Abstract

The ability of observers to discriminate between stimuli differing in orientation was measured using low contrast, foveally viewed stimuli. Detection and discrimination performance were measured simultaneously. A model is presented which permits bandwidths of orientation-tuned mechanisms to be estimated from the data. In a group of five observers, half-amplitude bandwidths varied from 10° to 20°.

© 1979 Optical Society of America

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References

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  1. D. H. Hubel and T. N. Wiesel, “Receptive fields, binocular interaction and functional architecture in the cat’s striate cortex,” J. Physiol. 160, 106–154 (1962).
  2. D. H. Hubel and T. N. Wiesel, “Sequence regularity and geometry of orientation columns in the monkey striate context,” J. Comp. Neurol. 158, 267–294 (1974).
    [CrossRef] [PubMed]
  3. R. W. Sekuler, “Spatial and temporal determinants of visual backward masking,” J. Exptl. Psychol. 70, 401−406 (1965).
    [CrossRef]
  4. F. W. Campbell and J. J. Kulikowski, “Orientational selectivity of the human visual system,” J. Physiol. 187, 437–445 (1966).
  5. C. Blakemore and J. Nachmias, “The orientation specificity of two visual after-effects,” J. Physiol. 213, 157–174 (1971).
  6. J. J. Kulikowski, R. Abadi, and P. E. King-Smith, “Orientational selectivity of grating and line detectors in human vision,” Vision Res. 13, 1479–1486 (1973).
    [CrossRef] [PubMed]
  7. J. P. Thomas, “Spatial summation in the fovea: asymmetrical effects of longer and shorter dimensions,” Vision Res. 18, 1023–1029 (1978).
    [CrossRef] [PubMed]
  8. J. Nachmias and A. Weber, “Discrimination of simple and complex gratings,” Vision. Res. 15, 217–223 (1975).
    [CrossRef] [PubMed]
  9. C. S. Furchner, J. P. Thomas, and F. W. Campbell, “Detection and discrimination of simple and complex patterns at low spatial frequencies,” Vision Res. 17, 827–836 (1977).
    [CrossRef] [PubMed]
  10. The term channel has been used with at least two different meanings. Sometimes, the term refers to a mechanism which responds to a distinctly limited portion of the visual field and then only to components lying within limited ranges of spatial frequency and orientation.11,12 Sometimes, the term channel refers to a matrix consisting of all such mechanisms which are tuned to a given band of spatial frequencies or orientations.13,14 The formulations presented here were developed using the first meaning. However, given two conditions, the individual mechanism and the matrix have the same bandwidth with respect to orientation. The two conditions are (i) the different mechanisms in the matrix have the properties described in the discussion of nonoptimal channels in the Appendix; and (ii) the responses of the individual mechanisms Ri are combined to form the matrix response, RM, by a process which can be described asRM=[∑Rip]1/p.The Euclidean distance model is a special case of this general combining rule.
  11. H. Mostafavi and D. J. Sakrison, “Structure and properties of asingle channel in the human visual system,” Vision Res. 16, 957–968 (1976).
    [CrossRef]
  12. H. R. Wilson and S. C. Giese, “Threshold visibility of frequency gradient patterns,” Vision Res. 17, 1177–1190 (1977).
    [CrossRef] [PubMed]
  13. N. Graham, “Visual detection of aperiodic spatial stimuli by probability summation among narrowband channels,” Vision Res. 17, 637–653 (1977).
    [CrossRef] [PubMed]
  14. M. B. Sachs, J. Nachmias, and J. G. Robson, “Spatial-frequency channels in human vision,” J. Opt. Soc. Am. 61, 1176–1186 (1971).
    [CrossRef] [PubMed]
  15. The development of this model and the related computer simulation were carried out by R. A. Barker. We thank him for permitting us to use his results.
  16. J. P. Thomas, “Model of the function of receptive fields in human vision,” Psychol. Rev. 77, 121–134 (1970).
    [CrossRef] [PubMed]
  17. J. P. Thomas and K. K. Shimamura, “Inhibitory interaction between visual pathways tuned to different orientations,” Vision Res. 15, 1373–1380 (1975).
    [CrossRef] [PubMed]
  18. J. P. Thomas and K. K. Shimamura, “Perception of size at the detection threshold: its accuracy and possible mechanisms,” Vision Res. 14, 535–543 (1974).
    [CrossRef] [PubMed]
  19. F. W. Campbell and J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. 197, 551–566 (1968).
  20. N. Graham and J. Nachmias, “Detection of grating patterns containing two spatial frequencies: a comparison of single-channel and multiple-channel models,” Vision Res. 11, 251–259 (1971).
    [CrossRef] [PubMed]
  21. P. E. King-Smith and J. J. Kulikowski, “The detection of gratings by independent activation of line detectors,” J. Physiol. 247, 237–271 (1975).
  22. C. F. Stromeyer and S. Klein, “Evidence against narrow-band spatial frequency channels in human vision: the detectability of frequency modulated gratings,” Vision Res. 15, 899–910 (1975).
    [CrossRef] [PubMed]
  23. P. H. Schiller, B. L. Finlay, and S. F. Volman, “Quantitative studies of single-cell properties in monkey striate cortex. II. Orientation specificity and ocular dominance,” J. Neurophysiol. 39, 1334–1351 (1976).
    [PubMed]

