Abstract

Although a great deal of experimental evidence supports the notion of a Reichardt correlator as a mechanism for biological motion detection, the correlator does not signal true image velocity. This study examines the accuracy with which realistic Reichardt correlators can provide velocity estimates in an organism’s natural visual environment. The predictable statistics of natural images imply a consistent correspondence between mean correlator response and velocity, allowing the otherwise ambiguous Reichardt correlator to act as a practical velocity estimator. Analysis and simulations suggest that processes commonly found in visual systems, such as prefiltering, response compression, integration, and adaptation, improve the reliability of velocity estimation and expand the range of velocities coded. Experimental recordings confirm our predictions of correlator response to broadband images.

© 2001 Optical Society of America

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    [PubMed]
  36. G. Nalbach, R. Hengstenberg, “The halteres of the blowfly Calliphora,” J. Comp. Physiol. A 175, 695–708 (1994).
    [CrossRef]
  37. S. Single, A. Borst, “Dendritic integration and its role in computing image velocity,” Science 281, 1848–1850 (1998).
    [CrossRef] [PubMed]
  38. K. Hausen, M. Egelhaaf, “Neural mechanisms of visualcourse control in insects,” in Facets of Vision, D. G. Stavenga, R. C. Hardie, eds. (Springer-Verlag, Heidelberg, 1989), pp. 391–424.
  39. M. Egelhaaf, W. Reichardt, “Dynamic response properties of movement detectors: theoretical analysis and electrophysiological investigation in the visual system of the fly,” Biol. Cybern. 56, 69–87 (1987).
    [CrossRef]
  40. D. W. Dong, J. J. Atick, “Statistics of natural time-varying images,” Network Comput. Neural Syst. 6, 345–358 (1995).
    [CrossRef]
  41. N. Franceschini, J. M. Pichon, C. Blanes, “From insect vision to robot vision,” Philos. Trans. R. Soc. London Ser. B 337, 283–294 (1992).
    [CrossRef]
  42. R. Sarpeshkar, W. Bair, C. Koch, “An analog VLSI chip for local velocity estimation based on Reichardt’s motion algorithm,” in Advances in Neural Information Processing Systems, S. Hanson, J. Cowan, L. Giles, eds. (Morgan Kauffman, San Mateo, Calif., 1993), Vol. 5, pp. 781–788.
  43. R. O. Dror, D. C. O’Carroll, S. B. Laughlin, “The role of natural image statistics in biological motion estimation,” Springer Lect. Notes Comput. Sci. 1811, 492–501 (2000).
    [CrossRef]
  44. M. F. Land, H. M. Eckert, “Maps of the acute zones of fly eyes,” J. Comp. Physiol. A 156, 525–538 (1985).
    [CrossRef]

2000 (2)

B. Tatler, D. C. O’Carroll, S. B. Laughlin, “Temperature and temporal resolving power of fly photoreceptors,” J. Comp. Physiol. A 186, 399–407 (2000).
[CrossRef] [PubMed]

R. O. Dror, D. C. O’Carroll, S. B. Laughlin, “The role of natural image statistics in biological motion estimation,” Springer Lect. Notes Comput. Sci. 1811, 492–501 (2000).
[CrossRef]

1999 (1)

R. A. Harris, D. C. O’Carroll, S. B. Laughlin, “Adaptation and the temporal delay filter of fly motion detectors,” Vision Res. 39, 2603–2613 (1999).
[CrossRef] [PubMed]

1998 (1)

S. Single, A. Borst, “Dendritic integration and its role in computing image velocity,” Science 281, 1848–1850 (1998).
[CrossRef] [PubMed]

1997 (3)

J. H. van Hateren, “Processing of natural time series of intensities by the visual system of the blowfly,” Vision Res. 37, 3407–3416 (1997).
[CrossRef]

D. C. O’Carroll, S. B. Laughlin, N. J. Bidwell, R. A. Harris, “Spatio-temporal properties of motion detectors matched to low image velocities in hovering insects,” Vision Res. 37, 3427–3439 (1997).
[CrossRef]

