Abstract

The oblique effect refers to a better perception of horizontal and vertical image features as compared with the perception at oblique angles. This effect can be observed in both animals and humans. Recent neurophysiological data suggest that the basis of this effect lies in the structure of the primary visual cortex, where more cortical area is devoted to processing contours with angles at horizontal and vertical orientations (cardinal orientations). It has been suggested that this cortical feature has developed according to the statistical properties of natural scenes. To examine this hypothesis in more detail, we established six image classes and categorized the images with respect to their semantical contents. From the images the oriented energy was calculated by using the corresponding power spectra. We defined simple measures for the degree (cardinal versus oblique energy ratio) and the skewness or anisotropy (aligned energy ratio) of the alignment of energy at horizontal and vertical orientations. Our results provide evidence that (1) alignment depends strongly on the environment, (2) the degree of alignment drops off characteristically at higher frequencies, and (3) in natural images there is on the average an anisotropy in the distribution of energy at the cardinal orientations (i.e., a difference between the amounts of vertical energy and horizontal energy). In light of our results, we further discuss whether the observed cortical anisotropy has its origin in phylogeny or ontogeny.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. P. Jones, L. A. Palmer, “An evaluation of the two-dimensional Gabor filter model of simple receptive fields in cat striate cortex,” J. Neurophysiol. 58, 1233–1258 (1987).
    [PubMed]
  2. S. Apelle, “Perception and discrimination as a function of stimulus orientation: the oblique effect in man and animals,” Psychol. Bull. 78, 226–278 (1972).
  3. I. P. Howard, Human Visual Orientation (Wiley, New York, 1982).
  4. B. A. Olshausen, D. J. Field, “Wavelet-like receptive fields emerge from a network that learns sparse codes for natural images,” Nature (London) 381, 607–609 (1996).
    [CrossRef]
  5. F. Sengpiel, P. Stawinski, T. Bonhoeffer, “Influence of experience on orientation maps in cat visual cortex,” Nature Neurosci. 2, 727–732 (1999).
    [CrossRef] [PubMed]
  6. B. Chapman, T. Bonhoeffer, “Overrepresentation of horizontal and vertical orientations preferences in developing ferret area 17,” Proc. Natl. Acad. Sci. USA 95, 2609–2614 (1998).
    [CrossRef]
  7. D. Coppola, L. White, D. Fitzpatrick, D. Purves, “Unequal representation of cardinal and oblique contours in ferret visual cortex,” Proc. Natl. Acad. Sci. USA 95, 2621–2623 (1998).
    [CrossRef] [PubMed]
  8. J. D. Schall, D. J. Vitek, A. G. Leventha, “Retinal constraints on orientation specificity in cat visual cortex,” J. Neurosci. 6, 823–836 (1986).
    [PubMed]
  9. D. C. Linden, R. W. Guillery, J. Cucchiaro, “The dorsal lateral geniculate nucleus of the normal ferret and its postnatal development,” J. Comp. Neurol. 203, 189–211 (1981).
    [CrossRef] [PubMed]
  10. M. I. Law, K. R. Zahs, M. P. Stryker, “Organization of the primary visual cortex (area 17) in the ferret,” J. Comp. Neurol. Power 278, 157–180 (1988).
    [CrossRef]
  11. E. Switkes, M. J. Mayer, J. A. Sloan, “Spatial frequency analysis of the visual environment: anisotropy and the carpentered environment hypothesis,” Vision Res. 18, 1393–1399 (1978).
    [CrossRef] [PubMed]
  12. D. Coppola, H. Purves, A. McCoy, D. Purves, “The distribution of oriented contours in the real world,” Proc. Natl. Acad. Sci. USA 95, 4002–4006 (1998).
    [CrossRef] [PubMed]
  13. A. van der Schaaf, J. H. van Hateren, “Modeling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759–2770 (1996).
    [CrossRef] [PubMed]
  14. D. J. Field, “Relations between the statistics of natural images and the response properties of cortical cells,” J. Opt. Soc. Am. A 14, 2379–2394 (1987).
    [CrossRef]
  15. http://hlab.phys.rug.nl/archive.html .
  16. F. W. Campbell, D. G. Green, “Optical and retinal factors affecting visual resolution,” J. Physiol. (London) 181, 576–593 (1965).
  17. J. H. van Hateren, A. van der Schaaf, “Independent component filters of natural images compared with simple cells in primary visual cortex,” Proc. R. Soc. London Ser. B 265, 359–366 (1998).
    [CrossRef]
  18. http://hlab.phys.rug.nl/imlib/peaks.html .
  19. D. L. Ruderman, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
    [CrossRef] [PubMed]
  20. M. C. Crair, D. C. Gillespie, M. P. Stryker, “The role of visual experience in the development of columns in cat visual cortex,” Science 279, 566–570 (1998).
    [CrossRef] [PubMed]

