Abstract

The classical receptive fields of simple cells in mammalian primary visual cortex demonstrate three cardinal response properties: (1) they do not respond to stimuli that are spatially homogeneous; (2) they respond best to stimuli in a preferred orientation (direction); and (3) they do not respond to stimuli in other, nonpreferred orientations (directions). We refer to these as the balanced field property, the maximum response direction property, and the zero response direction property, respectively. These empirically determined response properties are used to derive a complete characterization of elementary receptive field functions defined as products of a circularly symmetric weight function and a simple periodic carrier. Two disjoint classes of elementary receptive field functions result: the balanced Gabor class, a generalization of the traditional Gabor filter, and a bandlimited class whose Fourier transforms have compact support (i.e., are zero valued outside of a bounded range). The detailed specification of these two classes of receptive field functions from empirically based postulates may prove useful to neurophysiologists seeking to test alternative theories of simple cell receptive field structure and to computational neuroscientists seeking basis functions with which to model human vision.

© 2009 Optical Society of America

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    [CrossRef] [PubMed]

2007 (1)

J. M. Foley, S. Varadharajan, C. C. Koh, and M. C. Farias, “Detection of Gabor patterns of different sizes, shapes, phases and eccentricities,” Vision Res. 47, 85-107 (2007).
[CrossRef]

2005 (2)

X. Peng and D. C. Van Essen, “Peaked encoding of relative luminance in macaque areas V1 and V2,” J. Neurophysiol. 93, 1620-1632 (2005).
[CrossRef]

A. W. Roe, H. D. Lu, and C. P. Hung, “Cortical processing of a brightness illusion,” Proc. Natl. Acad. Sci. U.S.A. 102, 3869-3874 (2005).
[CrossRef] [PubMed]

2003 (1)

T. Wachtler, T. J. Sejnowski, and T. D. Albright, “Representation of color stimuli in awake macaque primary visual cortex,” Neuron 37, 681-691 (2003).
[CrossRef] [PubMed]

2002 (2)

T. D. Albright and G. R. Stoner, “Contextual influences on visual processing,” Annu. Rev. Neurosci. 25, 339-379 (2002).
[CrossRef] [PubMed]

D. L. Ringach, “Spatial structure and symmetry of simple-cell receptive fields in macaque primary visual cortex,” J. Neurophysiol. 88, 455-463 (2002).
[PubMed]

2001 (2)

G. Wallis, “Linear models of simple cells: correspondence to real cell responses and space spanning properties,” Spatial Vis. 14, 237-260 (2001).
[CrossRef]

M. Kinoshita and H. Komatsu, “Neural representation of the luminance and brightness of a uniform surface in the macaque primary visual cortex,” J. Neurophysiol. 86, 2559-2570 (2001).
[PubMed]

2000 (1)

B. A. Olshausen and D. J. Field, “Vision and the coding of natural images,” Am. Sci. 88, 238-245 (2000).

1999 (2)

U. Polat and C. W. Tyler, “What pattern the eye sees best,” Vision Res. 39, 887-895 (1999).
[CrossRef] [PubMed]

A. F. Rossi and M. A. Paradiso, “Neural correlates of perceived brightness in the retina, lateral geniculate nucleus, and striate cortex,” J. Neurosci. 19, 6145-6156 (1999).
[PubMed]

1998 (1)

S. P. MacEvoy, W. Kim, and M. A. Paradiso, “Integration of surface information in primary visual cortex,” Nat. Neurosci. 1, 616-620 (1998).
[CrossRef]

1997 (1)

N. Petkov and P. Kruizinga, “Computational models of visual neurons specialised in the detection of periodic and aperiodic oriented visual stimuli: bar and grating cells,” Biol. Cybern. 76, 83-96 (1997).
[CrossRef] [PubMed]

1996 (5)

B. A. Olshausen and D. J. Field, “Emergence of simple-cell receptive field properties by learning a sparse code for natural images,” Nature 381, 607-609 (1996).
[CrossRef] [PubMed]

T. S. Lee, “Image representation using 2D gabor wavelets,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 1-13 (1996).
[CrossRef]

M. Potzsch, N. Kruger, and C. von der Malsburg, “Improving object recognition by transforming Gabor filter responses,” Network Comput. Neural Syst. 7, 341-347 (1996).
[CrossRef]

A. F. Rossi, C. D. Rittenhouse, and M. A. Paradiso, “The representation of brightness in primary visual cortex,” Science 273, 1104-1107 (1996).
[CrossRef] [PubMed]

A. F. Rossi and M. A. Paradiso, “Temporal limits of brightness induction and mechanisms of brightness perception,” Vision Res. 36, 1391-1398 (1996).
[CrossRef] [PubMed]

1995 (1)

N. Petkov, “Biologically motivated computationally intensive approaches to image pattern recognition,” Future Gener. Comput. Syst. 11, 451-465 (1995).
[CrossRef]

1992 (1)

F. Heitger, L. Rosenthaler, R. von der Heydt, E. Peterhans, and O. Kubler, “Simulation of neural contour mechanisms: from simple to end-stopped cells,” Vision Res. 32, 963-981 (1992).
[CrossRef] [PubMed]

1991 (1)

1988 (1)

1987 (2)

D. J. Field, “Relations between the statistics of natural images and the response properties of cortical cells,” J. Opt. Soc. Am. A 4, 2379-2394 (1987).
[CrossRef] [PubMed]

J. P. Jones and 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]

1986 (2)

P. Heggelund, “Quantitative studies of the discharge fields of single cells in cat striate cortex,” J. Physiol. (London) 373, 277-292 (1986).

P. Heggelund, “Quantitative studies of enhancement and suppression zones in the receptive field of simple cells in cat striate cortex,” J. Physiol. (London) 373, 293-310 (1986).

