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3D locations of the object directly from in-line holograms using the Gabor transform Yan Zhang1, De-Xiang Zheng, Jing-Ling Shen, Cun-Lin Zhang Department of Physics, Capital Normal University Xisanhuan Beilu 105, Beijing 100037 China ABSTRACT In-line holography is a simple way for small object imaging. The Gabor transform is used for object characterization directly from in-line holograms. Three-dimensional locations of an object can be determined from the Gabor transform spectrum of its in-line hologram. The relationship between the Gabor angle and the location of the object is established, Computer simulations and experimental result have been presented to demonstrate the validity of this method for object charactering. The promising application of this method is particle field analysis. Keywords: Digital holography and the Gabor transform 1. INTRODUCTION Digital holography was proposed by Goodman et al. in 1960s1. From then on, this technology was constantly developed, allowing acquisition of three-dimensional information about an object by using a CCD camera and a computer. In–line digital holography 2 uses the same beam as the reference wave and object illumination simultaneously, its advantages lie in the simple experimental set-up, higher signal-to-noise rate (SNR), and providing a simple way for short wavelength imaging which is an ideal approach for high-resolution imaging, especially for some wavelengths for which there is no suitable lens or beam splitter. The most promising application of the in-line holography is particle field analysis, such as the detection and measurement of fog droplets, study of smoke, and research of other kinds of dynamic airborne aerosols 3. Besides these, its applications also involve deformation analysis 4, object contouring 5, microscopy 6, and particle measurement 7. With the improvement of the spatial resolution of CCD cameras and the increasing of computational performance of the personal computer, digital reconstruction of holograms has been introduced into practice, and many reconstruction methods have been explored for extracting information of small particles from the in-line hologram. The Wigner distribution function 8-10 is a time-frequency operator that used for the analysis of non-stationary signals, and yielding the time variation of the frequency components of a signal; it has been used to extract the information of the particle directly from the in-line holograms 3. Recently, the wavelet transform 11, 12 and the fractional Fourier transform13 are also used for analyzing the small particles and holograms reconstruction. The Gerchberg-Saxton (GS) 14, 15 and Yang-Gu (YG) 16 phase-retrieval algorithms are also employed as approaches for digital reconstruction, however, the prior information about the object should be known; furthermore, more processing time is cost due to the iterative property of the algorithms. The Gabor transform, which is also proposed by Gabor, is a time-frequency tool for signal processing, and can present both the frequency and time information of a signal simultaneously, but it has not been employed to extract information of the object from in-line holograms so far. In this work, the Gabor transform will be employed to characterize the diffraction pattern of the directly form in-line holograms. The relationship between the Gabor angle and the longitude distance of the object from the CCD is established. Computer simulations and experimental result are also presented. It is demonstrated that the Gabor transform is an effective mathematical tool for extracting information from in-line holograms. The presentation is organized as follows: in Section 2, we present the definition of the Gabor transform; in Section 3, we describe some computer simulations to show how to use the Gabor transform to extract information of the object from diffraction patterns; in Section 4, we present the experimental result abut 3D locations of the object by using the Gabor transform; and at last, we offer some concluding remarks in Section 5. 1 Further author information: (Send correspondence to Dr. Yan Zhang) Yan Zhang: Email: [email protected], Fax and Tel: 0086-10-68902178 116 Holography, Diffractive Optics, and Applications II, edited by Yunlong Sheng, Dahsiung Hsu, Chongxiu Yu, Byoungho Lee, Proceedings of SPIE Vol. 5636 (SPIE, Bellingham, WA, 2005) · 0277-786X/05/$15 · doi: 10.1117/12.570465 2. IN-LINE HOLOGRAPHY AND THE GABOR TRANSFORM In-line holograms record the far–field diffraction patterns of an object. The typical setup for in-line holograms recording is schematically shown in Fig. 1, the object with complex amplitude transmittance a(x, y) is coherently illuminated by a plane wave, and the light propagates along the z direction, a CCD located at distance d from the object records the diffraction pattern. Figure 1 Experimental setup for in-line hologram recording. According to the Fresnel approximation of the free space propagation, the complex field be given by: A( x, y ) = A(x, y) on the CCD plane can 1 i 2π d iπ exp( ) a ( x ', y ') × exp{ [( x − x ') 2 + ( y − y ') 2 ]}dx ' dy ', iλ d λ ∫∫ λd (1) where λ is the wavelength of the illuminating light, and d is the distance between the object and CCD camera. For the sake of the brevity, only one-dimensional (1D) case is discussed here, the two-dimensional (2D) case can be obtained directly. Thus Eq. (1) can be simplified as follows: A( x) = 1 iλ d exp( i 2π d λ ) ∫ a( x) × exp[ iπ ( x − x ' ) 2 ]dx′ . λd (2) The Gabor transform of a function A(x) is defined as 17: G ( x2 , v) = ∫ A( x1 )g ( x1 , x2 ) exp(−i 2π x1v )dx1 where v is the spatial frequency, and (3) g ( x1 , x 2 ) is a window function, usually, this function can be selected as a Gaussian function exp[−( x1 − x2 ) / b] , and b is Gaussian width, which play an important role in the description of the Gabor transform. The value of b should be selected to be suitable according to the size of object, and we choose b = 9 × 2000 µ m in this paper. 