1999 (1)

F. Sengpiel, P. Stawinski, T. Bonhoeffer, “Influence of experience on orientation maps in cat visual cortex,” Nature Neurosci. 2, 727–732 (1999).
[CrossRef] [PubMed]

1998 (5)

B. Chapman, T. Bonhoeffer, “Overrepresentation of horizontal and vertical orientations preferences in developing ferret area 17,” Proc. Natl. Acad. Sci. USA 95, 2609–2614 (1998).
[CrossRef]

D. Coppola, L. White, D. Fitzpatrick, D. Purves, “Unequal representation of cardinal and oblique contours in ferret visual cortex,” Proc. Natl. Acad. Sci. USA 95, 2621–2623 (1998).
[CrossRef] [PubMed]

D. Coppola, H. Purves, A. McCoy, D. Purves, “The distribution of oriented contours in the real world,” Proc. Natl. Acad. Sci. USA 95, 4002–4006 (1998).
[CrossRef] [PubMed]

J. H. van Hateren, A. van der Schaaf, “Independent component filters of natural images compared with simple cells in primary visual cortex,” Proc. R. Soc. London Ser. B 265, 359–366 (1998).
[CrossRef]

M. C. Crair, D. C. Gillespie, M. P. Stryker, “The role of visual experience in the development of columns in cat visual cortex,” Science 279, 566–570 (1998).
[CrossRef] [PubMed]

1996 (2)

B. A. Olshausen, D. J. Field, “Wavelet-like receptive fields emerge from a network that learns sparse codes for natural images,” Nature (London) 381, 607–609 (1996).
[CrossRef]

A. van der Schaaf, J. H. van Hateren, “Modeling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759–2770 (1996).
[CrossRef] [PubMed]

1994 (1)

D. L. Ruderman, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
[CrossRef] [PubMed]

1988 (1)

M. I. Law, K. R. Zahs, M. P. Stryker, “Organization of the primary visual cortex (area 17) in the ferret,” J. Comp. Neurol. Power 278, 157–180 (1988).
[CrossRef]

1987 (2)

J. P. Jones, L. A. Palmer, “An evaluation of the two-dimensional Gabor filter model of simple receptive fields in cat striate cortex,” J. Neurophysiol. 58, 1233–1258 (1987).
[PubMed]

D. J. Field, “Relations between the statistics of natural images and the response properties of cortical cells,” J. Opt. Soc. Am. A 14, 2379–2394 (1987).
[CrossRef]

1986 (1)

J. D. Schall, D. J. Vitek, A. G. Leventha, “Retinal constraints on orientation specificity in cat visual cortex,” J. Neurosci. 6, 823–836 (1986).
[PubMed]

1981 (1)

D. C. Linden, R. W. Guillery, J. Cucchiaro, “The dorsal lateral geniculate nucleus of the normal ferret and its postnatal development,” J. Comp. Neurol. 203, 189–211 (1981).
[CrossRef] [PubMed]

1978 (1)

E. Switkes, M. J. Mayer, J. A. Sloan, “Spatial frequency analysis of the visual environment: anisotropy and the carpentered environment hypothesis,” Vision Res. 18, 1393–1399 (1978).
[CrossRef] [PubMed]

1972 (1)

S. Apelle, “Perception and discrimination as a function of stimulus orientation: the oblique effect in man and animals,” Psychol. Bull. 78, 226–278 (1972).