S. Single, J. Haag, A. Borst, “Dendritic computation of direction selectivity and gain control in visual interneurons,” J. Neurosci. 17, 6023–6030 (1997).
[PubMed]

1996 (2)

M. V. Srinivasan, S. W. Zhang, M. Lehrer, T. S. Collett, “Honeybee navigation en route to the goal: visual flight control and odometry,” J. Exp. Biol. 199, 237–244 (1996).
[PubMed]

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

1995 (2)

J. K. Douglass, N. J. Strausfeld, “Visual motion detection circuits in flies: peripheral motion computation by identified small-field retinotopic neurons,” J. Neurosci. 15, 5596–5611 (1995).
[PubMed]

D. W. Dong, J. J. Atick, “Statistics of natural time-varying images,” Network Comput. Neural Syst. 6, 345–358 (1995).
[CrossRef]

1994 (4)

G. Nalbach, R. Hengstenberg, “The halteres of the blowfly Calliphora,” J. Comp. Physiol. A 175, 695–708 (1994).
[CrossRef]

S. B. Laughlin, “Matching coding, circuits, cells and molecules to signals: general principles of retinal design in the fly’s eye,” Prog. Retinal Res. 13, 165–195 (1994).
[CrossRef]

F. Wolf-Oberhollenzer, K. Kirschfeld, “Motion sensitivity in the nucleus of the basal optic root of the pigeon,” J. Neurophysiol. 71, 1559–1573 (1994).
[PubMed]

D. L. Ruderman, “The statistics of natural images,” Network Comput. Neural Syst. 5, 517–48 (1994).
[CrossRef]

1992 (4)

D. J. Tolhurst, Y. Tadmor, T. Chao, “Amplitude spectra of natural images,” Ophthalmol. Physiol. Opt. 12, 229–232 (1992).
[CrossRef]

G. A. Horridge, L. Marcelja, “On the existence of fast and slow directionally sensitive motion detector neurons in insects,” Proc. R. Soc. London, Ser. B 248, 47–54 (1992).
[CrossRef]

D. J. Heeger, “Normalization of cell responses in cat striate cortex,” Visual Neurosci. 9, 181–197 (1992).
[CrossRef]

N. Franceschini, J. M. Pichon, C. Blanes, “From insect vision to robot vision,” Philos. Trans. R. Soc. London Ser. B 337, 283–294 (1992).
[CrossRef]

1991 (1)

D. Osorio, “Mechanisms of early visual processing in the medulla of the locust optic lobe: how self-inhibition, spatial-pooling and signal rectification contribute to the properties of transient cells,” Visual Neurosci. 7, 345–355 (1991).
[CrossRef]

1989 (2)

1987 (4)

G. J. Burton, I. R. Moorhead, “Color and spatial structure in natural scenes,” Appl. Opt. 26, 157–170 (1987).
[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]

R. C. Emerson, M. C. Citron, W. J. Vaughn, S. A. Klein, “Nonlinear directionally selective subunits in complex cells of cat striate cortex,” J. Neurophysiol. 58, 33–65 (1987).
[PubMed]

M. Egelhaaf, W. Reichardt, “Dynamic response properties of movement detectors: theoretical analysis and electrophysiological investigation in the visual system of the fly,” Biol. Cybern. 56, 69–87 (1987).
[CrossRef]

1986 (1)

S. P. McKee, G. H. Silverman, K. Nakayama, “Precise velocity discrimination despite random variations in temporal frequency and contrast,” Vision Res. 26, 609–619 (1986).
[CrossRef] [PubMed]

1985 (4)

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

J. P. H. van Santen, G. Sperling, “Elaborated Reichardt detectors,” J. Opt. Soc. Am. A 2, 300–321 (1985).
[CrossRef] [PubMed]

T. Maddess, S. B. Laughlin, “Adaptation of the motion-sensitive neuron H1 is generated locally and governed by contrast frequency,” Proc. R. Soc. London Ser. B 225, 251–275 (1985).
[CrossRef]

