Abstract

The inference of three-dimensional camera motion parameters and the layout of a scene from image flows becomes particularly simple from a computational point of view if the scene contains depth variations. Under this condition, the differential image motion yields a simple estimate of the translational field lines at image locations corresponding to depth discontinuities in the scene. This in turn facilitates closed-form solutions of camera motion parameters and environmental depth. Our results may have relevance to human motion perception, which also seems to rely on depth variation in processing image motion.

© 1985 Optical Society of America

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References

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  1. H. von Helmholtz, Handbuch der Physiologischen Optik (Leopold Voss, Hamburg, 1896).
  2. J. J. Gibson, The Perception of the Visual World (Houghton Mifflin, Boston, 1950).
  3. J. J. Koenderink, A. J. van Doorn, “Invariant properties of the motion parallax field due to movement of rigid bodies relative to an observer,” Opt. Acta 22, 773–791 (1975).
    [CrossRef]
  4. H. C. Longuet-Higgins, K. Prazdny, “The interpretation of a moving retinal image,” Proc. R. Soc. London Ser. B 208, 385–397 (1980).
    [CrossRef]
  5. H. C. Longuet-Higgins, “A computer algorithm for reconstructing a scene from two projections,” Nature 293, 133–135 (1981).
    [CrossRef]
  6. R. Y. Tsai, T. S. Huang, “Uniqueness and estimation of three-dimensional motion parameters of rigid objects with curved surfaces,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 13–27 (1984).
    [CrossRef]
  7. K. Prazdny, “Determining the instantaneous direction of motion from optical flow generated by a curvilinearly moving observer,” Comput. Graphics Image Process. 17, 238–248 (1981).
    [CrossRef]
  8. S. Ullman, “The interpretation of structure from motion,” Proc. R. Soc. London Ser. B 203, 405–426 (1979).
    [CrossRef]
  9. D. T. Lawton, “Processing translational image sequences,” Comput. Vision Graphics Image Process. 22, 116–144 (1983).
    [CrossRef]
  10. L. Dreschler, H.-H. Nagel, “Volumetric model and 3D-trajectory of a moving car derived from monocular TV-frame sequences of a street scene,” Comput. Graphics Image Process. 20, 199–228 (1982).
    [CrossRef]
  11. D. B. Gennery, “Modelling the environment of an exploring vehicle by means of stereo vision,” Memo AIM-339, Stanford Artificial Intelligence Laboratory, Stanford, Calif. (1980).
  12. J. H. Rieger, “Information in optical flows induced by curved paths of observation,”J. Opt. Soc. Am. 73, 339–344 (1983).
    [CrossRef] [PubMed]
  13. J. H. Rieger, D. T. Lawton, in Proceedings of the ACM SIGGRAPH/SIGART Conference on Motion: Representation and Perception, Toronto, Canada, 1983 (Association for Computing Machinery, New York, 1983), pp. 33–41.
  14. D. N. Lee, P. E. Reddish, “Plummeting gannets: a paradigm of ecological optics,” Nature 293, 293–294 (1981).
    [CrossRef]
  15. D. N. Lee, “The optical flow field: the foundation of vision,” Phil. Trans. R. Soc. London Ser. B 290, 169–179 (1980).
    [CrossRef]
  16. H. Wagner, “Flow-field variables trigger landing in flies,” Nature 297, 147–148 (1982).
    [CrossRef]
  17. P. Anandan, “A confidence measure for correlation matching,”COINS Tech. Rep., Department of Computer and Information Science, University of Massachusetts, Amherst, Mass. (1984).
  18. F. Glazer, G. Reynolds, P. Anandan, “Scene matching by hierarchical correlation,” in Proceedings of IEEE Conference on Computer Vision Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 432–441.
  19. K. R. Llewellyn, “Visual guidance of locomotion,”J. Exp. Psychol. 91, 245–261 (1971).
    [CrossRef] [PubMed]
  20. I. R. Johnston, G. R. White, R. W. Cumming, “The role of optical expansion patterns in locomotor control,” Am. J. Psychol. 86, 311–324 (1973).
    [CrossRef] [PubMed]
  21. D. Regan, K. I. Beverley, “How do we avoid confounding the direction we are looking and the direction we are moving?” Science 215, 194–196 (1982).
    [CrossRef] [PubMed]
  22. A. J. van Doorn, J. J. Koenderink, “Temporal properties of the visual detectability of moving spatial white noise,” Exp. Brain Res. 45, 179–188 (1982).
    [PubMed]
  23. A. J. van Doorn, J. J. Koenderink, “Visibility of movement gradients,” Biol. Cybernet. 44, 167–175 (1982).
    [CrossRef]
  24. A. J. van Doorn, J. J. Koenderink, “Detectability of velocity gradients in moving random-dot patterns,” Vision Res. 23, 799–804 (1983).
    [CrossRef] [PubMed]
  25. K. Nakayama, “Differential motion hyperacuity under conditions of common image motion,” Vision Res. 21, 1475–1482 (1981).
    [CrossRef] [PubMed]
  26. S. P. McKee, “A local mechanism for differential velocity detection,” Vision Res. 21, 491–500 (1981).
    [CrossRef] [PubMed]
  27. R. Warren, “The perception of egomotion,”J. Exp. Psychol.: Human Percept. Perf. 2, 448–456 (1976).
    [CrossRef]
  28. J. E. Cutting, “Perceiving and recovering structure from events,” in Proceedings of the ACM SIGGRAPH/SIGART Conference on Motion: Representation and Perception, Toronto, Canada, 1983 (Association for Computing Machinery, New York, 1983), pp. 141–147.