1985 (2)

1983 (1)

A. B. Watson, H. B. Barlow, and J. G. Robson, “What does the eye see best?” Nature 302, 419-422 (1983).
[CrossRef] [PubMed]

1981 (1)

J. J. Kulikowski and P. O. Bishop, “Fourier analysis and spatial representation in the visual cortex,” Experientia 37, 160-163 (1981).
[CrossRef] [PubMed]

1980 (3)

D. Marr and E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London, Ser. B 207, 187-217 (1980).
[CrossRef]

S. Marcelja, “Mathematical description of the responses of simple cortical cells,” J. Opt. Soc. Am. 70, 1297-1300 (1980).
[CrossRef] [PubMed]

J. G. Daugman, “Two-dimensional spectral analysis of cortical receptive field profiles,” Vision Res. 20, 847-856 (1980).
[CrossRef] [PubMed]

1979 (1)

Y. Kayama, R. R. Riso, J. R. Bartlett, and R. W. Doty, “Luxotonic responses of units in macaque striate cortex,” J. Neurophysiol. 42(6), 1495-1517 (1979).
[PubMed]

1977 (1)

D. H. Hubel and T. N. Wiesel, “Ferrier lecture. Functional architecture of macaque monkey visual cortex,” Proc. R. Soc. London, Ser. B 198, 1-59 (1977).
[CrossRef]

1976 (1)

L. Maffei and A. Fiorentini, “The unresponsive regions of visual cortical receptive fields,” Vision Res. 16, 1131-1139 (1976).
[CrossRef] [PubMed]

1968 (1)

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

1946 (1)

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93, 429-457 (1946).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. Applied Mathematics Series 55 (National Bureau of Standards, 1972).

Albrecht, D. G.

R. L. DeValois, D. G. Albrecht, and L. G. Thorell, “Cortical cells: bar and edge detectors, or spatial frequency filters?” in Frontiers in Visual Science, S.J.Cool and E.L.Smith, eds. (Springer-Verlag, 1978), pp. 544-556.

Albright, T. D.

T. Wachtler, T. J. Sejnowski, and T. D. Albright, “Representation of color stimuli in awake macaque primary visual cortex,” Neuron 37, 681-691 (2003).
[CrossRef] [PubMed]

T. D. Albright and G. R. Stoner, “Contextual influences on visual processing,” Annu. Rev. Neurosci. 25, 339-379 (2002).
[CrossRef] [PubMed]

Anderson, S. J.

Barlow, H. B.

A. B. Watson, H. B. Barlow, and J. G. Robson, “What does the eye see best?” Nature 302, 419-422 (1983).
[CrossRef] [PubMed]

Bartlett, J. R.

Y. Kayama, R. R. Riso, J. R. Bartlett, and R. W. Doty, “Luxotonic responses of units in macaque striate cortex,” J. Neurophysiol. 42(6), 1495-1517 (1979).
[PubMed]

Bishop, P. O.

J. J. Kulikowski and P. O. Bishop, “Fourier analysis and spatial representation in the visual cortex,” Experientia 37, 160-163 (1981).
[CrossRef] [PubMed]

Blakeslee, B.

D. Cope, B. Blakeslee, and M. E. McCourt, “Simple cell response properties imply receptive field structure: Balanced Gabor and/or bandlimited field functions. Supplement. Appendices A, B, C and Figures 13-16.” http://hdl.handle.net/10365/5418.

Burr, D. C.

Cope, D.

D. Cope, B. Blakeslee, and M. E. McCourt, “Simple cell response properties imply receptive field structure: Balanced Gabor and/or bandlimited field functions. Supplement. Appendices A, B, C and Figures 13-16.” http://hdl.handle.net/10365/5418.

Daugman, J. G.

DeValois, K. K.

R. L. DeValois and K. K. DeValois, Spatial Vision (Oxford Univ. Press, 1988).

DeValois, R. L.

R. L. DeValois and K. K. DeValois, Spatial Vision (Oxford Univ. Press, 1988).

R. L. DeValois, D. G. Albrecht, and L. G. Thorell, “Cortical cells: bar and edge detectors, or spatial frequency filters?” in Frontiers in Visual Science, S.J.Cool and E.L.Smith, eds. (Springer-Verlag, 1978), pp. 544-556.

Doty, R. W.

Y. Kayama, R. R. Riso, J. R. Bartlett, and R. W. Doty, “Luxotonic responses of units in macaque striate cortex,” J. Neurophysiol. 42(6), 1495-1517 (1979).
[PubMed]

Farias, M. C.

J. M. Foley, S. Varadharajan, C. C. Koh, and M. C. Farias, “Detection of Gabor patterns of different sizes, shapes, phases and eccentricities,” Vision Res. 47, 85-107 (2007).
[CrossRef]

Field, D. J.

B. A. Olshausen and D. J. Field, “Vision and the coding of natural images,” Am. Sci. 88, 238-245 (2000).

B. A. Olshausen and D. J. Field, “Emergence of simple-cell receptive field properties by learning a sparse code for natural images,” Nature 381, 607-609 (1996).
[CrossRef] [PubMed]

D. J. Field, “Relations between the statistics of natural images and the response properties of cortical cells,” J. Opt. Soc. Am. A 4, 2379-2394 (1987).
[CrossRef] [PubMed]

Fiorentini, A.

L. Maffei and A. Fiorentini, “The unresponsive regions of visual cortical receptive fields,” Vision Res. 16, 1131-1139 (1976).
[CrossRef] [PubMed]

Foley, J. M.

J. M. Foley, S. Varadharajan, C. C. Koh, and M. C. Farias, “Detection of Gabor patterns of different sizes, shapes, phases and eccentricities,” Vision Res. 47, 85-107 (2007).
[CrossRef]

Gabor, D.

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93, 429-457 (1946).

Hawken, M. J.

Heggelund, P.