M. F. Land, H. M. Eckert, “Maps of the acute zones of fly eyes,” J. Comp. Physiol. A 156, 525–538 (1985).
[CrossRef]

1981 (1)

R. Payne, J. Howard, “Response of an insect photoreceptor: a simple log-normal model,” Nature 290, 415–416 (1981).
[CrossRef]

1976 (1)

T. Poggio, W. Reichardt, “Visual control of orientation behaviour in the fly. II. Towards the underlying neural interactions,” Q. Rev. Biophys. 9, 377–438 (1976).
[CrossRef] [PubMed]

1974 (1)

M. F. Land, T. S. Collett, “Chasing behaviour of houseflies (Fannia canicularis): a description and analysis,” J. Comp. Physiol. A 89, 331–357 (1974).
[CrossRef]

Adelson, E. H.

Atick, J. J.

D. W. Dong, J. J. Atick, “Statistics of natural time-varying images,” Network Comput. Neural Syst. 6, 345–358 (1995).
[CrossRef]

Bair, W.

R. Sarpeshkar, W. Bair, C. Koch, “An analog VLSI chip for local velocity estimation based on Reichardt’s motion algorithm,” in Advances in Neural Information Processing Systems, S. Hanson, J. Cowan, L. Giles, eds. (Morgan Kauffman, San Mateo, Calif., 1993), Vol. 5, pp. 781–788.

Bergen, J.

Bidwell, N. J.

D. C. O’Carroll, S. B. Laughlin, N. J. Bidwell, R. A. Harris, “Spatio-temporal properties of motion detectors matched to low image velocities in hovering insects,” Vision Res. 37, 3427–3439 (1997).
[CrossRef]

Blanes, C.

N. Franceschini, J. M. Pichon, C. Blanes, “From insect vision to robot vision,” Philos. Trans. R. Soc. London Ser. B 337, 283–294 (1992).
[CrossRef]

Borst, A.

S. Single, A. Borst, “Dendritic integration and its role in computing image velocity,” Science 281, 1848–1850 (1998).
[CrossRef] [PubMed]

S. Single, J. Haag, A. Borst, “Dendritic computation of direction selectivity and gain control in visual interneurons,” J. Neurosci. 17, 6023–6030 (1997).
[PubMed]

M. Egelhaaf, A. Borst, “Transient and steady-state response properties of movement detectors,” J. Opt. Soc. Am. A 6, 116–127 (1989).
[CrossRef] [PubMed]

M. Egelhaaf, A. Borst, W. Reichardt, “Computational structure of a biological motion-detection system as revealed by local detector analysis in the fly’s nervous system,” J. Opt. Soc. Am. A 6, 1070–1087 (1989).
[CrossRef] [PubMed]

M. Egelhaaf, A. Borst, “Movement detection in arthropods,” in Visual Motion and Its Role in the Stabilization of Gaze, J. Wallman, F. A. Miles, eds. (Elsevier, Amsterdam, 1993), pp. 53–77.

Burton, G. J.

Chao, T.

D. J. Tolhurst, Y. Tadmor, T. Chao, “Amplitude spectra of natural images,” Ophthalmol. Physiol. Opt. 12, 229–232 (1992).
[CrossRef]

Citron, M. C.

R. C. Emerson, M. C. Citron, W. J. Vaughn, S. A. Klein, “Nonlinear directionally selective subunits in complex cells of cat striate cortex,” J. Neurophysiol. 58, 33–65 (1987).
[PubMed]

Collett, T. S.

M. V. Srinivasan, S. W. Zhang, M. Lehrer, T. S. Collett, “Honeybee navigation en route to the goal: visual flight control and odometry,” J. Exp. Biol. 199, 237–244 (1996).
[PubMed]

M. F. Land, T. S. Collett, “Chasing behaviour of houseflies (Fannia canicularis): a description and analysis,” J. Comp. Physiol. A 89, 331–357 (1974).
[CrossRef]

Dong, D. W.

D. W. Dong, J. J. Atick, “Statistics of natural time-varying images,” Network Comput. Neural Syst. 6, 345–358 (1995).
[CrossRef]

Douglass, J. K.

