Abstract

Integral photography offers an interesting alternative to holography for recording and displaying 3-dimensional information. This paper derives the optimum size of the lenslet in the lenticular screen and derives a resolution limitation for integral photography which for conventional objects is comparable to the resolution of a TV picture. The number of resolvable spots needed to record an integral photograph is also computed and it is shown that this number is proportional to the fourth power of the (linear) resolution in the reconstruction.

© 1968 Optical Society of America

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References

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  1. G. Lippmann, J. Phys. théorique et appliquée 7, 821 (1908).
    [CrossRef]
  2. H. E. Ives, J. Opt. Soc. Am. 21, 171 (1931).
    [CrossRef]
  3. R. V. Pole, Appl. Phys. Letters 10, 20 (1967).
    [CrossRef]
  4. R. L. De Montebello, private communication.
  5. G. A. Campbell and R. M. Foster, Fourier Integrals for Practical Applications (D. Van Nostrand Company, Inc., New York, 1961).
  6. G. A. Fry, in Applied Optics and Optical Engineering, Vol. II, R. Kingslake, Ed. (Academic Press Inc., New York, 1965).
  7. If b=0 we obtain Nmax=∞, which, of course, cannot be true. This singularity occurs because then, according to Eq. (14), φopt=∞ and the approximations used for computing the spread function break down.

1967 (1)

R. V. Pole, Appl. Phys. Letters 10, 20 (1967).
[CrossRef]

1931 (1)

1908 (1)

G. Lippmann, J. Phys. théorique et appliquée 7, 821 (1908).
[CrossRef]

Campbell, G. A.

G. A. Campbell and R. M. Foster, Fourier Integrals for Practical Applications (D. Van Nostrand Company, Inc., New York, 1961).

De Montebello, R. L.

R. L. De Montebello, private communication.

Foster, R. M.

G. A. Campbell and R. M. Foster, Fourier Integrals for Practical Applications (D. Van Nostrand Company, Inc., New York, 1961).

Fry, G. A.

G. A. Fry, in Applied Optics and Optical Engineering, Vol. II, R. Kingslake, Ed. (Academic Press Inc., New York, 1965).

Ives, H. E.

Lippmann, G.

G. Lippmann, J. Phys. théorique et appliquée 7, 821 (1908).
[CrossRef]

Pole, R. V.

R. V. Pole, Appl. Phys. Letters 10, 20 (1967).
[CrossRef]

Appl. Phys. Letters (1)

R. V. Pole, Appl. Phys. Letters 10, 20 (1967).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. théorique et appliquée (1)

G. Lippmann, J. Phys. théorique et appliquée 7, 821 (1908).
[CrossRef]

Other (4)

R. L. De Montebello, private communication.

G. A. Campbell and R. M. Foster, Fourier Integrals for Practical Applications (D. Van Nostrand Company, Inc., New York, 1961).

G. A. Fry, in Applied Optics and Optical Engineering, Vol. II, R. Kingslake, Ed. (Academic Press Inc., New York, 1965).

If b=0 we obtain Nmax=∞, which, of course, cannot be true. This singularity occurs because then, according to Eq. (14), φopt=∞ and the approximations used for computing the spread function break down.

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Figures (5)

Fig. 1
Fig. 1

Recording an integral photograph.

Fig. 2
Fig. 2

Reconstruction of the real image from an integral photograph.

Fig. 3
Fig. 3

Geometry for making the integral photograph.

Fig. 4
Fig. 4

Space between lens and photographic emulsion c: image distance for central plane of the object c′: image distance for marginal plane of the object.

Fig. 5
Fig. 5

Solid curve: Fourier transform of rectangular window. Broken curve: gaussian approximation.

