Abstract

The disagreement between physical and psychophysical estimates of human optical performance is discussed. Recent measurements of the eye’s modulation transfer functions in white light for several pupil sizes are used to compare the eye with an ideal optical system in terms of normalized modulation transfer functions, point image profiles, and Strehl ratios. Several simple fundal-image profiles are derived from the measured modulation transfer functions, and the importance of these profiles to psychophysical measurements is discussed. Glare is considered as the extension of point spread functions to large angles; experimental measurements are compared with theories for the special case of an annular target.

© 1967 Optical Society of America

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References

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  14. F. W. Campbell and R. W. Gubisch, J. Physiol. (London) 186, 558 (1966).
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1966 (3)

F. W. Campbell and R. W. Gubisch, J. Physiol. (London) 186, 558 (1966).

R. W. Gubisch, Quart. J. Exp. Psychol. 18, 366 (1966).
[CrossRef]

W. A. H. Rushton and R. W. Gubisch, J. Opt. Soc. Am. 56, 104 (1966).
[CrossRef] [PubMed]

1965 (6)

T. N. Cornsweet and D. Y. Teller, J. Opt. Soc. Am. 55, 1303 (1965).
[CrossRef] [PubMed]

G. A. Fry, J. Opt. Soc. Am. 55, 333 (1965).
[CrossRef]

R. B. Joynson, L. J. Newson, and D. S. May, Quart. J. Exp. Psychol. 17, 209 (1965).
[CrossRef]

G. Westheimer, J. Physiol. (London) 181, 881 (1965).

V. D. Glezer, Vision Res. 5, 497 (1965).
[CrossRef] [PubMed]

F. W. Campbell and D. G. Green, J. Physiol. (London) 181, 576 (1965).

1963 (1)

1962 (3)

1961 (1)

E. G. Heinemann, J. Exp. Psychol. 61, 389 (1961).
[CrossRef] [PubMed]

1960 (2)

F. W. Campbell and G. Westheimer, J. Physiol. (London) 151, 285 (1960).

A. Arnulf and O. Dupuy, Compt. Rend., Paris 250, 2757 (1960).

1958 (3)

1957 (1)

H. H. Hopkins, Proc. Phys. Soc. (London) 70B, B449 (1957).

1956 (1)

E. L. O’Neill, J. Opt. Soc. Am. 46, 258 (1956).
[CrossRef]

1955 (1)

F. Flamant, Rev. Opt. 34, 433 (1955).

1954 (1)

1952 (1)

1947 (2)

A. Ivanoff, Rev. Opt. 26, 145 (1947); H. Hartridge, Phil. Trans. (London) 232B, 519 (1947); F. W. Campbell, Optica Acta 4, 157 (1957).
[CrossRef]

A. Maréchal, Rev. Opt. 26, 257 (1947).

1943 (1)

1938 (1)

C. Berger and F. Buchthal, Skand. Arch. Physiol. 78, 197 (1938).
[CrossRef]

1937 (1)

Y. LeGrand, Rev. Opt. 16, 201, 241 (1937).

1929 (1)

W. S. Stiles, Proc. Roy. Soc. (London) B104, 322 (1929).

1927 (1)

1926 (1)

1922 (1)

H. Hartridge, J. Physiol. (London) 57, 52 (1922).

1915 (1)

P. W. Cobb, Am. J. Physiol. 36, 335 (1915).

1903 (1)

Rayleigh, J. Roy. Microscop. Soc. (London) Pt. 1, 474 (1903).
[CrossRef]

1902 (1)

K. Strehl, Z. Instrumentenk. 22, 213 (1902).

Arnulf, A.

A. Arnulf and O. Dupuy, Compt. Rend., Paris 250, 2757 (1960).

Berger, C.

C. Berger and F. Buchthal, Skand. Arch. Physiol. 78, 197 (1938).
[CrossRef]

Boynton, R. M.

Buchthal, F.

C. Berger and F. Buchthal, Skand. Arch. Physiol. 78, 197 (1938).
[CrossRef]

Bush, W. R.

Campbell, F. W.

F. W. Campbell and R. W. Gubisch, J. Physiol. (London) 186, 558 (1966).

F. W. Campbell and D. G. Green, J. Physiol. (London) 181, 576 (1965).

G. Westheimer and F. W. Campbell, J. Opt. Soc. Am. 52, 1040 (1962).
[CrossRef] [PubMed]

F. W. Campbell and G. Westheimer, J. Physiol. (London) 151, 285 (1960).

Cobb, P. W.

P. W. Cobb, Am. J. Physiol. 36, 335 (1915).

Cornsweet, T. N.

Dupuy, O.