J. K. Douglass, N. J. Strausfeld, “Visual motion detection circuits in flies: peripheral motion computation by identified small-field retinotopic neurons,” J. Neurosci. 15, 5596–5611 (1995).
[PubMed]

Dror, R. O.

R. O. Dror, D. C. O’Carroll, S. B. Laughlin, “The role of natural image statistics in biological motion estimation,” Springer Lect. Notes Comput. Sci. 1811, 492–501 (2000).
[CrossRef]

R. O. Dror, “Accuracy of visual velocity estimation by Reichardt correlators,” Master’s thesis (University of Cambridge, Cambridge, UK, 1998).

Eckert, H. M.

M. F. Land, H. M. Eckert, “Maps of the acute zones of fly eyes,” J. Comp. Physiol. A 156, 525–538 (1985).
[CrossRef]

Egelhaaf, M.

M. Egelhaaf, A. Borst, “Transient and steady-state response properties of movement detectors,” J. Opt. Soc. Am. A 6, 116–127 (1989).
[CrossRef] [PubMed]

M. Egelhaaf, A. Borst, W. Reichardt, “Computational structure of a biological motion-detection system as revealed by local detector analysis in the fly’s nervous system,” J. Opt. Soc. Am. A 6, 1070–1087 (1989).
[CrossRef] [PubMed]

M. Egelhaaf, W. Reichardt, “Dynamic response properties of movement detectors: theoretical analysis and electrophysiological investigation in the visual system of the fly,” Biol. Cybern. 56, 69–87 (1987).
[CrossRef]

K. Hausen, M. Egelhaaf, “Neural mechanisms of visualcourse control in insects,” in Facets of Vision, D. G. Stavenga, R. C. Hardie, eds. (Springer-Verlag, Heidelberg, 1989), pp. 391–424.

M. Egelhaaf, A. Borst, “Movement detection in arthropods,” in Visual Motion and Its Role in the Stabilization of Gaze, J. Wallman, F. A. Miles, eds. (Elsevier, Amsterdam, 1993), pp. 53–77.

Emerson, R. C.

R. C. Emerson, M. C. Citron, W. J. Vaughn, S. A. Klein, “Nonlinear directionally selective subunits in complex cells of cat striate cortex,” J. Neurophysiol. 58, 33–65 (1987).
[PubMed]

Field, D. J.

Franceschini, N.

N. Franceschini, J. M. Pichon, C. Blanes, “From insect vision to robot vision,” Philos. Trans. R. Soc. London Ser. B 337, 283–294 (1992).
[CrossRef]

N. Franceschini, A. Riehle, A. le Nestour, “Directionally selective motion detection by insect neurons,” in Facets of Vision, D. G. Stavenga, R. C. Hardie, eds. (Springer-Verlag, Berlin, 1989), pp. 360–390.

Haag, J.

S. Single, J. Haag, A. Borst, “Dendritic computation of direction selectivity and gain control in visual interneurons,” J. Neurosci. 17, 6023–6030 (1997).
[PubMed]

Hardie, R. C.

R. C. Hardie, “Functional organization of the fly retina,” in Progress in Sensory Physiology, D. Ottoson, ed. (Springer-Verlag, Berlin, 1985), Vol. 5, pp. 1–80.

Harris, R. A.

R. A. Harris, D. C. O’Carroll, S. B. Laughlin, “Adaptation and the temporal delay filter of fly motion detectors,” Vision Res. 39, 2603–2613 (1999).
[CrossRef] [PubMed]

D. C. O’Carroll, S. B. Laughlin, N. J. Bidwell, R. A. Harris, “Spatio-temporal properties of motion detectors matched to low image velocities in hovering insects,” Vision Res. 37, 3427–3439 (1997).
[CrossRef]

Hausen, K.

K. Hausen, M. Egelhaaf, “Neural mechanisms of visualcourse control in insects,” in Facets of Vision, D. G. Stavenga, R. C. Hardie, eds. (Springer-Verlag, Heidelberg, 1989), pp. 391–424.