1965 (1)

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

Apelle, S.

S. Apelle, “Perception and discrimination as a function of stimulus orientation: the oblique effect in man and animals,” Psychol. Bull. 78, 226–278 (1972).

Bonhoeffer, T.

F. Sengpiel, P. Stawinski, T. Bonhoeffer, “Influence of experience on orientation maps in cat visual cortex,” Nature Neurosci. 2, 727–732 (1999).
[CrossRef] [PubMed]

B. Chapman, T. Bonhoeffer, “Overrepresentation of horizontal and vertical orientations preferences in developing ferret area 17,” Proc. Natl. Acad. Sci. USA 95, 2609–2614 (1998).
[CrossRef]

Campbell, F. W.

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

Chapman, B.

B. Chapman, T. Bonhoeffer, “Overrepresentation of horizontal and vertical orientations preferences in developing ferret area 17,” Proc. Natl. Acad. Sci. USA 95, 2609–2614 (1998).
[CrossRef]

Coppola, D.

D. Coppola, H. Purves, A. McCoy, D. Purves, “The distribution of oriented contours in the real world,” Proc. Natl. Acad. Sci. USA 95, 4002–4006 (1998).
[CrossRef] [PubMed]

D. Coppola, L. White, D. Fitzpatrick, D. Purves, “Unequal representation of cardinal and oblique contours in ferret visual cortex,” Proc. Natl. Acad. Sci. USA 95, 2621–2623 (1998).
[CrossRef] [PubMed]

Crair, M. C.

M. C. Crair, D. C. Gillespie, M. P. Stryker, “The role of visual experience in the development of columns in cat visual cortex,” Science 279, 566–570 (1998).
[CrossRef] [PubMed]

Cucchiaro, J.

D. C. Linden, R. W. Guillery, J. Cucchiaro, “The dorsal lateral geniculate nucleus of the normal ferret and its postnatal development,” J. Comp. Neurol. 203, 189–211 (1981).
[CrossRef] [PubMed]

Field, D. J.

B. A. Olshausen, D. J. Field, “Wavelet-like receptive fields emerge from a network that learns sparse codes for natural images,” Nature (London) 381, 607–609 (1996).
[CrossRef]

D. J. Field, “Relations between the statistics of natural images and the response properties of cortical cells,” J. Opt. Soc. Am. A 14, 2379–2394 (1987).
[CrossRef]

Fitzpatrick, D.

D. Coppola, L. White, D. Fitzpatrick, D. Purves, “Unequal representation of cardinal and oblique contours in ferret visual cortex,” Proc. Natl. Acad. Sci. USA 95, 2621–2623 (1998).
[CrossRef] [PubMed]

Gillespie, D. C.

M. C. Crair, D. C. Gillespie, M. P. Stryker, “The role of visual experience in the development of columns in cat visual cortex,” Science 279, 566–570 (1998).
[CrossRef] [PubMed]

Green, D. G.

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

Guillery, R. W.

D. C. Linden, R. W. Guillery, J. Cucchiaro, “The dorsal lateral geniculate nucleus of the normal ferret and its postnatal development,” J. Comp. Neurol. 203, 189–211 (1981).
[CrossRef] [PubMed]

Howard, I. P.

I. P. Howard, Human Visual Orientation (Wiley, New York, 1982).

Jones, J. P.

J. P. Jones, L. A. Palmer, “An evaluation of the two-dimensional Gabor filter model of simple receptive fields in cat striate cortex,” J. Neurophysiol. 58, 1233–1258 (1987).
[PubMed]

Law, M. I.

M. I. Law, K. R. Zahs, M. P. Stryker, “Organization of the primary visual cortex (area 17) in the ferret,” J. Comp. Neurol. Power 278, 157–180 (1988).
[CrossRef]

Leventha, A. G.

J. D. Schall, D. J. Vitek, A. G. Leventha, “Retinal constraints on orientation specificity in cat visual cortex,” J. Neurosci. 6, 823–836 (1986).
[PubMed]

Linden, D. C.

