Abstract

The geometry of binocular projection is analyzed in relation to the primate visual system. An oculomotor parameterization that includes the classical vergence and version angles is defined. It is shown that the epipolar geometry of the system is constrained by binocular coordination of the eyes. A local model of the scene is adopted in which depth is measured relative to a plane containing the fixation point. These constructions lead to an explicit parameterization of the binocular disparity field involving the gaze angles as well as the scene structure. The representation of visual direction and depth is discussed with reference to the relevant psychophysical and neurophysiological literature.

© 2008 Optical Society of America

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2006 (1)

A. J. Noest, R. Van Ee, and A. V. van den Berg, “Direct extraction of curvature-based metric shape from stereo by view-modulated receptive fields,” Biol. Cybern. 95, 455-486 (2006).
[CrossRef] [PubMed]

2004 (1)

J. C. A. Read and B. G. Cumming, “Understanding the cortical specialization for horizontal disparity,” Neural Comput. 16, 1983-2020 (2004).
[CrossRef] [PubMed]

2002 (2)

A. Glennerster, S. P. McKee, and M. D. Birch, “Evidence for surface-based processing of binocular disparity,” Curr. Biol. 12, 825-828 (2002).
[CrossRef] [PubMed]

S. J. D. Prince, B. G. Cumming, and A. J. Parker, “Range and mechanism of encoding of horizontal disparity in macaque V1,” J. Neurophysiol. 87, 209-221 (2002).
[PubMed]

2001 (3)

J. J. Koenderink, A. J. van Doorn, A. M. L. Kappers, and J. T. Todd, “Ambiguity and the 'mental eye' in pictorial relief,” Perception 30, 431-448 (2001).
[CrossRef] [PubMed]

E. M. Berends and C. J. Erkelens, “Strength of depth effects induced by three types of vertical disparity,” Vision Res. 41, 37-45 (2001).
[CrossRef] [PubMed]

B. G. Cumming and G. C. DeAngelis, “The physiology of stereopsis,” Annu. Rev. Neurosci. 24, 203-238 (2001).
[CrossRef] [PubMed]

1999 (3)

A. Anzai, I. Ohzawa, and R. D. Freeman, “Neural mechanisms for encoding binocular disparity: Receptive field position versus phase,” J. Neurophysiol. 82, 874-890 (1999).
[PubMed]

B. T. Backus, M. S. Banks, R. van Ee, and J. A. Crowell, “Horizontal and vertical disparity, eye position, and stereoscopic slant perception,” Vision Res. 39, 1143-1170 (1999).
[CrossRef] [PubMed]

B. G. Cumming and A. J. Parker, “Binocular neurons in V1 of awake monkeys are selective for absolute, not relative disparity,” J. Neurosci. 19, 5602-5618 (1999).
[PubMed]

1998 (3)

E. Brenner and W. J. M. van Damme, “Judging distance from ocular convergence,” Vision Res. 38, 493-498 (1998).
[CrossRef] [PubMed]

C. J. Erkelens and R. van Ee, “A computational model of depth perception based on headcentric disparity,” Vision Res. 38, 2999-3018 (1998).
[CrossRef] [PubMed]

M. J. Brooks, L. de Agapito, D. Q. Huynh, and L. Baumela, “Towards robust metric reconstruction via a dynamic uncalibrated stereo head,” Image and Vision Computing 16, 989-1002 (1998).
[CrossRef]

1997 (3)

H. Collewijn, C. J. Erkelens, and R. M. Steinman, “Trajectories of the human binocular fixation point during conjugate and non-conjugate gaze-shifts,” Vision Res. 37, 1049-1069 (1997).
[CrossRef] [PubMed]

W. Zhou and W. M. King, “Binocular eye movements not coordinated during REM sleep,” Exp. Brain Res. 117, 153-160 (1997).
[CrossRef]

D. Tweed, “Visual-motor optimization in binocular control,” Vision Res. 37, 1939-1951 (1997).
[CrossRef] [PubMed]

1996 (3)

Q.-T. Luong and T. Viéville, “Canonical representations for the geometries of multiple projective views,” Comput. Vis. Image Underst. 64, 193-229 (1996).
[CrossRef]

D. J. Fleet, H. Wagner, and D. J. Heeger, “Neural encoding of binocular disparity: Energy models, position shifts and phase shifts,” Vision Res. 36, 1839-1857 (1996).
[CrossRef] [PubMed]

A. Shashua and N. Navab, “Relative affine structure: Canonical model for 3-D from 2-D geometry and applications,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 873-883 (1996).
[CrossRef]

1995 (3)

J. Gårding, J. Porrill, J. E. W. Mayhew, and J. P. Frisby, “Stereopsis, vertical disparity and relief transformations,” Vision Res. 35, 703-722 (1995).
[CrossRef] [PubMed]

H. Ono and A. P. Mapp, “A re-statement and modification of Wells-Hering's laws of visual direction,” Perception 24, 237-252 (1995).
[CrossRef] [PubMed]

O. Faugeras, “Stratification of three-dimensional vision: projective, affine, and metric representations,” J. Opt. Soc. Am. A 12, 465-484 (1995).
[CrossRef]

1994 (1)

K. Hepp, “Oculomotor control: Listing's law and all that,” Curr. Opin. Neurobiol. 4, 862-868 (1994).
[CrossRef] [PubMed]

1993 (2)

B. J. Rogers and M. F. Bradshaw, “Vertical disparities, differential perspective and binocular stereopsis,” Nature (London) 361, 253-255 (1993).
[CrossRef]

L. J. Van Rijn and A. V. Van den Berg, “Binocular eye orientation during fixations: Listing's law extended to include eye vergence,” Vision Res. 33, 691-708 (1993).
[CrossRef] [PubMed]

1992 (2)

D. Mok, A. Ro, W. Cadera, J. D. Crawford, and T. Vilis, “Rotation of Listing's plane during vergence,” Vision Res. 32, 2055-2064 (1992).
[CrossRef] [PubMed]

J. J. Koenderink and A. J. van Doorn, “Second-order optic flow,” J. Opt. Soc. Am. A 9, 530-538 (1992).
[CrossRef]

1991 (1)

1990 (1)

D. Weinshall, “Qualitative depth from stereo, with applications,” Comput. Vis. Graph. Image Process. 49, 222-241 (1990).
[CrossRef]

1989 (1)