1984 (1)

R. Y. Tsai, T. S. Huang, “Uniqueness and estimation of three-dimensional motion parameters of rigid objects with curved surfaces,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 13–27 (1984).
[CrossRef]

1983 (3)

D. T. Lawton, “Processing translational image sequences,” Comput. Vision Graphics Image Process. 22, 116–144 (1983).
[CrossRef]

J. H. Rieger, “Information in optical flows induced by curved paths of observation,”J. Opt. Soc. Am. 73, 339–344 (1983).
[CrossRef] [PubMed]

A. J. van Doorn, J. J. Koenderink, “Detectability of velocity gradients in moving random-dot patterns,” Vision Res. 23, 799–804 (1983).
[CrossRef] [PubMed]

1982 (5)

H. Wagner, “Flow-field variables trigger landing in flies,” Nature 297, 147–148 (1982).
[CrossRef]

L. Dreschler, H.-H. Nagel, “Volumetric model and 3D-trajectory of a moving car derived from monocular TV-frame sequences of a street scene,” Comput. Graphics Image Process. 20, 199–228 (1982).
[CrossRef]

D. Regan, K. I. Beverley, “How do we avoid confounding the direction we are looking and the direction we are moving?” Science 215, 194–196 (1982).
[CrossRef] [PubMed]

A. J. van Doorn, J. J. Koenderink, “Temporal properties of the visual detectability of moving spatial white noise,” Exp. Brain Res. 45, 179–188 (1982).
[PubMed]

A. J. van Doorn, J. J. Koenderink, “Visibility of movement gradients,” Biol. Cybernet. 44, 167–175 (1982).
[CrossRef]

1981 (5)

D. N. Lee, P. E. Reddish, “Plummeting gannets: a paradigm of ecological optics,” Nature 293, 293–294 (1981).
[CrossRef]

K. Prazdny, “Determining the instantaneous direction of motion from optical flow generated by a curvilinearly moving observer,” Comput. Graphics Image Process. 17, 238–248 (1981).
[CrossRef]

H. C. Longuet-Higgins, “A computer algorithm for reconstructing a scene from two projections,” Nature 293, 133–135 (1981).
[CrossRef]

K. Nakayama, “Differential motion hyperacuity under conditions of common image motion,” Vision Res. 21, 1475–1482 (1981).
[CrossRef] [PubMed]

S. P. McKee, “A local mechanism for differential velocity detection,” Vision Res. 21, 491–500 (1981).
[CrossRef] [PubMed]

1980 (2)

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

D. N. Lee, “The optical flow field: the foundation of vision,” Phil. Trans. R. Soc. London Ser. B 290, 169–179 (1980).
[CrossRef]

