Abstract

Rotational transformation based on coordinate rotation in Fourier space is a useful technique for simulating wave field propagation between nonparallel planes. This technique is characterized by fast computation because the transformation only requires executing a fast Fourier transform twice and a single interpolation. It is proved that the formula of the rotational transformation mathematically satisfies the Helmholtz equation. Moreover, to verify the formulation and its usefulness in wave optics, it is also demonstrated that the transformation makes it possible to reconstruct an image on arbitrarily tilted planes from a wave field captured experimentally by using digital holography.

© 2008 Optical Society of America

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References

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2005 (3)

2004 (1)

L. Yu, Y. An, and L. Cai, “Numerical reconstruction of digital holography with changed viewing angles,” J. Electron. Imaging 13, 814-818 (2004).
[CrossRef]

2003 (1)

2001 (1)

2000 (1)

1998 (1)

1997 (1)

1993 (1)

1992 (1)

Alfieri, D.

An, Y.

L. Yu, Y. An, and L. Cai, “Numerical reconstruction of digital holography with changed viewing angles,” J. Electron. Imaging 13, 814-818 (2004).
[CrossRef]

Bianco, B.

Cai, L.

L. Yu, Y. An, and L. Cai, “Numerical reconstruction of digital holography with changed viewing angles,” J. Electron. Imaging 13, 814-818 (2004).
[CrossRef]

De Nicola, S.

Delen, N.

Ferraro, P.

Finizio, A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), chap. 3.10.

Hooker, B.

Leseberg, D.

Matsushima, K.

K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Appl. Opt. 44, 4607-4614 (2005).
[CrossRef] [PubMed]

K. Matsushima, “Exact hidden-surface removal in digitally synthetic full-parallax holograms,” Proc. SPIE 5742, 25-32(2005).
[CrossRef]

K. Matsushima, H. Schimmel, and F. Wyrowski, “Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves,” J. Opt. Soc. Am. A 20, 1755-1762 (2003).
[CrossRef]

T. Nakatsuji and K. Matsushima, “Free-viewpoint images captured using phase-shifting synthetic aperture digital holography,” Appl. Opt. 47 (2008), to be published.

Nakatsuji, T.

T. Nakatsuji and K. Matsushima, “Free-viewpoint images captured using phase-shifting synthetic aperture digital holography,” Appl. Opt. 47 (2008), to be published.

Patten, R.

Pierattini, G.

Schimmel, H.

Sheridan, J. T.

Tommasi, T.

Wyrowski, F.

Yamaguchi, I.

Yu, L.

L. Yu, Y. An, and L. Cai, “Numerical reconstruction of digital holography with changed viewing angles,” J. Electron. Imaging 13, 814-818 (2004).
[CrossRef]

Zhang, T.

Appl. Opt. (2)

J. Electron. Imaging (1)

L. Yu, Y. An, and L. Cai, “Numerical reconstruction of digital holography with changed viewing angles,” J. Electron. Imaging 13, 814-818 (2004).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Express (1)

Opt. Lett. (3)

Proc. SPIE (1)

K. Matsushima, “Exact hidden-surface removal in digitally synthetic full-parallax holograms,” Proc. SPIE 5742, 25-32(2005).
[CrossRef]

Other (2)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), chap. 3.10.

T. Nakatsuji and K. Matsushima, “Free-viewpoint images captured using phase-shifting synthetic aperture digital holography,” Appl. Opt. 47 (2008), to be published.

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Figures (12)

Fig. 1
Fig. 1

Definition of the coordinate systems and their rotation.

Fig. 2
Fig. 2

Example of sampling areas in three Fourier spaces: (a), (d) source, (b), (e) reference, and (c), (f) shifted reference Fourier spaces. The left-hand column (a)–(c) is the case of a spectrum sampled in an equidistant grid within the square area in the source space, whereas the right-hand column (d)–(f) is for equidistant and square sampling in a shifted reference space. This example is for T = R x ( 20 ° ) R y ( 30 ° ) .

Fig. 3
Fig. 3

Schematics for capturing the wave field from a tilted plane.

Fig. 4
Fig. 4

Photograph of the pattern printed on the planar object.

Fig. 5
Fig. 5

Schematic illustration of an aliasing-free zone.

Fig. 6
Fig. 6

Experimental setup for recording phase-shifting lensless Fourier digital holograms: L, lens; M, mirror; BS, beam splitter; SF, spatial filter.

Fig. 7
Fig. 7

Procedure for numerical reconstruction of images on the planar object from the complex image captured on the image sensor.

Fig. 8
Fig. 8

(a) Close-up photograph of the object and (b) the amplitude image | f ( x , y , z R ) | numerically reconstructed in the plane parallel to the image sensor.

Fig. 9
Fig. 9

Amplitude image of the wave field calculated in the tilted plane by rotational transformation.

