision (Freeman, San Francisco, Calif., 1982).We have recently become aware of some interesting theorems for the zero crossings of space–size plots by A. L. Yuille, T. Poggio, in “Scaling theorems for zero-crossings,” A. I. Memo 722,and “Fingerprint theorems for zero-crossings,” A. I. Memo 730 (Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, Mass., 1983).

N. McWhirter, Guiness Book of World Records (Bantam, New York, 1984), p. 33.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pp. 367–375.

H. E. Bedell, University of Houston, Houston, Tex. 77004 (personal communication).

F. H. C. Crick, D. C. Marr, T. Poggio, “An information-processing approach to understanding the visual cortex,” in The Organization of the Cerebral Cortex, F. O. Schmidt, ed. (MIT, Boston, Mass., 1981), pp. 505–533.

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Figures (14)

Fig. 1
Fig. 1

Schematic of the bisection stimulus near the optimal spacing of 1.3 min. The reference lines were presented continuously, and the test line flashed on for 0.6 sec (for subject SK) or 1.0 sec (for subject DL) in one of five positions (0, 1, or 2 pixels above or below the bisection point). For most of the experiments each pixel subtended 1.64 sec at the 10-m testing distance.

Fig. 2
Fig. 2

Bisection thresholds for subject DL as a function of the separation between the lines are shown by the filled triangles. The dashed line has a slope of 1, showing that threshold is a more-or-less constant fraction of the separation down to about 1.2 min. For still closer separations, thresholds increase rapidly. The filled circles show thresholds for bisection with the luminance cue removed by placing the test line adjacent to (see inset) the reference lines (bisection–no overlap). Note that at wide separations these thresholds are a factor of 2 higher than the overlapped condition and there is no sharp decrease in thresholds for small separations.

Fig. 3
Fig. 3

Effect of line length on bisection thresholds of DL. The separation was 1.3 min (filled circles) or 3 min (3’s). For the small separation, increasing line length resulted in a threefold reduction in thresholds, while thresholds for the 3′ separation decreased only slightly.

Fig. 4
Fig. 4

Bisection thresholds for SK as a function of separation. Open circles are for a luminance of 0.56 cd/m and were obtained under conditions identical to those for the data in Fig. 2. The other data were gathered under monocular viewing with a 3-mm artificial pupil, thinner lines (0.079 cd/m), and a duration of 0.3 sec. The filled circles, triangles, squares, and crosses correspond to line luminances of 0.079, 0.020, 0.0031, and 0.0016 cd/m, respectively. The dashed line and arrow associated with the lowest luminance curve indicates that bisection was impossible at small separations at this low luminance. Thus no threshold value could be obtained in spite of many attempts.

Fig. 5
Fig. 5

Schematic of the bisection stimulus shown in Fig. 1 but with added flanking lines at the optimal distance (1.2 min). The flanking lines were shown continuously and were equal in brightness, width, and length to the reference lines.

Fig. 6
Fig. 6

Bisection thresholds (arc sec) for SK (open circles) and DL (filled circles) as a function of the distance of the flanking lines. The inner separation was 1.3 min. The data at the far right-hand side are for the three-bar stimulus where flanks are not present.

Fig. 7
Fig. 7

Bisection thresholds for SK (open circles) and DL (filled circles) as a function of the distance of the flanking lines, similar to Fig. 6, except for inner separations varying from 2.0 to 0.8 min.

Fig. 8
Fig. 8

Eight replications (125 trials per run) of the bisection threshold for the optimal stimulus configuration (i.e., five lines: inner separation 1.3 min, outer separation 1.24 min). In these runs the test line was either 0 or 1 pixel above or below the bisection point. The observer responded up, down, or center. The dashed line shows the mean threshold (0.85 ± 0.04 arc sec).

Fig. 9
Fig. 9

Schematic representation of the dual space–size analysis. Shown here are pairs of symmetric and antisymmetric receptive fields of different sizes (increasing downward) at a variety of locations. In the quantitative calculations, the spatial sampling interval was 0.2 min, and the size of mechanisms varied in 0.125-octave steps.

Fig. 10
Fig. 10

Graphic representation of the viewprint calculations. A, First a new type of receptive field is introduced—the Cauchy functions. Symmetric and antisymmetric Cauchy functions are shown on the top row. C3, 2-octave bandwidth; C5, 1.5-octave bandwidth. B, Frequency tuning of the C3 and C5 Cauchy functions as normalized by the contrast-sensitivity function. The peaks of the functions are spaced by 0.5-octave intervals. The envelope of the functions gives the CSF. C, Contrast-response function (transducer function) is used to compute the response of the mechanisms to suprathreshold stimuli. The horizontal axis is in threshold units, where threshold is defined as the contrast that gives d′ = 1. The ordinate is d′ as measured at the vertical intercept of the ROC curve. D, Contrast jnd (Δs) shown on the ordinate is the change in contrast that produces a unity change in d′. For zero pedestal (s = 0), the contrast jnd equals unity, the threshold value. For a large pedestal (s ≫ 1) Weber’s law is obeyed. The reciprocal Weber fraction, the effective contrast, is shown by the dotted line and is discussed in Section 4 of Appendix A.