P. Heggelund, “Quantitative studies of the discharge fields of single cells in cat striate cortex,” J. Physiol. (London) 373, 277-292 (1986).

P. Heggelund, “Quantitative studies of enhancement and suppression zones in the receptive field of simple cells in cat striate cortex,” J. Physiol. (London) 373, 293-310 (1986).

Heitger, F.

F. Heitger, L. Rosenthaler, R. von der Heydt, E. Peterhans, and O. Kubler, “Simulation of neural contour mechanisms: from simple to end-stopped cells,” Vision Res. 32, 963-981 (1992).
[CrossRef] [PubMed]

Hildreth, E.

D. Marr and E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London, Ser. B 207, 187-217 (1980).
[CrossRef]

Hubel, D. H.

D. H. Hubel and T. N. Wiesel, “Ferrier lecture. Functional architecture of macaque monkey visual cortex,” Proc. R. Soc. London, Ser. B 198, 1-59 (1977).
[CrossRef]

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

Hung, C. P.

A. W. Roe, H. D. Lu, and C. P. Hung, “Cortical processing of a brightness illusion,” Proc. Natl. Acad. Sci. U.S.A. 102, 3869-3874 (2005).
[CrossRef] [PubMed]

Jones, J. P.

J. P. Jones and 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]

Kayama, Y.

Y. Kayama, R. R. Riso, J. R. Bartlett, and R. W. Doty, “Luxotonic responses of units in macaque striate cortex,” J. Neurophysiol. 42(6), 1495-1517 (1979).
[PubMed]

Kim, W.

S. P. MacEvoy, W. Kim, and M. A. Paradiso, “Integration of surface information in primary visual cortex,” Nat. Neurosci. 1, 616-620 (1998).
[CrossRef]

Kinoshita, M.

M. Kinoshita and H. Komatsu, “Neural representation of the luminance and brightness of a uniform surface in the macaque primary visual cortex,” J. Neurophysiol. 86, 2559-2570 (2001).
[PubMed]

Klein, S. A.

Knight, B. W.

J. D. Victor and B. W. Knight, “Simultaneously band and space limited functions in two dimensions, and receptive fields of visual neurons,” in Springer Applied Mathematical Sciences Series, E.Kaplan, J.Marsden, and K.R.Sreenivasan, eds. (Springer, 2003), pp. 375-420.

Koh, C. C.

J. M. Foley, S. Varadharajan, C. C. Koh, and M. C. Farias, “Detection of Gabor patterns of different sizes, shapes, phases and eccentricities,” Vision Res. 47, 85-107 (2007).
[CrossRef]

Komatsu, H.

M. Kinoshita and H. Komatsu, “Neural representation of the luminance and brightness of a uniform surface in the macaque primary visual cortex,” J. Neurophysiol. 86, 2559-2570 (2001).
[PubMed]

Kruger, N.

M. Potzsch, N. Kruger, and C. von der Malsburg, “Improving object recognition by transforming Gabor filter responses,” Network Comput. Neural Syst. 7, 341-347 (1996).
[CrossRef]

Kruizinga, P.

N. Petkov and P. Kruizinga, “Computational models of visual neurons specialised in the detection of periodic and aperiodic oriented visual stimuli: bar and grating cells,” Biol. Cybern. 76, 83-96 (1997).
[CrossRef] [PubMed]

Kubler, O.

F. Heitger, L. Rosenthaler, R. von der Heydt, E. Peterhans, and O. Kubler, “Simulation of neural contour mechanisms: from simple to end-stopped cells,” Vision Res. 32, 963-981 (1992).
[CrossRef] [PubMed]

Kulikowski, J. J.

J. J. Kulikowski and P. O. Bishop, “Fourier analysis and spatial representation in the visual cortex,” Experientia 37, 160-163 (1981).
[CrossRef] [PubMed]

Lee, T. S.

T. S. Lee, “Image representation using 2D gabor wavelets,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 1-13 (1996).
[CrossRef]

Levi, D. M.

Lu, H. D.

A. W. Roe, H. D. Lu, and C. P. Hung, “Cortical processing of a brightness illusion,” Proc. Natl. Acad. Sci. U.S.A. 102, 3869-3874 (2005).
[CrossRef] [PubMed]

MacEvoy, S. P.

S. P. MacEvoy, W. Kim, and M. A. Paradiso, “Integration of surface information in primary visual cortex,” Nat. Neurosci. 1, 616-620 (1998).
[CrossRef]

Maffei, L.

L. Maffei and A. Fiorentini, “The unresponsive regions of visual cortical receptive fields,” Vision Res. 16, 1131-1139 (1976).
[CrossRef] [PubMed]

Marcelja, S.

Marr, D.

D. Marr and E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London, Ser. B 207, 187-217 (1980).
[CrossRef]

McCourt, M. E.

D. Cope, B. Blakeslee, and M. E. McCourt, “Simple cell response properties imply receptive field structure: Balanced Gabor and/or bandlimited field functions. Supplement. Appendices A, B, C and Figures 13-16.” http://hdl.handle.net/10365/5418.

Olshausen, B. A.

B. A. Olshausen and D. J. Field, “Vision and the coding of natural images,” Am. Sci. 88, 238-245 (2000).

B. A. Olshausen and D. J. Field, “Emergence of simple-cell receptive field properties by learning a sparse code for natural images,” Nature 381, 607-609 (1996).
[CrossRef] [PubMed]

Palmer, L. A.

J. P. Jones and 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]

Paradiso, M. A.