B. Rogers and R. Cagenello, “Disparity curvature and the perception of three-dimensional surfaces,” Nature (London) 339, 135-137 (1989).
[CrossRef]

1988 (1)

B. Gillam, D. Chambers, and B. Lawergren, “The role of vertical disparity in the scaling of stereoscopic depth perception: An empirical and theoretical study,” Percept. Psychophys. 44, 473-484 (1988).
[CrossRef] [PubMed]

1985 (1)

G. J. Mitchison and S. P. McKee, “Interpolation in stereoscopic matching,” Nature (London) 315, 402-404 (1985).
[CrossRef]

1984 (1)

L. E. Mays, “Neural control of vergence eye movements: Convergence and divergence neurons in midbrain,” J. Neurophysiol. 51, 1091-1108 (1984).
[PubMed]

1982 (2)

J. E. W. Mayhew, “The interpretation of stereo-disparity information: The computation of surface orientation and depth,” Perception 11, 387-403 (1982).
[CrossRef] [PubMed]

J. E. W. Mayhew and H. C. Longuet-Higgins, “A computational model of binocular depth perception,” Nature (London) 297, 376-379 (1982).
[CrossRef]

1981 (1)

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

1980 (1)

J. M. Foley, “Binocular distance perception,” Psychol. Rev. 87, 411-434 (1980).
[CrossRef] [PubMed]

1979 (3)

J. D. Pettigrew, “Binocular visual processing in the owl's telencephalon,” Proc. R. Soc. London, Ser. B 204, 435-454 (1979).
[CrossRef]

M. L. Cooper and J. D. Pettigrew, “A neurophysiological determination of the vertical horopter in the cat and owl,” J. Comp. Neurol. 184, 1-25 (1979).
[CrossRef] [PubMed]

G. Westheimer, “Cooperative neural processes involved in stereoscopic acuity,” Exp. Brain Res. 36, 585-597 (1979).
[CrossRef] [PubMed]

1977 (1)

G. F. Poggio and B. Fischer, “Binocular interaction and depth sensitivity in striate and prestriate cortex of behaving rhesus monkeys,” J. Neurophysiol. 40, 1392-1407 (1977).
[PubMed]

1976 (2)

D. Marr and T. Poggio, “Cooperative computation of stereo disparity,” Science 194, 283-287 (1976).
[CrossRef] [PubMed]

J. J. Koenderink and A. J. van Doorn, “Geometry of binocular vision and a model for stereopsis,” Biol. Cybern. 21, 29-35 (1976).
[CrossRef] [PubMed]

1972 (1)

B. Julesz, “Cyclopean perception and neurophysiology,” Invest. Ophthalmol. 11, 540-548 (1972).
[PubMed]

1970 (1)

C. Blakemore, “The range and scope of binocular depth discrimination in man,” J. Physiol. (London) 211, 599-622 (1970).

1962 (1)

G. L. Walls, “The evolutionary history of eye movements,” Vision Res. 2, 69-80 (1962).
[CrossRef]

1961 (1)

C. Rashbass and G. Westheimer, “Disjunctive eye movements,” J. Physiol. (London) 159, 339-360 (1961).

Anandan, P.

J. R. Bergen, P. Anandan, K. J. Hanna, and R. Hingorani, “Hierarchical model-based motion estimation,” in Proceedings of the European Conference on Computer Vision (Springer-Verlag, 1992), pp. 237-252.

Anzai, A.

A. Anzai, I. Ohzawa, and R. D. Freeman, “Neural mechanisms for encoding binocular disparity: Receptive field position versus phase,” J. Neurophysiol. 82, 874-890 (1999).
[PubMed]

Armstrong, M.

M. Armstrong, A. Zisserman, and R. I. Hartley, “Self-calibration from image triplets,” in Proceedings of the European Conference on Computer Vision (Springer-Verlag, 1996), Vol. 1, pp. 3-16.

Backus, B. T.

B. T. Backus, M. S. Banks, R. van Ee, and J. A. Crowell, “Horizontal and vertical disparity, eye position, and stereoscopic slant perception,” Vision Res. 39, 1143-1170 (1999).
[CrossRef] [PubMed]

Banks, M. S.

B. T. Backus, M. S. Banks, R. van Ee, and J. A. Crowell, “Horizontal and vertical disparity, eye position, and stereoscopic slant perception,” Vision Res. 39, 1143-1170 (1999).
[CrossRef] [PubMed]

Baumela, L.

M. J. Brooks, L. de Agapito, D. Q. Huynh, and L. Baumela, “Towards robust metric reconstruction via a dynamic uncalibrated stereo head,” Image and Vision Computing 16, 989-1002 (1998).
[CrossRef]

Berends, E. M.

E. M. Berends and C. J. Erkelens, “Strength of depth effects induced by three types of vertical disparity,” Vision Res. 41, 37-45 (2001).
[CrossRef] [PubMed]

Bergen, J. R.

J. R. Bergen, P. Anandan, K. J. Hanna, and R. Hingorani, “Hierarchical model-based motion estimation,” in Proceedings of the European Conference on Computer Vision (Springer-Verlag, 1992), pp. 237-252.

Birch, M. D.

A. Glennerster, S. P. McKee, and M. D. Birch, “Evidence for surface-based processing of binocular disparity,” Curr. Biol. 12, 825-828 (2002).
[CrossRef] [PubMed]

Blakemore, C.

C. Blakemore, “The range and scope of binocular depth discrimination in man,” J. Physiol. (London) 211, 599-622 (1970).

Bradshaw, M. F.

B. J. Rogers and M. F. Bradshaw, “Vertical disparities, differential perspective and binocular stereopsis,” Nature (London) 361, 253-255 (1993).
[CrossRef]

Brenner, E.

E. Brenner and W. J. M. van Damme, “Judging distance from ocular convergence,” Vision Res. 38, 493-498 (1998).
[CrossRef] [PubMed]

Brooks, M. J.

M. J. Brooks, L. de Agapito, D. Q. Huynh, and L. Baumela, “Towards robust metric reconstruction via a dynamic uncalibrated stereo head,” Image and Vision Computing 16, 989-1002 (1998).
[CrossRef]

Cadera, W.

D. Mok, A. Ro, W. Cadera, J. D. Crawford, and T. Vilis, “Rotation of Listing's plane during vergence,” Vision Res. 32, 2055-2064 (1992).
[CrossRef] [PubMed]

Cagenello, R.