Heeger, D. J.

D. J. Heeger, “Normalization of cell responses in cat striate cortex,” Visual Neurosci. 9, 181–197 (1992).
[CrossRef]

Hengstenberg, R.

G. Nalbach, R. Hengstenberg, “The halteres of the blowfly Calliphora,” J. Comp. Physiol. A 175, 695–708 (1994).
[CrossRef]

Horridge, G. A.

G. A. Horridge, L. Marcelja, “On the existence of fast and slow directionally sensitive motion detector neurons in insects,” Proc. R. Soc. London, Ser. B 248, 47–54 (1992).
[CrossRef]

Howard, J.

R. Payne, J. Howard, “Response of an insect photoreceptor: a simple log-normal model,” Nature 290, 415–416 (1981).
[CrossRef]

James, A. C.

A. C. James, “White-noise studies in the fly Lamina,” Ph.D. thesis (Australian National University, Canberra, Australia, 1990).

Kirschfeld, K.

F. Wolf-Oberhollenzer, K. Kirschfeld, “Motion sensitivity in the nucleus of the basal optic root of the pigeon,” J. Neurophysiol. 71, 1559–1573 (1994).
[PubMed]

Klein, S. A.

R. C. Emerson, M. C. Citron, W. J. Vaughn, S. A. Klein, “Nonlinear directionally selective subunits in complex cells of cat striate cortex,” J. Neurophysiol. 58, 33–65 (1987).
[PubMed]

Koch, C.

R. Sarpeshkar, W. Bair, C. Koch, “An analog VLSI chip for local velocity estimation based on Reichardt’s motion algorithm,” in Advances in Neural Information Processing Systems, S. Hanson, J. Cowan, L. Giles, eds. (Morgan Kauffman, San Mateo, Calif., 1993), Vol. 5, pp. 781–788.

Land, M. F.

M. F. Land, H. M. Eckert, “Maps of the acute zones of fly eyes,” J. Comp. Physiol. A 156, 525–538 (1985).
[CrossRef]

M. F. Land, T. S. Collett, “Chasing behaviour of houseflies (Fannia canicularis): a description and analysis,” J. Comp. Physiol. A 89, 331–357 (1974).
[CrossRef]

Laughlin, S. B.

B. Tatler, D. C. O’Carroll, S. B. Laughlin, “Temperature and temporal resolving power of fly photoreceptors,” J. Comp. Physiol. A 186, 399–407 (2000).
[CrossRef] [PubMed]

R. O. Dror, D. C. O’Carroll, S. B. Laughlin, “The role of natural image statistics in biological motion estimation,” Springer Lect. Notes Comput. Sci. 1811, 492–501 (2000).
[CrossRef]

R. A. Harris, D. C. O’Carroll, S. B. Laughlin, “Adaptation and the temporal delay filter of fly motion detectors,” Vision Res. 39, 2603–2613 (1999).
[CrossRef] [PubMed]

D. C. O’Carroll, S. B. Laughlin, N. J. Bidwell, R. A. Harris, “Spatio-temporal properties of motion detectors matched to low image velocities in hovering insects,” Vision Res. 37, 3427–3439 (1997).
[CrossRef]

S. B. Laughlin, “Matching coding, circuits, cells and molecules to signals: general principles of retinal design in the fly’s eye,” Prog. Retinal Res. 13, 165–195 (1994).
[CrossRef]

T. Maddess, S. B. Laughlin, “Adaptation of the motion-sensitive neuron H1 is generated locally and governed by contrast frequency,” Proc. R. Soc. London Ser. B 225, 251–275 (1985).
[CrossRef]

le Nestour, A.

N. Franceschini, A. Riehle, A. le Nestour, “Directionally selective motion detection by insect neurons,” in Facets of Vision, D. G. Stavenga, R. C. Hardie, eds. (Springer-Verlag, Berlin, 1989), pp. 360–390.

Lehrer, M.