A. F. Rossi and M. A. Paradiso, “Neural correlates of perceived brightness in the retina, lateral geniculate nucleus, and striate cortex,” J. Neurosci. 19, 6145-6156 (1999).
[PubMed]

S. P. MacEvoy, W. Kim, and M. A. Paradiso, “Integration of surface information in primary visual cortex,” Nat. Neurosci. 1, 616-620 (1998).
[CrossRef]

A. F. Rossi and M. A. Paradiso, “Temporal limits of brightness induction and mechanisms of brightness perception,” Vision Res. 36, 1391-1398 (1996).
[CrossRef] [PubMed]

A. F. Rossi, C. D. Rittenhouse, and M. A. Paradiso, “The representation of brightness in primary visual cortex,” Science 273, 1104-1107 (1996).
[CrossRef] [PubMed]

Parker, A. J.

Peng, X.

X. Peng and D. C. Van Essen, “Peaked encoding of relative luminance in macaque areas V1 and V2,” J. Neurophysiol. 93, 1620-1632 (2005).
[CrossRef]

Peterhans, E.

F. Heitger, L. Rosenthaler, R. von der Heydt, E. Peterhans, and O. Kubler, “Simulation of neural contour mechanisms: from simple to end-stopped cells,” Vision Res. 32, 963-981 (1992).
[CrossRef] [PubMed]

Petkov, N.

N. Petkov and P. Kruizinga, “Computational models of visual neurons specialised in the detection of periodic and aperiodic oriented visual stimuli: bar and grating cells,” Biol. Cybern. 76, 83-96 (1997).
[CrossRef] [PubMed]

N. Petkov, “Biologically motivated computationally intensive approaches to image pattern recognition,” Future Gener. Comput. Syst. 11, 451-465 (1995).
[CrossRef]

Polat, U.

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[CrossRef] [PubMed]

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[CrossRef]

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Other (5)

J. D. Victor and B. W. Knight, “Simultaneously band and space limited functions in two dimensions, and receptive fields of visual neurons,” in Springer Applied Mathematical Sciences Series, E.Kaplan, J.Marsden, and K.R.Sreenivasan, eds. (Springer, 2003), pp. 375-420.

D. Cope, B. Blakeslee, and M. E. McCourt, “Simple cell response properties imply receptive field structure: Balanced Gabor and/or bandlimited field functions. Supplement. Appendices A, B, C and Figures 13-16.” http://hdl.handle.net/10365/5418.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. Applied Mathematics Series 55 (National Bureau of Standards, 1972).

R. L. DeValois, D. G. Albrecht, and L. G. Thorell, “Cortical cells: bar and edge detectors, or spatial frequency filters?” in Frontiers in Visual Science, S.J.Cool and E.L.Smith, eds. (Springer-Verlag, 1978), pp. 544-556.

R. L. DeValois and K. K. DeValois, Spatial Vision (Oxford Univ. Press, 1988).

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Figures (12)

Fig. 1
Fig. 1

Simple balanced Gabor weight functions, g ( r ) g ( 0 ) vs. r, (left) and Hankel transforms, H g ( ρ ) vs. ρ, (right) for four exponents γ R = 0.75 (dashed), 1.5 (dotted), 3.0 (solid), 6.0 (near-solid). As the exponent γ R increases, the weight g ( r ) broadens and the Hankel transform H g ( ρ ) narrows. Note that the weight functions are strictly positive and monotonic and are not oscillatory.

Fig. 2
Fig. 2

Left panels: comparison of traditional cosine-type Gabor receptive field function (near-solid) with the corresponding simple balanced Gabor receptive field function (dashed) with identical Gaussian components, that is, with identical weight functions (envelopes, solid). Right panels: corresponding Fourier transforms. All receptive field functions are cosine-type. The imbalance of the traditional Gabor is indicated by the nonzero values of its Fourier transform at the origin. Comparisons are for γ R = 0.75 , 1.5 , 3.0 , 6.0 . Note that the traditional and simple balanced Gabor receptive field functions become increasingly similar (i.e., the balancing parameter approaches 0) as γ R increases (i.e., as effective spatial frequency bandwidth decreases), becoming virtually identical for γ R > 3 .

Fig. 3
Fig. 3

Left panels: orientation max-response functions (in degrees). Right panels: spatial frequency max-response functions for simple balanced Gabor receptive field functions as they vary with carrier spatial phase: 0 deg (cosine-type, dashed), 45 deg (mixed-type, dotted) and 90 deg (sine-type, solid). Response curves are for γ R = 0.75 , 1.5 , 3.0 , 6.0 . Note that as exponent γ R increases, max-response curves become independent of carrier spatial phase ϕ R and that sine-type receptive fields always give larger responses than cosine-type receptive fields with corresponding parameters. The cosine-type spatial frequency max-response matches the Fourier transform of Fig. 2 (differing only by a scale factor).

Fig. 4
Fig. 4

Nonsimple balanced Gabor weight functions, g ( r ) g ( 0 ) vs. r, (left) and Hankel transforms, H g ( ρ ) vs. ρ, (right) for four exponents γ R = 0.75 , 1.5 , 3.0 , 6.0 (with ψ R = 0 ). Each plot shows three curves. The curves plot weight functions when c R is set to a positive (solid) or negative (dotted) boundary value. These extreme curves (solid and dotted) are interchanged relative to the corresponding extreme curves for ψ R = π 2 (see Fig. 7 below). The dashed curve replots the weight function of the simple balanced Gabor ( c R = 0 ) for comparison. Note that nonsimple Gabor weight functions exhibit an oscillatory behavior and can take on negative values. The departure from the simple balanced Gabor weight function is most pronounced for large values of γ R . As the exponent γ R increases, the weight g ( r ) broadens and the Hankel transform H g ( ρ ) narrows.