1978 (1)

J. P. Thomas, “Spatial summation in the fovea: asymmetrical effects of longer and shorter dimensions,” Vision Res. 18, 1023–1029 (1978).
[CrossRef] [PubMed]

1977 (3)

C. S. Furchner, J. P. Thomas, and F. W. Campbell, “Detection and discrimination of simple and complex patterns at low spatial frequencies,” Vision Res. 17, 827–836 (1977).
[CrossRef] [PubMed]

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

N. Graham, “Visual detection of aperiodic spatial stimuli by probability summation among narrowband channels,” Vision Res. 17, 637–653 (1977).
[CrossRef] [PubMed]

1976 (2)

H. Mostafavi and D. J. Sakrison, “Structure and properties of asingle channel in the human visual system,” Vision Res. 16, 957–968 (1976).
[CrossRef]

P. H. Schiller, B. L. Finlay, and S. F. Volman, “Quantitative studies of single-cell properties in monkey striate cortex. II. Orientation specificity and ocular dominance,” J. Neurophysiol. 39, 1334–1351 (1976).
[PubMed]

1975 (4)

J. P. Thomas and K. K. Shimamura, “Inhibitory interaction between visual pathways tuned to different orientations,” Vision Res. 15, 1373–1380 (1975).
[CrossRef] [PubMed]

P. E. King-Smith and J. J. Kulikowski, “The detection of gratings by independent activation of line detectors,” J. Physiol. 247, 237–271 (1975).

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

J. Nachmias and A. Weber, “Discrimination of simple and complex gratings,” Vision. Res. 15, 217–223 (1975).
[CrossRef] [PubMed]

1974 (2)

D. H. Hubel and T. N. Wiesel, “Sequence regularity and geometry of orientation columns in the monkey striate context,” J. Comp. Neurol. 158, 267–294 (1974).
[CrossRef] [PubMed]

J. P. Thomas and K. K. Shimamura, “Perception of size at the detection threshold: its accuracy and possible mechanisms,” Vision Res. 14, 535–543 (1974).
[CrossRef] [PubMed]

1973 (1)

J. J. Kulikowski, R. Abadi, and P. E. King-Smith, “Orientational selectivity of grating and line detectors in human vision,” Vision Res. 13, 1479–1486 (1973).
[CrossRef] [PubMed]

1971 (3)

C. Blakemore and J. Nachmias, “The orientation specificity of two visual after-effects,” J. Physiol. 213, 157–174 (1971).