2 3. COMPUTER SIMULATIONS A one-dimensional particle is considered as the original object in following computer simulations and its size is 9.0 × 20 µm . The parameters of the system are selected as follows: The wavelength of the illuminating light is λ = 0.532 µm , the width of the CCD window is 9.0 × 1024 µm , and the number of the pixels is 1024. The distance between the particle and the CCD is selected as d = 140mm ,. The particle is located in the middle of the input plane. In calculations, the new hologram A( x) − mean[ A( x )] is used to replace the original hologram A( x ) , for the aim of eliminating the unimportant direct current, where mean[ A( x)] expresses the mean value of the A( x ) . Proc. of SPIE Vol. 5636 117 Figure 2(a) represents the hologram of the particle, which is recorded by the CCD camera, and Fig. 2(b) describes the corresponding Gabor transform spectrum G ( x2 , v) of the new hologram. The direct current (the bright line which parallels to the abscissa) is quite weak due to the pretreatment of the hologram. As shown in Fig. 2(b), the angle between the two bright lines is defined as the Gabor angle, and denoted by θ . It can be found that tan(θ / 2) has a good linear relationship with distance d , as shown by the discrete points in Fig. 3. The linear fitted curve has also been plotted in the same figure, the relationship between the distance d and tan(θ / 2) can be expressed as: (4) tan(θ / 2) = 2.80 × 10 −6 d + 2.28 × 10 −3. 1.1 0.05 0.04 0.03 0.02 -1 Frequency (µm ) Intensity (Arb.Unit) 1.05 1 0.95 0.01 0 -0.01 -0.02 -0.03 0.9 -0.04 -0.05 0.85 -5000 0 -4000 -3000 -2000 -1000 5000 0 1000 2000 3000 4000 X2 (µm) X1 (µm) Figure 2 (a) Hologram of a particle. (b) The Gabor transform spectrum of (a). 1.8 data linear 1.6 1.4 tan(θ/2) 1.2 1 0.8 0.6 0.4 0.2 0 0 1 2 3 d (µm) 4 5 6 5 x 10 Figure 3 Relationship between d and θ . Thus the distance d can be determined by measuring the Gabor angle in the Gabor transform spectrum of the hologram. The hologram of three different particles located at different place along the x coordinate is represented in Fig. 4(a), the sizes of particles are 9.0 × 8µm , 9.0 × 14µm , 9.0 × 20 µm from left to right, respectively. The distance between the particles and the CCD plane is d = 200mm . The corresponding Gabor transform spectrum is shown in Fig. 4(b), which can be easily found that the Gabor angle in this figure is larger than that in the Fig. 2(b). It can also be found that the positions of the particles correspond to the X coordinates of the corresponding cross points in the Gabor spectrum of its hologram. The intensity of the Gabor spectrum is different due to the different size, different position, and different absorption of the particles. Furthermore, the size of the particles can be determined from the Gabor spectrum of the hologram. 118 Proc. of SPIE Vol. 5636 4. EXPERIMENTAL RESULT The experiment has also been carried out to demonstrate the application of the Gabor transform. The setup is the same as it shown in Fig.1, the wavelength of the illuminating light is λ = 0.532 µ m , the CCD used in the experiment has 1018× 992 pixels with pixel size of 9.0 µm × 9.0µm , only a part of record hologram is used for processing. The recording distance between the object and the CCD is about 140 mm , and the digital hologram is transmitted to a computer via a frame grabber that performs 10 bits digitization. A hologram of a fiber recorded by the CCD is shown in Fig 5(a), however, there are many interference annulus caused by the tiny dust on the surface of the collimating lens, the illuminating light is not the ideal plane wave. As shown by the black line in Fig. 5(a), only 1D signal along the x direction is extracted as the location information of the fiber. The corresponding Gabor transform spectrum is shown in Fig. 5(b). The location of the fiber in the x direction can be directly extracted from the cross of the highlights. In order to determine the location of the fiber along z direction, the Gabor angle is measured, and the value tan(θ / 2) = 0.40 is calculated, according to the Eq. (4), the distance d can be obtained as d = 143µ m , which is in good agreement with experimental value d = 140 µm . 0.05 0.04 0.02 -1 Frequency (µm ) 0.03 0.01 0 -0.01 -0.02 -0.03 -0.04 -0.05 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 X2 (µm) Figure 4 (a) 1D hologram of three particles. From left to right, the sizes of the particles are 9.0 × 8µm , 9.0 × 14µm , and 9.0 × 20 µm , respectively, and (b) the Gabor transform spectrum of (a). Figure 5 (a) The experiment obtained hologram of a fiber and (b) the Gabor transform spectrum of the cross line in (a). Proc. of SPIE Vol. 5636 119 This result demonstrates that this method can effectively extract the location information of the object from the in-line hologram. However, the SNR is not as optimal as expected. This is due to that the input wave is not a standard plane wave which influences the SNR. Carefully collimating the input wave, clearing the optical elements, cooling the CCD, and summing up several frames of holograms can effectively reduce the noise. Generally speaking, it can be concluded that the location of the object in the z direction can be determined by calculating the Gabor angle well. 5. CONCLUSION The Gabor transform has been adapted for charactering small particles directly from its in-line hologram. For a given Gabor angle measured from the Gabor transform spectrum of the diffraction patterns recorded by a CCD camera, the three-dimensional locations of the object can be determined, i.e. the position of the particle in the x and y directions and the distance between the object and the CCD plane. The signal to noise rate can be improved by using CCD cameras with smaller pixel size and higher dynamical rang. Experimental result has also been presented to shown the validity of this method. 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