Fig. 5
Fig. 5

Normalized balanced Gabor receptive field functions (cosine-type), λ R 2 R ( x 1 , 0 ) 2 π g ( 0 ) vs. x 1 λ R , and associated weight functions (left) for γ R = 0.75 , 1.5 , 3.0 , 6.0 ( ψ R = 0 ) , together with their Fourier transforms (right). Weight functions (left) are in thinner gray curves. Each plot shows c R = 0 (simple balanced Gabor, dashed) and c R = ± γ R 2 ( γ R 2 + 4 π 2 ) for max (solid) and min (dotted) bounds of this cofficient. For the Fourier transforms, note that as γ R increases, (1) the transform narrows (consistent with decreasing bandwidth) and (2) the range of variation of the curves increases (consistent with the increased bounds for c R ). Note the variation in the Fourier transforms for small values of s 1 (low spatial frequency). There is no apparent corresponding variation in the receptive field functions for small γ R , indicating the importance of the large-scale receptive field structure.

Fig. 6
Fig. 6

Orientation max-response functions in degrees (left) and spatial frequency max-response functions (right) for nonsimple balanced Gabor receptive field functions for γ R = 0.75 , 1.5 , 3.0 , 6.0 ( ψ R = 0 ) . Each plot shows c R = 0 (simple balanced Gabor, dashed) and c R = ± γ R 2 ( γ R 2 + 4 π 2 ) for max (solid) and min (dotted) bounds of this coefficient. Three curves are plotted for each c R corresponding to receptive field phase ϕ R = 0 , 45 , 90 deg , respectively, cosine-type, mixed-type, sine-type. As γ R increases, the dependence of both orientation and spatial frequency response on field phase ϕ R ( c R fixed) decreases, becoming virtually independent at γ R = 6.0 . The cosine-type spatial frequency max-response function matches the Fourier transform of Fig. 5 (up to scaling).

Fig. 7
Fig. 7

Nonsimple balanced Gabor weight functions, g ( r ) g ( 0 ) vs. r, (left) and Hankel transforms, H g ( ρ ) vs. ρ, (right) for four exponents γ R = 0.75 , 1.5 , 3.0 , 6.0 (with ψ R = π 2 ). Each plot shows three curves. The curves plot weight functions when c R is set to a positive (solid) or negative (dotted) boundary value. These extreme curves (solid and dotted) are interchanged relative to the corresponding extreme curves for ψ R = 0 (Fig. 4). The dashed curve replots the weight function of the simple balanced Gabor ( c R = 0 ) for comparison. Note that nonsimple Gabor weight functions exhibit an oscillatory behavior and can take on negative values. The departure from the simple balanced Gabor weight function is most pronounced for large values of γ R . As the exponent γ R increases, the weight g ( r ) broadens and the Hankel transform H g ( ρ ) narrows.

Fig. 8
Fig. 8

Four pairs of plots showing normalized balanced Gabor receptive field functions (cosine-type), λ R 2 R ( x 1 , 0 ) 2 π g ( 0 ) vs. x 1 λ R , and associated weight functions (left) with their Fourier transforms (right) for γ R = 0.75 , 1.5 , 3.0 , 6.0 ( ψ R = π 2 ) . Weight functions (left) are in thinner gray curves. Each plot shows c R = 0 (simple balanced Gabor, dashed) and c R = ± γ R 2 ( γ R 2 + 4 π 2 ) for max (solid) and min (dotted) bounds of this coefficient. For the Fourier tranforms, note that as γ R increases, (1) the transform narrows (consistent with decreasing bandwidth) and (2) the range of variation of the curves increases (consistent with the increased bounds for c R ). Note the variation in the Fourier transforms for small values of s 1 seen in Fig. 5 ( ψ R = 0 ) is here most prominent for s 1 slightly less than 1.0. Note the field phase reversal for γ R = 6.0 (solid).

Fig. 9
Fig. 9

Orientation max-response functions in degrees (left) and spatial frequency max-response functions (right) for nonsimple balanced Gabor receptive field functions for γ R = 0.75 , 1.5, 3.0, 6.0 ( ψ R = π 2 ) . Each plot shows c R = 0 (simple balanced Gabor, dashed) and c R = ± γ R 2 ( γ R 2 + 4 π 2 ) for max (solid) and min (dotted) bounds of this coefficient. Three curves are plotted for each c R corresponding to receptive field phase ϕ R = 0 , 45 , 90 deg , respectively cosine-type, mixed-type, sine-type. As γ R increases, the dependence of both orientation and spatial frequency response on receptive field phase ϕ R ( c R fixed) decreases, becoming virtually independent at γ R = 6.0 . The cosine-type spatial frequency max-response function matches the Fourier transform of Fig. 7 (up to scaling).

Fig. 10
Fig. 10

Bandlimited weight functions for b ( r ) b ( 0 ) vs. r, (left) and Hankel transforms, H b ( ρ ) vs. ρ, (right) for support parameter values s R = 1.0 , 0.85,0.7,0.5. Each plot shows three curves that plot weight functions when the order of the Bessel weight ν R is set to 2.0 (dotted), 3.5 (dashed), and 5.0 (solid). Note that Bessel weight functions exhibit oscillatory behavior and take on negative values. As the support parameter s R decreases, the weight b ( r ) broadens and the Hankel transform H b ( ρ ) narrows.

Fig. 11
Fig. 11

Bandlimited Bessel receptive field functions, λ R 2 R ( x 1 , 0 ) 2 π b ( 0 ) vs. x 1 λ R , (left), and their Fourier transforms, F R ( s 1 , 0 ) vs. log 10 ( λ R s 1 ) (right), for support parameter values s R = 1.0 , 0.85, 0.7, 0.5. Each plot shows three curves corresponding to Bessel weights of order ν R = 2.0 (dotted), 3.5 (dashed), and 5.0 (solid). Unlike balanced Gabor functions, the bandlimited receptive field function and its weight function are equal at the origin and, for the Fourier transform, the transform max always occurs for λ P = λ R . Like balanced Gabor receptive field functions, the bandlimited receptive field function can show phase reversal (note s R = 0.7 ). Note that the bandwidth narrows as s R decreases and also narrows for fixed s R and increasing order ν R . of the Bessel weight.