Fig. 10
Fig. 10

Amplitude images (a) | f ( x , y , z R ) | in the parallel plane and (b) | f ˆ ( x ˆ , y ˆ , 0 ) | in the tilted plane reconstructed by using rotational transformation for two-axis rotation. The planar object is rotated at 60 ° around the x axis prior to rotation at 30 ° around the y axis.

Fig. 11
Fig. 11

Procedure for compensation for tilting a sensor surface by rotational transformation.

Fig. 12
Fig. 12

Tomographic imaging of a transparent specimen by rotational transformation. P n and S n are planes parallel and nonparallel to the image sensor, respectively.

Tables (1)

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Table 1 Rotation Matrices

Equations (24)

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r ˆ = Tr ,
r = T 1 r ˆ .
T = R ξ ( θ ξ ) R η ( θ η ) ,
R ξ 1 ( θ ξ ) = R ξ ( θ ξ ) = R ξ t ( θ ξ ) ,
T 1 = T t .
T 1 = [ a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 ] ,
T = [ a 1 a 4 a 7 a 2 a 5 a 8 a 3 a 6 a 9 ] .
f ( x , y , z ) = + F ( u , v ) exp [ i 2 π { u x + v y + ( λ 2 u 2 v 2 ) 1 / 2 z } ] d u d v = F 1 { F ( u , v ) exp [ i 2 π w ( u , v ) z } ,
w ( u , v ) = ( λ 2 u 2 v 2 ) 1 / 2 ,
f ˆ ( x ˆ , y ˆ , 0 ) = f ( a 1 x ˆ + a 2 y ˆ , a 4 x ˆ + a 5 y ˆ , a 7 x ˆ + a 8 y ˆ ) = F ( u , v ) exp [ { i 2 π [ ( a 1 u + a 4 v + a 7 w ) x ˆ + ( a 2 u + a 5 v + a 8 w ) y ˆ ] } ] d u d v ,
f ˆ ( x ˆ , y ˆ , 0 ) = F ( α ( u ˆ , v ˆ ) , β ( u ˆ , v ˆ ) ) exp [ i 2 π ( u ˆ x ˆ + v ˆ y ˆ ) ] d u d v ,
u = α ( u ˆ , v ˆ ) a 1 u ˆ + a 2 v ˆ + a 3 w ˆ ( u ˆ , v ˆ ) , v = β ( u ˆ , v ˆ ) a 4 u ˆ + a 5 v ˆ + a 6 w ˆ ( u ˆ , v ˆ ) .
J ( u ˆ , v ˆ ) = α u ˆ β v ˆ α v ˆ β u ˆ .
f ˆ ( x ˆ , y ˆ , 0 ) = F ˆ ( u ˆ , v ˆ ) exp [ i 2 π ( u ˆ x ˆ + v ˆ y ˆ ) ] d u ˆ d v ˆ = F 1 { F ˆ ( u ˆ , v ˆ ) } ,
F ˆ ( u ˆ , v ˆ ) = F ( a 1 u ˆ + a 2 v ˆ + a 3 w ˆ ( u ˆ , v ˆ ) , a 4 u ˆ + a 5 v ˆ + a 6 w ˆ ( u ˆ , v ˆ ) ) | J ( u ˆ , v ˆ ) | ,
J ( u ˆ , v ˆ ) = ( a 2 a 6 a 3 a 5 ) u ˆ w ˆ ( u ˆ , v ˆ ) + ( a 3 a 4 a 1 a 6 ) v ˆ w ˆ ( u ˆ , v ˆ ) + ( a 1 a 5 a 2 a 4 ) .
F ˆ ( u ˆ , v ˆ ) = F ˆ ( u ˜ + u ˆ 0 , v ˜ + v ˆ 0 ) = F ˜ ( u ˜ , v ˜ ) .
u = α ( u ˜ + u ˆ 0 , v ˜ + v ˆ 0 ) ,
v = β ( u ˜ + u ˆ 0 , v ˜ + v ˆ 0 ) .
F 1 { F ˜ ( u ˜ , v ˜ ) } = F 1 { F ˆ ( u ˜ + u ˆ 0 , v ˜ + v ˆ 0 ) } = F 1 { F ˆ ( u ˜ , v ˜ ) } exp [ i 2 π ( u ˆ 0 x ˆ + v ˆ 0 y ˆ ) ] .
f ˆ ( x ˆ , y ˆ , 0 ) = F 1 { F ˜ ( u ˜ , v ˜ ) } exp [ i 2 π ( u ˆ 0 x ˆ + v ˆ 0 y ˆ ) ] .
F ˜ ( u ˜ , v ˜ ) F ( α ( u ˜ + u ˆ 0 , v ˜ + v ˆ 0 ) , β ( u ˜ + u ˆ 0 , v ˜ + v ˆ 0 ) ) | J ( u ˜ + u ˆ 0 , v ˜ + v ˆ 0 ) | .
w 4 λ d 16 δ 2 λ 2 ,
f ( x , y , z = z R ) = F { g ( x 0 , y 0 ) } u = x / λ z R , v = y / λ z R ,

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