A. Arnulf and O. Dupuy, Compt. Rend., Paris 250, 2757 (1960).

Enoch, J. M.

Flamant, F.

F. Flamant, Rev. Opt. 34, 433 (1955).

Fry, G. A.

Glezer, V. D.

V. D. Glezer, Vision Res. 5, 497 (1965).
[CrossRef] [PubMed]

Green, D. G.

F. W. Campbell and D. G. Green, J. Physiol. (London) 181, 576 (1965).

Gubisch, R. W.

W. A. H. Rushton and R. W. Gubisch, J. Opt. Soc. Am. 56, 104 (1966).
[CrossRef] [PubMed]

F. W. Campbell and R. W. Gubisch, J. Physiol. (London) 186, 558 (1966).

R. W. Gubisch, Quart. J. Exp. Psychol. 18, 366 (1966).
[CrossRef]

Hartridge, H.

H. Hartridge, J. Physiol. (London) 57, 52 (1922).

Heinemann, E. G.

E. G. Heinemann, J. Exp. Psychol. 61, 389 (1961).
[CrossRef] [PubMed]

Helmholtz, H.

H. Helmholtz, Handbook of Physiological Optics, translation edited by J. P. C. Southall (Dover Publications, Inc., New York, 1962), Vol. 2, p. 36.

H. Helmholtz, Popular Scientific Lectures edited by M. Kline (Dover Publications, Inc., New York, 1962), p. 116.

Higgins, G. C.

Holladay, L. L.

Hopkins, H. H.

H. H. Hopkins, Proc. Phys. Soc. (London) 70B, B449 (1957).

Ivanoff, A.

A. Ivanoff, Rev. Opt. 26, 145 (1947); H. Hartridge, Phil. Trans. (London) 232B, 519 (1947); F. W. Campbell, Optica Acta 4, 157 (1957).
[CrossRef]

Jones, R. C.

Joynson, R. B.

R. B. Joynson, L. J. Newson, and D. S. May, Quart. J. Exp. Psychol. 17, 209 (1965).
[CrossRef]

Krauskopf, J. K.

Lamberts, R. L.

LeGrand, Y.

Y. LeGrand, Rev. Opt. 16, 201, 241 (1937).

Y. LeGrand, Optique Physiologique (Éditions de la Revue d’Optique, Paris, 1956), Vol. 3, p. 52.

Leibowitz, H.

Linfoot, E. H.

E. H. Linfoot, Fourier Methods in Optical Image Evaluation (Focal Press, London and New York, 1964), p. 82.

Maréchal, A.

A. Maréchal, Rev. Opt. 26, 257 (1947).

Martin, L. C.

L. C. Martin, Technical Optics (Pitman and Sons, London, 1954), Vol. 2, p. 230.

May, D. S.

R. B. Joynson, L. J. Newson, and D. S. May, Quart. J. Exp. Psychol. 17, 209 (1965).
[CrossRef]

Moon, P.

Newson, L. J.

R. B. Joynson, L. J. Newson, and D. S. May, Quart. J. Exp. Psychol. 17, 209 (1965).
[CrossRef]

O’Neill, E. L.

E. L. O’Neill, J. Opt. Soc. Am. 46, 258 (1956).
[CrossRef]

Picht, J.

J. Picht, Optik 15, 83 (1958).

Ratliff, F.

F. Ratliff, Mach Bands (Holden-Day, San Francisco, 1966), Ch. 4.

Rayleigh,

Rayleigh, J. Roy. Microscop. Soc. (London) Pt. 1, 474 (1903).
[CrossRef]

Röhler, R.

R. Röhler, Vision Res. 2, 391 (1962).
[CrossRef]

Rushton, W. A. H.

Spencer, D. E.

Stiles, W. S.

W. S. Stiles, Proc. Roy. Soc. (London) B104, 322 (1929).

Strehl, K.

K. Strehl, Z. Instrumentenk. 22, 213 (1902).

Tansley, K.

K. Tansley, Vision in Vertebrates (Science Paperbacks, London, 1965), p. 107.

Teller, D. Y.

Vos, J. J.

J. J. Vos, On Mechanisms of Glare, thesis, Utrecht (1963).

Westheimer, G.

G. Westheimer, J. Physiol. (London) 181, 881 (1965).

G. Westheimer, J. Opt. Soc. Am. 53, 86 (1963).
[CrossRef] [PubMed]

G. Westheimer and F. W. Campbell, J. Opt. Soc. Am. 52, 1040 (1962).
[CrossRef] [PubMed]

F. W. Campbell and G. Westheimer, J. Physiol. (London) 151, 285 (1960).

Wolfe, R. N.