Fig. 12
Fig. 12

Orientation max-response functions in degrees (left) and spatial frequency max-response functions (right) for bandlimited Bessel receptive field functions as they vary with s R = 1.0 , 0.85, 0.7, 0.5, and ν R = 2.0 (dotted), 3.5 (dashed), and 5.0 (solid). Unlike balanced Gabor receptive field functions, the max-response of bandlimited receptive field functions is independent of the receptive field phase ϕ R . As support parameter s R values decrease, the receptive field function narrows in effective bandwidth. For a particular support parameter value, as the order of the Bessel weight ν R increases, the receptive field function narrows in effective bandwidth.

Equations (107)

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R p ( t ) R × R R ( x ) p ( x ; t ) d x .
R 0 R × R R ( x ) d x .
p ( x ) 1 + c P cos ( 2 π d ( α P ) x λ P ϕ P ) ,
d ( α P ) x = cos ( α P ) x 1 + sin ( α P ) x 2 ,
F R ( s 1 , s 2 ) R × R exp ( 2 π i ( s 1 x 1 + s 2 x 2 ) ) R ( x 1 , x 2 ) d x 1 d x 2 .
R 0 = F R ( 0 , 0 ) ,
R p = R × R R ( x ) ( 1 + c P cos ( 2 π d ( α P ) x λ P ϕ P ) ) d x = R 0 + c P Re [ e i ϕ P F R ( 1 λ P cos ( α P ) , 1 λ P sin ( α P ) ) ] ,
M ( c P , α P , λ P ) max ϕ P ( R p ) = max ϕ P { R 0 + c P Re [ e i ϕ P F R ( 1 λ P cos ( α P ) , 1 λ P sin ( α P ) ) ] } = R 0 + | c P | | F R ( 1 λ P cos ( α P ) , 1 λ P sin ( α P ) ) | .
M ( c P , α P , λ P ) = M ( | c P | , α P , λ P ) = M ( | c P | , α P ± π , λ P ) .
R 0 = F R ( 0 , 0 ) = 0 .
R 0 + c ZR Re [ e i ϕ P F R ( 1 λ P cos ( α ZR ) , 1 λ P sin ( α ZR ) ) ] = 0
F R ( 1 λ P cos ( α ZR ) , 1 λ P sin ( α ZR ) ) = 0 .
M ( c MR , α P , λ P ) = | c MR | | F R ( 1 λ P cos ( α P ) , 1 λ P sin ( α P ) ) | .
α MR + ζ 0 α P α MR + π 2
F R ( 1 λ P cos ( α P ) , 1 λ P sin ( α P ) ) 0
| F R ( 1 λ P cos ( α P ) , 1 λ P sin ( α P ) ) |
| F R ( 1 λ P cos ( α P ) , 1 λ P sin ( α P ) ) |
R ( x ) 2 π λ R 2 q ( 2 π λ R x ) ( cos ( 2 π λ R d ( α R ) x ϕ R ) b R ) ,
q ( r ) is C ( 0 , + ) ,
q ( r ) r is absolutely integrable on ( 0 , + ) ,
and 0 q ( r ) r d r = 1 .
If f ( x 1 , x 2 ) = g ( r ) where r = ( x 1 2 + x 2 2 ) 1 2 ,
then F f ( s 1 , s 2 ) = 2 π H g ( 2 π ρ ) where ρ = ( s 1 2 + s 2 2 ) 1 2 ,
H f ( ρ ) = 0 J 0 ( ρ r ) f ( r ) r d r and f ( r ) = 0 J 0 ( ρ r ) H f ( ρ ) ρ d ρ .
H q ( ρ ) is C [ 0 , + ) , H q ( 0 ) = 1 , and H q ( + ) = 0 .
s R = ( cos ( α R ) λ R , sin ( α R ) λ R ) , s P = ( cos ( α P ) λ P , sin ( α P ) λ P )
s ± s R = 1 λ R ( λ R s 1 ± cos ( α R ) ) 2 + ( λ R s 2 ± sin ( α R ) ) 2 ,
s P ± s R = 1 λ R 1 ± 2 cos ( α P α R ) λ R λ P + λ R 2 λ P 2 .
F R ( s 1 , s 2 ) = 1 2 ( e + i ϕ R H q ( λ R s + s R ) + e i ϕ R H q ( λ R s s R ) ) b R H q ( λ R s ) .
R 0 = cos ( ϕ R ) H q ( 1 ) b R H q ( 0 ) .
R p = R 0 + c P 2 cos ( ϕ P + ϕ R ) H q ( λ R s P + s R ) + c P 2 cos ( ϕ P ϕ R ) H q ( λ R s P s R ) b R c P cos ( ϕ P ) H q ( λ R λ P ) .
M ( c P , α P , λ P ) = max ϕ P ( R p ) = R 0 + 1 2 | c P | N ( α P , λ P ) ,
N ( α P , λ P ) 2 | F R ( cos ( α P ) λ P , sin ( α P ) λ P ) | 0
N ( α P , λ P ) 2 = [ cos ( ϕ R ) ( H q ( λ R s P + s R ) + H q ( λ R s P s R ) ) 2 b R H q ( λ R λ P ) ] 2 + [ sin ( ϕ R ) ( H q ( λ R s P + s R ) H q ( λ R s P s R ) ) ] 2 .
