Abstract

We describe necessary and sufficient conditions for image transformations so that surfaces refracting or reflecting collimated beams exist. The surfaces are given by explicit formulas. When these conditions are not satisfied, we show that a surface can be found so that it minimizes the L2 distance between its gradient and the image transformation.

© 2011 Optical Society of America

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References

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  1. R. A. Hicks and R. Bajcsy, “Reflective surfaces as computational sensors,” Image Vis. Comput. 19, 773–777 (2001).
    [CrossRef]
  2. R. A. Hicks and R. K. Perline, “Blind-spot problem for motor vehicles,” Appl. Opt. 44, 3893–3897 (2005).
    [CrossRef] [PubMed]
  3. D. L. Shealy, “Geometrical Methods,” in Laser Beam Shaping: Theory and Techniques, F.M.Dickey and S.C.Holswade, eds. (Marcel Dekker, 2000), pp. 163–213.
  4. C. E. Gutiérrez and Qingbo Huang, “The refractor problem in reshaping light beams,” Arch. Ration. Mech. Anal. 193, 423–443 (2009).
    [CrossRef]
  5. C. E. Gutiérrez, “Reflection, refraction and the Legendre transform,” J. Opt. Soc. Am. A 28, 284–289 (2011).
    [CrossRef]

2011

2009

C. E. Gutiérrez and Qingbo Huang, “The refractor problem in reshaping light beams,” Arch. Ration. Mech. Anal. 193, 423–443 (2009).
[CrossRef]

2005

2001

R. A. Hicks and R. Bajcsy, “Reflective surfaces as computational sensors,” Image Vis. Comput. 19, 773–777 (2001).
[CrossRef]

Bajcsy, R.

R. A. Hicks and R. Bajcsy, “Reflective surfaces as computational sensors,” Image Vis. Comput. 19, 773–777 (2001).
[CrossRef]

Gutiérrez, C. E.

C. E. Gutiérrez, “Reflection, refraction and the Legendre transform,” J. Opt. Soc. Am. A 28, 284–289 (2011).
[CrossRef]

C. E. Gutiérrez and Qingbo Huang, “The refractor problem in reshaping light beams,” Arch. Ration. Mech. Anal. 193, 423–443 (2009).
[CrossRef]

Hicks, R. A.

R. A. Hicks and R. K. Perline, “Blind-spot problem for motor vehicles,” Appl. Opt. 44, 3893–3897 (2005).
[CrossRef] [PubMed]

R. A. Hicks and R. Bajcsy, “Reflective surfaces as computational sensors,” Image Vis. Comput. 19, 773–777 (2001).
[CrossRef]

Huang, Qingbo

C. E. Gutiérrez and Qingbo Huang, “The refractor problem in reshaping light beams,” Arch. Ration. Mech. Anal. 193, 423–443 (2009).
[CrossRef]

Perline, R. K.

Shealy, D. L.

D. L. Shealy, “Geometrical Methods,” in Laser Beam Shaping: Theory and Techniques, F.M.Dickey and S.C.Holswade, eds. (Marcel Dekker, 2000), pp. 163–213.

Appl. Opt.

Arch. Ration. Mech. Anal.

C. E. Gutiérrez and Qingbo Huang, “The refractor problem in reshaping light beams,” Arch. Ration. Mech. Anal. 193, 423–443 (2009).
[CrossRef]

Image Vis. Comput.

R. A. Hicks and R. Bajcsy, “Reflective surfaces as computational sensors,” Image Vis. Comput. 19, 773–777 (2001).
[CrossRef]

J. Opt. Soc. Am. A

Other

D. L. Shealy, “Geometrical Methods,” in Laser Beam Shaping: Theory and Techniques, F.M.Dickey and S.C.Holswade, eds. (Marcel Dekker, 2000), pp. 163–213.

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Figures (2)

Fig. 1
Fig. 1

Lens problem.

Fig. 2
Fig. 2

Two views of the refracting surface with κ = 2 / 3 and β = 2 .