M. V. Srinivasan, S. W. Zhang, M. Lehrer, T. S. Collett, “Honeybee navigation en route to the goal: visual flight control and odometry,” J. Exp. Biol. 199, 237–244 (1996).
[PubMed]

Maddess, T.

T. Maddess, S. B. Laughlin, “Adaptation of the motion-sensitive neuron H1 is generated locally and governed by contrast frequency,” Proc. R. Soc. London Ser. B 225, 251–275 (1985).
[CrossRef]

Marcelja, L.

G. A. Horridge, L. Marcelja, “On the existence of fast and slow directionally sensitive motion detector neurons in insects,” Proc. R. Soc. London, Ser. B 248, 47–54 (1992).
[CrossRef]

McKee, S. P.

S. P. McKee, G. H. Silverman, K. Nakayama, “Precise velocity discrimination despite random variations in temporal frequency and contrast,” Vision Res. 26, 609–619 (1986).
[CrossRef] [PubMed]

Moorhead, I. R.

Nakayama, K.

S. P. McKee, G. H. Silverman, K. Nakayama, “Precise velocity discrimination despite random variations in temporal frequency and contrast,” Vision Res. 26, 609–619 (1986).
[CrossRef] [PubMed]

Nalbach, G.

G. Nalbach, R. Hengstenberg, “The halteres of the blowfly Calliphora,” J. Comp. Physiol. A 175, 695–708 (1994).
[CrossRef]

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N. Franceschini, A. Riehle, A. le Nestour, “Directionally selective motion detection by insect neurons,” in Facets of Vision, D. G. Stavenga, R. C. Hardie, eds. (Springer-Verlag, Berlin, 1989), pp. 360–390.

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Appl. Opt. (1)

Biol. Cybern. (1)

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J. Neurophysiol. (2)

F. Wolf-Oberhollenzer, K. Kirschfeld, “Motion sensitivity in the nucleus of the basal optic root of the pigeon,” J. Neurophysiol. 71, 1559–1573 (1994).
[PubMed]

R. C. Emerson, M. C. Citron, W. J. Vaughn, S. A. Klein, “Nonlinear directionally selective subunits in complex cells of cat striate cortex,” J. Neurophysiol. 58, 33–65 (1987).
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S. Single, J. Haag, A. Borst, “Dendritic computation of direction selectivity and gain control in visual interneurons,” J. Neurosci. 17, 6023–6030 (1997).
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[PubMed]

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Nature (1)

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Philos. Trans. R. Soc. London Ser. B (1)

N. Franceschini, J. M. Pichon, C. Blanes, “From insect vision to robot vision,” Philos. Trans. R. Soc. London Ser. B 337, 283–294 (1992).
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[CrossRef] [PubMed]

Science (1)

S. Single, A. Borst, “Dendritic integration and its role in computing image velocity,” Science 281, 1848–1850 (1998).
[CrossRef] [PubMed]

Springer Lect. Notes Comput. Sci. (1)

R. O. Dror, D. C. O’Carroll, S. B. Laughlin, “The role of natural image statistics in biological motion estimation,” Springer Lect. Notes Comput. Sci. 1811, 492–501 (2000).
[CrossRef]

Vision Res. (5)

J. H. van Hateren, “Processing of natural time series of intensities by the visual system of the blowfly,” Vision Res. 37, 3407–3416 (1997).
[CrossRef]

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

R. A. Harris, D. C. O’Carroll, S. B. Laughlin, “Adaptation and the temporal delay filter of fly motion detectors,” Vision Res. 39, 2603–2613 (1999).
[CrossRef] [PubMed]

D. C. O’Carroll, S. B. Laughlin, N. J. Bidwell, R. A. Harris, “Spatio-temporal properties of motion detectors matched to low image velocities in hovering insects,” Vision Res. 37, 3427–3439 (1997).
[CrossRef]

S. P. McKee, G. H. Silverman, K. Nakayama, “Precise velocity discrimination despite random variations in temporal frequency and contrast,” Vision Res. 26, 609–619 (1986).
[CrossRef] [PubMed]

Visual Neurosci. (2)

D. J. Heeger, “Normalization of cell responses in cat striate cortex,” Visual Neurosci. 9, 181–197 (1992).
[CrossRef]

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[CrossRef]

Other (10)

Information is available from David C. O’Carroll, Department of Zoology, University of Washington, Box 351800, Seattle, Wash. 98195-1800; davidoc@u.washington.edu.