b R = cos ( ϕ R ) H q ( 1 ) .
cos ( ϕ R ) ( H q ( λ R s ZR + s R ) + H q ( λ R s ZR s R ) ) = 2 b R H q ( λ R λ P ) ,
sin ( ϕ R ) ( H q ( λ R s ZR + s R ) H q ( λ R s ZR s R ) ) = 0 ,
b R = cos ( ϕ R ) H q ( 1 ) .
H q ( 1 2 ρ cos ( ζ 0 ) + ρ 2 ) + H q ( 1 + 2 ρ cos ( ζ 0 ) + ρ 2 ) = 2 H q ( 1 ) H q ( ρ )
0 < ζ R | α ZR α R | π 2 .
f ( a ; y ) = e c y F ( a y ) + e c y F ( a + y ) 2 F ( a )
( cos ( ϕ R ) ) 2 ( e c y F ( a y ) + e c y F ( a + y ) 2 F ( a ) ) 2 + ( sin ( ϕ R ) ) 2 ( e c y F ( a y ) e c y F ( a + y ) ) 2
e c y F ( y ) = e c a F ( a ) K ( a ) ( y a ) on | y a | δ a for some constant K ( a ) > 0 .
N ( α P , λ P ) 2 = exp ( c ( 1 + λ R 2 λ P 2 ) ) T F ( ϕ R , c , 1 + λ R 2 λ P 2 ; 2 cos ( α P α R ) λ R λ P ) ,
T F ( ϕ R , c , a ; y ) ( cos ( ϕ R ) ) 2 ( e c y F ( a y ) + e c y F ( a + y ) 2 F ( a ) ) 2 + ( sin ( ϕ R ) ) 2 ( e c y F ( a y ) e c y F ( a + y ) ) 2 .
N ( α P , λ P ) = H q ( λ R s P s R ) ,
N ( α P , λ P ) = | H q ( λ R s P s R ) H q ( λ R s P + s R ) |
g ( r ) is C ( 0 , + ) ,
g ( r ) r is absolutely integrable on ( 0 , + ) ,
and 0 g ( r ) r d r = 1 ,
H g ( ρ ) is C [ 0 , + ) , H g ( 0 ) = 1 , and H g ( + ) = 0
T G ( γ R , a ; y ) e γ R y G ( a y ) + e γ R y G ( a + y ) 2 G ( a )
e c y F ( y ) = e c a F ( a ) K ( a ) ( y a ) on | y a | δ a
e γ R ( y + 1 ) G ( y + 1 ) = e γ R e γ R y G ( y ) .
T G ( γ R , a ; y + 1 ) = e γ R T G ( γ R , a ; y ) + ( e γ R e γ R ) e γ R y G ( y ) .
Γ ( γ R ) { G ( y ) : G ( y ) satisfies conditions ( a ) , ( b ) , ( c ) for balanced Gabor weights with exponent γ R }
H 3 ( ρ ) = e γ 2 ρ 2 ( α 1 G 1 ( ρ 2 ) + α 2 G 2 ( ρ 2 ) )
R BG ( x ) = 2 π λ R 2 g ( 2 π λ R x ) ( cos ( 2 π λ R d ( α R ) x ϕ R ) b R ) ,
F R ( s 1 , s 2 ) = 1 2 exp ( γ R ( λ R 2 s 2 + 1 ) ) [ e + i ϕ R exp ( 2 γ R λ R d ( α R ) s ) G ( λ R 2 s 2 + 2 λ R d ( α R ) s ) + e i ϕ R exp ( + 2 γ R λ R d ( α R ) s ) G ( λ R 2 s 2 2 λ R d ( α R ) s ) 2 cos ( ϕ R ) G ( λ R 2 s 2 ) ] .
C ± ( α P α R ) exp ( ± 2 γ R λ R d ( α R ) s P ) G ( λ R 2 s P 2 2 λ R d ( α R ) s P ) = exp ( ± 2 γ R cos ( α P α R ) λ R λ P ) G ( λ R 2 λ P 2 2 cos ( α P α R ) λ R λ P ) .
R p = c P 2 exp ( γ R ( 1 + λ R 2 λ P 2 ) ) [ cos ( ϕ P ϕ R ) ( C + ( α P α R ) G ( λ R 2 λ P 2 ) ) + cos ( ϕ P + ϕ R ) ( C ( α P α R ) G ( λ R 2 λ P 2 ) ) ] .
M ( c P , α P , λ P ) = max ϕ P ( R p ) = 1 2 | c P | N ( α P , λ P ) ,
N ( α P , λ P ) 2 = exp ( 2 γ R ( λ R 2 λ P 2 + 1 ) ) [ cos ( ϕ R ) 2 N c 2 + sin ( ϕ R ) 2 N s 2 ] ,
N c C + ( α P α R ) + C ( α P α R ) 2 G ( λ R 2 λ P 2 ) ,
N s C + ( α P α R ) C ( α P α R ) .
N s 2 N c 2 = 4 ( C ( α P α R ) + G ( λ R 2 λ P 2 ) ) ( C + ( α P α R ) + G ( λ R 2 λ P 2 ) ) ,
exp ( γ R z ) G ( z + a ) and exp ( + γ R z ) G ( z + a )
z ( r ) = 1 2 c + 2 π i k c 2 + ( 2 π k ) 2 exp ( r 2 4 c + 2 π i k c 2 + ( 2 π k ) 2 ) ,
H z ( ρ ) = e c ρ 2 + 2 π i k ρ 2 .
q ( r ) = k = + g k 2 c + 2 π i k c 2 + ( 2 π k ) 2 exp ( r 2 4 c + 2 π i k c 2 + ( 2 π k ) 2 ) ,
H q ( ρ ) = e c ρ 2 k = + g k e + 2 π i k ρ 2 .
R ( x ) = 1 2 π σ R 2 exp ( x 2 2 σ R 2 ) cos ( 2 π d ( α R ) x λ R ϕ R ) ,