Am. J. Physiol. (1)

P. W. Cobb, Am. J. Physiol. 36, 335 (1915).

Compt. Rend., Paris (1)

A. Arnulf and O. Dupuy, Compt. Rend., Paris 250, 2757 (1960).

J. Exp. Psychol. (1)

E. G. Heinemann, J. Exp. Psychol. 61, 389 (1961).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (14)

J. Physiol. (London) (5)

G. Westheimer, J. Physiol. (London) 181, 881 (1965).

F. W. Campbell and D. G. Green, J. Physiol. (London) 181, 576 (1965).

F. W. Campbell and R. W. Gubisch, J. Physiol. (London) 186, 558 (1966).

H. Hartridge, J. Physiol. (London) 57, 52 (1922).

F. W. Campbell and G. Westheimer, J. Physiol. (London) 151, 285 (1960).

J. Roy. Microscop. Soc. (London) (1)

Rayleigh, J. Roy. Microscop. Soc. (London) Pt. 1, 474 (1903).
[CrossRef]

Optik (1)

J. Picht, Optik 15, 83 (1958).

Proc. Phys. Soc. (London) (1)

H. H. Hopkins, Proc. Phys. Soc. (London) 70B, B449 (1957).

Proc. Roy. Soc. (London) (1)

W. S. Stiles, Proc. Roy. Soc. (London) B104, 322 (1929).

Quart. J. Exp. Psychol. (2)

R. B. Joynson, L. J. Newson, and D. S. May, Quart. J. Exp. Psychol. 17, 209 (1965).
[CrossRef]

R. W. Gubisch, Quart. J. Exp. Psychol. 18, 366 (1966).
[CrossRef]

Rev. Opt. (4)

A. Maréchal, Rev. Opt. 26, 257 (1947).

A. Ivanoff, Rev. Opt. 26, 145 (1947); H. Hartridge, Phil. Trans. (London) 232B, 519 (1947); F. W. Campbell, Optica Acta 4, 157 (1957).
[CrossRef]

F. Flamant, Rev. Opt. 34, 433 (1955).

Y. LeGrand, Rev. Opt. 16, 201, 241 (1937).

Skand. Arch. Physiol. (1)

C. Berger and F. Buchthal, Skand. Arch. Physiol. 78, 197 (1938).
[CrossRef]

Vision Res. (2)

R. Röhler, Vision Res. 2, 391 (1962).
[CrossRef]

V. D. Glezer, Vision Res. 5, 497 (1965).
[CrossRef] [PubMed]

Z. Instrumentenk. (1)

K. Strehl, Z. Instrumentenk. 22, 213 (1902).

Other (8)

E. H. Linfoot, Fourier Methods in Optical Image Evaluation (Focal Press, London and New York, 1964), p. 82.

Y. LeGrand, Optique Physiologique (Éditions de la Revue d’Optique, Paris, 1956), Vol. 3, p. 52.

K. Tansley, Vision in Vertebrates (Science Paperbacks, London, 1965), p. 107.

F. Ratliff, Mach Bands (Holden-Day, San Francisco, 1966), Ch. 4.

J. J. Vos, On Mechanisms of Glare, thesis, Utrecht (1963).

L. C. Martin, Technical Optics (Pitman and Sons, London, 1954), Vol. 2, p. 230.

H. Helmholtz, Popular Scientific Lectures edited by M. Kline (Dover Publications, Inc., New York, 1962), p. 116.

H. Helmholtz, Handbook of Physiological Optics, translation edited by J. P. C. Southall (Dover Publications, Inc., New York, 1962), Vol. 2, p. 36.

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Figures (12)

Fig. 1
Fig. 1

Modulation transfer functions of the human eye in white light, with 2.4 mm, 3.8 mm, and 6.6 mm artificial pupils. The results are summarized from Campbell and Gubisch14 and represent the average of three subjects. The curves lie in order of pupil diameter, with the highest curve corresponding to the smallest pupil. Dashed line gives MTF for 1.5 mm pupil.

Fig. 2
Fig. 2

Point-image profiles for the human eye in white light at eight pupil sizes, computed from formula (3) using the modulation transfer functions summarized in Fig. 1. Computed points occur at 0.1 min of arc intervals. Dotted line is profile formed by an ideal achromatic system in light of the same spectral composition and with the same aperture.

Fig. 3
Fig. 3

Point-image profile of the human eye (filled circles) in white light with a 1.5-mm pupil, displayed in semilog coordinates. Plain curve gives the image profile formed by an ideal achromatic system in white light with a 1.5-mm aperture. As the ideal curve consists of many superimposed monochromatic point-diffraction images, it never reaches zero illuminance as each of its components does cyclically.