1979 (1)

S. Ullman, “The interpretation of structure from motion,” Proc. R. Soc. London Ser. B 203, 405–426 (1979).
[CrossRef]

1976 (1)

R. Warren, “The perception of egomotion,”J. Exp. Psychol.: Human Percept. Perf. 2, 448–456 (1976).
[CrossRef]

1975 (1)

J. J. Koenderink, A. J. van Doorn, “Invariant properties of the motion parallax field due to movement of rigid bodies relative to an observer,” Opt. Acta 22, 773–791 (1975).
[CrossRef]

1973 (1)

I. R. Johnston, G. R. White, R. W. Cumming, “The role of optical expansion patterns in locomotor control,” Am. J. Psychol. 86, 311–324 (1973).
[CrossRef] [PubMed]

1971 (1)

K. R. Llewellyn, “Visual guidance of locomotion,”J. Exp. Psychol. 91, 245–261 (1971).
[CrossRef] [PubMed]

Anandan, P.

P. Anandan, “A confidence measure for correlation matching,”COINS Tech. Rep., Department of Computer and Information Science, University of Massachusetts, Amherst, Mass. (1984).

F. Glazer, G. Reynolds, P. Anandan, “Scene matching by hierarchical correlation,” in Proceedings of IEEE Conference on Computer Vision Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 432–441.

Beverley, K. I.

D. Regan, K. I. Beverley, “How do we avoid confounding the direction we are looking and the direction we are moving?” Science 215, 194–196 (1982).
[CrossRef] [PubMed]

Cumming, R. W.

I. R. Johnston, G. R. White, R. W. Cumming, “The role of optical expansion patterns in locomotor control,” Am. J. Psychol. 86, 311–324 (1973).
[CrossRef] [PubMed]

Cutting, J. E.

J. E. Cutting, “Perceiving and recovering structure from events,” in Proceedings of the ACM SIGGRAPH/SIGART Conference on Motion: Representation and Perception, Toronto, Canada, 1983 (Association for Computing Machinery, New York, 1983), pp. 141–147.

Dreschler, L.

L. Dreschler, H.-H. Nagel, “Volumetric model and 3D-trajectory of a moving car derived from monocular TV-frame sequences of a street scene,” Comput. Graphics Image Process. 20, 199–228 (1982).
[CrossRef]

Gennery, D. B.

D. B. Gennery, “Modelling the environment of an exploring vehicle by means of stereo vision,” Memo AIM-339, Stanford Artificial Intelligence Laboratory, Stanford, Calif. (1980).

Gibson, J. J.

J. J. Gibson, The Perception of the Visual World (Houghton Mifflin, Boston, 1950).

Glazer, F.

F. Glazer, G. Reynolds, P. Anandan, “Scene matching by hierarchical correlation,” in Proceedings of IEEE Conference on Computer Vision Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 432–441.

Huang, T. S.

R. Y. Tsai, T. S. Huang, “Uniqueness and estimation of three-dimensional motion parameters of rigid objects with curved surfaces,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 13–27 (1984).
[CrossRef]

Johnston, I. R.

I. R. Johnston, G. R. White, R. W. Cumming, “The role of optical expansion patterns in locomotor control,” Am. J. Psychol. 86, 311–324 (1973).
[CrossRef] [PubMed]

Koenderink, J. J.

A. J. van Doorn, J. J. Koenderink, “Detectability of velocity gradients in moving random-dot patterns,” Vision Res. 23, 799–804 (1983).
[CrossRef] [PubMed]

A. J. van Doorn, J. J. Koenderink, “Temporal properties of the visual detectability of moving spatial white noise,” Exp. Brain Res. 45, 179–188 (1982).
[PubMed]

A. J. van Doorn, J. J. Koenderink, “Visibility of movement gradients,” Biol. Cybernet. 44, 167–175 (1982).
[CrossRef]

J. J. Koenderink, A. J. van Doorn, “Invariant properties of the motion parallax field due to movement of rigid bodies relative to an observer,” Opt. Acta 22, 773–791 (1975).
[CrossRef]

Lawton, D. T.