F R ( s ) = exp ( 2 π 2 σ R 2 ( s 2 + 1 λ R 2 ) ) [ cos ( ϕ R ) cosh ( 4 π 2 σ R 2 d ( α R ) s λ R ) i sin ( ϕ R ) sinh ( 4 π 2 σ R 2 d ( α R ) s λ R ) ] .
R × R R ( x ) d x = cos ( ϕ R ) exp ( 2 π 2 σ R 2 λ R 2 ) .
g ( r ) = 1 2 γ R exp ( r 2 4 γ R ) with H g ( ρ ) = exp ( γ R ρ 2 ) .
R SBG ( x ) = 2 π λ R 2 g ( 2 π λ R x ) ( cos ( 2 π λ R d ( α R ) x ϕ R ) cos ( ϕ R ) e γ R ) ,
F R ( s ) = exp ( γ R λ R 2 ( s 2 + 1 λ R 2 ) ) [ cos ( ϕ R ) ( cosh ( 2 γ R λ R d ( α R ) s ) 1 ) i sin ( ϕ R ) sinh ( 2 γ R λ R d ( α R ) s ) ] .
R p = c P 2 exp ( γ R ( 1 + λ R 2 λ P 2 ) ) [ cos ( ϕ P ϕ R ) ( exp ( + 2 γ R cos ( α P α R ) λ R λ P ) 1 ) + cos ( ϕ P + ϕ R ) ( exp ( 2 γ R cos ( α P α R ) λ R λ P ) 1 ) ] .
N ( α P , λ P ) 2 = 4 exp ( 2 γ R ( λ R 2 λ P 2 + 1 ) ) [ cos ( ϕ R ) 2 ( cosh ( 2 γ R cos ( α P α R ) λ R λ P ) 1 ) 2 + sin ( ϕ R ) 2 ( sinh ( 2 γ R cos ( α P α R ) λ R λ P ) ) 2 ] .
g ( r ) = 1 2 γ R ( 1 + c R cos ( ψ R ) ) exp ( r 2 4 γ R ) + c R γ R 2 ( 1 + c R cos ( ψ R ) ) ( γ R 2 + 4 π 2 ) exp ( γ R r 2 4 ( γ R 2 + 4 π 2 ) ) ( cos ( π r 2 2 ( γ R 2 + 4 π 2 ) + ψ R ) + 2 π γ R sin ( π r 2 2 ( γ R 2 + 4 π 2 ) + ψ R ) ) ,
H g ( ρ ) = e γ R ρ 2 G ( ρ 2 ) where G ( y ) = 1 + c R cos ( 2 π y ψ R ) 1 + c R cos ( ψ R ) ,
| c R | γ R 2 γ R 2 + 4 π 2 .
g ( r ) = 1 2 γ R ( 1 + c R cos ( ψ R ) ) exp ( γ R r 2 4 ( γ R 2 + 4 π 2 ) ) [ exp ( π 2 r 2 γ R ( γ R 2 + 4 π 2 ) ) + c R γ R 2 γ R 2 + 4 π 2 ( cos ( π r 2 2 ( γ R 2 + 4 π 2 ) + ψ R ) + 2 π γ R sin ( π r 2 2 ( γ R 2 + 4 π 2 ) + ψ R ) ) ] .
R BG ( x ) = 2 π λ R 2 g ( 2 π λ R | x | ) ( cos ( 2 π λ R d ( α R ) x ϕ R ) cos ( ϕ R ) e γ R ) ,
b ( r ) is C ( 0 , + ) ,
b ( r ) r is absolutely integrable on ( 0 , + ) ,
and 0 b ( r ) r d r = 1 ,
H b ( ρ ) is C [ 0 , + ) , H b ( 0 ) = 1 , and H b ( + ) = 0
R BL ( x ) = 2 π λ R 2 b ( 2 π λ R x ) cos ( 2 π λ R d ( α R ) x ϕ R ) .
F R ( s 1 , s 2 ) = 1 2 ( e + i ϕ R H b ( λ R s + s R ) + e i ϕ R H b ( λ R s s R ) ) .
R p = c P 2 cos ( ϕ P ϕ R ) H b ( λ R s P s R ) .
M ( c P , α P , λ P ) = max ϕ P ( R p ) = 1 2 | c P | N ( α P , λ P ) ,
N ( α P , λ P ) = H b ( λ R s P s R ) .
cos ( α P α R ) δ λ R λ P cos ( α P α R ) + δ ,
cos ( α P α R ) 1 2 ( λ R λ P + cos ( ζ R ) 2 λ P λ R ) ,
λ R λ P = cos ( ζ R ) < 1 ,
| sin ( α P α R 2 ) | 1 2 sin ( ζ R )
b ( r ) = 0 1 J 0 ( ρ r ) H b ( ρ ) ρ d ρ = k = 0 ( r 2 4 ) k ( k ! ) 2 0 s R H b ( ρ ) ρ 2 k + 1 d ρ
H b ( ν R ; ρ ) ( ( 1 ρ 2 ) ν R 1 for 0 ρ < 1 0 for 1 ρ ) ,
b ( ν R ; r ) 2 ν R 1 Γ ( ν R ) r ν R J ν R ( r ) = 0 1 J 0 ( ρ r ) ( 1 ρ 2 ) ν R 1 ρ d ρ .
b ( r ) = s R 2 b ( ν R ; s R r ) and H b ( ρ ) = H b ( ν R ; ρ s R ) .
R B ( x ) = 2 π s R 2 λ R 2 b ( ν R ; 2 π s R λ R x ) cos ( 2 π d ( α R ) x λ R ϕ R ) ,
R p = c P 2 cos ( ϕ P ϕ R ) H b ( ν R ; λ R s R s P s R )
= c P 2 cos ( ϕ P ϕ R ) s R 2 ν R 2 D ν R 1
when D s R 2 sin ( α P α R ) 2 ( λ R λ P cos ( α P α R ) ) 2 > 0 , = 0 otherwise ,
N ( α P , λ P ) = H b ( ν R ; λ R s R s P s R ) = 1 s R 2 ν R 2 D ν R 1 when D > 0 = 0 otherwise .
R ( x ) 2 π λ R 2 q ( 2 π λ R x ) p ( d ( α R ) x λ R ) .

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