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Nonlinear filtering method of zero-order term suppression for improving the image quality in off-axis holography
Optics Communications 315 (2014) 232–237
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Nonlinear filtering method of zero-order term suppression for improving the image quality in off-axis holography Zhonghong Ma a, Yong Yang a,n, Qi Ge b, Lijun Deng a, Zhenxin Xu c, Xuna Sun c a Institute of Modern Optics, Nankai University, The Key Laboratory of Optical Information Science and Technology of the Education Ministry of China, Tianjin 300071, China b College of Zhonghuan Information, Tianjin University of Technology, Tianjin 300071, China c School of Electronical Engineering, Tianjin University of Technology, Tianjin 300072, China
art ic l e i nf o
a b s t r a c t
Article history: Received 20 September 2013 Received in revised form 31 October 2013 Accepted 12 November 2013 Available online 22 November 2013
In order to alleviate the crosstalk between the reconstructed image and the zero-order image in off-axis digital holography, this paper proposes a novel nonlinear filtering method to suppress the zero-order image. Its feasibility in eliminating the zero-order term is verified by the theoretical analyses, and its robustness is confirmed by applying the appropriately designed nonlinear filter to both a Fresnel hologram and an image plane hologram. Experimental results show that the image reconstructed from the proposed method has the characteristics of enhanced image contrast and improved signal-to-noise ratio. This method also accelerated the speed of processing via a simple calculation procedure. & 2013 Elsevier B.V. All rights reserved.
Keywords: Digital holography Digital image processing Image reconstruction techniques Nonlinear filtering method Zero-order term elimination
1. Introduction The off-axis holography [1] is used originally to separate the zero-order image and the two conjugate images called the virtual image and the real image. Confined by the pixel size [2] and pixel number [3] of the digital recording device, the zero-order image inevitably becomes a blurring to the recorded sample. It is necessary to eliminate the zero-order term for increasing the recorded bandwidth of the sample, and therefore improving the reconstructed image quality [4–7]. Several methods have been proposed to achieve this purpose in off-axis digital holography. By using experimental procedures, such as phase-shifting digital holography [8–12], the zero-order diffraction is effectively eliminated. However, it requires a precise vibration-free optical table during the hologram recording procedure. Since the multi-frame holograms are required [13], it also has some difficulties in dealing with the dynamic real-time detection of a moving target. Therefore the numerical suppression method [14] which needs neither multi-frame holograms nor high precision device is applied to digital holography. Many numerical methods have specific applicable conditions in coping with the zero-order term elimination. For example, the spatial filter window designed in Ref. [15] is only applicable to convolutional
n
Corresponding author. Tel.: þ 86 13072041093. E-mail address: [email protected] (Y. Yang).
0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.11.021
reconstruction of the hologram. A nonlinear reconstruction technique proposed by Pavillon et al. [16,17] is able to eliminate the zero-order term successfully, but it requires a larger reference amplitude than that of the object, and the image order needs to be confined in one quadrant of the Fourier spectrum. In this paper, a novel nonlinear filtering method for numerically eliminating the zero-order term in off-axis digital holography is proposed. In this method, an appropriate nonlinear filter is designed and applied to the spectra of the logarithmically processed hologram. Followed by the inverse Fourier transform and exponential operation, a new hologram free from zero-order term blurring is obtained. Since the major blurring caused by zero-order image is filtered, both the contrast and the quality of the image reconstructed from the new hologram are improved. Due to the simple calculation procedure and the requirement of no more than one hologram acquisition, the method has the potential capability of dynamic real-time detection. It's worth mentioning that the novel nonlinear filtering method requires neither the specific Fourier spectrum distribution nor the certain intensity ratio between the object and the reference waves. 2. Theoretical analyses The flowchart of a nonlinear filtering method is shown in Fig. 1, in which f (x, y) represents the input hologram and g (x, y) represents the processed hologram free from the zero-order term
Z. Ma et al. / Optics Communications 315 (2014) 232–237
233
Fig. 1. The flowchart of nonlinear filtering method.