Equations (35)

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k κ v = λ N ,
λ = k · N κ v · N .
v · N = 1 κ 2 ( 1 ( k · N ) 2 ) ,
λ = k · N κ 1 κ 2 ( 1 ( k · N ) 2 ) .
k · N = 1 1 + | D u ( x , y ) | 2 ,
| D u ( x , y ) | κ 1 κ 2 .
α = d = | t ( x , y ) ( x , y ) | 2 + | m u ( x , y ) | 2 ,
( t ( x , y ) ( x , y ) , m u ( x , y ) ) = d v ,
t ( x , y ) ( x , y ) = d λ κ D u ( x , y ) 1 + | D u ( x , y ) | 2 , m u ( x , y ) = d κ ( 1 λ 1 + | D u ( x , y ) | 2 ) .
D u ( x , y ) = t ( x , y ) ( x , y ) d κ m + u ( x , y ) .
D u ( x , y ) = ( x , y ) m g ( x , y ) m u ( x , y ) 1 κ | m g ( x , y ) ( x , y ) | 2 + | m u ( x , y ) | 2 ,
D u ( x , y ) = g ( x , y ) 1 κ | g ( x , y ) | 2 + 1 1 .
( g 1 ( x , y ) 1 κ | g ( x , y ) | 2 + 1 1 ) y = ( g 2 ( x , y ) 1 κ | g ( x , y ) | 2 + 1 1 ) x .
u ( x , y ) = a x β s 1 κ | g ( s , y ) | 2 + 1 1 d s + b y β t 1 κ | g ( a , t ) | 2 + 1 1 d t .
u ( x , y ) = ψ ( β 2 ( x 2 + y 2 ) ) ψ ( β 2 ( a 2 + b 2 ) ) ,
ψ ( r ) = κ β r + 1 + κ 2 β log ( r + 1 κ )
u ( x ) = g ( x ) 1 κ | g ( x ) | 2 + 1 1 ,
u ( x ) = a x g ( t ) 1 κ | g ( t ) | 2 + 1 1 d t ,
v = k 2 ( k · N ) N = ( 2 1 + | D u ( x , y ) | 2 u x ( x , y ) , 2 1 + | D u ( x , y ) | 2 u y ( x , y ) , 1 2 1 + | D u ( x , y ) | 2 ) ,
( x , y , u ( x , y ) ) + d v = ( g 1 ( x , y ) , m , g 2 ( x , y ) )
d = ( x g 1 ( x , y ) ) 2 + ( y m ) 2 + ( u ( x , y ) g 2 ( x , y ) ) 2 .
D u ( x , y ) = ( g 1 ( x , y ) x , m y ) d g 2 ( x , y ) + u ( x , y ) .
D u ( x , y ) = m ( h 1 ( x , y ) ( x / m ) , 1 ( y / m ) ) m ( h 2 ( x , y ) + u ( x , y ) m + ( x m h 1 ( x , y ) ) 2 + ( y m 1 ) 2 + ( u ( x , y ) m h 2 ( x , y ) ) 2 ) ,
D u ( x , y ) = ( h 1 ( x , y ) , 1 ) 1 + h 1 ( x , y ) 2 + h 2 ( x , y ) 2 h 2 ( x , y ) .
( h 1 1 + h 1 2 + h 2 2 h 2 ) y = ( 1 1 + h 1 2 + h 2 2 h 2 ) x .
u ( x , y ) = a x h 1 ( s , y ) 1 + h 1 ( s , y ) 2 + h 2 ( s , y ) 2 h 2 ( s , y ) d s + b y 1 1 + h 1 ( a , t ) 2 + h 2 ( a , t ) 2 h 2 ( a , t ) d t .
u ( x ) = m x d g ( x ) + u ( x ) ,
u ( x ) = 1 1 + h ( x ) 2 h ( x ) ,
u ( x ) = 0 x 1 1 + h ( t ) 2 h ( t ) d t .
I ( u ) = Ω | D u ( x ) F ( x ) | 2 d x
Δ u = D · F , in     Ω ,
I ( v ) = Ω ( D ϕ + D u F ) · ( D ϕ + D u F ) d x = 2 Ω ( D ϕ ) · ( D u F ) d x + Ω D ϕ · D ϕ d x + Ω ( D u F ) · ( D u F ) d x .
Ω D ϕ · D u d x = Ω ( Δ u ) ϕ d x ,
Ω D ϕ · F d x = Ω ϕ ( D · F ) d x ,
I ( v ) = Ω D ϕ · D ϕ d x + I ( u ) I ( u ) ,

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