Abstract

An object viewed via reflection from a mirrored surface is often perceived by the observer to be located behind the mirror’s surface. The image of this object behind the mirror is known as its virtual image. Conventional methods for determining the location and shape of a virtual image for non-planar mirrors are complex and impractical unless both the observer and object are near the optical axis. We have developed a technique designed to be simple and practical for determining the location of a virtual image in a non-planar mirror far from the optical axis. Results using this technique were compared with known results from geometric optics for an object point on the optical axis of a parabola and for an object point imaged off the optical axis of a spherical mirror. These results were also in agreement with experimental measurements for a hemispherical mirror viewed at large angles with respect to its optical axis. This technique has applications for display devices or imaging tools utilizing curved, mirrored surfaces.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Looking into the water with oblique head tilting: revision of the aerial binocular imaging of underwater objects

Gábor Horváth, Krisztián Buchta, and Dezsö Varjú
J. Opt. Soc. Am. A 20(6) 1120-1131 (2003)

Multifocal planes head-mounted displays

Jannick P. Rolland, Myron W. Krueger, and Alexei Goon
Appl. Opt. 39(19) 3209-3215 (2000)

References

  • View by:
  • |
  • |
  • |

  1. L. Baltrueaitis, Anamorphic Art (Harry N. Abrams, Inc. Publishers, 1977).
  2. O. Faugeras, Panoramic Vision: Sensors, Theory, and Applications, Ryad Benosman, and Sing Bing Kang, eds. (Springer Science & Business Media, 2013).
  3. D. Zhao, B. Su, G. Chen, and H. Liao, “360 degree viewable floating autostereoscopic display using integral photography and multiple semitransparent mirrors,” Opt. Express 23(8), 9812–9823 (2015).
    [Crossref] [PubMed]
  4. G. Chen, C. Ma, D. Zhao, Z. Fan, and H. Liao, “Crosstalk-free 360-Degree Viewable 3D Display based on Pyramidal Mirrors and Diaphragms,” in Digital Holography and Three Dimensional Imaging (Optical Society of America, 2015), paper DW3A–7.
  5. “Flight simulator visual display system.” U.S. Patent 3,904,289, issued September 9, 1975.
  6. W. J. Smith, Modern Optical Engineering (McGraw-Hill, 2008), Chap. 16.
  7. R. E. Fischer, Optical System Design (McGraw-Hill, 2000), Chapter 9.
  8. M. Teittinen, “Depth Cues in the Human Visual System,” http://www.hitl.washington.edu/research/knowledge_base/virtual-worlds/EVE/III.A.1.c.DepthCues.html
  9. T. Okoshi, Three-dimensional imaging techniques (Elsevier, 2012).
  10. C. A. Levin and R. N. Haber, “Visual angle as a determinant of perceived interobject distance,” Percept. Psychophys. 54(2), 250–259 (1993).
    [Crossref] [PubMed]
  11. S. J. Watt, K. Akeley, M. O. Ernst, and M. S. Banks, “Focus cues affect perceived depth,” J. Vision 5(10), 834–862 (2005).
    [Crossref] [PubMed]
  12. D. M. Hoffman, A. R. Girshick, K. Akeley, and M. S. Banks, “Vergence-accommodation conflicts hinder visual performance and cause visual fatigue,” J. Vision 8(3), 1–30 (2008).
    [Crossref] [PubMed]
  13. J. L. Hunt, B. G. Nickel, and C. Gigault, “Anamorphic Images,” Am. J. Phys. 68(3), 232–237 (2000).
    [Crossref]
  14. M. O. Ernst and M. S. Banks, “Humans integrate visual and haptic information in a statistically optimal fashion,” Nature 415(6870), 429–433 (2002).
    [Crossref] [PubMed]
  15. D. Murra and P. Di Lazzaro, “Analytical treatment and experiments of the virtual image of cone mirrors,” Appl. Phys. B 117(1), 145–150 (2014).
    [Crossref]
  16. G. Monk, Light: Principles and Experiments, (McGraw Hill Book Company Inc., 1937) Appendix III.
  17. M. Herzberger, Modern Geometrical Optics, (Robert E. Krieger Publishing Company, 1980), Chap. 2.
  18. F. Jenkins and H. White, Fundamentals of Optics, 4th Edition (McGraw-Hill, 1976), Chap. 6.
  19. J. Mrovlje and D. Vrančić, “Distance measuring based on stereoscopic pictures,” In Proceedings of the 9th International PhD Workshop on Systems and Control, M. Gašperin and B. Pregelj, ed. (Institut Jožef Stefan, 2008), pp.1–6.
  20. E. Trucco and A. Verri, Introductory Techniques for 3-D computer vision (Prentice Hall, 1998).
  21. J. G. Fryer and D. C. Brown, “Lens distortion for close-range photogrammetry,” Photogramm. Eng. Remote Sensing 52(1), 51–58 (1986).

2015 (1)

2014 (1)

D. Murra and P. Di Lazzaro, “Analytical treatment and experiments of the virtual image of cone mirrors,” Appl. Phys. B 117(1), 145–150 (2014).
[Crossref]

2008 (1)

D. M. Hoffman, A. R. Girshick, K. Akeley, and M. S. Banks, “Vergence-accommodation conflicts hinder visual performance and cause visual fatigue,” J. Vision 8(3), 1–30 (2008).
[Crossref] [PubMed]

2005 (1)

S. J. Watt, K. Akeley, M. O. Ernst, and M. S. Banks, “Focus cues affect perceived depth,” J. Vision 5(10), 834–862 (2005).
[Crossref] [PubMed]

2002 (1)

M. O. Ernst and M. S. Banks, “Humans integrate visual and haptic information in a statistically optimal fashion,” Nature 415(6870), 429–433 (2002).
[Crossref] [PubMed]

2000 (1)

J. L. Hunt, B. G. Nickel, and C. Gigault, “Anamorphic Images,” Am. J. Phys. 68(3), 232–237 (2000).
[Crossref]

1993 (1)

C. A. Levin and R. N. Haber, “Visual angle as a determinant of perceived interobject distance,” Percept. Psychophys. 54(2), 250–259 (1993).
[Crossref] [PubMed]

1986 (1)

J. G. Fryer and D. C. Brown, “Lens distortion for close-range photogrammetry,” Photogramm. Eng. Remote Sensing 52(1), 51–58 (1986).

Akeley, K.

D. M. Hoffman, A. R. Girshick, K. Akeley, and M. S. Banks, “Vergence-accommodation conflicts hinder visual performance and cause visual fatigue,” J. Vision 8(3), 1–30 (2008).
[Crossref] [PubMed]

S. J. Watt, K. Akeley, M. O. Ernst, and M. S. Banks, “Focus cues affect perceived depth,” J. Vision 5(10), 834–862 (2005).
[Crossref] [PubMed]

Banks, M. S.

D. M. Hoffman, A. R. Girshick, K. Akeley, and M. S. Banks, “Vergence-accommodation conflicts hinder visual performance and cause visual fatigue,” J. Vision 8(3), 1–30 (2008).
[Crossref] [PubMed]

S. J. Watt, K. Akeley, M. O. Ernst, and M. S. Banks, “Focus cues affect perceived depth,” J. Vision 5(10), 834–862 (2005).
[Crossref] [PubMed]

M. O. Ernst and M. S. Banks, “Humans integrate visual and haptic information in a statistically optimal fashion,” Nature 415(6870), 429–433 (2002).
[Crossref] [PubMed]

Brown, D. C.

J. G. Fryer and D. C. Brown, “Lens distortion for close-range photogrammetry,” Photogramm. Eng. Remote Sensing 52(1), 51–58 (1986).

Chen, G.

Di Lazzaro, P.

D. Murra and P. Di Lazzaro, “Analytical treatment and experiments of the virtual image of cone mirrors,” Appl. Phys. B 117(1), 145–150 (2014).
[Crossref]

Ernst, M. O.

S. J. Watt, K. Akeley, M. O. Ernst, and M. S. Banks, “Focus cues affect perceived depth,” J. Vision 5(10), 834–862 (2005).
[Crossref] [PubMed]

M. O. Ernst and M. S. Banks, “Humans integrate visual and haptic information in a statistically optimal fashion,” Nature 415(6870), 429–433 (2002).
[Crossref] [PubMed]

Fryer, J. G.

J. G. Fryer and D. C. Brown, “Lens distortion for close-range photogrammetry,” Photogramm. Eng. Remote Sensing 52(1), 51–58 (1986).

Gigault, C.

J. L. Hunt, B. G. Nickel, and C. Gigault, “Anamorphic Images,” Am. J. Phys. 68(3), 232–237 (2000).
[Crossref]

Girshick, A. R.

D. M. Hoffman, A. R. Girshick, K. Akeley, and M. S. Banks, “Vergence-accommodation conflicts hinder visual performance and cause visual fatigue,” J. Vision 8(3), 1–30 (2008).
[Crossref] [PubMed]

Haber, R. N.

C. A. Levin and R. N. Haber, “Visual angle as a determinant of perceived interobject distance,” Percept. Psychophys. 54(2), 250–259 (1993).
[Crossref] [PubMed]

Hoffman, D. M.

D. M. Hoffman, A. R. Girshick, K. Akeley, and M. S. Banks, “Vergence-accommodation conflicts hinder visual performance and cause visual fatigue,” J. Vision 8(3), 1–30 (2008).
[Crossref] [PubMed]

Hunt, J. L.

J. L. Hunt, B. G. Nickel, and C. Gigault, “Anamorphic Images,” Am. J. Phys. 68(3), 232–237 (2000).
[Crossref]

Levin, C. A.

C. A. Levin and R. N. Haber, “Visual angle as a determinant of perceived interobject distance,” Percept. Psychophys. 54(2), 250–259 (1993).
[Crossref] [PubMed]

Liao, H.

Murra, D.

D. Murra and P. Di Lazzaro, “Analytical treatment and experiments of the virtual image of cone mirrors,” Appl. Phys. B 117(1), 145–150 (2014).
[Crossref]

Nickel, B. G.

J. L. Hunt, B. G. Nickel, and C. Gigault, “Anamorphic Images,” Am. J. Phys. 68(3), 232–237 (2000).
[Crossref]

Su, B.

Watt, S. J.

S. J. Watt, K. Akeley, M. O. Ernst, and M. S. Banks, “Focus cues affect perceived depth,” J. Vision 5(10), 834–862 (2005).
[Crossref] [PubMed]

Zhao, D.

Am. J. Phys. (1)

J. L. Hunt, B. G. Nickel, and C. Gigault, “Anamorphic Images,” Am. J. Phys. 68(3), 232–237 (2000).
[Crossref]

Appl. Phys. B (1)

D. Murra and P. Di Lazzaro, “Analytical treatment and experiments of the virtual image of cone mirrors,” Appl. Phys. B 117(1), 145–150 (2014).
[Crossref]

J. Vision (2)

S. J. Watt, K. Akeley, M. O. Ernst, and M. S. Banks, “Focus cues affect perceived depth,” J. Vision 5(10), 834–862 (2005).
[Crossref] [PubMed]

D. M. Hoffman, A. R. Girshick, K. Akeley, and M. S. Banks, “Vergence-accommodation conflicts hinder visual performance and cause visual fatigue,” J. Vision 8(3), 1–30 (2008).
[Crossref] [PubMed]

Nature (1)

M. O. Ernst and M. S. Banks, “Humans integrate visual and haptic information in a statistically optimal fashion,” Nature 415(6870), 429–433 (2002).
[Crossref] [PubMed]

Opt. Express (1)

Percept. Psychophys. (1)

C. A. Levin and R. N. Haber, “Visual angle as a determinant of perceived interobject distance,” Percept. Psychophys. 54(2), 250–259 (1993).
[Crossref] [PubMed]

Photogramm. Eng. Remote Sensing (1)

J. G. Fryer and D. C. Brown, “Lens distortion for close-range photogrammetry,” Photogramm. Eng. Remote Sensing 52(1), 51–58 (1986).

Other (13)

L. Baltrueaitis, Anamorphic Art (Harry N. Abrams, Inc. Publishers, 1977).

O. Faugeras, Panoramic Vision: Sensors, Theory, and Applications, Ryad Benosman, and Sing Bing Kang, eds. (Springer Science & Business Media, 2013).

G. Monk, Light: Principles and Experiments, (McGraw Hill Book Company Inc., 1937) Appendix III.

M. Herzberger, Modern Geometrical Optics, (Robert E. Krieger Publishing Company, 1980), Chap. 2.

F. Jenkins and H. White, Fundamentals of Optics, 4th Edition (McGraw-Hill, 1976), Chap. 6.

J. Mrovlje and D. Vrančić, “Distance measuring based on stereoscopic pictures,” In Proceedings of the 9th International PhD Workshop on Systems and Control, M. Gašperin and B. Pregelj, ed. (Institut Jožef Stefan, 2008), pp.1–6.

E. Trucco and A. Verri, Introductory Techniques for 3-D computer vision (Prentice Hall, 1998).

G. Chen, C. Ma, D. Zhao, Z. Fan, and H. Liao, “Crosstalk-free 360-Degree Viewable 3D Display based on Pyramidal Mirrors and Diaphragms,” in Digital Holography and Three Dimensional Imaging (Optical Society of America, 2015), paper DW3A–7.

“Flight simulator visual display system.” U.S. Patent 3,904,289, issued September 9, 1975.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, 2008), Chap. 16.

R. E. Fischer, Optical System Design (McGraw-Hill, 2000), Chapter 9.

M. Teittinen, “Depth Cues in the Human Visual System,” http://www.hitl.washington.edu/research/knowledge_base/virtual-worlds/EVE/III.A.1.c.DepthCues.html

T. Okoshi, Three-dimensional imaging techniques (Elsevier, 2012).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1 Rays from an object point reflecting from a spherical mirror. Reflected rays are back-propagated into the sphere, creating a virtual ray caustic indicated by dotted curves. In (a) and (b), rays from a distant and nearby object point, respectively, reflect from a spherical surface. In (c) and (d), two different observers perceive the virtual image point to be at two different locations lying on different portions of the ray caustic.
Fig. 2
Fig. 2 Two light rays depicted reflecting from the osculating circle used to approximate the surface curvature of a mirror (a). In (b), a closer look at the relevant angles made between lines connecting the object point, virtual image point, and points of reflection on the mirror’s surface.
Fig. 3
Fig. 3 Ray tracing simulating depicting the virtual image of an object reflecting from a parabolic mirror with a focal length of 1 m. Near the optical axis, depicted in (a), the virtual image can be readily predicted with Gaussian optics (illustrated using traditional rays taught in introductory physics courses), however, observed far from the optical axis, depicted in (b), the virtual image deviates from these predictions.
Fig. 4
Fig. 4 Primary ray reflecting from a mirrored surface to an observer, depicted in the tangential plane, sagittal plane, and an arbitrary plane
Fig. 5
Fig. 5 Virtual surface of a planar screen reflecting from a hemispherical mirror. In blue, virtual image points calculated based on ray-tracing performed in Zemax. In red, simulated virtual image points based on semi-analytical ray-tracing method. Points where a solution was not found in Zemax are not shown.
Fig. 6
Fig. 6 Object points, depicted by green squares, create virtual image points when viewed reflected from a spherical mirror. Virtual image points determined using a semi-analytical ray tracing solution are shown by small red points. Nearly identical virtual image points, depicted by blue circles, were found using numerical ray-tracing performed in Zemax.
Fig. 7
Fig. 7 Stereoscopic measurements for pinhole cameras.
Fig. 8
Fig. 8 Image of display screen reflecting from the spherical mirror in a dark room (a) and in a well-lit room (b). The display screen is displaying a black image with an array of lighted, green pixels. The contrast was increased by the same amount in both pictures to enhance the visibility of the pixels.
Fig. 9
Fig. 9 Side (a) and front (b) views of measured virtual surface points (purple) along with corresponding simulated virtual surface points (red). Only a portion of the hemispherical mirror is shown for clarity.

Tables (1)

Tables Icon

Table 1 Radial distortion coefficients for camera calibration

Equations (21)

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

R= ( (x') 2 + (y') 2 ) 3\2 | x'y''y'x'' |
R= (1+4 a 2 x 2 ) 3/2 2a
L= E 2tan(dγ+ dβ 2 )
D i = Rdγcos(θ+ 3 2 dγ+dβ) 2dγ+dβ
D i = Rdγcos(θ) 2dγ+dβ
dβ= R D o dγcos(θ)
D i = D o Rcos(θ) 2 D o +Rcos(θ)
1 D i = 2 R + 1 D o
D i = D o Rcos(θ) 2 D o +Rcos(θ)
cos(θ)cos(α)
D o D o cos(θ) cos(α)
D i D i cos(θ) cos(α)
D i = D o R cos 2 (α) 2 D o cos(θ)+R cos 2 (α)
e ^ = ( E ^ u ^ 1 ) u ^ 1 +( E ^ u ^ 2 ) u ^ 2 ( E ^ u ^ 1 ) 2 + ( E ^ u ^ 2 ) 2
α= tan 1 ( D 0 r e ^ D 0 cos(θ) )
D o = D i R cos 2 (α) R cos 2 (α)2 D i Rcos(θ)
L= B tan(θ+ϕ)tan(θ)
x L f L = B L L , x R f R = B L L
B L B R =B=L( x R f R x L f L )
L=B ( x R f R x L f L ) 1
r'=r(1+a r 2 +b r 4 +c r 6 +d r 8 )

Metrics