B. Rogers and R. Cagenello, “Disparity curvature and the perception of three-dimensional surfaces,” Nature (London) 339, 135-137 (1989).
[CrossRef]

Carpenter, R. H. S.

R. H. S. Carpenter, Movements of the Eyes (Pion, 1988).

Chambers, D.

B. Gillam, D. Chambers, and B. Lawergren, “The role of vertical disparity in the scaling of stereoscopic depth perception: An empirical and theoretical study,” Percept. Psychophys. 44, 473-484 (1988).
[CrossRef] [PubMed]

Collewijn, H.

H. Collewijn, C. J. Erkelens, and R. M. Steinman, “Trajectories of the human binocular fixation point during conjugate and non-conjugate gaze-shifts,” Vision Res. 37, 1049-1069 (1997).
[CrossRef] [PubMed]

Cooper, M. L.

M. L. Cooper and J. D. Pettigrew, “A neurophysiological determination of the vertical horopter in the cat and owl,” J. Comp. Neurol. 184, 1-25 (1979).
[CrossRef] [PubMed]

Crawford, J. D.

D. Mok, A. Ro, W. Cadera, J. D. Crawford, and T. Vilis, “Rotation of Listing's plane during vergence,” Vision Res. 32, 2055-2064 (1992).
[CrossRef] [PubMed]

Crowell, J. A.

B. T. Backus, M. S. Banks, R. van Ee, and J. A. Crowell, “Horizontal and vertical disparity, eye position, and stereoscopic slant perception,” Vision Res. 39, 1143-1170 (1999).
[CrossRef] [PubMed]

Cumming, B. G.

J. C. A. Read and B. G. Cumming, “Understanding the cortical specialization for horizontal disparity,” Neural Comput. 16, 1983-2020 (2004).
[CrossRef] [PubMed]

S. J. D. Prince, B. G. Cumming, and A. J. Parker, “Range and mechanism of encoding of horizontal disparity in macaque V1,” J. Neurophysiol. 87, 209-221 (2002).
[PubMed]

B. G. Cumming and G. C. DeAngelis, “The physiology of stereopsis,” Annu. Rev. Neurosci. 24, 203-238 (2001).
[CrossRef] [PubMed]

B. G. Cumming and A. J. Parker, “Binocular neurons in V1 of awake monkeys are selective for absolute, not relative disparity,” J. Neurosci. 19, 5602-5618 (1999).
[PubMed]

de Agapito, L.

M. J. Brooks, L. de Agapito, D. Q. Huynh, and L. Baumela, “Towards robust metric reconstruction via a dynamic uncalibrated stereo head,” Image and Vision Computing 16, 989-1002 (1998).
[CrossRef]

DeAngelis, G. C.

B. G. Cumming and G. C. DeAngelis, “The physiology of stereopsis,” Annu. Rev. Neurosci. 24, 203-238 (2001).
[CrossRef] [PubMed]

Erkelens, C. J.

E. M. Berends and C. J. Erkelens, “Strength of depth effects induced by three types of vertical disparity,” Vision Res. 41, 37-45 (2001).
[CrossRef] [PubMed]

C. J. Erkelens and R. van Ee, “A computational model of depth perception based on headcentric disparity,” Vision Res. 38, 2999-3018 (1998).
[CrossRef] [PubMed]

H. Collewijn, C. J. Erkelens, and R. M. Steinman, “Trajectories of the human binocular fixation point during conjugate and non-conjugate gaze-shifts,” Vision Res. 37, 1049-1069 (1997).
[CrossRef] [PubMed]

Faugeras, O.

Fischer, B.

G. F. Poggio and B. Fischer, “Binocular interaction and depth sensitivity in striate and prestriate cortex of behaving rhesus monkeys,” J. Neurophysiol. 40, 1392-1407 (1977).
[PubMed]

Fleet, D. J.

D. J. Fleet, H. Wagner, and D. J. Heeger, “Neural encoding of binocular disparity: Energy models, position shifts and phase shifts,” Vision Res. 36, 1839-1857 (1996).
[CrossRef] [PubMed]

Foley, J. M.

J. M. Foley, “Binocular distance perception,” Psychol. Rev. 87, 411-434 (1980).
[CrossRef] [PubMed]

Freeman, R. D.

A. Anzai, I. Ohzawa, and R. D. Freeman, “Neural mechanisms for encoding binocular disparity: Receptive field position versus phase,” J. Neurophysiol. 82, 874-890 (1999).
[PubMed]

Frisby, J. P.

J. Gårding, J. Porrill, J. E. W. Mayhew, and J. P. Frisby, “Stereopsis, vertical disparity and relief transformations,” Vision Res. 35, 703-722 (1995).
[CrossRef] [PubMed]

Gårding, J.

J. Gårding, J. Porrill, J. E. W. Mayhew, and J. P. Frisby, “Stereopsis, vertical disparity and relief transformations,” Vision Res. 35, 703-722 (1995).
[CrossRef] [PubMed]

Gillam, B.

B. Gillam, D. Chambers, and B. Lawergren, “The role of vertical disparity in the scaling of stereoscopic depth perception: An empirical and theoretical study,” Percept. Psychophys. 44, 473-484 (1988).
[CrossRef] [PubMed]

Glennerster, A.

A. Glennerster, S. P. McKee, and M. D. Birch, “Evidence for surface-based processing of binocular disparity,” Curr. Biol. 12, 825-828 (2002).
[CrossRef] [PubMed]

Hanna, K. J.

J. R. Bergen, P. Anandan, K. J. Hanna, and R. Hingorani, “Hierarchical model-based motion estimation,” in Proceedings of the European Conference on Computer Vision (Springer-Verlag, 1992), pp. 237-252.

Hartley, R. I.

R. I. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge U. Press, 2000).

M. Armstrong, A. Zisserman, and R. I. Hartley, “Self-calibration from image triplets,” in Proceedings of the European Conference on Computer Vision (Springer-Verlag, 1996), Vol. 1, pp. 3-16.

Heeger, D. J.

D. J. Fleet, H. Wagner, and D. J. Heeger, “Neural encoding of binocular disparity: Energy models, position shifts and phase shifts,” Vision Res. 36, 1839-1857 (1996).
[CrossRef] [PubMed]

Hepp, K.

K. Hepp, “Oculomotor control: Listing's law and all that,” Curr. Opin. Neurobiol. 4, 862-868 (1994).
[CrossRef] [PubMed]

Hering, E.

