Abstract

A new method for recovery of heading from motion is developed on the basis of Longuet-Higgins and Prazdny’s algorithm [Proc. R. Soc. London Ser. B 208, 385 (1980)]. In the algorithm a radial virtual flow field is generated and the difference between the original velocity field and the virtual radial field is computed. The difference vectors, which are directed to the heading point in the projected plane, allow us to estimate the direction of heading. The simulations of the algorithm were performed, and it was shown that the method estimates the direction of heading accurately.

© 2000 Optical Society of America

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References

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  1. A. R. Bruss, B. K. P. Horn, “Passive navigation,” Comput. Graph. Image Process. 21, 3–20 (1983).
    [CrossRef]
  2. H. C. Longuet-Higgins, K. Prazdny, “The interpretation of moving retinal images,” Proc. R. Soc. London Ser. B 208, 385–397 (1980).
    [CrossRef]
  3. K. Kanatani, “3-D interpretation of optical flow by renormalization,” Int. J. Comput. Vis. 11, 267–282 (1993).
    [CrossRef]
  4. C. Tomasi, J. Shi, “Direction of heading from image deformations,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Ca., 1993), pp. 422–427.
  5. J. H. Rieger, D. T. Lawton, “Processing differential image motion,” J. Opt. Soc. Am. A 2, 354–360 (1985).
    [CrossRef] [PubMed]
  6. J. M. H. Beusmans, “Computing the direction of heading from affine image flow,” Biol. Cybern. 70, 123–136 (1993).
    [CrossRef] [PubMed]
  7. D. J. Heeger, A. Jepson, “Visual perception of three-dimensional motion,” Neural Comput. 2, 129–137 (1990).
    [CrossRef]
  8. D. J. Heeger, A. Jepson, “Subspace methods for recovering rigid motion. I: Algorithm and implementation,” Int. J. Comput. Vis. 7, 95–117 (1992).
    [CrossRef]
  9. E. C. Hildreth, “Recovering heading for visually-guided navigation,” Vision Res. 32, 1177–1192 (1992).
    [CrossRef] [PubMed]
  10. M. Hanada, Y. Ejima, “A model of human heading judgement in forward motion,” Vision Res. 40, 243–263 (2000)
    [CrossRef] [PubMed]
  11. D. Regan, S. Kashal, “Monocular discrimination of the direction of motion in depth,” Vision Res. 34, 163–177 (1994).
    [CrossRef] [PubMed]
  12. J. A. Perrone, L. S. Stone, “A model of self-motion estimation within primate extrastriate visual cortex,” Vision Res. 34, 2917–2938 (1994).
    [CrossRef] [PubMed]
  13. W. H. Warren, D. Hannon, “Eye movement and optical flow,” J. Opt. Soc. Am. A 7, 160–169 (1990).
    [CrossRef] [PubMed]
  14. C. S. Royden, J. A. Crowell, M. S. Banks, “Estimating heading during eye movements,” Vision Res. 34, 3197–3214 (1994).
    [CrossRef] [PubMed]

2000

M. Hanada, Y. Ejima, “A model of human heading judgement in forward motion,” Vision Res. 40, 243–263 (2000)
[CrossRef] [PubMed]

1994

D. Regan, S. Kashal, “Monocular discrimination of the direction of motion in depth,” Vision Res. 34, 163–177 (1994).
[CrossRef] [PubMed]

J. A. Perrone, L. S. Stone, “A model of self-motion estimation within primate extrastriate visual cortex,” Vision Res. 34, 2917–2938 (1994).
[CrossRef] [PubMed]

C. S. Royden, J. A. Crowell, M. S. Banks, “Estimating heading during eye movements,” Vision Res. 34, 3197–3214 (1994).
[CrossRef] [PubMed]

1993

K. Kanatani, “3-D interpretation of optical flow by renormalization,” Int. J. Comput. Vis. 11, 267–282 (1993).
[CrossRef]

J. M. H. Beusmans, “Computing the direction of heading from affine image flow,” Biol. Cybern. 70, 123–136 (1993).
[CrossRef] [PubMed]

1992

D. J. Heeger, A. Jepson, “Subspace methods for recovering rigid motion. I: Algorithm and implementation,” Int. J. Comput. Vis. 7, 95–117 (1992).
[CrossRef]

E. C. Hildreth, “Recovering heading for visually-guided navigation,” Vision Res. 32, 1177–1192 (1992).
[CrossRef] [PubMed]

1990

D. J. Heeger, A. Jepson, “Visual perception of three-dimensional motion,” Neural Comput. 2, 129–137 (1990).
[CrossRef]

W. H. Warren, D. Hannon, “Eye movement and optical flow,” J. Opt. Soc. Am. A 7, 160–169 (1990).
[CrossRef] [PubMed]

1985

1983

A. R. Bruss, B. K. P. Horn, “Passive navigation,” Comput. Graph. Image Process. 21, 3–20 (1983).
[CrossRef]

1980

H. C. Longuet-Higgins, K. Prazdny, “The interpretation of moving retinal images,” Proc. R. Soc. London Ser. B 208, 385–397 (1980).
[CrossRef]

Banks, M. S.