D. T. Lawton, “Processing translational image sequences,” Comput. Vision Graphics Image Process. 22, 116–144 (1983).
[CrossRef]

J. H. Rieger, D. T. Lawton, in Proceedings of the ACM SIGGRAPH/SIGART Conference on Motion: Representation and Perception, Toronto, Canada, 1983 (Association for Computing Machinery, New York, 1983), pp. 33–41.

Lee, D. N.

D. N. Lee, P. E. Reddish, “Plummeting gannets: a paradigm of ecological optics,” Nature 293, 293–294 (1981).
[CrossRef]

D. N. Lee, “The optical flow field: the foundation of vision,” Phil. Trans. R. Soc. London Ser. B 290, 169–179 (1980).
[CrossRef]

Llewellyn, K. R.

K. R. Llewellyn, “Visual guidance of locomotion,”J. Exp. Psychol. 91, 245–261 (1971).
[CrossRef] [PubMed]

Longuet-Higgins, H. C.

H. C. Longuet-Higgins, “A computer algorithm for reconstructing a scene from two projections,” Nature 293, 133–135 (1981).
[CrossRef]

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

McKee, S. P.

S. P. McKee, “A local mechanism for differential velocity detection,” Vision Res. 21, 491–500 (1981).
[CrossRef] [PubMed]

Nagel, H.-H.

L. Dreschler, H.-H. Nagel, “Volumetric model and 3D-trajectory of a moving car derived from monocular TV-frame sequences of a street scene,” Comput. Graphics Image Process. 20, 199–228 (1982).
[CrossRef]

Nakayama, K.

K. Nakayama, “Differential motion hyperacuity under conditions of common image motion,” Vision Res. 21, 1475–1482 (1981).
[CrossRef] [PubMed]

Prazdny, K.

K. Prazdny, “Determining the instantaneous direction of motion from optical flow generated by a curvilinearly moving observer,” Comput. Graphics Image Process. 17, 238–248 (1981).
[CrossRef]

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

Reddish, P. E.

D. N. Lee, P. E. Reddish, “Plummeting gannets: a paradigm of ecological optics,” Nature 293, 293–294 (1981).
[CrossRef]

Regan, D.

D. Regan, K. I. Beverley, “How do we avoid confounding the direction we are looking and the direction we are moving?” Science 215, 194–196 (1982).
[CrossRef] [PubMed]

Reynolds, G.

F. Glazer, G. Reynolds, P. Anandan, “Scene matching by hierarchical correlation,” in Proceedings of IEEE Conference on Computer Vision Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 432–441.

Rieger, J. H.

J. H. Rieger, “Information in optical flows induced by curved paths of observation,”J. Opt. Soc. Am. 73, 339–344 (1983).
[CrossRef] [PubMed]

J. H. Rieger, D. T. Lawton, in Proceedings of the ACM SIGGRAPH/SIGART Conference on Motion: Representation and Perception, Toronto, Canada, 1983 (Association for Computing Machinery, New York, 1983), pp. 33–41.

Tsai, R. Y.

R. Y. Tsai, T. S. Huang, “Uniqueness and estimation of three-dimensional motion parameters of rigid objects with curved surfaces,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 13–27 (1984).
[CrossRef]

Ullman, S.

S. Ullman, “The interpretation of structure from motion,” Proc. R. Soc. London Ser. B 203, 405–426 (1979).
[CrossRef]

van Doorn, A. J.

A. J. van Doorn, J. J. Koenderink, “Detectability of velocity gradients in moving random-dot patterns,” Vision Res. 23, 799–804 (1983).
[CrossRef] [PubMed]

A. J. van Doorn, J. J. Koenderink, “Visibility of movement gradients,” Biol. Cybernet. 44, 167–175 (1982).
[CrossRef]

A. J. van Doorn, J. J. Koenderink, “Temporal properties of the visual detectability of moving spatial white noise,” Exp. Brain Res. 45, 179–188 (1982).
[PubMed]

J. J. Koenderink, A. J. van Doorn, “Invariant properties of the motion parallax field due to movement of rigid bodies relative to an observer,” Opt. Acta 22, 773–791 (1975).
[CrossRef]

von Helmholtz, H.