E. Hering, The Theory of Binocular Vision (Englemann, 1868), edited and translated, B.Bridgeman and L.Stark (Plenum Press, 1977).

Hingorani, R.

J. R. Bergen, P. Anandan, K. J. Hanna, and R. Hingorani, “Hierarchical model-based motion estimation,” in Proceedings of the European Conference on Computer Vision (Springer-Verlag, 1992), pp. 237-252.

Huynh, D. Q.

M. J. Brooks, L. de Agapito, D. Q. Huynh, and L. Baumela, “Towards robust metric reconstruction via a dynamic uncalibrated stereo head,” Image and Vision Computing 16, 989-1002 (1998).
[CrossRef]

Julesz, B.

B. Julesz, “Cyclopean perception and neurophysiology,” Invest. Ophthalmol. 11, 540-548 (1972).
[PubMed]

B. Julesz, Foundations of Cyclopean Perception (U. Chicago Press, 1971).

Kappers, A. M. L.

J. J. Koenderink, A. J. van Doorn, A. M. L. Kappers, and J. T. Todd, “Ambiguity and the 'mental eye' in pictorial relief,” Perception 30, 431-448 (2001).
[CrossRef] [PubMed]

King, W. M.

W. Zhou and W. M. King, “Binocular eye movements not coordinated during REM sleep,” Exp. Brain Res. 117, 153-160 (1997).
[CrossRef]

Koenderink, J. J.

J. J. Koenderink, A. J. van Doorn, A. M. L. Kappers, and J. T. Todd, “Ambiguity and the 'mental eye' in pictorial relief,” Perception 30, 431-448 (2001).
[CrossRef] [PubMed]

J. J. Koenderink and A. J. van Doorn, “Second-order optic flow,” J. Opt. Soc. Am. A 9, 530-538 (1992).
[CrossRef]

J. J. Koenderink and A. J. van Doorn, “Affine structure from motion,” J. Opt. Soc. Am. A 8, 377-385 (1991).
[CrossRef] [PubMed]

J. J. Koenderink and A. J. van Doorn, “Geometry of binocular vision and a model for stereopsis,” Biol. Cybern. 21, 29-35 (1976).
[CrossRef] [PubMed]

Lawergren, B.

B. Gillam, D. Chambers, and B. Lawergren, “The role of vertical disparity in the scaling of stereoscopic depth perception: An empirical and theoretical study,” Percept. Psychophys. 44, 473-484 (1988).
[CrossRef] [PubMed]

Longuet-Higgins, H. C.

J. E. W. Mayhew and H. C. Longuet-Higgins, “A computational model of binocular depth perception,” Nature (London) 297, 376-379 (1982).
[CrossRef]

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

Luong, Q.-T.

Q.-T. Luong and T. Viéville, “Canonical representations for the geometries of multiple projective views,” Comput. Vis. Image Underst. 64, 193-229 (1996).
[CrossRef]

Mapp, A. P.

H. Ono and A. P. Mapp, “A re-statement and modification of Wells-Hering's laws of visual direction,” Perception 24, 237-252 (1995).
[CrossRef] [PubMed]

Marr, D.

D. Marr and T. Poggio, “Cooperative computation of stereo disparity,” Science 194, 283-287 (1976).
[CrossRef] [PubMed]

Maybank, S.

S. Maybank, Theory of Reconstruction from Image Motion (Springer-Verlag, 1993).
[CrossRef]

Maybank, S. J.

S. J. Maybank and P. F. Sturm, “MDL, collineations and the fundamental matrix,” in Proceedings of the 10th British Machine Vision Conference (The British Machine Vision Association and Society for Pattern Recognition, 1999), pp. 53-62.

Mayhew, J. E. W.

J. Gårding, J. Porrill, J. E. W. Mayhew, and J. P. Frisby, “Stereopsis, vertical disparity and relief transformations,” Vision Res. 35, 703-722 (1995).
[CrossRef] [PubMed]

J. E. W. Mayhew and H. C. Longuet-Higgins, “A computational model of binocular depth perception,” Nature (London) 297, 376-379 (1982).
[CrossRef]

J. E. W. Mayhew, “The interpretation of stereo-disparity information: The computation of surface orientation and depth,” Perception 11, 387-403 (1982).
[CrossRef] [PubMed]

Mays, L. E.

L. E. Mays, “Neural control of vergence eye movements: Convergence and divergence neurons in midbrain,” J. Neurophysiol. 51, 1091-1108 (1984).
[PubMed]

McKee, S. P.

A. Glennerster, S. P. McKee, and M. D. Birch, “Evidence for surface-based processing of binocular disparity,” Curr. Biol. 12, 825-828 (2002).
[CrossRef] [PubMed]

G. J. Mitchison and S. P. McKee, “Interpolation in stereoscopic matching,” Nature (London) 315, 402-404 (1985).
[CrossRef]

Mitchison, G. J.

G. J. Mitchison and S. P. McKee, “Interpolation in stereoscopic matching,” Nature (London) 315, 402-404 (1985).
[CrossRef]

Mok, D.

D. Mok, A. Ro, W. Cadera, J. D. Crawford, and T. Vilis, “Rotation of Listing's plane during vergence,” Vision Res. 32, 2055-2064 (1992).
[CrossRef] [PubMed]

Navab, N.

A. Shashua and N. Navab, “Relative affine structure: Canonical model for 3-D from 2-D geometry and applications,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 873-883 (1996).
[CrossRef]

Noest, A. J.

A. J. Noest, R. Van Ee, and A. V. van den Berg, “Direct extraction of curvature-based metric shape from stereo by view-modulated receptive fields,” Biol. Cybern. 95, 455-486 (2006).
[CrossRef] [PubMed]

Ogle, K. N.

K. N. Ogle, Researches in Binocular Vision (W. B. Saunders, 1950).

Ohzawa, I.

A. Anzai, I. Ohzawa, and R. D. Freeman, “Neural mechanisms for encoding binocular disparity: Receptive field position versus phase,” J. Neurophysiol. 82, 874-890 (1999).
[PubMed]

Ono, H.

H. Ono and A. P. Mapp, “A re-statement and modification of Wells-Hering's laws of visual direction,” Perception 24, 237-252 (1995).
[CrossRef] [PubMed]

Parker, A. J.