C. S. Royden, J. A. Crowell, M. S. Banks, “Estimating heading during eye movements,” Vision Res. 34, 3197–3214 (1994).
[CrossRef] [PubMed]

Beusmans, J. M. H.

J. M. H. Beusmans, “Computing the direction of heading from affine image flow,” Biol. Cybern. 70, 123–136 (1993).
[CrossRef] [PubMed]

Bruss, A. R.

A. R. Bruss, B. K. P. Horn, “Passive navigation,” Comput. Graph. Image Process. 21, 3–20 (1983).
[CrossRef]

Crowell, J. A.

C. S. Royden, J. A. Crowell, M. S. Banks, “Estimating heading during eye movements,” Vision Res. 34, 3197–3214 (1994).
[CrossRef] [PubMed]

Ejima, Y.

M. Hanada, Y. Ejima, “A model of human heading judgement in forward motion,” Vision Res. 40, 243–263 (2000)
[CrossRef] [PubMed]

Hanada, M.

M. Hanada, Y. Ejima, “A model of human heading judgement in forward motion,” Vision Res. 40, 243–263 (2000)
[CrossRef] [PubMed]

Hannon, D.

Heeger, D. J.

D. J. Heeger, A. Jepson, “Subspace methods for recovering rigid motion. I: Algorithm and implementation,” Int. J. Comput. Vis. 7, 95–117 (1992).
[CrossRef]

D. J. Heeger, A. Jepson, “Visual perception of three-dimensional motion,” Neural Comput. 2, 129–137 (1990).
[CrossRef]

Hildreth, E. C.

E. C. Hildreth, “Recovering heading for visually-guided navigation,” Vision Res. 32, 1177–1192 (1992).
[CrossRef] [PubMed]

Horn, B. K. P.

A. R. Bruss, B. K. P. Horn, “Passive navigation,” Comput. Graph. Image Process. 21, 3–20 (1983).
[CrossRef]

Jepson, A.

D. J. Heeger, A. Jepson, “Subspace methods for recovering rigid motion. I: Algorithm and implementation,” Int. J. Comput. Vis. 7, 95–117 (1992).
[CrossRef]

D. J. Heeger, A. Jepson, “Visual perception of three-dimensional motion,” Neural Comput. 2, 129–137 (1990).
[CrossRef]

Kanatani, K.

K. Kanatani, “3-D interpretation of optical flow by renormalization,” Int. J. Comput. Vis. 11, 267–282 (1993).
[CrossRef]

Kashal, S.

D. Regan, S. Kashal, “Monocular discrimination of the direction of motion in depth,” Vision Res. 34, 163–177 (1994).
[CrossRef] [PubMed]

Lawton, D. T.

Longuet-Higgins, H. C.

H. C. Longuet-Higgins, K. Prazdny, “The interpretation of moving retinal images,” Proc. R. Soc. London Ser. B 208, 385–397 (1980).
[CrossRef]

Perrone, J. A.

J. A. Perrone, L. S. Stone, “A model of self-motion estimation within primate extrastriate visual cortex,” Vision Res. 34, 2917–2938 (1994).
[CrossRef] [PubMed]

Prazdny, K.

H. C. Longuet-Higgins, K. Prazdny, “The interpretation of moving retinal images,” Proc. R. Soc. London Ser. B 208, 385–397 (1980).
[CrossRef]

Regan, D.

D. Regan, S. Kashal, “Monocular discrimination of the direction of motion in depth,” Vision Res. 34, 163–177 (1994).
[CrossRef] [PubMed]

Rieger, J. H.

Royden, C. S.

C. S. Royden, J. A. Crowell, M. S. Banks, “Estimating heading during eye movements,” Vision Res. 34, 3197–3214 (1994).
[CrossRef] [PubMed]

Shi, J.

C. Tomasi, J. Shi, “Direction of heading from image deformations,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Ca., 1993), pp. 422–427.

Stone, L. S.

J. A. Perrone, L. S. Stone, “A model of self-motion estimation within primate extrastriate visual cortex,” Vision Res. 34, 2917–2938 (1994).
[CrossRef] [PubMed]

Tomasi, C.

C. Tomasi, J. Shi, “Direction of heading from image deformations,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Ca., 1993), pp. 422–427.

Warren, W. H.

Biol. Cybern.

J. M. H. Beusmans, “Computing the direction of heading from affine image flow,” Biol. Cybern. 70, 123–136 (1993).
[CrossRef] [PubMed]

Comput. Graph. Image Process.

A. R. Bruss, B. K. P. Horn, “Passive navigation,” Comput. Graph. Image Process. 21, 3–20 (1983).
[CrossRef]

Int. J. Comput. Vis.

D. J. Heeger, A. Jepson, “Subspace methods for recovering rigid motion. I: Algorithm and implementation,” Int. J. Comput. Vis. 7, 95–117 (1992).
[CrossRef]

K. Kanatani, “3-D interpretation of optical flow by renormalization,” Int. J. Comput. Vis. 11, 267–282 (1993).
[CrossRef]

J. Opt. Soc. Am. A

Neural Comput.