M. B. Sachs, J. Nachmias, and J. G. Robson, “Spatial-frequency channels in human vision,” J. Opt. Soc. Am. 61, 1176–1186 (1971).
[CrossRef] [PubMed]

N. Graham and J. Nachmias, “Detection of grating patterns containing two spatial frequencies: a comparison of single-channel and multiple-channel models,” Vision Res. 11, 251–259 (1971).
[CrossRef] [PubMed]

1970 (1)

J. P. Thomas, “Model of the function of receptive fields in human vision,” Psychol. Rev. 77, 121–134 (1970).
[CrossRef] [PubMed]

1968 (1)

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

1966 (1)

F. W. Campbell and J. J. Kulikowski, “Orientational selectivity of the human visual system,” J. Physiol. 187, 437–445 (1966).

1965 (1)

R. W. Sekuler, “Spatial and temporal determinants of visual backward masking,” J. Exptl. Psychol. 70, 401−406 (1965).
[CrossRef]

1962 (1)

D. H. Hubel and T. N. Wiesel, “Receptive fields, binocular interaction and functional architecture in the cat’s striate cortex,” J. Physiol. 160, 106–154 (1962).

Abadi, R.

J. J. Kulikowski, R. Abadi, and P. E. King-Smith, “Orientational selectivity of grating and line detectors in human vision,” Vision Res. 13, 1479–1486 (1973).
[CrossRef] [PubMed]

Blakemore, C.

C. Blakemore and J. Nachmias, “The orientation specificity of two visual after-effects,” J. Physiol. 213, 157–174 (1971).

Campbell, F. W.

C. S. Furchner, J. P. Thomas, and F. W. Campbell, “Detection and discrimination of simple and complex patterns at low spatial frequencies,” Vision Res. 17, 827–836 (1977).
[CrossRef] [PubMed]

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

F. W. Campbell and J. J. Kulikowski, “Orientational selectivity of the human visual system,” J. Physiol. 187, 437–445 (1966).

Finlay, B. L.

P. H. Schiller, B. L. Finlay, and S. F. Volman, “Quantitative studies of single-cell properties in monkey striate cortex. II. Orientation specificity and ocular dominance,” J. Neurophysiol. 39, 1334–1351 (1976).
[PubMed]

Furchner, C. S.

C. S. Furchner, J. P. Thomas, and F. W. Campbell, “Detection and discrimination of simple and complex patterns at low spatial frequencies,” Vision Res. 17, 827–836 (1977).
[CrossRef] [PubMed]

Giese, S. C.

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

Graham, N.

N. Graham, “Visual detection of aperiodic spatial stimuli by probability summation among narrowband channels,” Vision Res. 17, 637–653 (1977).
[CrossRef] [PubMed]

N. Graham and J. Nachmias, “Detection of grating patterns containing two spatial frequencies: a comparison of single-channel and multiple-channel models,” Vision Res. 11, 251–259 (1971).
[CrossRef] [PubMed]

Hubel, D. H.

D. H. Hubel and T. N. Wiesel, “Sequence regularity and geometry of orientation columns in the monkey striate context,” J. Comp. Neurol. 158, 267–294 (1974).
[CrossRef] [PubMed]

D. H. Hubel and T. N. Wiesel, “Receptive fields, binocular interaction and functional architecture in the cat’s striate cortex,” J. Physiol. 160, 106–154 (1962).

King-Smith, P. E.

P. E. King-Smith and J. J. Kulikowski, “The detection of gratings by independent activation of line detectors,” J. Physiol. 247, 237–271 (1975).

J. J. Kulikowski, R. Abadi, and P. E. King-Smith, “Orientational selectivity of grating and line detectors in human vision,” Vision Res. 13, 1479–1486 (1973).
[CrossRef] [PubMed]

Klein, S.

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

Kulikowski, J. J.

P. E. King-Smith and J. J. Kulikowski, “The detection of gratings by independent activation of line detectors,” J. Physiol. 247, 237–271 (1975).

J. J. Kulikowski, R. Abadi, and P. E. King-Smith, “Orientational selectivity of grating and line detectors in human vision,” Vision Res. 13, 1479–1486 (1973).
[CrossRef] [PubMed]

F. W. Campbell and J. J. Kulikowski, “Orientational selectivity of the human visual system,” J. Physiol. 187, 437–445 (1966).

Mostafavi, H.

