Abstract

The primary purpose of this paper is to propose a comprehensive optimum-design theory of lenticular-sheet three-dimensional pictures. The proposed theory features the use of depth resolution of a 3–D image as the measure of the 3–D picture quality. The optimum parameters in the picture taking process, the optimum lens pitch and the depth-resolution limitation, are discussed. The obtained results are also applicable to a specific type of integral photography and to the projection-type 3–D display including projection-type holography. It is found that the optimum pitch of the lens sheet or the lens-type direction selective screen ranges between 0.1 mm and 0.5 mm in most cases, whereas it ranges between 0.2 mm and 1 mm for the triple-mirror screen.

© 1971 Optical Society of America

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References

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  1. C. B. Burckhardt, J. Opt. Soc. Am. 58, 71 (1968).
    [CrossRef]
  2. The term lens sheet in the title is used generically in this paper for both LS and IP.
  3. T. Okoshi, A. Yano, Y. Fukumori, Appl. Opt. 10, 482 (1971).
    [CrossRef] [PubMed]
  4. T. Okoshi, A. Yano, Opt. Commun. 3, 85(1971).
    [CrossRef]
  5. The symbols are chosen to facilitate the comparison with related papers such as Refs. 1, 2, and 3.
  6. In practice, some part of the image (usually 10–40%) is reconstructed in front of the lenticular sheet. It is only for mathematical simplicity that in Fig. 2 the whole image is assumed to be reconstructed behind the sheet.
  7. An alternative, probably more reasonable definition will be Δ′=(Δs2+Δd2)12. The definition in the text (algebraic sum) is used here only for its simplicity. The difference from the second definition only makes the estimation of the over-all resolution a little more pessimistic. It is easy to show that the ratio (Δ/Δ′) is between 1 and √2.
  8. C. B. Burckhardt, R. J. Collier, E. T. Doherty, Appl. Opt. 7, 627 (1968).
    [CrossRef] [PubMed]
  9. In a triple mirror without curvature, the six reflected components are in the same phase with each other for any direction of incidence. Therefore, the theoretical diffraction limitation is smaller than in a curved triple mirror.

1971

1968

Appl. Opt.

J. Opt. Soc. Am.

Opt. Commun.

T. Okoshi, A. Yano, Opt. Commun. 3, 85(1971).
[CrossRef]

Other

The symbols are chosen to facilitate the comparison with related papers such as Refs. 1, 2, and 3.

In practice, some part of the image (usually 10–40%) is reconstructed in front of the lenticular sheet. It is only for mathematical simplicity that in Fig. 2 the whole image is assumed to be reconstructed behind the sheet.

An alternative, probably more reasonable definition will be Δ′=(Δs2+Δd2)12. The definition in the text (algebraic sum) is used here only for its simplicity. The difference from the second definition only makes the estimation of the over-all resolution a little more pessimistic. It is easy to show that the ratio (Δ/Δ′) is between 1 and √2.

The term lens sheet in the title is used generically in this paper for both LS and IP.

In a triple mirror without curvature, the six reflected components are in the same phase with each other for any direction of incidence. Therefore, the theoretical diffraction limitation is smaller than in a curved triple mirror.

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Figures (16)

Fig. 1
Fig. 1

(a) Primary picture taking using N cameras. (b) Secondary picture taking using N projectors.

Fig. 2
Fig. 2

Symbols used in the derivation of the scaling rule.

Fig. 3
Fig. 3

Expansion and compression of the image depth.

Fig. 4
Fig. 4

(a) The flipping observed for the spots reproduced inside the lenticular sheet. (b) The defocus observed for the spots reproduced inside the lenticular sheet.

Fig. 5
Fig. 5

Factors affecting the lateral resolution upon the lenticular sheet shown as functions of d2. The optimum value of d2 and the obtainable resolution are also illustrated.

Fig. 6
Fig. 6

Light flux generated at a spot upon the emulsion.

Fig. 7
Fig. 7

(a) Symbols used in the analysis of the diffraction. (b) Angular spread due to diffraction.

Fig. 8
Fig. 8

Aberration in a cylindrical-spherical lenslet.

Fig. 9
Fig. 9

Spread of the focus due to aberration.

Fig. 10
Fig. 10

Angular spread due to aberration.

Fig. 11
Fig. 11

Depth-resolution limitation due to a finite lens pitch.

Fig. 12
Fig. 12

Depth-resolution limitation due to a finite directivity of the lens sheet.

Fig. 13
Fig. 13

Over-all depth resolution NR as a function of the lens pitch and directivity for a2 = 1 m and B2 = ∞. The broken line is the symmetry axis. The contours of NR for other values of a2 and B2 can be obtained by translations.