N. Franceschini, A. Riehle, A. le Nestour, “Directionally selective motion detection by insect neurons,” in Facets of Vision, D. G. Stavenga, R. C. Hardie, eds. (Springer-Verlag, Berlin, 1989), pp. 360–390.

R. O. Dror, “Accuracy of visual velocity estimation by Reichardt correlators,” Master’s thesis (University of Cambridge, Cambridge, UK, 1998).

A. W. Snyder, “The physics of vision in compound eyes,” in Comparative Physiology and Evolution of Vision in Invertebrates: Invertebrate Photoreceptors, Vol. 6A of Handbook of Sensory Physiology, H. Autrum, ed. (Springer-Verlag, Berlin, 1979), pp. 225–313.
[CrossRef]

R. Sarpeshkar, W. Bair, C. Koch, “An analog VLSI chip for local velocity estimation based on Reichardt’s motion algorithm,” in Advances in Neural Information Processing Systems, S. Hanson, J. Cowan, L. Giles, eds. (Morgan Kauffman, San Mateo, Calif., 1993), Vol. 5, pp. 781–788.

M. Egelhaaf, A. Borst, “Movement detection in arthropods,” in Visual Motion and Its Role in the Stabilization of Gaze, J. Wallman, F. A. Miles, eds. (Elsevier, Amsterdam, 1993), pp. 53–77.

R. C. Hardie, “Functional organization of the fly retina,” in Progress in Sensory Physiology, D. Ottoson, ed. (Springer-Verlag, Berlin, 1985), Vol. 5, pp. 1–80.

W. Reichardt, “Autocorrelation, a principle for the evaluation of sensory information by the central nervous system,” in Sensory Communication, A. Rosenblith, ed. (MIT Press, Cambridge, Mass.1961), pp. 303–317.

A. C. James, “White-noise studies in the fly Lamina,” Ph.D. thesis (Australian National University, Canberra, Australia, 1990).

K. Hausen, M. Egelhaaf, “Neural mechanisms of visualcourse control in insects,” in Facets of Vision, D. G. Stavenga, R. C. Hardie, eds. (Springer-Verlag, Heidelberg, 1989), pp. 391–424.

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Figures (11)

Fig. 1
Fig. 1

(a) Minimal Reichardt correlator with a potential input luminance signal, in this case a sinusoid with amplitude C and wavelength λ=1/fs moving at velocity v. (b) A more realistic correlator model, considered in Section 4, includes spatial prefilters, S’s; temporal prefilters, T’s; compressive nonlinearities, ρ’s and θ; temporal integration, M and spatial summation, Σ.

Fig. 2
Fig. 2

Velocity response curves for the simple correlator model of Fig. 1(a). (a) Response curves for horizontal sinusoidal gratings of 100% contrast at two different spatial frequencies; (b) response curves for the natural images shown in Fig. 3; (c) the same four curves, normalized so that their maximum values are identical.

Fig. 3
Fig. 3

Examples of the natural images used in simulations throughout this work. Images a and b are panoramic images that represent the visual scenes in locations where Episyrphus chooses to hover. Images were obtained by using a video camera with a linear CCD element, then digitized, calibrated to real luminance units, and mosaicked into panoramas spanning a full circle horizontally and approximately 23° vertically (one spatial degree corresponds to 4.64 pixels). Images c and d are samples of the image set acquired by David Tolhurst12 with a still camera and then digitized and corrected for luminance nonlinearities in the film. Images in this set measure 256 pixels both horizontally and vertically, with approximately 10 pixels to 1°.

