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OPTICS LETTERS / Vol. 29, No. 14 / July 15, 2004
Algorithm for reconstruction of digital holograms with adjustable magnification Fucai Zhang and Ichirou Yamaguchi Department of Electronics, Gunma University, Kiryu, Gunma 376-8515, Japan
L. P. Yaroslavsky Department of Interdisciplinary Studies, Tel Aviv University, Tel Aviv 69978, Israel Received February 25, 2004 A new algorithm that allows for reconstruction of digital holograms with adjustable magnification is proposed. The algorithm involves two reconstruction steps implemented by a conventional single Fourier-transform algorithm. The advantages of the algorithm lie in its adaptability to various object sizes and recording distances as well as in its capability to maintain the pitch of a reconstructed image, independent of the reconstruction distance and wavelength for objects larger than a CCD. The feasibility of the algorithm is demonstrated by experiments. The algorithm is especially useful for reconstructing color holograms and for metrological applications. © 2004 Optical Society of America OCIS codes: 090.1760, 100.3010, 090.0090.
Image reconstruction from electronically recorded holograms has attracted considerable attention during the past several years. So far, a number of reconstruction algorithms have been developed. Among them the single Fourier-transform- (SFT-) based algorithm and the convolution-based (CV) algorithm are most commonly used.1 – 3 The SFT algorithm is fast and can be used with objects larger than a CCD. However, variation of the pitch of the reconstructed image as result of a change in the reconstruction depth poses problems in applications such as reconstruction of color holograms4 and particle sizing.5 In contrast, the CV algorithm keeps the pitch of the reconstructed image the same as that of the CCD. However, it is applicable only to objects that are smaller than the CCD. In addition, when the CV algorithm is used for objects much smaller than a CCD, it degrades the image quality, since the image is represented by only a small number of pixels. In Refs. 2 and 4 the CV algorithm was extended to large objects by zero padding the holograms before reconstruction. This approach, however, led to an increase in computational load. In this Letter we present a cascaded algorithm for reconstruction of digital holograms with a variable factor. We present a one-dimensional analysis for conciseness. For Fresnel holograms the reconstruction process is described by a Fresnel integral that allows for interpretation from a different point of view. First, the integral can be treated as a conventional Fourier transform. If we denote the complex amplitude of the object beam at the CCD by U 共x兲, then the reconstructed complex amplitude is given by3
UI 共X, Z兲 苷 exp共2ipX 2 兾lZ兲
Z
CCD and the observation plane, respectively. Second, the Fresnel integral can also be interpreted as a description of a linear system with an impulse response of exp共2ipx2 兾lZ兲. Therefore rewriting Eq. (1) in the spatial frequency domain gives us UI 共j, Z兲 苷 U 共j兲exp共iplZj 2 兲 ,
(2)
where j is the spatial frequency and U 共j兲 and UI 共j, Z兲 are the Fourier transforms of U 共x兲 and UI 共X, Z兲, respectively. Here, some constant is neglected. From linear system theory it is possible to decompose the system into a cascade of two subsystems: U1 共j, z1 兲 苷 U 共j兲exp共iplz1 j 2 兲 ,
(3a)
U1 共j, Z兲 苷 U1 共j兲exp共iplz2 j 兲 ,
(3b)
2
with z1 1 z2 苷 Z. If we transform Eqs. (3a) and (3b) back into the spatial domain and represent them in the form of Eq. (1), f inally, we obtain a representation for UI 共X, Z兲 involving two integrals:
U 共x兲exp共2ipx2 兾lZ兲
3 exp共2ipXx兾lZ兲dx .