D. C. Linden, R. W. Guillery, J. Cucchiaro, “The dorsal lateral geniculate nucleus of the normal ferret and its postnatal development,” J. Comp. Neurol. 203, 189–211 (1981).
[CrossRef] [PubMed]

Mayer, M. J.

E. Switkes, M. J. Mayer, J. A. Sloan, “Spatial frequency analysis of the visual environment: anisotropy and the carpentered environment hypothesis,” Vision Res. 18, 1393–1399 (1978).
[CrossRef] [PubMed]

McCoy, A.

D. Coppola, H. Purves, A. McCoy, D. Purves, “The distribution of oriented contours in the real world,” Proc. Natl. Acad. Sci. USA 95, 4002–4006 (1998).
[CrossRef] [PubMed]

Olshausen, B. A.

B. A. Olshausen, D. J. Field, “Wavelet-like receptive fields emerge from a network that learns sparse codes for natural images,” Nature (London) 381, 607–609 (1996).
[CrossRef]

Palmer, L. A.

J. P. Jones, L. A. Palmer, “An evaluation of the two-dimensional Gabor filter model of simple receptive fields in cat striate cortex,” J. Neurophysiol. 58, 1233–1258 (1987).
[PubMed]

Purves, D.

D. Coppola, H. Purves, A. McCoy, D. Purves, “The distribution of oriented contours in the real world,” Proc. Natl. Acad. Sci. USA 95, 4002–4006 (1998).
[CrossRef] [PubMed]

D. Coppola, L. White, D. Fitzpatrick, D. Purves, “Unequal representation of cardinal and oblique contours in ferret visual cortex,” Proc. Natl. Acad. Sci. USA 95, 2621–2623 (1998).
[CrossRef] [PubMed]

Purves, H.

D. Coppola, H. Purves, A. McCoy, D. Purves, “The distribution of oriented contours in the real world,” Proc. Natl. Acad. Sci. USA 95, 4002–4006 (1998).
[CrossRef] [PubMed]

Ruderman, D. L.

D. L. Ruderman, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
[CrossRef] [PubMed]

Schall, J. D.

J. D. Schall, D. J. Vitek, A. G. Leventha, “Retinal constraints on orientation specificity in cat visual cortex,” J. Neurosci. 6, 823–836 (1986).
[PubMed]

Sengpiel, F.

F. Sengpiel, P. Stawinski, T. Bonhoeffer, “Influence of experience on orientation maps in cat visual cortex,” Nature Neurosci. 2, 727–732 (1999).
[CrossRef] [PubMed]

Sloan, J. A.

E. Switkes, M. J. Mayer, J. A. Sloan, “Spatial frequency analysis of the visual environment: anisotropy and the carpentered environment hypothesis,” Vision Res. 18, 1393–1399 (1978).
[CrossRef] [PubMed]

Stawinski, P.

F. Sengpiel, P. Stawinski, T. Bonhoeffer, “Influence of experience on orientation maps in cat visual cortex,” Nature Neurosci. 2, 727–732 (1999).
[CrossRef] [PubMed]

Stryker, M. P.

M. C. Crair, D. C. Gillespie, M. P. Stryker, “The role of visual experience in the development of columns in cat visual cortex,” Science 279, 566–570 (1998).
[CrossRef] [PubMed]

M. I. Law, K. R. Zahs, M. P. Stryker, “Organization of the primary visual cortex (area 17) in the ferret,” J. Comp. Neurol. Power 278, 157–180 (1988).
[CrossRef]

Switkes, E.

E. Switkes, M. J. Mayer, J. A. Sloan, “Spatial frequency analysis of the visual environment: anisotropy and the carpentered environment hypothesis,” Vision Res. 18, 1393–1399 (1978).
[CrossRef] [PubMed]

van der Schaaf, A.

J. H. van Hateren, A. van der Schaaf, “Independent component filters of natural images compared with simple cells in primary visual cortex,” Proc. R. Soc. London Ser. B 265, 359–366 (1998).
[CrossRef]

A. van der Schaaf, J. H. van Hateren, “Modeling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759–2770 (1996).
[CrossRef] [PubMed]

van Hateren, J. H.