blurring; Ln and exp denote the logarithm and the exponential operation respectively; the DFT and DFT 1 indicate the Fourier transform and inverse Fourier transform respectively; F corresponds to the Fourier spectrum of the logarithmically processed hologram and H(u, v) corresponds to the properly designed nonlinear filter. According to the principle that an image intensity is jointly determined by the image illumination and the surface reflection properties [18], the nonlinear filtering method treats a hologram f (x,y) as a product of the incident component i (x, y) and the reflection component r (x, y), as is shown in Eq. (1). For a given hologram f (x, y), the incident component corresponding to the image illumination is found to have a slight fluctuation from place to place, therefore its Fourier spectrum is mainly concentrated on the low frequency domain. Different from the incident component, the reflection component corresponds to the detail of the image, thus the spectrum is mainly concentrated on the high frequency domain. f ðx; yÞ ¼ iðx; yÞ U rðx; yÞ;
ð1Þ
The two components i (x, y) and r (x, y) are separated and meanwhile the multiplication is converted into an addition after taking the logarithm of both sides of Eq. (1), as is shown in Eq. (2). Moreover, the logarithm operation is helpful for improving the gray scale of the hologram and narrowing the brightness scope of the hologram [19]. ln ½f ðx; yÞ ¼ ln ½rðx; yÞ þ ln ½iðx; yÞ:
ð2Þ
Applying Fourier transforms on both sides of Eq. (2), one obtains the Fourier spectrum of the logarithmically processed hologram: ℑf ln ½f ðx; yÞg ¼ ℑf ln ½rðx; yÞg þ ℑf ln ½iðx; yÞg:
ð3Þ
Eq. (3) can be further simplified as: Fðu; vÞ ¼ Rðu; vÞ þ Iðu; vÞ;
ð4Þ
where F(u,v), R(u,v) and I(u,v) are the Fourier transform of ln[f(x,y)], ln[r(x,y)] and ln[i(x,y)] respectively. The spectrum I(u,v) corresponding to the illumination intensity of the hologram mainly distributes in the low frequency domain, and the spectrum R(u,v) corresponding to the reflection property mainly distributes in the high frequency domain. Such spectral distribution provides the probability of both eliminating the low frequency spectrum and meanwhile enhancing the high frequency spectrum by employing an appropriately designed nonlinear filter. In designing an appropriate filter H, the problems are how to enhance the intensity of 7 1 order diffraction and how to suppress the zero-order diffraction disturbance as much as possible. In order to achieve such purpose, a filter is designed with totally different functions on the high and low frequency spectrum, as shown in Fig. 2.
Fig. 2. Schematic diagram of the nonlinear filter function.
The filter function can be mathematically written as Eq. (5): 2
Hðu; vÞ ¼ ðr H r L Þ½1 e cðD
ðu;vÞ=ÞD20
þ rL
ð5Þ
where rL and rH values determine what kinds of effects do the filter have on the low-frequency and high-frequency components; c is the parameter used to control the sharpness of the filter function and D0 is the cut-off frequency of the filter; D(u, v) is the frequency at the point of (u, v). Under the condition of rL o1 and rH 41, the filter function is certain to achieve the purpose of suppressing the low frequency component and enhancing the high-frequency component. Under the situation that the above parameters were taken the following specific values: rL ¼0, rH ¼ 1, c ¼0.5, the filter can be considered as an inverse Gaussian filter. The recorded digital hologram on the CCD plane can be written as [20]: 2 2 f ðx; yÞ ¼ I H ¼ ðψ O þ ψ R Þðψ O þ ψ R Þn ¼ ψ O þ ψ R þ ψ O ψ n R þψ n O ψ R ð6Þ where ΨO and ΨR are respectively the complex amplitude of the object wave and the reference