S. J. D. Prince, B. G. Cumming, and A. J. Parker, “Range and mechanism of encoding of horizontal disparity in macaque V1,” J. Neurophysiol. 87, 209-221 (2002).
[PubMed]

B. G. Cumming and A. J. Parker, “Binocular neurons in V1 of awake monkeys are selective for absolute, not relative disparity,” J. Neurosci. 19, 5602-5618 (1999).
[PubMed]

Pettigrew, J. D.

M. L. Cooper and J. D. Pettigrew, “A neurophysiological determination of the vertical horopter in the cat and owl,” J. Comp. Neurol. 184, 1-25 (1979).
[CrossRef] [PubMed]

J. D. Pettigrew, “Binocular visual processing in the owl's telencephalon,” Proc. R. Soc. London, Ser. B 204, 435-454 (1979).
[CrossRef]

J. D. Pettigrew, “Evolution of binocular vision,” in Visual Neuroscience, J.D.Pettigrew, K.J.Sanderson, and W.R.Levick, eds. 208-22 (Cambridge U. Press, 1986).

Poggio, G. F.

G. F. Poggio and B. Fischer, “Binocular interaction and depth sensitivity in striate and prestriate cortex of behaving rhesus monkeys,” J. Neurophysiol. 40, 1392-1407 (1977).
[PubMed]

Poggio, T.

D. Marr and T. Poggio, “Cooperative computation of stereo disparity,” Science 194, 283-287 (1976).
[CrossRef] [PubMed]

Porrill, J.

J. Gårding, J. Porrill, J. E. W. Mayhew, and J. P. Frisby, “Stereopsis, vertical disparity and relief transformations,” Vision Res. 35, 703-722 (1995).
[CrossRef] [PubMed]

Prince, S. J. D.

S. J. D. Prince, B. G. Cumming, and A. J. Parker, “Range and mechanism of encoding of horizontal disparity in macaque V1,” J. Neurophysiol. 87, 209-221 (2002).
[PubMed]

Rashbass, C.

C. Rashbass and G. Westheimer, “Disjunctive eye movements,” J. Physiol. (London) 159, 339-360 (1961).

Read, J. C. A.

J. C. A. Read and B. G. Cumming, “Understanding the cortical specialization for horizontal disparity,” Neural Comput. 16, 1983-2020 (2004).
[CrossRef] [PubMed]

Ro, A.

D. Mok, A. Ro, W. Cadera, J. D. Crawford, and T. Vilis, “Rotation of Listing's plane during vergence,” Vision Res. 32, 2055-2064 (1992).
[CrossRef] [PubMed]

Rogers, B.

B. Rogers and R. Cagenello, “Disparity curvature and the perception of three-dimensional surfaces,” Nature (London) 339, 135-137 (1989).
[CrossRef]

Rogers, B. J.

B. J. Rogers and M. F. Bradshaw, “Vertical disparities, differential perspective and binocular stereopsis,” Nature (London) 361, 253-255 (1993).
[CrossRef]

Shashua, A.

A. Shashua and N. Navab, “Relative affine structure: Canonical model for 3-D from 2-D geometry and applications,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 873-883 (1996).
[CrossRef]

Steinman, R. M.

H. Collewijn, C. J. Erkelens, and R. M. Steinman, “Trajectories of the human binocular fixation point during conjugate and non-conjugate gaze-shifts,” Vision Res. 37, 1049-1069 (1997).
[CrossRef] [PubMed]

Sturm, P. F.

S. J. Maybank and P. F. Sturm, “MDL, collineations and the fundamental matrix,” in Proceedings of the 10th British Machine Vision Conference (The British Machine Vision Association and Society for Pattern Recognition, 1999), pp. 53-62.

Todd, J. T.

J. J. Koenderink, A. J. van Doorn, A. M. L. Kappers, and J. T. Todd, “Ambiguity and the 'mental eye' in pictorial relief,” Perception 30, 431-448 (2001).
[CrossRef] [PubMed]

Tweed, D.

D. Tweed, “Visual-motor optimization in binocular control,” Vision Res. 37, 1939-1951 (1997).
[CrossRef] [PubMed]

Tyler, C. W.

C. W. Tyler, “The horopter and binocular fusion,” in Vision and Visual Disorders, Vol. 9, Binocular Vision, D.Regan, ed. (MacMillan, 1991), pp. 19-37.

van Damme, W. J. M.

E. Brenner and W. J. M. van Damme, “Judging distance from ocular convergence,” Vision Res. 38, 493-498 (1998).
[CrossRef] [PubMed]

van den Berg, A. V.

A. J. Noest, R. Van Ee, and A. V. van den Berg, “Direct extraction of curvature-based metric shape from stereo by view-modulated receptive fields,” Biol. Cybern. 95, 455-486 (2006).
[CrossRef] [PubMed]

L. J. Van Rijn and A. V. Van den Berg, “Binocular eye orientation during fixations: Listing's law extended to include eye vergence,” Vision Res. 33, 691-708 (1993).
[CrossRef] [PubMed]

van Doorn, A. J.

J. J. Koenderink, A. J. van Doorn, A. M. L. Kappers, and J. T. Todd, “Ambiguity and the 'mental eye' in pictorial relief,” Perception 30, 431-448 (2001).
[CrossRef] [PubMed]

J. J. Koenderink and A. J. van Doorn, “Second-order optic flow,” J. Opt. Soc. Am. A 9, 530-538 (1992).
[CrossRef]

J. J. Koenderink and A. J. van Doorn, “Affine structure from motion,” J. Opt. Soc. Am. A 8, 377-385 (1991).
[CrossRef] [PubMed]

J. J. Koenderink and A. J. van Doorn, “Geometry of binocular vision and a model for stereopsis,” Biol. Cybern. 21, 29-35 (1976).
[CrossRef] [PubMed]

Van Ee, R.

A. J. Noest, R. Van Ee, and A. V. van den Berg, “Direct extraction of curvature-based metric shape from stereo by view-modulated receptive fields,” Biol. Cybern. 95, 455-486 (2006).
[CrossRef] [PubMed]

B. T. Backus, M. S. Banks, R. van Ee, and J. A. Crowell, “Horizontal and vertical disparity, eye position, and stereoscopic slant perception,” Vision Res. 39, 1143-1170 (1999).
[CrossRef] [PubMed]

C. J. Erkelens and R. van Ee, “A computational model of depth perception based on headcentric disparity,” Vision Res. 38, 2999-3018 (1998).
[CrossRef] [PubMed]

Van Rijn, L. J.