J. H. van Hateren, A. van der Schaaf, “Independent component filters of natural images compared with simple cells in primary visual cortex,” Proc. R. Soc. London Ser. B 265, 359–366 (1998).
[CrossRef]

A. van der Schaaf, J. H. van Hateren, “Modeling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759–2770 (1996).
[CrossRef] [PubMed]

Vitek, D. J.

J. D. Schall, D. J. Vitek, A. G. Leventha, “Retinal constraints on orientation specificity in cat visual cortex,” J. Neurosci. 6, 823–836 (1986).
[PubMed]

White, L.

D. Coppola, L. White, D. Fitzpatrick, D. Purves, “Unequal representation of cardinal and oblique contours in ferret visual cortex,” Proc. Natl. Acad. Sci. USA 95, 2621–2623 (1998).
[CrossRef] [PubMed]

Zahs, K. R.

M. I. Law, K. R. Zahs, M. P. Stryker, “Organization of the primary visual cortex (area 17) in the ferret,” J. Comp. Neurol. Power 278, 157–180 (1988).
[CrossRef]

J. Comp. Neurol. (1)

D. C. Linden, R. W. Guillery, J. Cucchiaro, “The dorsal lateral geniculate nucleus of the normal ferret and its postnatal development,” J. Comp. Neurol. 203, 189–211 (1981).
[CrossRef] [PubMed]

J. Comp. Neurol. Power (1)

M. I. Law, K. R. Zahs, M. P. Stryker, “Organization of the primary visual cortex (area 17) in the ferret,” J. Comp. Neurol. Power 278, 157–180 (1988).
[CrossRef]

J. Neurophysiol. (1)

J. P. Jones, L. A. Palmer, “An evaluation of the two-dimensional Gabor filter model of simple receptive fields in cat striate cortex,” J. Neurophysiol. 58, 1233–1258 (1987).
[PubMed]

J. Neurosci. (1)

J. D. Schall, D. J. Vitek, A. G. Leventha, “Retinal constraints on orientation specificity in cat visual cortex,” J. Neurosci. 6, 823–836 (1986).
[PubMed]

J. Opt. Soc. Am. A (1)

J. Physiol. (London) (1)

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

Nature (London) (1)

B. A. Olshausen, D. J. Field, “Wavelet-like receptive fields emerge from a network that learns sparse codes for natural images,” Nature (London) 381, 607–609 (1996).
[CrossRef]

Nature Neurosci. (1)

F. Sengpiel, P. Stawinski, T. Bonhoeffer, “Influence of experience on orientation maps in cat visual cortex,” Nature Neurosci. 2, 727–732 (1999).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

D. L. Ruderman, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. USA (3)

D. Coppola, H. Purves, A. McCoy, D. Purves, “The distribution of oriented contours in the real world,” Proc. Natl. Acad. Sci. USA 95, 4002–4006 (1998).
[CrossRef] [PubMed]

B. Chapman, T. Bonhoeffer, “Overrepresentation of horizontal and vertical orientations preferences in developing ferret area 17,” Proc. Natl. Acad. Sci. USA 95, 2609–2614 (1998).
[CrossRef]

D. Coppola, L. White, D. Fitzpatrick, D. Purves, “Unequal representation of cardinal and oblique contours in ferret visual cortex,” Proc. Natl. Acad. Sci. USA 95, 2621–2623 (1998).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. B (1)

J. H. van Hateren, A. van der Schaaf, “Independent component filters of natural images compared with simple cells in primary visual cortex,” Proc. R. Soc. London Ser. B 265, 359–366 (1998).
[CrossRef]

Psychol. Bull. (1)

S. Apelle, “Perception and discrimination as a function of stimulus orientation: the oblique effect in man and animals,” Psychol. Bull. 78, 226–278 (1972).