wave. By simple transformation, the intensity distribution of the hologram can be seen as the product of the incident component In and the reflection component Re, as shown in Eq. (7). ψ ψ I H ¼ jψ R j2 ð1 þ O Þð1 þ O Þn ¼ I n Re ; ð7Þ ψR ψR According to the above mentioned analysis, the nonlinear filtering method is applicable to suppress the zero-order term and to enhance the high-order term. By taking the logarithm on both sides of Eq. (7), the hologram with compressed luminance ranges can be obtained, as shown in Eq. (8): ψ ψ lnðI H Þ ¼ lnðjψ R j2 Þ þ ln ½ð1 þ O Þð1 þ O Þn ð8Þ ψR ψR Then applying the Fourier transforms on both sides of Eq. (8) and the Fourier spectra of the logarithmically processed hologram can be obtained, as shown in Eq. (9): ψ ℑf ln ½I H gðu;vÞ ¼ ℑf ln ½jψ R j2 gðu;vÞ þ ℑ ln ½ð1 þ O Þ ψR ðu;vÞ ψO n þℑ ln ½ð1 þ Þ ð9Þ ψR ðu;vÞ The low frequency spectrum is suppressed and the high frequency spectrum is enhanced after applying the designed nonlinear filter to the input hologram. Following the subsequent inverse Fourier transform and the exponential operation, the new
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hologram with eliminated zero-order term are obtained. gðx; yÞ ¼ I H New ¼ expfℑ
1
½ℑf ln ½I H gðu;vÞ Hðu; vÞg
ð10Þ
The reconstructed image from such hologram has the characteristics of improved image contrast and high signal-to-noise ratio, as confirmed by both the simulation and the experimental results.
3. Experimental results and parameter optimization
is 3.75 μm 3.75 μm) is used to capture the hologram which is formed by the interference between the diffracted light of the sample and the reference light. The Fresnel hologram of a letter A with a height of 0.8 cm is shown in Fig. 4(a). The corresponding Fourier spectra and the directly reconstructed image are shown in Fig. 4(b and c), respectively. The nonlinear filter with optimized parameters of D0 ¼367, rL ¼0.001, rH ¼ 1.3, c¼2 is applied to the logarithmically operated hologram of Fig. 4(a), with the final processed hologram
The schematic of the experimental set-up is shown in Fig. 3, in which a He–Ne laser (wavelength: 633 nm, power: 5 mW) is used as the light source. A commercially available CCD camera (chameleon CMLN-13S2M, 1280 960 pixels, and the individual pixel size
Fig. 3. The schematic diagram of the experimental setup. (BE: beam expander, BS: beam splitter, M: reflection mirror)
Fig. 5. Comparison of the image noise in different methods.
Fig. 4. A comparison of reconstruction is made between the traditional and the proposed method. (a) The experimentally obtained Fresnel hologram; (b) the Fourier spectrum of (a); (c) the directly reconstructed results of the original hologram (a); (d) the Fresnel hologram processed by the proposed method; (e) the Fourier spectrum of (d); (f) the reconstructed results of the processed hologram (d); (g) the hologram processed by a Laplacian operator; (h): the Fourier spectrum of (g); (i) the reconstructed result of the hologram (g).
Z. Ma et al. / Optics Communications 315 (2014) 232–237
235
Fig. 6. Optimization of the parameters. (a) The directly reconstructed image of a dice hologram; (b) reconstructed image with zero-order term eliminated by the proposed method; (c) the curve of RMS values of the first quadrant of (a) versus the cutoff frequency of D0; (d) the curve of RMS values of the first quadrant of (a) versus the parameter rH.
Fig. 7. The reconstructed image obtained in different filter parameters of D0. (a) The reconstructed image obtained in an optimized value of D0 ¼ 367; (b) the reconstructed image in the case of D0 ¼ 267; (c) the reconstructed image in the case of D0 ¼ 467.