L. J. Van Rijn and A. V. Van den Berg, “Binocular eye orientation during fixations: Listing's law extended to include eye vergence,” Vision Res. 33, 691-708 (1993).
[CrossRef] [PubMed]

Viéville, T.

Q.-T. Luong and T. Viéville, “Canonical representations for the geometries of multiple projective views,” Comput. Vis. Image Underst. 64, 193-229 (1996).
[CrossRef]

Vilis, T.

D. Mok, A. Ro, W. Cadera, J. D. Crawford, and T. Vilis, “Rotation of Listing's plane during vergence,” Vision Res. 32, 2055-2064 (1992).
[CrossRef] [PubMed]

von Helmholtz, H. L. F.

H. L. F. von Helmholtz, Treatise on Physiological Optics, Vol. 3, 3rd ed., 1910 (Optical Society of America, 1925), translated by J. P. C. Southall.

Wagner, H.

D. J. Fleet, H. Wagner, and D. J. Heeger, “Neural encoding of binocular disparity: Energy models, position shifts and phase shifts,” Vision Res. 36, 1839-1857 (1996).
[CrossRef] [PubMed]

Walls, G. L.

G. L. Walls, “The evolutionary history of eye movements,” Vision Res. 2, 69-80 (1962).
[CrossRef]

Weinshall, D.

D. Weinshall, “Qualitative depth from stereo, with applications,” Comput. Vis. Graph. Image Process. 49, 222-241 (1990).
[CrossRef]

Westheimer, G.

G. Westheimer, “Cooperative neural processes involved in stereoscopic acuity,” Exp. Brain Res. 36, 585-597 (1979).
[CrossRef] [PubMed]

C. Rashbass and G. Westheimer, “Disjunctive eye movements,” J. Physiol. (London) 159, 339-360 (1961).

Zhou, W.

W. Zhou and W. M. King, “Binocular eye movements not coordinated during REM sleep,” Exp. Brain Res. 117, 153-160 (1997).
[CrossRef]

Zisserman, A.

M. Armstrong, A. Zisserman, and R. I. Hartley, “Self-calibration from image triplets,” in Proceedings of the European Conference on Computer Vision (Springer-Verlag, 1996), Vol. 1, pp. 3-16.

R. I. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge U. Press, 2000).

Annu. Rev. Neurosci. (1)

B. G. Cumming and G. C. DeAngelis, “The physiology of stereopsis,” Annu. Rev. Neurosci. 24, 203-238 (2001).
[CrossRef] [PubMed]

Biol. Cybern. (2)

J. J. Koenderink and A. J. van Doorn, “Geometry of binocular vision and a model for stereopsis,” Biol. Cybern. 21, 29-35 (1976).
[CrossRef] [PubMed]

A. J. Noest, R. Van Ee, and A. V. van den Berg, “Direct extraction of curvature-based metric shape from stereo by view-modulated receptive fields,” Biol. Cybern. 95, 455-486 (2006).
[CrossRef] [PubMed]

Comput. Vis. Graph. Image Process. (1)

D. Weinshall, “Qualitative depth from stereo, with applications,” Comput. Vis. Graph. Image Process. 49, 222-241 (1990).
[CrossRef]

Comput. Vis. Image Underst. (1)

Q.-T. Luong and T. Viéville, “Canonical representations for the geometries of multiple projective views,” Comput. Vis. Image Underst. 64, 193-229 (1996).
[CrossRef]

Curr. Biol. (1)

A. Glennerster, S. P. McKee, and M. D. Birch, “Evidence for surface-based processing of binocular disparity,” Curr. Biol. 12, 825-828 (2002).
[CrossRef] [PubMed]

Curr. Opin. Neurobiol. (1)

K. Hepp, “Oculomotor control: Listing's law and all that,” Curr. Opin. Neurobiol. 4, 862-868 (1994).
[CrossRef] [PubMed]

Exp. Brain Res. (2)

G. Westheimer, “Cooperative neural processes involved in stereoscopic acuity,” Exp. Brain Res. 36, 585-597 (1979).
[CrossRef] [PubMed]

W. Zhou and W. M. King, “Binocular eye movements not coordinated during REM sleep,” Exp. Brain Res. 117, 153-160 (1997).
[CrossRef]

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

A. Shashua and N. Navab, “Relative affine structure: Canonical model for 3-D from 2-D geometry and applications,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 873-883 (1996).
[CrossRef]

Image and Vision Computing (1)

M. J. Brooks, L. de Agapito, D. Q. Huynh, and L. Baumela, “Towards robust metric reconstruction via a dynamic uncalibrated stereo head,” Image and Vision Computing 16, 989-1002 (1998).
[CrossRef]

Invest. Ophthalmol. (1)

B. Julesz, “Cyclopean perception and neurophysiology,” Invest. Ophthalmol. 11, 540-548 (1972).
[PubMed]

J. Comp. Neurol. (1)

M. L. Cooper and J. D. Pettigrew, “A neurophysiological determination of the vertical horopter in the cat and owl,” J. Comp. Neurol. 184, 1-25 (1979).
[CrossRef] [PubMed]

J. Neurophysiol. (4)

L. E. Mays, “Neural control of vergence eye movements: Convergence and divergence neurons in midbrain,” J. Neurophysiol. 51, 1091-1108 (1984).
[PubMed]

A. Anzai, I. Ohzawa, and R. D. Freeman, “Neural mechanisms for encoding binocular disparity: Receptive field position versus phase,” J. Neurophysiol. 82, 874-890 (1999).
[PubMed]

G. F. Poggio and B. Fischer, “Binocular interaction and depth sensitivity in striate and prestriate cortex of behaving rhesus monkeys,” J. Neurophysiol. 40, 1392-1407 (1977).
[PubMed]

S. J. D. Prince, B. G. Cumming, and A. J. Parker, “Range and mechanism of encoding of horizontal disparity in macaque V1,” J. Neurophysiol. 87, 209-221 (2002).
[PubMed]

J. Neurosci. (1)

B. G. Cumming and A. J. Parker, “Binocular neurons in V1 of awake monkeys are selective for absolute, not relative disparity,” J. Neurosci. 19, 5602-5618 (1999).
[PubMed]