H. Mostafavi and D. J. Sakrison, “Structure and properties of asingle channel in the human visual system,” Vision Res. 16, 957–968 (1976).
[CrossRef]

Nachmias, J.

J. Nachmias and A. Weber, “Discrimination of simple and complex gratings,” Vision. Res. 15, 217–223 (1975).
[CrossRef] [PubMed]

C. Blakemore and J. Nachmias, “The orientation specificity of two visual after-effects,” J. Physiol. 213, 157–174 (1971).

N. Graham and J. Nachmias, “Detection of grating patterns containing two spatial frequencies: a comparison of single-channel and multiple-channel models,” Vision Res. 11, 251–259 (1971).
[CrossRef] [PubMed]

M. B. Sachs, J. Nachmias, and J. G. Robson, “Spatial-frequency channels in human vision,” J. Opt. Soc. Am. 61, 1176–1186 (1971).
[CrossRef] [PubMed]

Robson, J. G.

M. B. Sachs, J. Nachmias, and J. G. Robson, “Spatial-frequency channels in human vision,” J. Opt. Soc. Am. 61, 1176–1186 (1971).
[CrossRef] [PubMed]

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

Sachs, M. B.

Sakrison, D. J.

H. Mostafavi and D. J. Sakrison, “Structure and properties of asingle channel in the human visual system,” Vision Res. 16, 957–968 (1976).
[CrossRef]

Schiller, P. H.

P. H. Schiller, B. L. Finlay, and S. F. Volman, “Quantitative studies of single-cell properties in monkey striate cortex. II. Orientation specificity and ocular dominance,” J. Neurophysiol. 39, 1334–1351 (1976).
[PubMed]

Sekuler, R. W.

R. W. Sekuler, “Spatial and temporal determinants of visual backward masking,” J. Exptl. Psychol. 70, 401−406 (1965).
[CrossRef]

Shimamura, K. K.

J. P. Thomas and K. K. Shimamura, “Inhibitory interaction between visual pathways tuned to different orientations,” Vision Res. 15, 1373–1380 (1975).
[CrossRef] [PubMed]

J. P. Thomas and K. K. Shimamura, “Perception of size at the detection threshold: its accuracy and possible mechanisms,” Vision Res. 14, 535–543 (1974).
[CrossRef] [PubMed]

Stromeyer, C. F.

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

Thomas, J. P.

J. P. Thomas, “Spatial summation in the fovea: asymmetrical effects of longer and shorter dimensions,” Vision Res. 18, 1023–1029 (1978).
[CrossRef] [PubMed]

C. S. Furchner, J. P. Thomas, and F. W. Campbell, “Detection and discrimination of simple and complex patterns at low spatial frequencies,” Vision Res. 17, 827–836 (1977).
[CrossRef] [PubMed]

J. P. Thomas and K. K. Shimamura, “Inhibitory interaction between visual pathways tuned to different orientations,” Vision Res. 15, 1373–1380 (1975).
[CrossRef] [PubMed]

J. P. Thomas and K. K. Shimamura, “Perception of size at the detection threshold: its accuracy and possible mechanisms,” Vision Res. 14, 535–543 (1974).
[CrossRef] [PubMed]

J. P. Thomas, “Model of the function of receptive fields in human vision,” Psychol. Rev. 77, 121–134 (1970).
[CrossRef] [PubMed]

Volman, S. F.

P. H. Schiller, B. L. Finlay, and S. F. Volman, “Quantitative studies of single-cell properties in monkey striate cortex. II. Orientation specificity and ocular dominance,” J. Neurophysiol. 39, 1334–1351 (1976).
[PubMed]

Weber, A.

J. Nachmias and A. Weber, “Discrimination of simple and complex gratings,” Vision. Res. 15, 217–223 (1975).
[CrossRef] [PubMed]

Wiesel, T. N.

D. H. Hubel and T. N. Wiesel, “Sequence regularity and geometry of orientation columns in the monkey striate context,” J. Comp. Neurol. 158, 267–294 (1974).
[CrossRef] [PubMed]

D. H. Hubel and T. N. Wiesel, “Receptive fields, binocular interaction and functional architecture in the cat’s striate cortex,” J. Physiol. 160, 106–154 (1962).