Equations (45)

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A ( w ) = K ξ = - φ / 2 + φ / 2 exp ( j k ξ 2 / 2 c ) exp ( - j k R ) d ξ .
A ( w ) = K ξ = - φ / 2 + φ / 2 exp ( j k ξ 2 / 2 c ) × exp [ - ( i k / 2 c ) ( w 2 - 2 w ξ + ξ 2 ) ] d ξ .
c = c + Δ .
Δ = ( c 2 / a 2 ) b .
A ( w ) = K ξ = - φ / 2 + φ / 2 exp ( - j π b ξ 2 / λ a 2 ) exp ( j 2 π w ξ / λ c ) d ξ ,
v = w / λ c ,
A ( v ) = K ξ = - φ / 2 + φ / 2 exp ( - j π b ξ 2 / λ a 2 ) exp ( j 2 π v ξ ) d ξ .
y 1 = sin ( π v φ ) / ( π v φ ) .
y 2 = C exp ( j π λ a 2 v 2 / b ) ,
y 1 exp ( - 2.04 v 2 φ 2 ) .
A ( v ) = y 1 * y 2 = exp [ - π v 2 2.04 φ 2 λ a 2 ( π λ a 2 - 2.04 j b φ 2 ) π 2 λ 2 a 4 + 4.16 b 2 φ 4 ] ,
I ( v ) = A ( v ) A * ( v ) = exp [ - π 2 v 2 4.08 φ 2 λ 2 a 4 π 2 λ 2 a 4 + 4.16 b 2 φ 4 ] .
I ( w ) = exp [ - w 2 4.08 π 2 φ 2 λ 2 a 4 λ 2 c 2 ( π 2 λ 2 a 4 + 4.16 b 2 φ 4 ) ] .
φ opt = 1.24 a ( λ / b ) 1 2 .
I ( w ) = exp ( - 4 w 2 / u I 2 ) .
u I 2 = 0.415 φ 2 b 2 c 2 a 4 + 0.981 λ 2 c 2 φ 2 .
u I = 1.13 ( c / a ) ( b λ ) 1 2             for             φ = φ opt .
y 0 = exp ( - 4 x 2 / u 0 2 )
u 0 = 1.13 ( b λ ) 1 2 .
y tot = exp ( - 4 x 2 / u tot 2 )
u tot = ( 2 ) 1 2 u 0 = 1.59 ( b λ ) 1 2 .
δ = u tot / a = 1.59 ( b λ ) 1 2 / a .
B / a ψ ,
δ = 1.59 ψ ( λ b ) 1 2 / B .
N max = ψ / δ = B / 1.59 ( λ b ) 1 2 .
tan ( ψ max / 2 ) = φ / 2 c ,
ψ max = 2 arc tan ( φ / 2 c ) φ / c .
y I I = K exp ( - 4 x 2 / u I I 2 ) .
u I I 2 = 3 u I 2 = 1.25 φ 2 b 2 c 2 a 4 + 2.95 λ 2 c 2 φ 2 ,
u final 2 = 4 ( a / c ) 2 u I 2 = 1.66 b 2 φ 2 a 2 + 3.93 λ 2 a 2 φ 2 .
N = B / u final ,
P = D 2 / u I I 2 ,
P = D 2 / u I I 2 = 4 a 2 D 2 / 3 u final 2 c 2 ,
P = ( 4 a 2 D 2 N 4 / 3 c 2 B 4 ) u final 2 .
a / c = B / φ .
P = 2.21 D 2 N 4 b 2 a 2 B 2 ( 1 + 2.37 λ 2 a 4 φ 4 b 2 ) .
Ω ( D / a ) ,
P = 2.21 Ω 2 N 4 b 2 B 2 ( 1 + 2.37 λ 2 a 4 φ 4 b 2 ) .
P b = 0 = 5.24 Ω 2 N 4 λ 2 a 4 / B 2 φ 4 .
P b = 0 = 5.24 Ω 2 N 4 λ 2 a 4 / B 2 D 4 .
u final 2 , b = 0 = 3.93 λ 2 a 2 φ 2 = 3.93 λ 2 a 2 D 2 ,
N b = 0 2 = B 2 D 2 / 3.93 λ 2 a 2 .
P b = 0 = 4 3 N 2 .
P = 2.21 Ω 2 N 4 b 2 B 2 ,             for             N N max .
P = 4.41 Ω 2 N 4 b 2 B 2 ,             for             N = N max .