H. von Helmholtz, Handbuch der Physiologischen Optik (Leopold Voss, Hamburg, 1896).

Wagner, H.

H. Wagner, “Flow-field variables trigger landing in flies,” Nature 297, 147–148 (1982).
[CrossRef]

Warren, R.

R. Warren, “The perception of egomotion,”J. Exp. Psychol.: Human Percept. Perf. 2, 448–456 (1976).
[CrossRef]

White, G. R.

I. R. Johnston, G. R. White, R. W. Cumming, “The role of optical expansion patterns in locomotor control,” Am. J. Psychol. 86, 311–324 (1973).
[CrossRef] [PubMed]

Am. J. Psychol. (1)

I. R. Johnston, G. R. White, R. W. Cumming, “The role of optical expansion patterns in locomotor control,” Am. J. Psychol. 86, 311–324 (1973).
[CrossRef] [PubMed]

Biol. Cybernet. (1)

A. J. van Doorn, J. J. Koenderink, “Visibility of movement gradients,” Biol. Cybernet. 44, 167–175 (1982).
[CrossRef]

Comput. Graphics Image Process. (2)

K. Prazdny, “Determining the instantaneous direction of motion from optical flow generated by a curvilinearly moving observer,” Comput. Graphics Image Process. 17, 238–248 (1981).
[CrossRef]

L. Dreschler, H.-H. Nagel, “Volumetric model and 3D-trajectory of a moving car derived from monocular TV-frame sequences of a street scene,” Comput. Graphics Image Process. 20, 199–228 (1982).
[CrossRef]

Comput. Vision Graphics Image Process. (1)

D. T. Lawton, “Processing translational image sequences,” Comput. Vision Graphics Image Process. 22, 116–144 (1983).
[CrossRef]

Exp. Brain Res. (1)

A. J. van Doorn, J. J. Koenderink, “Temporal properties of the visual detectability of moving spatial white noise,” Exp. Brain Res. 45, 179–188 (1982).
[PubMed]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

R. Y. Tsai, T. S. Huang, “Uniqueness and estimation of three-dimensional motion parameters of rigid objects with curved surfaces,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 13–27 (1984).
[CrossRef]

J. Exp. Psychol. (1)

K. R. Llewellyn, “Visual guidance of locomotion,”J. Exp. Psychol. 91, 245–261 (1971).
[CrossRef] [PubMed]

J. Exp. Psychol.: Human Percept. Perf. (1)

R. Warren, “The perception of egomotion,”J. Exp. Psychol.: Human Percept. Perf. 2, 448–456 (1976).
[CrossRef]

J. Opt. Soc. Am. (1)

Nature (3)

D. N. Lee, P. E. Reddish, “Plummeting gannets: a paradigm of ecological optics,” Nature 293, 293–294 (1981).
[CrossRef]

H. Wagner, “Flow-field variables trigger landing in flies,” Nature 297, 147–148 (1982).
[CrossRef]

H. C. Longuet-Higgins, “A computer algorithm for reconstructing a scene from two projections,” Nature 293, 133–135 (1981).
[CrossRef]

Opt. Acta (1)

J. J. Koenderink, A. J. van Doorn, “Invariant properties of the motion parallax field due to movement of rigid bodies relative to an observer,” Opt. Acta 22, 773–791 (1975).
[CrossRef]

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

D. N. Lee, “The optical flow field: the foundation of vision,” Phil. Trans. R. Soc. London Ser. B 290, 169–179 (1980).
[CrossRef]

Proc. R. Soc. London Ser. B (2)

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

S. Ullman, “The interpretation of structure from motion,” Proc. R. Soc. London Ser. B 203, 405–426 (1979).
[CrossRef]

Science (1)

D. Regan, K. I. Beverley, “How do we avoid confounding the direction we are looking and the direction we are moving?” Science 215, 194–196 (1982).
[CrossRef] [PubMed]

Vision Res. (3)

A. J. van Doorn, J. J. Koenderink, “Detectability of velocity gradients in moving random-dot patterns,” Vision Res. 23, 799–804 (1983).
[CrossRef] [PubMed]

K. Nakayama, “Differential motion hyperacuity under conditions of common image motion,” Vision Res. 21, 1475–1482 (1981).
[CrossRef] [PubMed]

S. P. McKee, “A local mechanism for differential velocity detection,” Vision Res. 21, 491–500 (1981).
[CrossRef] [PubMed]

Other (7)

J. E. Cutting, “Perceiving and recovering structure from events,” in Proceedings of the ACM SIGGRAPH/SIGART Conference on Motion: Representation and Perception, Toronto, Canada, 1983 (Association for Computing Machinery, New York, 1983), pp. 141–147.