Wilson, H. R.

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

J. Comp. Neurol. (1)

D. H. Hubel and T. N. Wiesel, “Sequence regularity and geometry of orientation columns in the monkey striate context,” J. Comp. Neurol. 158, 267–294 (1974).
[CrossRef] [PubMed]

J. Exptl. Psychol. (1)

R. W. Sekuler, “Spatial and temporal determinants of visual backward masking,” J. Exptl. Psychol. 70, 401−406 (1965).
[CrossRef]

J. Neurophysiol. (1)

P. H. Schiller, B. L. Finlay, and S. F. Volman, “Quantitative studies of single-cell properties in monkey striate cortex. II. Orientation specificity and ocular dominance,” J. Neurophysiol. 39, 1334–1351 (1976).
[PubMed]

J. Opt. Soc. Am. (1)

J. Physiol. (5)

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

P. E. King-Smith and J. J. Kulikowski, “The detection of gratings by independent activation of line detectors,” J. Physiol. 247, 237–271 (1975).

F. W. Campbell and J. J. Kulikowski, “Orientational selectivity of the human visual system,” J. Physiol. 187, 437–445 (1966).

C. Blakemore and J. Nachmias, “The orientation specificity of two visual after-effects,” J. Physiol. 213, 157–174 (1971).

D. H. Hubel and T. N. Wiesel, “Receptive fields, binocular interaction and functional architecture in the cat’s striate cortex,” J. Physiol. 160, 106–154 (1962).

Psychol. Rev. (1)

J. P. Thomas, “Model of the function of receptive fields in human vision,” Psychol. Rev. 77, 121–134 (1970).
[CrossRef] [PubMed]

Vision Res. (10)

J. P. Thomas and K. K. Shimamura, “Inhibitory interaction between visual pathways tuned to different orientations,” Vision Res. 15, 1373–1380 (1975).
[CrossRef] [PubMed]

J. P. Thomas and K. K. Shimamura, “Perception of size at the detection threshold: its accuracy and possible mechanisms,” Vision Res. 14, 535–543 (1974).
[CrossRef] [PubMed]

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

N. Graham and J. Nachmias, “Detection of grating patterns containing two spatial frequencies: a comparison of single-channel and multiple-channel models,” Vision Res. 11, 251–259 (1971).
[CrossRef] [PubMed]

J. J. Kulikowski, R. Abadi, and P. E. King-Smith, “Orientational selectivity of grating and line detectors in human vision,” Vision Res. 13, 1479–1486 (1973).
[CrossRef] [PubMed]

J. P. Thomas, “Spatial summation in the fovea: asymmetrical effects of longer and shorter dimensions,” Vision Res. 18, 1023–1029 (1978).
[CrossRef] [PubMed]

C. S. Furchner, J. P. Thomas, and F. W. Campbell, “Detection and discrimination of simple and complex patterns at low spatial frequencies,” Vision Res. 17, 827–836 (1977).
[CrossRef] [PubMed]

H. Mostafavi and D. J. Sakrison, “Structure and properties of asingle channel in the human visual system,” Vision Res. 16, 957–968 (1976).
[CrossRef]

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

N. Graham, “Visual detection of aperiodic spatial stimuli by probability summation among narrowband channels,” Vision Res. 17, 637–653 (1977).
[CrossRef] [PubMed]

Vision. Res. (1)

J. Nachmias and A. Weber, “Discrimination of simple and complex gratings,” Vision. Res. 15, 217–223 (1975).
[CrossRef] [PubMed]

Other (2)

The term channel has been used with at least two different meanings. Sometimes, the term refers to a mechanism which responds to a distinctly limited portion of the visual field and then only to components lying within limited ranges of spatial frequency and orientation.11,12 Sometimes, the term channel refers to a matrix consisting of all such mechanisms which are tuned to a given band of spatial frequencies or orientations.13,14 The formulations presented here were developed using the first meaning. However, given two conditions, the individual mechanism and the matrix have the same bandwidth with respect to orientation. The two conditions are (i) the different mechanisms in the matrix have the properties described in the discussion of nonoptimal channels in the Appendix; and (ii) the responses of the individual mechanisms Ri are combined to form the matrix response, RM, by a process which can be described asRM=[∑Rip]1/p.The Euclidean distance model is a special case of this general combining rule.