Fig. 4
Fig. 4

Variation of the Strehl ratios for the human eye in white light as a function of pupil diameter. The Strehl ratio as a measure of optical quality is the ratio of the maximum illuminance in the point image formed by a real system, to the maximum illuminance in the central core of the point diffraction image formed by an ideal optical system working at the same aperture and with the same spectral composition of light.

Fig. 5
Fig. 5

Profiles of the images of edges formed by the human eye at three pupil sizes, computed from formula (4). The oblique lines on the right indicate the maximum illuminance gradient for each curve, which occurs at the geometrical position of the edge.

Fig. 6
Fig. 6

A comparison of edge spreads for the human eye with a 3-mm pupil for several cases. (– – –) ideal achromatic optical system in white light; ■ Hartridge’s estimate for the human eye, supposedly including diffraction and chromatic aberration; (— · — · —) theoretical optical system with the same chromatic difference of focus as the human eye; ● measurement of the human eye by Campbell and Gubisch, including spherical and chromatic aberration.

Fig. 7
Fig. 7

Bar-image profiles for the human eye, computed from formula (6) for four different bar widths. Solid lines denote profiles obtained with 2.4-mm pupil; dashed lines, those with 6.6-mm pupil. Brackets indicate geometrical positions of bar edges. Only the right halves of the profiles are shown.

Fig. 8
Fig. 8

Disk-image profiles for the human eye, computed from formula (7) for four different disk diameters. Solid lines denote profiles obtained with a 2.4-mm pupil; dashed lines, those with a 6.6-mm pupil. Brackets indicate the geometrical positions of the disk edges. Only the right halves of the profiles are shown.

Fig. 9
Fig. 9

Variation of illuminance at the center of a disk image on the retina as the source decreases in size but not luminance. Symbols: ● 0 2.4-mm pupil, ■ 6.6-mm pupil. The curves for both pupil sizes approach a slope of +2 at the smallest target diameters shown; this indicates that no further change in the profile of the image occurs as the target constricts, only the image illuminance decreases.

Fig. 10
Fig. 10

Annulus-image profiles for the human eye with a 2.4-mm pupil. Number above each curve gives annulus inner diameter in min of arc. In three cases the target annulus is 1 min of arc wide; the profile given for an inner diameter equal to zero is the profile for a 1 min of arc diameter disk.

Fig. 11
Fig. 11

Illuminance variation at the centers of the retinal images of annuli as annulus inner diameter increases. Open symbols were computed from formula (12) for infinitesimally thin rings. Solid symbols were derived from Fig. 9 for annuli with infinite outer diameters. Circles show results with 2.4-mm pupil, square plot results for 6.6 mm.

Fig. 12
Fig. 12

Comparison of estimates of scattered light in the human eye, using annuli. Solid curves are taken directly from Fig. 11 for annuli of infinite extent, for both 2.4- and 6.6-mm pupils. Dashed curves were computed from the modified “veiling luminance” formula (13). Curve A was computed from the formula given by Moon and Spencer.41 Open symbols are psychophysical measurements of Heinemann38 (○), Rushton and Gubisch40 (△), and Cornsweet and Teller39 (□). Filled circles plot psychophysical measurements made under the same experimental conditions as those used to derive the modulation transfer functions in Fig. 1.

Equations (13)

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I ( x , y ) = - - F ( u , v ) T ( u , v ) e 2 π i ( u x + v y ) d u d v
I ( r ) = 2 π - F ( ω ) T ( ω ) J 0 ( 2 π ω r ) ω d ω .
I ( r ) = 2 π - T ( ω ) J 0 ( 2 π ω r ) ω d ω .
I ( x ) = - x L ( l ) d l .
F ( ω ) = sin ( 2 π ω a ) / 2 π ω a .
I ( x ) = - sin ( 2 π ω a ) 2 π ω a T ( ω ) cos ( 2 π ω x ) d ω .
F ( ω ) = J 1 ( 2 π ω b ) / π ω b
I ( r ) = 2 b - J 1 ( 2 π ω b ) T ( ω ) J 0 ( 2 π ω r ) d ω ,
F ( ω ) = 1 ( 1 - η 2 ) [ J 1 ( 2 π ω r 2 ) r 2 - η 2 J 1 ( 2 π ω r 1 ) r 1 ] ,
F ( ω ) = 0 δ ( R ) J 0 ( 2 π ω r ) r d r .
F ( ω ) = J 0 ( 2 π ω R ) .
I ( r ) = 2 π - J 0 ( 2 π ω R ) T ( ω ) J 0 ( 2 π ω r ) ω d ω .
I ( r ) = a 2 / ( a 2 + r 2 )