Science (1)

M. C. Crair, D. C. Gillespie, M. P. Stryker, “The role of visual experience in the development of columns in cat visual cortex,” Science 279, 566–570 (1998).
[CrossRef] [PubMed]

Vision Res. (2)

A. van der Schaaf, J. H. van Hateren, “Modeling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759–2770 (1996).
[CrossRef] [PubMed]

E. Switkes, M. J. Mayer, J. A. Sloan, “Spatial frequency analysis of the visual environment: anisotropy and the carpentered environment hypothesis,” Vision Res. 18, 1393–1399 (1978).
[CrossRef] [PubMed]

Other (3)

I. P. Howard, Human Visual Orientation (Wiley, New York, 1982).

http://hlab.phys.rug.nl/archive.html .

http://hlab.phys.rug.nl/imlib/peaks.html .

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

(a) Single club (representing a weighted integration area in the energy plane) shown at orientation ϕ=0° (corresponding to vertical image features) and ϕ=90° (corresponding to horizontal image features) and (b) full “club swing” in the spatial-frequency plane (optical representation). The effective image size is 1023×1023pixels.

Fig. 2
Fig. 2

Logarithmic average power spectra for each class in a pseudocolor representation hue–saturation–value color model. The corresponding classes are (a) city, (b) man, (c) natural, (d) straight, (e) grass, and (f) sky.

Fig. 3
Fig. 3

Oriented energy (OE) as calculated by the club integration scheme (CIS) by using the direct (i.e., original) averaged energy spectrum (ES). The corresponding classes are (a) city, (b) man, (c) natural, (d) straight, (e) grass, and (f) sky.

Fig. 4
Fig. 4

Frequency dependence of the OE calculated from the mean power spectrum (ES) of each class. This was done by using the CIS with isotropic bloblike integration areas. The frequency axis was sampled linearly. Blobs at different frequencies were simply rescaled versions of each other (with bigger blobs at higher spatial frequencies). For reasons of visualization, the energy axis has been rescaled with a factor proportional to the square of the spatial frequency. The corresponding classes are (a) city, (b) man, (c) natural, (d) straight, (e) grass, and (f) sky.

Fig. 5
Fig. 5

Frequency dependence (part 1) of the cardinal versus oblique energy ratio (COR) (left column) and the aligned energy ratio (AER) (right column), calculated from the frequency-dependent OE as shown in Fig. 4. Note that the results for natural and straight are summarized in one plot. In (e) the dotted curve shows the COR for straight and the solid curve the COR for natural. The dashed horizontal lines refer to the mean value of AER(σ=1.5), as obtained from uniform-distributed noise. The dotted horizontal lines are the associated absolute deviations. Note the sign reversal in (f). Corresponding classes are (a), (b) city, (c), (d) man, and (e), (f) natural and straight.

Fig. 6
Fig. 6

Frequency dependence (part 2) of the COR (left column) and the AER (right column). See also the caption to Fig. 5. Corresponding classes are (a), (b) grass and (c), (d) sky. In (c) the region enclosed by the two dotted lines corresponds to ΔCORcrit.

Fig. 7
Fig. 7

(a) Cardinal vs. oblique energy ratio COR and (b) AER calculated from the mean ES for each class. Triangles: original ES, squares: k2-windowed OE, circles: logarithmic ES. See also the caption to Fig. 5.

Tables (2)

Tables Icon

Table 1 Number of Images Fulfilling CORΔCORcrit While Using the Raw ES (Direct), the k2-Windowed ES, and the Logarithmic ES

Tables Icon

Table 2 Number of Images Fulfilling AERΔAERcrit While Using the Unmodified ES (Direct), the k2-Windowed ES, and the Logarithmic ES

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

E˜α(σ)(ϕ)=Eα(ψ)12πσexp-(ϕ-ψ)22σ2,
Ecardinal=E˜α(σc)(0°)+E˜α(σc)(90°)
Eoblique=E˜α(σo)(45°)+E˜α(σo)(135°).
COR(σ)=Ecardinal-EobliqueEcardinal+Eoblique.
AER(σ)=E˜α(0°)-E˜α(90°)E˜α(0°)+E˜α(90°).

Metrics