J. Opt. Soc. Am. A (3)

J. Physiol. (London) (2)

C. Rashbass and G. Westheimer, “Disjunctive eye movements,” J. Physiol. (London) 159, 339-360 (1961).

C. Blakemore, “The range and scope of binocular depth discrimination in man,” J. Physiol. (London) 211, 599-622 (1970).

Nature (London) (5)

B. Rogers and R. Cagenello, “Disparity curvature and the perception of three-dimensional surfaces,” Nature (London) 339, 135-137 (1989).
[CrossRef]

B. J. Rogers and M. F. Bradshaw, “Vertical disparities, differential perspective and binocular stereopsis,” Nature (London) 361, 253-255 (1993).
[CrossRef]

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

G. J. Mitchison and S. P. McKee, “Interpolation in stereoscopic matching,” Nature (London) 315, 402-404 (1985).
[CrossRef]

J. E. W. Mayhew and H. C. Longuet-Higgins, “A computational model of binocular depth perception,” Nature (London) 297, 376-379 (1982).
[CrossRef]

Neural Comput. (1)

J. C. A. Read and B. G. Cumming, “Understanding the cortical specialization for horizontal disparity,” Neural Comput. 16, 1983-2020 (2004).
[CrossRef] [PubMed]

Percept. Psychophys. (1)

B. Gillam, D. Chambers, and B. Lawergren, “The role of vertical disparity in the scaling of stereoscopic depth perception: An empirical and theoretical study,” Percept. Psychophys. 44, 473-484 (1988).
[CrossRef] [PubMed]

Perception (3)

J. E. W. Mayhew, “The interpretation of stereo-disparity information: The computation of surface orientation and depth,” Perception 11, 387-403 (1982).
[CrossRef] [PubMed]

J. J. Koenderink, A. J. van Doorn, A. M. L. Kappers, and J. T. Todd, “Ambiguity and the 'mental eye' in pictorial relief,” Perception 30, 431-448 (2001).
[CrossRef] [PubMed]

H. Ono and A. P. Mapp, “A re-statement and modification of Wells-Hering's laws of visual direction,” Perception 24, 237-252 (1995).
[CrossRef] [PubMed]

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

J. D. Pettigrew, “Binocular visual processing in the owl's telencephalon,” Proc. R. Soc. London, Ser. B 204, 435-454 (1979).
[CrossRef]

Psychol. Rev. (1)

J. M. Foley, “Binocular distance perception,” Psychol. Rev. 87, 411-434 (1980).
[CrossRef] [PubMed]

Science (1)

D. Marr and T. Poggio, “Cooperative computation of stereo disparity,” Science 194, 283-287 (1976).
[CrossRef] [PubMed]

Vision Res. (11)

H. Collewijn, C. J. Erkelens, and R. M. Steinman, “Trajectories of the human binocular fixation point during conjugate and non-conjugate gaze-shifts,” Vision Res. 37, 1049-1069 (1997).
[CrossRef] [PubMed]

D. Mok, A. Ro, W. Cadera, J. D. Crawford, and T. Vilis, “Rotation of Listing's plane during vergence,” Vision Res. 32, 2055-2064 (1992).
[CrossRef] [PubMed]

L. J. Van Rijn and A. V. Van den Berg, “Binocular eye orientation during fixations: Listing's law extended to include eye vergence,” Vision Res. 33, 691-708 (1993).
[CrossRef] [PubMed]

D. Tweed, “Visual-motor optimization in binocular control,” Vision Res. 37, 1939-1951 (1997).
[CrossRef] [PubMed]

D. J. Fleet, H. Wagner, and D. J. Heeger, “Neural encoding of binocular disparity: Energy models, position shifts and phase shifts,” Vision Res. 36, 1839-1857 (1996).
[CrossRef] [PubMed]

E. Brenner and W. J. M. van Damme, “Judging distance from ocular convergence,” Vision Res. 38, 493-498 (1998).
[CrossRef] [PubMed]

G. L. Walls, “The evolutionary history of eye movements,” Vision Res. 2, 69-80 (1962).
[CrossRef]

J. Gårding, J. Porrill, J. E. W. Mayhew, and J. P. Frisby, “Stereopsis, vertical disparity and relief transformations,” Vision Res. 35, 703-722 (1995).
[CrossRef] [PubMed]

B. T. Backus, M. S. Banks, R. van Ee, and J. A. Crowell, “Horizontal and vertical disparity, eye position, and stereoscopic slant perception,” Vision Res. 39, 1143-1170 (1999).
[CrossRef] [PubMed]

E. M. Berends and C. J. Erkelens, “Strength of depth effects induced by three types of vertical disparity,” Vision Res. 41, 37-45 (2001).
[CrossRef] [PubMed]

C. J. Erkelens and R. van Ee, “A computational model of depth perception based on headcentric disparity,” Vision Res. 38, 2999-3018 (1998).
[CrossRef] [PubMed]

Other (12)

S. Maybank, Theory of Reconstruction from Image Motion (Springer-Verlag, 1993).
[CrossRef]

M. Armstrong, A. Zisserman, and R. I. Hartley, “Self-calibration from image triplets,” in Proceedings of the European Conference on Computer Vision (Springer-Verlag, 1996), Vol. 1, pp. 3-16.

R. H. S. Carpenter, Movements of the Eyes (Pion, 1988).

B. Julesz, Foundations of Cyclopean Perception (U. Chicago Press, 1971).

J. D. Pettigrew, “Evolution of binocular vision,” in Visual Neuroscience, J.D.Pettigrew, K.J.Sanderson, and W.R.Levick, eds. 208-22 (Cambridge U. Press, 1986).

H. L. F. von Helmholtz, Treatise on Physiological Optics, Vol. 3, 3rd ed., 1910 (Optical Society of America, 1925), translated by J. P. C. Southall.

E. Hering, The Theory of Binocular Vision (Englemann, 1868), edited and translated, B.Bridgeman and L.Stark (Plenum Press, 1977).

K. N. Ogle, Researches in Binocular Vision (W. B. Saunders, 1950).

J. R. Bergen, P. Anandan, K. J. Hanna, and R. Hingorani, “Hierarchical model-based motion estimation,” in Proceedings of the European Conference on Computer Vision (Springer-Verlag, 1992), pp. 237-252.

C. W. Tyler, “The horopter and binocular fusion,” in Vision and Visual Disorders, Vol. 9, Binocular Vision, D.Regan, ed. (MacMillan, 1991), pp. 19-37.