Fig. 4
Fig. 4

Velocity response curves predicted by Eq. (2) for row power spectra P(fs)=fs-(1+η). Simulated velocity response curves from Fig. 2(c) are shown by thin dotted curves for comparison. Predicted peak response velocities are 32, 35, and 40°/s for η=-0.25, 0, and 0.25, respectively. Predicted curves have been normalized to a maximum value of 1.0 through multiplication by 1.26, 1.0, and 0.92, respectively.

Fig. 5
Fig. 5

Horizontal power spectral densities of the images in Fig. 3. Each spectrum is an average of power spectral densities of the rows that constitute the image. Images c and d have higher maximum and nonzero minimum frequencies because Tolhurst’s images are more finely sampled and span a smaller portion of the horizontal visual field than the panoramic images. Images a and b roll off in power at frequencies above 1.2 cycles/° as a result of averaging in the image acquisition process. Fortunately, the sampling lattice of the fly’s photoreceptors has a Nyquist frequency near 0.5 cycle/°, and optical blur effects in the fly’s eye reject almost all spatial frequency content above 1 cycle/° (see Subsection 4.B.).

Fig. 6
Fig. 6

(a) Time response of a simple correlator to image b of Fig. 3, moving at 19.6°/s. The dotted curve indicates the mean response level. (b) Relative error curves for the four natural images of Fig. 3.

Fig. 7
Fig. 7

Effect of low-pass spatial prefiltering due to optical blurring. (a) Simulated velocity response curves for the natural image of Fig. 3b with and without the spatial prefilter, compared with the analytical predictions of Eq. (3) and assuming a horizontal image power spectrum P(fs)=fs-1.1. These curves are normalized to have a maximum value of 1.0. The simulation results closely match the analytical predictions. (b) Relative error curves with and without the effects of optic blurring.

Fig. 8
Fig. 8

Effect of high-pass temporal prefiltering by an LMC with impulse response as described in the text. Other model parameters, including optical blurring, are as in Fig. 7. (a) Simulated and analytically predicted velocity response curves in the presence and in the absence of LMC temporal filters, normalized to have maximum value 1.0. (b) Simulated relative error curves in the presence and in the absence of the temporal prefilters.

Fig. 9
Fig. 9

Effect of contrast saturation on velocity response and error curves for images a and b of Fig. 3. The correlator model includes the spatial and temporal prefilters of Fig. 8. Mean responses for images a and b are expressed in identical units.

Fig. 10
Fig. 10

Effect of spatial integration on relative error for image b of Fig. 3, with and without contrast saturation. Model parameters are as in Fig. 9. We integrated outputs of an array of correlators arranged on an 8×18 grid with 1° separation between correlators, corresponding to a small HS cell. For simplicity, we used a square rather than a hexagonal grid. Integration of 144 independent outputs would lead to a 12-fold reduction in noise. Because the error signals in different correlators are correlated, relative error decreases by a smaller factor of 3 to 5.

Fig. 11
Fig. 11

(a) Velocity response curves for the model of Fig. 8 for random-texture images of various densities, as predicted by Eq. (3) from analytically computed power spectra. Each curve is normalized to a maximum value of 1.0. (b) Temporally averaged response of HS neurons with equatorial receptive fields to textures of corresponding densities, as a function of velocity. Each curve represents the mean and the standard error (SE) of the responses of five or six neurons, as indicated by the legend. Membrane potential was originally measured in millivolts, but the curves presented here have been normalized for comparison.

Equations (8)

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

R(t)=C22πτftft2+1/(2πτ)2sin(2πfsΔϕ),
R¯=12πτ0P(fs) fsv(fsv)2+1/(2πτ)2sin(2πfsΔϕ)dfs,
R¯(v)=12πτ0[S2(fs)P(fs)sin(2πfsΔϕ)]×T2(fsv) fsv(fsv)2+1/(2πτ)2dfs.
p(t)=exp-[loge(t/tp)]22σ2,
ρ(C)=tanh(sC),
R¯(v)=k-F(fs)W(fsv)d(log10 fs),
F(fs)=fsS2(fs)P(fs)sin(2πfsΔϕ),
W(fsv)=1τ T2(fsv) fsv(fsv)2+1/(2πτ)2,

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