H. von Helmholtz, Handbuch der Physiologischen Optik (Leopold Voss, Hamburg, 1896).

J. J. Gibson, The Perception of the Visual World (Houghton Mifflin, Boston, 1950).

P. Anandan, “A confidence measure for correlation matching,”COINS Tech. Rep., Department of Computer and Information Science, University of Massachusetts, Amherst, Mass. (1984).

F. Glazer, G. Reynolds, P. Anandan, “Scene matching by hierarchical correlation,” in Proceedings of IEEE Conference on Computer Vision Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 432–441.

J. H. Rieger, D. T. Lawton, in Proceedings of the ACM SIGGRAPH/SIGART Conference on Motion: Representation and Perception, Toronto, Canada, 1983 (Association for Computing Machinery, New York, 1983), pp. 33–41.

D. B. Gennery, “Modelling the environment of an exploring vehicle by means of stereo vision,” Memo AIM-339, Stanford Artificial Intelligence Laboratory, Stanford, Calif. (1980).

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Figures (4)

Fig. 1
Fig. 1

Camera model employed in the text. The image plane is at a unit distance along the z axis, which sets the focal length to unity. O is the nodal point of the camera’s lens, Pi and Pj are visible details in the scene.

Fig. 2
Fig. 2

Effects of depth variation. The family of curves corresponds to window sizes of ≈ 1.5, 3.0, 4.5, and 6.0 deg. The flow field was induced by an observer motion composed of a translation of 1 unit along (−1, 1, 1) and a rotation of 6 deg about (0, 1, 0). The nearer surface was at a depth of z1 = 10 units.

Fig. 3
Fig. 3

Effects of errors in the flow field. The parameters were the same as those in the previous experiment (Fig. 2) except for a constant depth variation of Δz/z1 = 2.0. The summed magnitudes of the added random components are expressed as percentages of the summed magnitudes of the correct flow vectors.

Fig. 4
Fig. 4

(a)–(h) The main steps in processing an image sequence. Note that the flow fields shown above are of reduced density and are normalized by a common factor.

Tables (2)

Tables Icon

Table 1 Effects of Depth Variation on the Distribution of Difference Vectors i = 1 N λ i Small / λ i Large N

Tables Icon

Table 2 Effects of Depth Variation on the Error of li i = 1 N [ ( r ˜ i - r ˜ T ) · e i Small / r ˜ i - r ˜ T ] N

Equations (9)

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

u = ( x ˙ - x ˜ z ˙ , y ˙ - y ˜ z ˙ ) / z .
r ˙ = - v - ω × r .
u T = ( x ˜ v z - v x , y ˜ v z - v y ) / z
u R = [ x ˜ y ˜ ω x - ( x ˜ 2 + 1 ) ω y + y ˜ ω z , ( y ˜ 2 + 1 ) ω x - x ˜ y ˜ ω y - x ˜ ω z ] .
Δ i j u R = u j R - u i R = ω z ( d y , - d x ) + ( y ˜ j ω x - x ˜ j ω y ) ( d x , d y ) + ( d y ω x - d x ω y ) r ˜ i .
Δ i j u T = u j T - u i T = v z z j [ r ˜ j - r ˜ i + Δ i j z z i ( r ˜ T - r ˜ i ) ] ,
Δ i j u = [ v z Δ i j z z i z j ( r ˜ T - r ˜ i ) ] Signal + [ v z z j ( r ˜ j - r ˜ i ) + Δ i j u R ] Noise .
i = 1 N λ i Small / λ i Large N
i = 1 N [ ( r ˜ i - r ˜ T ) · e i Small / r ˜ i - r ˜ T ] N

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