Abstract

Sixty years ago, August Sonnefeld of Zeiss reported on observations with experimental telescopes. The goal of his investigation was to determine the ideal amount of distortion applied to optical instruments that are used in combination with the human eye. His studies were inconclusive and partially contradictory. We have picked up this problem once again, adopting a modern point of view about the human imaging process, and supported by computer graphics. Based on experiments with Helmholtz checkerboards, we argue that human imaging introduces a certain amount of barrel distortion, which has to be counterbalanced through the implementation of an equally strong pincushion distortion into the binocular design. We discuss in detail how this approach is capable of eliminating the globe effect of the panning binocular and how the residual pincushion distortion affects the image when the eye is pointing off-center. Our results support the binocular designer in optimizing his instrument for its intended mode of application, and may help binocular users and astronomers better understand their tools.

© 2009 Optical Society of America

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References

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  1. A. Sonnefeld, “Über die Verzeichnung bei optischen Instrumenten, die in Verbindung mit dem rollenden Auge gebraucht werden,” Deutsche Optische Wochenschrift 13, 97-99 (1949).
  2. A. Whitwell, “On the sine, the tangent and the angle conditions,” The Opt. 48, 149-153 (1914).
  3. H. Tscherning, “Moyens de controle de verres de lunettes et de systèmes optiques en général,” Kgl. Danske Vid. Selsk. Math. Fys. Medd. 9, 3-29 (1918).
  4. E. Weiss, “Analytische Darstellung des Brillenproblems für sphärische Einzellinsen,” Central Ztg. Optik Mechanik 41, 321-325 (1920).
  5. H. Boegehold, “Treue Darstellung und Verzeichnung bei optischen Instrumenten,” Naturwiss. 9, 273-280 (1921).
    [CrossRef]
  6. H. Slevogt, “Zur Definition der Verzeichnung bei optischen Instrumenten für den subjektiven Gebrauch,” Optik (Stuttgart) 1, 358-367 (1946).
  7. H. v. Helmholtz, Handbuch der physiologischen Optik, Vol. 3, v.Kries, ed. (Hamburg und Leibzig, 1910).
  8. A. Barre and A. Flocon, Curvilinear Perspective, R.Hansen, trans. and ed. (Univ. California Press, 1987).
  9. A. H. J. Oomes, J. J. Koenderink, A. J. Doorn, and H. de Ridder, “What are the uncurved lines in our visual field? A fresh look at Helmholtz's checkerboard,” Perception 38, 1284-1294 (2009).
    [CrossRef] [PubMed]
  10. www.holgermerlitz. de/globe/distortion.html.
  11. I. P. Howard and B. J. Rogers, Binocular Vision and Stereopsis (Oxford Univ. Press, 1995).
  12. B. Rogers and K. Brecher, “Straight lines, 'uncurved lines', and Helmholtz's 'great circles on the celestial sphere',” Perception 36, 1275-1289 (2007).
    [CrossRef]
  13. M. Wagner, The Geometries of Visual Space (Erlbaum, 2006).

2009 (1)

A. H. J. Oomes, J. J. Koenderink, A. J. Doorn, and H. de Ridder, “What are the uncurved lines in our visual field? A fresh look at Helmholtz's checkerboard,” Perception 38, 1284-1294 (2009).
[CrossRef] [PubMed]

2007 (1)

B. Rogers and K. Brecher, “Straight lines, 'uncurved lines', and Helmholtz's 'great circles on the celestial sphere',” Perception 36, 1275-1289 (2007).
[CrossRef]

1949 (1)

A. Sonnefeld, “Über die Verzeichnung bei optischen Instrumenten, die in Verbindung mit dem rollenden Auge gebraucht werden,” Deutsche Optische Wochenschrift 13, 97-99 (1949).

1946 (1)

H. Slevogt, “Zur Definition der Verzeichnung bei optischen Instrumenten für den subjektiven Gebrauch,” Optik (Stuttgart) 1, 358-367 (1946).

1921 (1)

H. Boegehold, “Treue Darstellung und Verzeichnung bei optischen Instrumenten,” Naturwiss. 9, 273-280 (1921).
[CrossRef]

1920 (1)

E. Weiss, “Analytische Darstellung des Brillenproblems für sphärische Einzellinsen,” Central Ztg. Optik Mechanik 41, 321-325 (1920).

1918 (1)

H. Tscherning, “Moyens de controle de verres de lunettes et de systèmes optiques en général,” Kgl. Danske Vid. Selsk. Math. Fys. Medd. 9, 3-29 (1918).

1914 (1)

A. Whitwell, “On the sine, the tangent and the angle conditions,” The Opt. 48, 149-153 (1914).

Barre, A.

A. Barre and A. Flocon, Curvilinear Perspective, R.Hansen, trans. and ed. (Univ. California Press, 1987).

Boegehold, H.