D. J. Heeger, A. Jepson, “Visual perception of three-dimensional motion,” Neural Comput. 2, 129–137 (1990).
[CrossRef]

Proc. R. Soc. London Ser. B

H. C. Longuet-Higgins, K. Prazdny, “The interpretation of moving retinal images,” Proc. R. Soc. London Ser. B 208, 385–397 (1980).
[CrossRef]

Vision Res.

E. C. Hildreth, “Recovering heading for visually-guided navigation,” Vision Res. 32, 1177–1192 (1992).
[CrossRef] [PubMed]

M. Hanada, Y. Ejima, “A model of human heading judgement in forward motion,” Vision Res. 40, 243–263 (2000)
[CrossRef] [PubMed]

D. Regan, S. Kashal, “Monocular discrimination of the direction of motion in depth,” Vision Res. 34, 163–177 (1994).
[CrossRef] [PubMed]

J. A. Perrone, L. S. Stone, “A model of self-motion estimation within primate extrastriate visual cortex,” Vision Res. 34, 2917–2938 (1994).
[CrossRef] [PubMed]

C. S. Royden, J. A. Crowell, M. S. Banks, “Estimating heading during eye movements,” Vision Res. 34, 3197–3214 (1994).
[CrossRef] [PubMed]

Other

C. Tomasi, J. Shi, “Direction of heading from image deformations,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Ca., 1993), pp. 422–427.

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Figures (7)

Fig. 1
Fig. 1

Center of outflow. The center of outflow is the point that achieves the least-squares sum of d in (a). Examples of the center of outflow are shown in (b) and (c).

Fig. 2
Fig. 2

Examples of a flow field and the radial virtual flow. The flow field and the radial virtual flow are sketched in (a). Note that the flow is not actual one. The original flow and the virtual flow in Eq. (6) correspond to the projected motion of the original 3-D point and a frontoparallel plane with an average depth of original sampling points’ depth, as shown in (b). The difference vectors of the original velocity and virtual velocity are oriented to the heading point (U/W, V/W) in the image.

Fig. 3
Fig. 3

Results in the ground condition. Results of the simulation of the proposed algorithm in the ground condition are shown. The horizontal axis represents simulated heading (U/W), and the vertical axis represents heading estimated by the algorithm. Each point denotes the result of each trial.

Fig. 4
Fig. 4

Results in the cloud condition. Results of the simulation of the proposed algorithm in the ground condition are shown. (a) The horizontal axis represents the horizontal component of simulated heading (U/W), and the vertical axis represents the value estimated by the algorithm. (b) The horizontal axis represents the horizontal component of simulated heading (V/W), and the vertical axis represents the value estimated by the algorithm. Each point denotes the result of each trial.

Fig. 5
Fig. 5

Effects of the number of dots. Simulated heading and an estimated heading were divided into a horizontal component (U/W) and a vertical one (V/W). Slopes and correlation coefficients obtained by regression analyses conducted for each component are shown. The horizontal axis indicates the number of input dots.

Fig. 6
Fig. 6

Effects of the rotation rate. Simulated heading and an estimated heading were divided into a horizontal component (U/W) and a vertical one (V/W). Slopes and correlation coefficients obtained by regression analyses conducted for each component are shown. The horizontal axis indicates the absolute value of pitch (A) and yaw (B) rate.

Fig. 7
Fig. 7

Analysis of split double plane. (a) Two planes with different depth are positioned in the upper and lower fields, respectively. Focus 1 and 2 indicate the focus of the flow for each plane. The center of outflow is the best-fitting intersection of the line passing through all velocity vectors. The center of outflow is located between the two focuses. (b) The 3-D structure corresponding to the flow field in (a) is shown. The virtual radial flow corresponds to the image motion of the frontoparallel plane with average depth of the sampling points’ depth.

Equations (24)

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

X˙=-U-BZ+CY,
Y˙=-V-CX+AZ,
Z˙=-W-AY+BX.
x=X/Z,
y=Y/Z.
u=(-U+xW)Z-B+Cy+Axy-Bx2,
v=(-V+yW)Z-Cx+A+Ay2-Bxy.
Ce=1Nc |xi|>Tcxor|yi|>Tcy uiyi-vixixi2+yi2,
Ce=1Ncv |xi|>Tcx-vixi,
uri=(xi-xc)/τ,
vri=(yi-yc)/τ,
τ=1N i=0Nxi2+yi2ui2+vi21/2,
u=(-U+xW)Z-B,
v=(-V+yW)Z+A.
u=(-U+xW)Z0-B,
v=(-V+yW)Z0+A.
0=(-U+xcW)Z0-B,
0=(-V+ycW)Z0+A.
u=xWZ0-xc WZ0(x-xc)/τ,
ν=yWZ0-yc WZ0(y-yc)/τ.
udi=ui-uri,
vdi=vi-vri.
AVW-yc/τ,
B-UW-xc/τ.

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