(1)
where l is the wavelength, Z is the reconstruction distance, and x and X denote the coordinates at the 0146-9592/04/141668-03$15.00/0
Fig. 1. Schematic illustration of the DBFT algorithm. © 2004 Optical Society of America
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∏ ∑ N UI 关l兴 苷 N 21 exp ip共m1 1 m2 兲 2
µ ∂ ∂ µ X2 Z X1 2 Z UI 共X, Z兲 苷 exp 2ip exp 2ip lz2 lz1 z2 ∂ ∂ µ XX1 Z x2 3 exp 2ip U 共x兲exp 2ip lz2 lz1
# ∂ µ N 2 3 exp 2ip l 2 共m1 2 m兲m1 兾共mN 兲 2 ipl 2
∂ µ X1 x dxdX1 , 3 exp i2p lz1
3
∑
µ
(4)
where X1 is the coordinate at the intermediate plane (refer to Fig. 1). The numerical implementation of Eqs. (1) and (4) includes replacement of the continuous complex amplitudes with their sampled values and calculation of the Fourier integral with a discrete Fourier transform. Although the analytical forms are equivalent, their discrete forms exhibit quite different characteristics, especially in terms of scaling performance. Suppose that the CCD sensor has N pixels in the x direction with a pitch pH (where the subscript H is the hologram plane) and introduce a focal parameter m 苷 NpH 2 兾lZ. Then the numerical implementation of Eq. (1) (the SFT algorithm) takes the form
3
∑
µ
N U 关m兴exp 2ipm m 2 3 2 m苷0
∂2
兾N
DX2 苷 jz2 兾z1 jpH 苷 pH j共m1 2 m兲兾mj .
(8)
From Eq. (8) it is easy to see that one can control the pitch by adjusting z1 . The dependence of the reconstructed image pitch on wavelength and reconstruction distance can be eliminated if we set (9)
where SH 苷 NpH is the side length of the CCD in the x direction; Sview is the desired side length of the reconstruction view. Specifically, if m1 is equal to zero, then Eq. (7) can be reduced to
∏
(5)
∂ ∑ µ ∏ N 21 1 X N 2 兾共Nm0 兲 exp ip n 2 UI 关l兴 苷 N n苷0 2
where m 苷 sign共m兲 and U 关m兴 is the sampled version of U 共x兲, which is assumed to be derived from the optical holograms by the phase-shifting technique.3,6 Here the sampling interval is assumed to be positive in both domains and m is used to take into account the propagation direction of the wave front. The introduction of N兾2 in both domains can keep the origin in the focal plane invariant with respect to the reconstruction distance.7 The pitch in the observation plane is equal to lZ兾NpH 苷 pH 兾m. We need to keep the side length of object SO smaller than that of the view def ined by Sv 苷 lZ兾pH to avoid the aliasing effect. Two other auxiliary focal parameters are then introduced: m2 苷 NDX1 2 兾lz2 ,
(7)
Now, the pitch at the observation plane equals
m1 苷 共1 6 Sview 兾SH 兲m ,
µ ∂µ ∂ ∏ ∑ N N l2 兾N , 3 exp 2ipm m 2 2 2
m1 苷 NpH 2 兾lz1 ,
∂ µ ∑ ∏ N 2 U 关m兴exp 2ipm1 m 2 兾N 2 ipm 2 m苷0
N 21 X
3 exp共2ip m ¯ 1 mn兾N兲exp共2ipm2 nl兾N兲 .
∂ ∑ µ ∏ N 2 兾共Nm兲 UI 关l兴 苷 N 21兾2 exp 2ip l 2 2 N 21 X
∂ Ω µ æ N 2 exp 2ip n 2 兾关N共m1 2 m兲兴 2 n苷0
N 21 X
∂ ∏ µ N 兾N U 关m兴exp 2ipm n 2 3 2 m苷0 ∑
NX 21
µ ∂ ∏ N 3 exp 22ipl n 2 兾N , 2 ∑
(10)
(6)
where DX1 苷 pH 兾m1 is the sampling pitch at the intermediate plane. Then the numerical implementation of Eq. (4), hereafter referred to as the double Fresnel-transform algorithm (DBFT), is given by
Fig. 2. text.
Experimental setup.