H. Boegehold, “Treue Darstellung und Verzeichnung bei optischen Instrumenten,” Naturwiss. 9, 273-280 (1921).
[CrossRef]

Brecher, K.

B. Rogers and K. Brecher, “Straight lines, 'uncurved lines', and Helmholtz's 'great circles on the celestial sphere',” Perception 36, 1275-1289 (2007).
[CrossRef]

de Ridder, H.

A. H. J. Oomes, J. J. Koenderink, A. J. Doorn, and H. de Ridder, “What are the uncurved lines in our visual field? A fresh look at Helmholtz's checkerboard,” Perception 38, 1284-1294 (2009).
[CrossRef] [PubMed]

Doorn, A. J.

A. H. J. Oomes, J. J. Koenderink, A. J. Doorn, and H. de Ridder, “What are the uncurved lines in our visual field? A fresh look at Helmholtz's checkerboard,” Perception 38, 1284-1294 (2009).
[CrossRef] [PubMed]

Flocon, A.

A. Barre and A. Flocon, Curvilinear Perspective, R.Hansen, trans. and ed. (Univ. California Press, 1987).

Helmholtz, H. v.

H. v. Helmholtz, Handbuch der physiologischen Optik, Vol. 3, v.Kries, ed. (Hamburg und Leibzig, 1910).

Howard, I. P.

I. P. Howard and B. J. Rogers, Binocular Vision and Stereopsis (Oxford Univ. Press, 1995).

Koenderink, J. J.

A. H. J. Oomes, J. J. Koenderink, A. J. Doorn, and H. de Ridder, “What are the uncurved lines in our visual field? A fresh look at Helmholtz's checkerboard,” Perception 38, 1284-1294 (2009).
[CrossRef] [PubMed]

Oomes, A. H. J.

A. H. J. Oomes, J. J. Koenderink, A. J. Doorn, and H. de Ridder, “What are the uncurved lines in our visual field? A fresh look at Helmholtz's checkerboard,” Perception 38, 1284-1294 (2009).
[CrossRef] [PubMed]

Rogers, B.

B. Rogers and K. Brecher, “Straight lines, 'uncurved lines', and Helmholtz's 'great circles on the celestial sphere',” Perception 36, 1275-1289 (2007).
[CrossRef]

Rogers, B. J.

I. P. Howard and B. J. Rogers, Binocular Vision and Stereopsis (Oxford Univ. Press, 1995).

Slevogt, H.

H. Slevogt, “Zur Definition der Verzeichnung bei optischen Instrumenten für den subjektiven Gebrauch,” Optik (Stuttgart) 1, 358-367 (1946).

Sonnefeld, A.

A. Sonnefeld, “Über die Verzeichnung bei optischen Instrumenten, die in Verbindung mit dem rollenden Auge gebraucht werden,” Deutsche Optische Wochenschrift 13, 97-99 (1949).

Tscherning, H.

H. Tscherning, “Moyens de controle de verres de lunettes et de systèmes optiques en général,” Kgl. Danske Vid. Selsk. Math. Fys. Medd. 9, 3-29 (1918).

Wagner, M.

M. Wagner, The Geometries of Visual Space (Erlbaum, 2006).

Weiss, E.

E. Weiss, “Analytische Darstellung des Brillenproblems für sphärische Einzellinsen,” Central Ztg. Optik Mechanik 41, 321-325 (1920).

Whitwell, A.

A. Whitwell, “On the sine, the tangent and the angle conditions,” The Opt. 48, 149-153 (1914).

Central Ztg. Optik Mechanik (1)

E. Weiss, “Analytische Darstellung des Brillenproblems für sphärische Einzellinsen,” Central Ztg. Optik Mechanik 41, 321-325 (1920).

Deutsche Optische Wochenschrift (1)

A. Sonnefeld, “Über die Verzeichnung bei optischen Instrumenten, die in Verbindung mit dem rollenden Auge gebraucht werden,” Deutsche Optische Wochenschrift 13, 97-99 (1949).

Kgl. Danske Vid. Selsk. Math. Fys. Medd. (1)

H. Tscherning, “Moyens de controle de verres de lunettes et de systèmes optiques en général,” Kgl. Danske Vid. Selsk. Math. Fys. Medd. 9, 3-29 (1918).

Naturwiss. (1)

H. Boegehold, “Treue Darstellung und Verzeichnung bei optischen Instrumenten,” Naturwiss. 9, 273-280 (1921).
[CrossRef]

Optik (Stuttgart) (1)

H. Slevogt, “Zur Definition der Verzeichnung bei optischen Instrumenten für den subjektiven Gebrauch,” Optik (Stuttgart) 1, 358-367 (1946).