is shown in Fig. 4(d). The nonlinear filter is proved to be a powerful tool in eliminating the zero-order term, as is confirmed by both the Fourier spectrum and the reconstructed image in Fig. 4 (e and f) respectively. Here a hologram (Fig. 4(g)) processed by Laplacian operator [21] is provided as a comparison. The zeroorder term in Fig. 4(h) can be eliminated effectively, but a reduction in the signal-to-noise ratio is exposed as well. Fig. 4 (i) is reconstructed from the Laplace method with a factorized thickness of the letter A. The calculated root-mean-square (rms) value of the framed part of Fig. 4(i) is 0.4141, while the corresponding rms value of Fig. 4(f) is only 0.1810. The corresponding gray values of the solid lines in Fig. 4(c, f and i) are plotted in Fig. 5. Compared with the directly reconstructed image, the reconstructed image processed by the nonlinear filtering method greatly reduces the zero-order noise. As can be seen from Fig. 5, the image noise processed by nonlinear filtering method is lower than that of the Laplace method.
Fig. 8. The variation of PSNR verses the intensity ratio between reference wave and object wave.
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Z. Ma et al. / Optics Communications 315 (2014) 232–237
Fig. 9. Nonlinear filtering method used to reconstruct the hologram with phase information. (a) The original image plane hologram of normal human hepatic cell; (b) the Fourier transform of the logarithmically processed hologram (a) A and B denote the first order and the high order spectra respectively; (c) the reconstructed phase information of original hologram; (d) the processed hologram by nonlinear filtering method; (e) the Fourier transform of the logarithmically processed hologram (d) A′ and B ′ denote the first order and the high order spectra respectively; (f) the reconstructed phase information of processed hologram.
In order to study what kinds of effects the filter parameters have on the quality of the reconstructed image, a dice hologram with the reconstructed image located in a particular quadrant are used for studying the performance of the nonlinear filters with different values of D0 and rH conveniently. The root-mean-square (RMS) value, which should have minimum value when the image has the best uniformity, is used as a quantitative evaluation of the image quality of the reconstructed image. The RMS value of the first quadrant of Fig. 6(b) which does not contain the reconstructed object should have the minimum value when the most appropriate filtering parameters are chosen. From the curves shown in Fig. 6(c and d), we can see that the reconstructed image has the lowest noise level when the filtering parameters are chosen as D0 ¼367 and rH ¼1.3. For a properly designed D0 of 367 pixels, the reconstructed image has the highest image quality, as shown in Fig. 7(a). Fig. 7 (b) shows that a filter with smaller D0 of 267 pixels is not able to eliminate the zero-order term completely. Although a filter with larger D0 of 467 pixels eliminates the zero-order term completely, as shown in Fig. 7(c), the overall image noise is higher than that of the image in Fig. 7(a). In the references by Pavillon et al. [16,17] demonstrated that the reconstructed image quality is highly depended on the intensity ratio between the reference wave and the object wave. The peak-signal-to-noise ratio (PSNR) [22] between the reconstructed image and the original image in our method is also calculated as the intensity ratio between the reference wave and the object wave changes. As is shown in Fig. 8, the PSNR has relatively constant values which would be a great advantage when there is a difficulty in determining the relative intensity of the reference wave. The robustness is also confirmed by applying the proposed method to sample with phase information. An image plane hologram of a normal human hepatic cell is recorded, as shown in Fig. 9(a). Its Fourier spectrum, as shown in Fig. 9(b), comprises the zero-order term, first order term and high order term. The high order term is caused by a slight saturation of CCD camera. For comparison, Fig. 9(d) shows the hologram without the zero-order
term, which is processed by the proposed method, and its Fourier spectrum is shown in Fig. 9(e), where only the first order spectrum and the high order spectrum can be distinguished. Results indicate that the zero-order term can be eliminated effectively, and both the first order spectrum and the high-order spectrum can be enhanced through comparison of patterns between region A and B (in Fig. 9(b)) and its counterpart region A′ and B′ (in Fig. 9(e)). Comparison between the reconstructed phase information of original hologram and the processed hologram (Fig. 9(c and f)) indicates that the proposed method reduces the noise level, as can be seen obviously from the marked elliptical region.
4. Conclusions In this paper, a properly designed nonlinear filter manifests its superiority in eliminating the zero-order term during the reconstruction of a Fresnel digital hologram and an image plane hologram. