Abstract

To an ultra-wide-angle and panoramic optical system, the aberrations of point object at any field angle are separated into two types: the aperture-ray aberrations of off-axis point object and the chief-ray aberrations. A simple form of the triangular formulae of tracing an oblique-incidence ray is derived to calculate the chief-ray parameters and their aberrations; moreover, the aperture-ray aberrations of an off-axis point object are analyzed with the plane-symmetric aberration theory. Based on the two types of aberrations, we present a merit function for ultra-wide-angle and panoramic optical systems; the optimization program with the differential-evolution algorithm is then developed. To validate the optimization method we finally optimize a fish-eye lens and a catadioptric omnidirectional imaging system.

© 2012 Optical Society of America

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References

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  1. C. B. Martin, “Design issue of a hyper-field fisheye lens,” Proc. SPIE 5524, 84–392 (2004).
    [CrossRef]
  2. I. Powell, “Panoramic lens,” Appl. Opt. 33, 7356–7361(1994).
    [CrossRef]
  3. J. Kumler and M. Bauer, “Fisheye lens designs and their relative performance,” Proc. SPIE 4093, 360–369(2000).
    [CrossRef]
  4. J. S. Chahl and M. V. Srinvasan, “Reflective surfaces for panoramic imaging,” Appl. Opt. 36, 8275–8285 (1997).
    [CrossRef]
  5. G.-I. Kweon, K. T. Kim, G. H. Kim, and H.-S. Kim, “Folded catadioptric panoramic lens with equidistance projection scheme,” Appl. Opt. 44, 2759–2767 (2005).
    [CrossRef]
  6. S.-S. Lin and R. Bajcsy, “Single-viewpoint, catadioptric cone mirror omnidirectional imaging theory and analysis,” J. Opt. Soc. Am. 23, 2997–3015 (2006).
    [CrossRef]
  7. L.-P. Wang, L.-C. Zhang, F.-Y. He, and C.-S. Jin, “Design of aspheric mirror for panoramic imaging system using multi-population genetic algorithm,” Opt. Precis. Eng. 17, 1020–1025 (2009).
    [CrossRef]
  8. M. Born and E. Wolf, Principle of Optics7th ed. (Cambridge University, 2005).
  9. W. J. Smith, Modern Lens Design (McGraw-Hill, 1992).
  10. Y.-Z. Wand, Fisheye Lens Optics (Science, 2006).
  11. S. Baker, “A theory of single-viewpoint catadioptric image formation,” Int. J. Comput. Vis. 35, 175–196 (1999).
    [CrossRef]
  12. R. Swaminathan, M. D. Grossberg, and S. K. Nayer, “Non-single viewpoint catadioptric cameras: geometry and analysis,” Int. J. Comput. Vis. 66, 211–229 (2006).
    [CrossRef]
  13. H. Noda, T. Namioka, and M. Seya, “Geometrical theory of the grating,” J. Opt. Soc. Am. 64, 1031–1036 (1974).
    [CrossRef]
  14. T. Namioka, M. Koike, and D. Content, “Geometric theory of the ellipsoidal grating,” Appl. Opt. 33, 7261–7274 (1994).
    [CrossRef]
  15. T. Namioka, M. Koike, and S. Masui, “Geometric theory for the design of multielement optical system,” Opt. Precis. Eng.9, 459–467 (2001).
  16. M. P. Chrisp, “Aberrations of holographic toroidal grating systems,” Appl. Opt. 22, 1508–1518 (1983).
    [CrossRef]
  17. B. D. Stone and G. W. Forbes, “Second-order design methods for definitive studies of plane-symmetric, two-mirror systems,” J. Opt. Soc. Am. A 11, 3292–3307 (1994).
    [CrossRef]
  18. L.-J. Lu, “Aberration theory of plane-symmetric grating systems,” J. Synchrotron Radiat. 15, 399–410(2008).
    [CrossRef]
  19. L.-J. Lu and Z.-Y. Deng, “Geometric characteristics of aberrations of plane-symmetric optical systems,” Appl. Opt. 48, 6946–6960 (2009).
    [CrossRef]
  20. L.-J. Lu and D.-L. Lin, “Aberrations of plane-symmetric multi-element optical systems,” Optik 121, 1198–1218(2010).
    [CrossRef]
  21. H. Jung, Y. Lee, P. Yoon, and J. Kim, “Radial distortion refinement by inverse mapping-based extrapolation,” in 18th International Conference on Pattern Recognition, Vol. 1 (IEEE, 2006), pp. 675–678.
    [CrossRef]
  22. R. Muller, “Fish-eye lens system,” U.S. patent 4,525,038 (25June1985).
  23. CODE V, Optical Design Program, Reference Manual (Optical Research Associates, 2003).

2010 (1)

L.-J. Lu and D.-L. Lin, “Aberrations of plane-symmetric multi-element optical systems,” Optik 121, 1198–1218(2010).
[CrossRef]

2009 (2)

L.-J. Lu and Z.-Y. Deng, “Geometric characteristics of aberrations of plane-symmetric optical systems,” Appl. Opt. 48, 6946–6960 (2009).
[CrossRef]

L.-P. Wang, L.-C. Zhang, F.-Y. He, and C.-S. Jin, “Design of aspheric mirror for panoramic imaging system using multi-population genetic algorithm,” Opt. Precis. Eng. 17, 1020–1025 (2009).
[CrossRef]

2008 (1)

L.-J. Lu, “Aberration theory of plane-symmetric grating systems,” J. Synchrotron Radiat. 15, 399–410(2008).
[CrossRef]

2006 (2)

R. Swaminathan, M. D. Grossberg, and S. K. Nayer, “Non-single viewpoint catadioptric cameras: geometry and analysis,” Int. J. Comput. Vis. 66, 211–229 (2006).
[CrossRef]

S.-S. Lin and R. Bajcsy, “Single-viewpoint, catadioptric cone mirror omnidirectional imaging theory and analysis,” J. Opt. Soc. Am. 23, 2997–3015 (2006).
[CrossRef]

2005 (1)

2004 (1)

C. B. Martin, “Design issue of a hyper-field fisheye lens,” Proc. SPIE 5524, 84–392 (2004).
[CrossRef]

2000 (1)

J. Kumler and M. Bauer, “Fisheye lens designs and their relative performance,” Proc. SPIE 4093, 360–369(2000).
[CrossRef]

1999 (1)

S. Baker, “A theory of single-viewpoint catadioptric image formation,” Int. J. Comput. Vis. 35, 175–196 (1999).
[CrossRef]

1997 (1)

1994 (3)

1983 (1)

1974 (1)

Bajcsy, R.

S.-S. Lin and R. Bajcsy, “Single-viewpoint, catadioptric cone mirror omnidirectional imaging theory and analysis,” J. Opt. Soc. Am. 23, 2997–3015 (2006).
[CrossRef]

Baker, S.

S. Baker, “A theory of single-viewpoint catadioptric image formation,” Int. J. Comput. Vis. 35, 175–196 (1999).
[CrossRef]

Bauer, M.

J. Kumler and M. Bauer, “Fisheye lens designs and their relative performance,” Proc. SPIE 4093, 360–369(2000).
[CrossRef]

Chahl, J. S.

Chrisp, M. P.

Content, D.

Deng, Z.-Y.

Forbes, G. W.

Grossberg, M. D.

R. Swaminathan, M. D. Grossberg, and S. K. Nayer, “Non-single viewpoint catadioptric cameras: geometry and analysis,” Int. J. Comput. Vis. 66, 211–229 (2006).
[CrossRef]

He, F.-Y.

L.-P. Wang, L.-C. Zhang, F.-Y. He, and C.-S. Jin, “Design of aspheric mirror for panoramic imaging system using multi-population genetic algorithm,” Opt. Precis. Eng. 17, 1020–1025 (2009).
[CrossRef]

Jin, C.-S.

L.-P. Wang, L.-C. Zhang, F.-Y. He, and C.-S. Jin, “Design of aspheric mirror for panoramic imaging system using multi-population genetic algorithm,” Opt. Precis. Eng. 17, 1020–1025 (2009).
[CrossRef]

Jung, H.

H. Jung, Y. Lee, P. Yoon, and J. Kim, “Radial distortion refinement by inverse mapping-based extrapolation,” in 18th International Conference on Pattern Recognition, Vol. 1 (IEEE, 2006), pp. 675–678.
[CrossRef]

Kim, G. H.

Kim, H.-S.

Kim, J.

H. Jung, Y. Lee, P. Yoon, and J. Kim, “Radial distortion refinement by inverse mapping-based extrapolation,” in 18th International Conference on Pattern Recognition, Vol. 1 (IEEE, 2006), pp. 675–678.
[CrossRef]

Kim, K. T.

Koike, M.

T. Namioka, M. Koike, and D. Content, “Geometric theory of the ellipsoidal grating,” Appl. Opt. 33, 7261–7274 (1994).
[CrossRef]

T. Namioka, M. Koike, and S. Masui, “Geometric theory for the design of multielement optical system,” Opt. Precis. Eng.9, 459–467 (2001).

Kumler, J.

J. Kumler and M. Bauer, “Fisheye lens designs and their relative performance,” Proc. SPIE 4093, 360–369(2000).
[CrossRef]

Kweon, G.-I.

Lee, Y.

H. Jung, Y. Lee, P. Yoon, and J. Kim, “Radial distortion refinement by inverse mapping-based extrapolation,” in 18th International Conference on Pattern Recognition, Vol. 1 (IEEE, 2006), pp. 675–678.
[CrossRef]

Lin, D.-L.

L.-J. Lu and D.-L. Lin, “Aberrations of plane-symmetric multi-element optical systems,” Optik 121, 1198–1218(2010).
[CrossRef]

Lin, S.-S.

S.-S. Lin and R. Bajcsy, “Single-viewpoint, catadioptric cone mirror omnidirectional imaging theory and analysis,” J. Opt. Soc. Am. 23, 2997–3015 (2006).
[CrossRef]

Lu, L.-J.

L.-J. Lu and D.-L. Lin, “Aberrations of plane-symmetric multi-element optical systems,” Optik 121, 1198–1218(2010).
[CrossRef]

L.-J. Lu and Z.-Y. Deng, “Geometric characteristics of aberrations of plane-symmetric optical systems,” Appl. Opt. 48, 6946–6960 (2009).
[CrossRef]

L.-J. Lu, “Aberration theory of plane-symmetric grating systems,” J. Synchrotron Radiat. 15, 399–410(2008).
[CrossRef]

Martin, C. B.

C. B. Martin, “Design issue of a hyper-field fisheye lens,” Proc. SPIE 5524, 84–392 (2004).
[CrossRef]

Masui, S.

T. Namioka, M. Koike, and S. Masui, “Geometric theory for the design of multielement optical system,” Opt. Precis. Eng.9, 459–467 (2001).

Muller, R.

R. Muller, “Fish-eye lens system,” U.S. patent 4,525,038 (25June1985).

Namioka, T.

Nayer, S. K.

R. Swaminathan, M. D. Grossberg, and S. K. Nayer, “Non-single viewpoint catadioptric cameras: geometry and analysis,” Int. J. Comput. Vis. 66, 211–229 (2006).
[CrossRef]

Noda, H.

Powell, I.

Seya, M.

Smith, W. J.

W. J. Smith, Modern Lens Design (McGraw-Hill, 1992).

Srinvasan, M. V.

Stone, B. D.

Swaminathan, R.

R. Swaminathan, M. D. Grossberg, and S. K. Nayer, “Non-single viewpoint catadioptric cameras: geometry and analysis,” Int. J. Comput. Vis. 66, 211–229 (2006).
[CrossRef]

Wand, Y.-Z.

Y.-Z. Wand, Fisheye Lens Optics (Science, 2006).

Wang, L.-P.

L.-P. Wang, L.-C. Zhang, F.-Y. He, and C.-S. Jin, “Design of aspheric mirror for panoramic imaging system using multi-population genetic algorithm,” Opt. Precis. Eng. 17, 1020–1025 (2009).
[CrossRef]

Yoon, P.

H. Jung, Y. Lee, P. Yoon, and J. Kim, “Radial distortion refinement by inverse mapping-based extrapolation,” in 18th International Conference on Pattern Recognition, Vol. 1 (IEEE, 2006), pp. 675–678.
[CrossRef]

Zhang, L.-C.

L.-P. Wang, L.-C. Zhang, F.-Y. He, and C.-S. Jin, “Design of aspheric mirror for panoramic imaging system using multi-population genetic algorithm,” Opt. Precis. Eng. 17, 1020–1025 (2009).
[CrossRef]

Appl. Opt. (6)

Int. J. Comput. Vis. (2)

S. Baker, “A theory of single-viewpoint catadioptric image formation,” Int. J. Comput. Vis. 35, 175–196 (1999).
[CrossRef]

R. Swaminathan, M. D. Grossberg, and S. K. Nayer, “Non-single viewpoint catadioptric cameras: geometry and analysis,” Int. J. Comput. Vis. 66, 211–229 (2006).
[CrossRef]

J. Opt. Soc. Am. (2)

H. Noda, T. Namioka, and M. Seya, “Geometrical theory of the grating,” J. Opt. Soc. Am. 64, 1031–1036 (1974).
[CrossRef]

S.-S. Lin and R. Bajcsy, “Single-viewpoint, catadioptric cone mirror omnidirectional imaging theory and analysis,” J. Opt. Soc. Am. 23, 2997–3015 (2006).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Synchrotron Radiat. (1)

L.-J. Lu, “Aberration theory of plane-symmetric grating systems,” J. Synchrotron Radiat. 15, 399–410(2008).
[CrossRef]

Opt. Precis. Eng. (1)

L.-P. Wang, L.-C. Zhang, F.-Y. He, and C.-S. Jin, “Design of aspheric mirror for panoramic imaging system using multi-population genetic algorithm,” Opt. Precis. Eng. 17, 1020–1025 (2009).
[CrossRef]

Optik (1)

L.-J. Lu and D.-L. Lin, “Aberrations of plane-symmetric multi-element optical systems,” Optik 121, 1198–1218(2010).
[CrossRef]

Proc. SPIE (2)

J. Kumler and M. Bauer, “Fisheye lens designs and their relative performance,” Proc. SPIE 4093, 360–369(2000).
[CrossRef]

C. B. Martin, “Design issue of a hyper-field fisheye lens,” Proc. SPIE 5524, 84–392 (2004).
[CrossRef]

Other (7)

M. Born and E. Wolf, Principle of Optics7th ed. (Cambridge University, 2005).

W. J. Smith, Modern Lens Design (McGraw-Hill, 1992).

Y.-Z. Wand, Fisheye Lens Optics (Science, 2006).

H. Jung, Y. Lee, P. Yoon, and J. Kim, “Radial distortion refinement by inverse mapping-based extrapolation,” in 18th International Conference on Pattern Recognition, Vol. 1 (IEEE, 2006), pp. 675–678.
[CrossRef]

R. Muller, “Fish-eye lens system,” U.S. patent 4,525,038 (25June1985).

CODE V, Optical Design Program, Reference Manual (Optical Research Associates, 2003).

T. Namioka, M. Koike, and S. Masui, “Geometric theory for the design of multielement optical system,” Opt. Precis. Eng.9, 459–467 (2001).

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Figures (11)

Fig. 1.
Fig. 1.

Optical scheme shows the chief ray AiOiOi+1Oi+2¯ passing one reflecting and two refracting surfaces. The normal of surface k at the point Ok (k=i+1,i+2) intersects the optical axis at Ck; optical surface k intersects the optical axis at Dk; ωk1 means the field angle of surface k in object space, and αk, βk the angles of incidence and reflection or refraction.

Fig. 2.
Fig. 2.

Preceding part of a fish-eye-lens system, and a chief ray is traced conversely from the aperture stop to determine its initial position.

Fig. 3.
Fig. 3.

Chief ray passes the last optical surface and intersects with the image plane at a distance of h from the optical axis.

Fig. 4.
Fig. 4.

Optical scheme of the chief ray from a finite object plane and refracted by the first optical surface.

Fig. 5.
Fig. 5.

Chief ray is reflected by a quadrics of revolution; it also shows the coordinate system of xyz, whose origin is at the intersection point of the chief ray to the optical surface.

Fig. 6.
Fig. 6.

Flow diagram of calculation of the merit function.

Fig. 7.
Fig. 7.

Optical scheme of fish-eye lens (U.S. patent 4,525,038) [22].

Fig. 8.
Fig. 8.

Aperture-ray aberrations of the fish-eye lens (U.S. patent 4,525,038) obtained with the aberration expressions discussed in Subsection 4.B using (a) the optical parameters of the reference design and (b) the optical parameters from our optimization. The field angles are shown on the right side of each row.

Fig. 9.
Fig. 9.

Optical scheme of a catadioptric omnidirectional imaging system; the camera lens is a modified Tessar design.

Fig. 10.
Fig. 10.

Optical scheme and the optical spacing of the camera lens of the catadioptric omnidirectional imaging system.

Fig. 11.
Fig. 11.

Aperture-ray aberrations of the discussed catadioptric imaging system obtained with the aberration expressions discussed in Subsection 4.B using (a) the optical parameters of the initial design and (b) the optical parameters from the optimization. The field angles are shown on the right side of each row.

Tables (5)

Tables Icon

Table 1. Optical Parameters of Fish-Eye Lens (mm)

Tables Icon

Table 2. Values of Merit Function and Its Components (×103)

Tables Icon

Table 3. Optical Parameters of Camera Lens of Catadioptric Omnidirectional Imaging System (mm)

Tables Icon

Table 4. Optical Parameters of Panoramic Mirror (mm)

Tables Icon

Table 5. Values of Merit Function and Its components (×103)

Equations (49)

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Q=i=1nϵi(Qx(i)2+Qy(i)2+Qs(i)2+μiQc(i)2),
Qx(i)2=1WLW/2W/2L/2L/2(xx¯)2dxdy,Qy(i)2=1WLW/2W/2L/2L/2y2dxdy,x¯=1WLW/2W/2L/2L/2xdxdy,
MiC¯i=sinβisinωiCiOi¯,
MiC¯i+1=CiD¯i+DiD¯i+1+Di+1C¯i+1MiC¯i=Γi+1Γi+di+sinβisinωiρi.
sinαi+1=MiC¯i+1ρi+1sinωi.
sinαi+1=Γi+1+diΓiρi+1sinωi+ρiρi+1sinβi,
ωi=ωi1+βiαi=ω0+i=1i(βiαi).
βi+1=sin1(nini+1sinαi+1).
sinα1=Γ1S0ρ1sinω0,
d¯k=ρksin(ωk1αk)ρk+1sin(ωkαk+1)sinωk.
S0=Γ1ρ1sinα1sinω0.
MgCg¯=sinβgsinωgρl.
h=MgO¯tanωg=tanωg(ρgsinβgsinωgΓg+r0)
nλ2=1+B1λ2λ2C1+B2λ2λ2C2+B3λ2λ2C3,
ωl=f(ω0)
ω0=arctan(h0S0d0).
h=g(h0).
x2+y2=a1z+a2z2.
KO=a1+2a2z0*2x0*.
tanθ=1KO=2x0*a1+2a2z0*.
cosθ=a1+2a2·z0*(a1+2a2·z0*)2+4x0*2,sinθ=2x0*(a1+2a2·z0*)2+4x0*2.
{x=x·cosθ+z·sinθ+x0*,y=y,z=x·sinθ+z·cosθ+z0*.
z=c2,0x2+c0,2y2+c3,0x3+c1,2xy2+c4,0x4+c0,4y4+c2,2x2y2,
c2,0=a12B3,c3,0=4Aa12x0*B6,c4,0=a12(a12C2+16A2x0*2)B9,c0,4=C2B5,c02=1B,c1,2=4Ax0*B4,c2,2=2B7(a12C2+8A2x0*2),
A=(a1+2a2·z0*)·(1+a2),B=4x0*2(1+a2)+a12,C=4x0*2a2·(a1+2a2·z0*)2.
Γ=C1O¯=z0*x0*tanθ=a12+(1+a2)z0*,
ρ=OC¯1=B2=12c02.
{z0*=a1+2s0tan2ω0±a12+4s0(a1+a2s0)tan2ω02(a2tan2ω0),x0*=tanω0(z0*s0).
s0(k+1)=z0*(k)dkx0*(k)tanω0(k+1).
W=n1(w300x3+w120xy2+w400x4+w220x2y2+w040y4),
wij0=n0/1Mij0(α,rm,rs,0)+Mij0(β,rm,rs,0),
2c2,0(n0cosα+n1cosβ)(n0cos2αrm+n1cos2βrm)=0,
2c0,2(n0cosα+n1cosβ)(n0rs+n1rs)=0.
W=k=1gW(k)=k=1gij4nkwij0(k)xkiykj=ngij4wij0xgiygj,(i+j4),
wij0=k=1g1nk/gwij0(k)Ak|giBk|gj+wij0(g),
Ak|g=rm(k)rm(k+1)rm(g1)rm(k+1)rm(k+2)rm(g)cosαk+1cosαk+2cosαgcosβkcosβk+1cosβg1,
Bk|g=rs(k)rs(k+1)rs(g1)rs(k+1)rs(k+2)rs(g).
{x=1cosωg(d100xg+d200xg2+d020yg2+d300xg3+d120xgyg2),y=h010yg+h110xgyg+h210xg2yg+h030yg3,
d100=Λmcosβ,
d200=3r0w300cosβ+Λmsinβ(cosβrmc2,0)Λmcos2βrmtanωl,
d300=4rcw400cosβ3tanβ(1+rcrm2rcc2,0cosβ)w3003(12Λm)w300tanωg+Λm(cos(2β)c2,0rm+cosβsin2βrm2sinβc3,0)+Λmcos2βrm(cosβtanωg2sinβrm+2tanβc2,0)tanωg,
d020=rcw120cosβΛmsinβc0,2,
d120=2rcw220cosβtanβ(1+rcrm+2rcrs2rcc2,0cosβ)w120(12Λm)w120tanωg+Λm(cos(2β)c0,2rmsinβc1,2)+Λmsin(2β)c0,2rmtanωg,
h010=Λs,
h110=2rcw120+ΛssinβrsΛmcosβrstanωg,
h210=2rcw2202sinβw120+6tanβrcc0,2w3003rcw300cosβrstanωg+2Λmcosβw120tanωg+Λsrs(cosβc2,0+sin2βrs)+(ΛmΛs)cos2β2rmrs+Λmcosβrs(cosβtanωgsinβrmsinβrs+tanβc2,0)tanωg,
h030=4rcw040+2tanβrcc0,2w120rcw120cosβrstanωg+c0,2rs(Λscosβ+Λmsinβtanωg),
Λm=rmrcrm,Λs=rsrcrs,
rc=r0Γ+ρ cos (ωgβ) cos ωg.

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