Fig. 11
Fig. 11

Upper panel: The symmetric receptive fields whose characteristics are given in Table 1 are plotted. Curves C3 and C5 are Cauchy functions; G4 and G6 are Gabor function whose variances are 1/4 and 1/6; G″ is the second derivative of a Gaussian. The tic marks indicate σ = ±1 for the Cauchy distribution. Middle panel: Antisymmetric receptive fields for the Cauchy and Gabor functions. Lower panel: Spatial-frequency profiles of the three filter types. Two profiles are drawn for the Gabor functions since the symmetric and antisymmetric filters have different tuning. The horizontal axis is on an octave scale, with the tick marks indicating octave intervals. In all plots the vertical axis is linear.

Fig. 12
Fig. 12

Viewprints for three-line stimuli for the C3 mechanism. The interline separation was 4.0 arc min. The two panels correspond to two offsets of the test line. Lower panel: Threshold offset (0.08 min, corresponding to a 1/50 Weber fraction). Upper panel: A half-threshold (0.04-min) offset. The shading represents one bit of absolute phase (the sign). The contour lines are the iso-d′ lines of the contrast-response function as indicated by the numbers by the lines. The clusters of slightly smaller numbers are the differential response and indicate the discriminability of the offset. Negative differential-response values’ are indicated by the reversed contrast numbers.

Fig. 13
Fig. 13

Viewprints for three-line stimuli with different separations for the C3 and C5 mechanisms. Each horizontal grouping of two viewprints is for a different three-line stimulus with interline distances of 1.0, 1.2, 1.5, and 2.5 min. The test lines were offset by the threshold levels of 0.05, 0.026, 0.03, and 0.05-min. Other details are the same as in Fig. 12. The bottom panel is a perspective view of the d′ response to the three-line stimulus with 2.5-min separations and 3-sec offset. The dashed lines can be used for calibration since their horizontal positions coincides with the stimuli, and the vertical extent of each dash corresponds to d′ = 2.

Fig. 14
Fig. 14

Viewprints for five-line stimuli. For each viewprint, the inner line separation is 1.3 min and the offset is 0.02 min. The separations between flanking and reference lines (see Fig. 5 for nomenclature) are 1.0, 1.2, 1.5, and 2.0 min and ∞ (a three-line stimulus). Other details as in Fig. 12. The presence of the flanks is seen to increase the differential response of the viewprints, in agreement with the experimental results shown in Fig. 6.

Tables (1)

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Table 1 Characteristics of Filtersa

Equations (18)

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C n ( f ) = f n exp ( f ) .
S n 1 ( t ) = Re ( 1 + i t ) n = cos n ( θ ) cos ( n θ ) , A n 1 ( t ) = Im ( 1 + i t ) n = cos n ( θ ) sin ( n θ ) ,
S n = σ / n d A n 1 / d x , A n = σ / n d S n 1 / d x .
S 0 = 1 / D , A 0 = t / D , S 1 = ( 1 t 2 ) / D 2 A 1 = 2 t / D 2 , S 2 = ( 1 3 t 2 ) / D 3 , A 2 = t ( 3 t 2 ) / D 3 , S 3 = ( 1 6 t 2 + t 4 ) / D 4 , A 3 = t ( 4 4 t 2 ) / D 4 , S 4 = ( 1 10 t 2 + 5 t 4 ) / D 5 , A 4 = t ( 5 10 t 2 + t 4 ) / D 5 , S 5 = ( 1 15 t 2 + 15 t 4 t 6 ) / D 6 , A 5 = t ( 6 20 t 2 + 6 t 4 ) / D 6 ,
CSF ( f ) = A ( f ρ ) m exp ( f ρ ) ,
S ( σ , f ) = CSF ( f ) C n ( σ f ) .
S ( σ , f ) = N ( σ ) C n ( σ f ) .
CSF ( f t ) = N ( σ ) C n ( σ f t ) .
d = ln ( 1 + S r / W ) / ln ( 1 + 1 / W ) ,
d = 2.5 ln [ 1 + ( SS 2 + SA 2 ) / 2 ] .
CSF ( f ) = A ( ρ f ) 1 / 2 exp ( ρ f ) ,
cos ( f x ) + * sin ( f x ) sin ( m x ) cos [ f x sin ( m x ) ] .
c [ cos ( f x ) + 0.1 cos ( f x + θ ) ] .
d = 5 ln S ( σ f ) ,
S ( σ f ) = ( σ f ) n exp ( σ f ) .
Δ d = 5 Δ S / S = 5 Δ f ( n / f σ ) .
Δ d = 5 n ( Δ f / f ) ( p f ) / p .
Δ f / f = 0.1 / ( n + 1 ) .

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