Fig. 14
Fig. 14

Realizable region in the pβ diagram. Point P′ gives the optimum design. Point Q gives the optimum design in the case β = βD. The symbol ⊕ denotes a convolution.

Fig. 15
Fig. 15

Six effective reflecting areas of a triple mirror.

Fig. 16
Fig. 16

Diffraction limitations of the LS, IP, and CTM (curved triple-mirror screen). The measured directivities are also shown for comparison.

Tables (3)

Tables Icon

Table I Factors Affecting Directivity and Their Dependence on Lens Pitch p

Tables Icon

Table II Optimum Lens Pitch and Maximum Resolution of Lenticular Sheet and Triple-Mirror Screena

Tables Icon

Table III Measured Directivity (Half-intensity Spread) of Lenticular Sheetsa

Equations (35)

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α = d 2 / d 1 ,
β = f 2 / f 1 ,
γ = a 2 / a 1 ,
b 2 / a 2 = ( b 1 / a 1 ) · { 1 / [ ( b 1 / a 1 ) ( K - 1 ) + K ] } .
w 2 / a 2 = ( w 1 / a 1 ) · ( 1 / β ) ,
( b 2 / w 2 ) = ( b 1 / w 1 ) · { β / [ ( b 1 / a 1 ) ( K - 1 ) + K ] } .
K = α β / γ .
F = d 2 b 2 / ( b 2 + a 2 ) .
Δ f = f 1 - f 1 f 1 2 b 1 / a 1 ( a 1 + b 1 ) ,             δ = D 1 Δ f / f 1 .
k 1 = D 1 / d 1 ,             k 2 = D 2 / d 2 .
S = ( a 2 / f 2 ) δ = ( a 2 / f 2 ) [ ( D 1 / f 1 ) Δ f ] k 1 d 2 b 2 / ( a 2 + b 2 ) = k 1 F .
G = s a 2 / f 2 s w 2 / h = s w 2 / m d 2             ( m = h / d 2 < 1 ) .
δ D = 1.22 f 1 λ / D 1 ,
R = 2 a 2 δ D / f 2 = 1.22 2 λ a 2 f 1 / k 1 d 1 f 2 ,
R = 1.22 2 a 2 2 / a 1 k 1 d 2 .
d 2 opt = ( s w 2 ( a 2 + b 2 ) / m b 2 ) 1 2
( S 2 + G 2 ) min 1 2 = ( 2 s w 2 b 2 / m ( a 2 + b 2 ) ] 1 2 .
A ( x ) A 1 [ sin ( π p θ / λ ) / ( π p θ / λ ) ] A 1 [ sin ( π p x / t λ ) / ( π p x / t λ ) ] ,
A ( x ) = A 1 exp [ - ( 2 x / w D ) 2 ] ,
w D = 2 t λ / ( 2.04 ) 1 2 p .
β D = 2 w D n / t = [ 2 2 / ( 2.04 ) 1 2 ] ( λ / p ) 2.0 λ / p ,
β E = n w E / t = n w E M / p             ( M = p / t ) ,
β L = D 2 / a 2 = k 2 d 2 / a 2 .
Δ s 2 p ( a 2 + y 2 ) 2 / a 2 p 0 .
Δ d = 2 β y 2 ( a 2 + y 2 ) / P 0 ,
Δ = Δ s + Δ d = ( 2 p a 2 / P 0 ) ( 1 + Y 2 ) { 1 + Y 2 [ 1 + ( β a 2 / p ) ] } ,
N R = 0 b 2 ( 1 / Δ ) d y 2 = 0 B 2 ( a 2 / Δ ) d Y 2 = ( P 0 / 2 a 2 β ) log { 1 + ( a 2 β / p ) · [ B 2 / ( 1 + B 2 ) ] } ,
β D p - 1 ,             β E p - 1 ,             β A p 0 ,             and             β L p 0 .
p opt = 0.35 mm             and             N R max = 6.5.
β = β D = k λ / p             ( k = constant )
p opt = { ( k λ a 2 / 3.92 ) [ B 2 / ( 1 + B 2 ) ] } 1 2 .
N R max = log 4.92 ( P 0 p opt / 2 k λ a 2 ) = 0.0251 { ( 1 / k λ a 2 ) × [ B 2 / ( 1 + B 2 ) ] } 1 2 ( all lengths in m ) .
p opt = 0.71 ( λ b 2 ) 1 2
D = ( 2 / 3 ) / 4 5 ( π 1 2 ) p = 0.287 p ,
β D = [ 2 × 1.22 / ( 2.04 ) 1 2 ] ( λ / D ) = 1.70 λ / D 5.9 λ / p ,

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