The development of this model and the related computer simulation were carried out by R. A. Barker. We thank him for permitting us to use his results.

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Figures (6)

FIG. 1
FIG. 1

The relationship between the probability of detecting a grating, p(D), and the probability of identifying its orientation, p(I). The stimulus was a 5 c/deg grating. In (a), the orientations to be discriminated were 3° apart; in (b) they differed by 10°. The broken line is the best fitting straight line. The solid line is the best fitting line which is forced through the origin (0.5, 0.5). The two lines are coincident in (a). Observer: KS.

FIG. 2
FIG. 2

I/D slope as a function of the difference in orientation to be discriminated: 5-c/deg grating stimulus. ● KS, ○ JT, ▪ RB, ▴ JG.

FIG. 3
FIG. 3

I/D slope as a function of the difference in orientation to be discriminated: line stimulus. ● vertical condition, ○ oblique condition. Observer: KS.

FIG. 4
FIG. 4

Same as Fig. 3. Observer: MS.

FIG. 5
FIG. 5

Theoretical relationship between I/D slope and difference in orientation to be discriminated. The difference in orientation is expressed as a fraction of the total bandwidth of a single channel. Solid line, Euclidean model, closely spaced channels; dashed line, Euclidean model, widely spaced channels; ● probability summation model.

FIG. 6
FIG. 6

Alternative sensitivity functions for an individual channel. S is the sensitivity of channel i to a stimulus at orientation θ. As explained in the text, the relative bandwidths of the functions were determined by fitting the three functions to the experimental data. Absolute values for θ are not given because bandwidth varies significantly from one observer to another. Heavy line, cosine function, S = cos |θ|; light line, triangular function, S = 1 − |θ/θ′|; dashed line, Concave function, S = 1 − sin|θ|. θ′ is the orientation at which sensitivity falls to zero; the equations hold only for orientations between −θ′ and +θ′.

Tables (1)

Tables Icon

TABLE I Half-amplitude bandwidths for different observers. Estimates were obtained using the Euclidean distance model and assuming cosine sensitivity functions for the individual channels

Equations (11)

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d x , y = [ i = 1 n j = 1 2 ( R ijx R ijy ) 2 ] 1 / 2 ,
R ijx = S i x C j x ,
Z ( D A ) = d A 1 , A 2 / υ A 1 , A 2 ,
Z ( D B ) = d B 1 , B 2 / υ B 1 , B 2 .
z ( I ) = d A 1 , B 1 / υ A 1 , B 1 = d A 2 , B 2 / υ A 2 , B 2 ,
Z ( I ) / Z ( D ) = d A 1 , B 1 / d A 1 , A 2 = etc .
d A 1 , B 1 d A 1 , A 2 = [ i = 1 n j = 1 2 ( R ijA 1 R ijB 1 ) 2 ] 1 / 2 [ i = 1 n j = 1 2 ( R ijA 1 R ijA 2 ) 2 ] 1 / 2
= [ i = 1 n j = 1 2 ( C j A 1 S i A C j B 1 S i B ) 2 ] 1 / 2 [ i = 1 n j = 1 2 ( C j A 1 S i A C j A 2 S i A ) 2 ] 1 / 2 .
[ i = 1 n ( C 1 A 1 S i A C 1 B 1 S i B ) 2 ] 1 / 2 [ i = 1 n ( C 1 A 1 S i A ) 2 + i = 1 n ( C 2 A 2 S i A ) 2 ] 1 / 2 .
d A 1 , B 1 d A 1 , A 2 = [ i = 1 n ( S i A S i B ) 2 ] 1 / 2 [ 2 i = 1 n S i A 2 ] 1 / 2 .
RM=[Rip]1/p.