Perception (2)

A. H. J. Oomes, J. J. Koenderink, A. J. Doorn, and H. de Ridder, “What are the uncurved lines in our visual field? A fresh look at Helmholtz's checkerboard,” Perception 38, 1284-1294 (2009).
[CrossRef] [PubMed]

B. Rogers and K. Brecher, “Straight lines, 'uncurved lines', and Helmholtz's 'great circles on the celestial sphere',” Perception 36, 1275-1289 (2007).
[CrossRef]

The Opt. (1)

A. Whitwell, “On the sine, the tangent and the angle conditions,” The Opt. 48, 149-153 (1914).

Other (5)

www.holgermerlitz. de/globe/distortion.html.

I. P. Howard and B. J. Rogers, Binocular Vision and Stereopsis (Oxford Univ. Press, 1995).

H. v. Helmholtz, Handbuch der physiologischen Optik, Vol. 3, v.Kries, ed. (Hamburg und Leibzig, 1910).

A. Barre and A. Flocon, Curvilinear Perspective, R.Hansen, trans. and ed. (Univ. California Press, 1987).

M. Wagner, The Geometries of Visual Space (Erlbaum, 2006).

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Figures (8)

Fig. 1
Fig. 1

Helmholtz checkerboards with various different amounts of pincushion distortion generated using Eq. (6). The distortion parameter k was defined in Eq. (4). To generate these images, an undistorted checkerboard pattern was imaged using telescopes with 10 × and 7° true field of view.

Fig. 2
Fig. 2

Flat surface (wall) is imaged into the image space by the binocular. During this process, the object angle A is transformed into the apparent angle a through Eq. (4). Any radial distance to the center of field corresponds to the tangent of that angle.

Fig. 3
Fig. 3

Checkerboards of Fig. 1 in visual space, i.e., as they appear to the observer’s eye when viewed close up and with the direction of view pointing to the center of the checkerboard. The transformation was carried out using the visual imaging Eq. (8) assuming a distortion parameter l = 0.6 .

Fig. 4
Fig. 4

Characteristic drift ratio Γ, Eq. (16), for different degrees of distortion k as a function of the object angle A. Here, we assume a binocular with m = 10 × and 7° true field of view. The visual distortion parameter was assumed to be l = 0.6 .

Fig. 5
Fig. 5

Visual space and the rolling eye. Here, the direction of view (cross) is offset 20° below the center of field. A visual distortion parameter of l = 0.6 was assumed.

Fig. 6
Fig. 6

Distortion properties of Sonnefeld’s experimental telescope with low power m = 1.5 and huge true field of view of 90° as it is perceived in visual space. Only the angle condition offers an almost undistorted image. A visual distortion parameter l = 0.6 was assumed.

Fig. 7
Fig. 7

Distortion parameter k that would be required to fully eliminate the barrel distortion in visual space as a function of the power m and assuming a visual distortion of l = 0.6 .

Fig. 8
Fig. 8

Curvature of the visual space as a function of the visual distortion parameter l. For l = 0 , the space is spherical, for l = 1 it is flat (Euclidean). The curves are parameterized by the radius vector ρ ( A ) .

Equations (24)

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tan ( a ) tan ( A ) = m ( tangent condition )
a A = m ( angle condition )
tan ( a 2 ) tan ( A 2 ) = m ( circle condition ) ,
tan ( k a ) tan ( k A ) = m ,
a = ( 1 k ) arctan [ m tan ( k A ) ] .
tan ( a ) = tan { ( 1 k ) arctan [ m tan ( k A ) ] } .
tan ( a ) = m tan ( A ) ,
y = ( 1 l ) tan ( l a ) ,
y = ( 1 l ) tan { ( l k ) arctan [ m tan ( k A ) ] } .
y = ( 1 l ) tan ( l A ) ,
y = ( m l ) tan ( l A ) .
y = ( m l ) tan ( l A ) m tan ( A ) .
y ̇ = m A ̇ cos 2 ( k A ) cos 2 { ( l k ) arctan [ m tan ( k A ) ] } [ 1 + m 2 tan 2 ( k A ) ] ,
y ̇ ( 0 ) = m A ̇ .
Γ y ̇ ( A ) y ̇ ( 0 ) ,
Γ = cos 2 ( k A ) cos 2 { ( l k ) arctan [ m tan ( k A ) ] } [ 1 + m 2 tan 2 ( k A ) ] .
Γ = 1 cos 2 ( l A ) ,
Γ = 1 cos 2 ( l m A ) ,
Γ = 1 cos 2 ( l a ) .
Γ = 1 cos 2 ( l A ) ,
l 1 tan ( l A ) tan ( A ) < 1 ,
d y = d d A ( 1 l tan ( l A ) ) d A = d A cos 2 ( l A ) .
( d y ) 2 = ρ 2 ( d A ) 2 + ( d ρ ) 2 .
d | ρ | d A = ± 1 cos 4 ( l A ) ρ 2 ,

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