R. I. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge U. Press, 2000).

S. J. Maybank and P. F. Sturm, “MDL, collineations and the fundamental matrix,” in Proceedings of the 10th British Machine Vision Conference (The British Machine Vision Association and Society for Pattern Recognition, 1999), pp. 53-62.

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Figures (6)

Fig. 1
Fig. 1

Visual directions. A visual plane V α is defined by the optical centers c ¯ l and c ¯ r together with the fixation point p ¯ 0 . The visual directions v, v l , and v r lie in this plane, which has an elevation angle α. The scene coordinates are located at the cyclopean point c ¯ b = ( 0 , 0 , 0 ) T , such that V 0 coincides with the x, z plane.

Fig. 2
Fig. 2

Binocular coordinates. The fixation point p ¯ 0 in the visual plane V α is shown. The optical centers are indicated by c ¯ l and c ¯ r . The azimuth angles β and β l are positive in this example, whereas β r is negative. The cyclopean range of the fixation point is ρ.

Fig. 3
Fig. 3

Vergence geometry. The Vieth-Müller circle is defined by the positions of the optical centers c ¯ l and c ¯ r together with the fixation point p ¯ 0 . The forward ( z > 0 ) are of the circle intersects the midsagittal plane at the point c ¯ a . The vergence angle δ is inscribed at p ¯ 0 by c ¯ l and c ¯ r . The same angle is inscribed at all other points on the circle, including c ¯ a .

Fig. 4
Fig. 4

Version geometry. The points c ¯ a and p ¯ 0 inscribe the version angle ϵ at an optical center c ¯ that is located on the backward ( z < 0 ) are of the Vieth-Müller circle. The same angle is inscribed at c ¯ l and c ¯ r . It follows that as p ¯ 0 is fixated, c ¯ a lies in the same visual direction from each eye. Furthermore, the triangle defined by c ¯ l , c ¯ r , and c ¯ a is isosceles, so the point c ¯ a is at the same distance from each eye.

Fig. 5
Fig. 5

Construction of the epipolar geometry. Point q l is given, so the epipolar line in I l is u l = q l × e l . This line intersects the image a of the midline horopter in I l at q a = a × u l . The point q a is on a, and is therefore fixed, having the same coordinates q a in I r . It follows that the epipolar line in I r is u r = e r × q a . The location of q r that corresponds to q l is unknown, but it must lie on u r . The Vieth-Müller circle determined by the fixation point p ¯ 0 is shown in the figure, as is the midline horopter, which passes through points c ¯ a and q ¯ a .

Fig. 6
Fig. 6

Geometry of cyclopean parallax. The fixation plane P is defined by the fixation point and is parallel to the cyclopean image plane. Any point p c defines a cyclopean ray that intersects the fixation plane at p ¯ and the scene at q ¯ . The scene point q ¯ has depth s with respect to P . The predicted image projections of q ¯ are at p l and p r . The true projections q l and q r are displaced along the corresponding epipolar lines u l and u r , respectively. The displacement can be parameterized by s as described in the text.

Equations (50)

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b = c ¯ r c ¯ l ,
c ¯ b = 1 2 ( c ¯ l + c ¯ r ) .
c ¯ l = ( 1 2 , 0 , 0 ) T , c ¯ r = ( 1 2 , 0 , 0 ) T .
p l R l ( q ¯ c ¯ l ) ,
p ¯ 0 = ρ v .
v = ( sin β , sin α cos β , cos α cos β ) T ,
ρ = p ¯ 0 ,
v = p ¯ 0 ρ ,
tan β l = tan β + sec β 2 ρ , tan β r = tan β sec β 2 ρ .
R l = [ cos β l 0 sin β l 0 1 0 sin β l 0 cos β l ] .
γ l ( α , β l ) = γ r ( α , β r ) = 0 ,
δ = β l β r ,
ϵ = 1 2 ( β l + β r ) .
ζ = 1 2 cot δ ,
η = 1 2 csc δ .
x 2 + ( z ζ ) 2 = η 2 z 0 .
c ¯ a = ( 0 , 0 , ζ + η ) T ,
q ̃ a = c ¯ a + ( 0 , y , 0 ) T .
c a = 1 2 csc ( δ 2 ) ( sin ϵ , 0 , cos ϵ ) T ,
q a = c a + ( 0 , y , 0 ) T .
a ( cos ϵ , 0 , sin ϵ ) T .
u r e r × [ a × ( e l × q l ) ] .
[ w ] × = [ 0 z y z 0 x y x 0 ]
E [ e r ] × [ a ] × [ e l ] × ,
q r T E q l = 0 ,
e l ( cos β l , 0 , sin β l ) T , e r ( cos β r , 0 , sin β r ) T .
E [ 0 sin β r 0 sin β l 0 cos β l 0 cos β r 0 ] .
q l = p l + t l ( s ) d l , q r = p r + t r ( s ) d r .
P = { p ¯ : v T ( p ¯ p ¯ 0 ) = 0 } .
ρ = v T p ¯ 0 ,
s = v T ( q ¯ p ¯ 0 ) .
z c = v T q ¯ = ρ + s .
p ¯ = ρ R T p c ,
q ¯ = z c R T p c .
ρ l p l = ρ R l R T p c + 1 2 e l ,
z l q l = z c R l R T p c + 1 2 e l ,
λ l = ( R l R T p c ) 3 = x c sin ( β l β ) + cos ( β l β ) ,
μ l = ( 1 2 e l ) 3 = 1 2 sin β l .
d l = 1 κ l ( μ l p l 1 2 e l ) ,
ρ l = λ l ρ + μ l ,
z l = λ l z c + μ l ,
μ l = ρ l z c ρ z l z c ρ .
ρ z l q l = ρ l z c p l ( z c ρ ) 1 2 e l .
ρ z l ( q l p l ) = ( ρ l z c ρ z l ) p l ( z c ρ ) 1 2 e l = ( z c ρ ) ( ρ l z c ρ z l z c ρ p l 1 2 e l ) = ( z c ρ ) ( μ l p l 1 2 e l ) ,
q l p l = z c ρ ρ z l ( μ l p l 1 2 e l ) = κ l ( z c ρ ) ρ z l d l = κ l s ρ z l d l ,
q l = p l + t l ( s ) d l ,
with t l ( s ) = κ l ( s ρ ) λ l ( ρ + s ) + μ l .
p r = H p l
H = R r ( I b v T w l ) R l T .
q r = H p l + t r ( s ) d r .

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