Abbreviations def ined in
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OPTICS LETTERS / Vol. 29, No. 14 / July 15, 2004
Fig. 3. Images reconstructed by four algorithms at various reconstruction distances.
which is the conventional CV algorithm. Therefore the CV algorithm is a special case of the DBFT algorithm in which the intermediate plane is located at infinity. The physical interpretation of Eq. (7) is illustrated in Fig. 1: The complex field in the CCD plane propagates first to an intermediate plane at a distance z1 and then to the observation plane at a distance z2 . Distance z1 need not be positive and can have an absolute value larger than Z. The described DBFT algorithm was verif ied experimentally via reconstruction of optical holograms. The experimental setup is shown in Fig. 2. A He –Ne laser beam was coupled into a 50兾50 optical fiber directional coupler (FC). The beam from one fiber ref lected by the beam splitter (BS) was used to illuminate the test object. The test object, composed of bars printed on a white paper sheet, was tilted with an angle of 13± to the normal of the CCD plane. The separation between bars was 3 mm. The light from the other fiber ref lected by a mirror (M1) mounted on a piezoelectric transducer (PZT) was directed onto the CCD as a reference beam. The CCD had 1024 3 1024 pixels with a pitch of 6.45 mm 3 6.45 mm. Four holograms with phase steps of p兾2 were recorded. The complex amplitude U 关m兴 was calculated from the holograms by the four-step phase-shifting algorithm. The reconstruction algorithm was implemented in Matlab and took 2.7 s with the SFT algorithm and 3.3 s with the DBFT algorithm on a Pentium IV 1.6-GHz computer. For the SFT algorithm the side lengths of the views corresponding to reconstruction distances of 100, 125, and 150 mm were 9.8, 12.26, and 14.72 mm. For the
DBFT algorithm the side length of the view, Sview , was set to 1 cm. Distances z1 corresponding to the whole reconstruction distances 100, 125, and 150 mm were 194, 242.6, and 291.2 mm. Reconstruction with the CV algorithm with and without zero padding was also carried out. Since the object was larger than the CCD 共6.6 mm 3 6.6 mm兲, the image reconstructed by the CV algorithm was distorted, as shown in the top row of Fig. 3. In the implementation with zero padding (CVZP), the holograms were first padded to 2048 3 2048 pixels, which resulted in a view size of 1.32 cm 3 1.32 cm. The calculation in this case took 55 s. Figure 3 clearly shows that the images obtained with the DBFT algorithm keep the same scale at the focal planes indicated by the vertical dashed lines. The DBFT algorithm also proved to perform well for the reconstruction of color holograms used in Ref. 4. In conclusion, a two-stage reconstruction algorithm has been proposed for digital holography. Through the adjustment of the distance parameter in the first stages, it is possible to control the pitch of the reconstructed images, independent of distance and wavelength, even for objects larger than a CCD without any computational penalty. In particular, the algorithm facilitates combination of the three-color components for color digital holograms. Furthermore, in metrological applications the algorithm makes it possible to measure small deformations by directly subtracting phase maps of two images with different reconstruction distances. For instance, in the investigation of the deformation of a small object, unavoidable longitudinal motion leads to a larger variation in the reconstruction distance, especially when a microscope objective with high magnification is employed.8 When the SFT algorithm is used to reconstruct well-focused images, the scales will be different. Consequently, it does not allow extraction of quantitative information on the deformation by direct subtraction of the phase maps. The proposed algorithm eliminates such diff iculty, as it offers new potentials in digital microscopic holographic interferometry. The suggested two-stage algorithm can also be extended further to multiple stages that may add to its f lexibility and applicability. F. Zhang’s e-mail address is
[email protected]. References 1. U. Schnars, J. Opt. Soc. Am. A 11, 2011 (1994). 2. T. Kreis, M. Adams, and W. Jüptner, Proc. SPIE 3098, 224 (1997). 3. I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, Appl. Opt. 40, 6177 (2001). 4. I. Yamaguchi, T. Matsumura, and J. Kato, Opt. Lett. 27, 1108 (2002). 5. G. Pan and H. Meng, Appl. Opt. 42, 827 (2003). 6. I. Yamaguchi and T. Zhang, Opt. Lett. 22, 1268 (1997). 7. L. P. Yaroslavsky, Digital Holography and Digital Image Processing (Kluwer Academic, Boston, Mass., 2004), pp. 199– 209. 8. P. Ferraro, G. Coppola, S. Nicola, A. Finizio, and G. Pierattini, Opt. Lett. 28, 1257 (2003).
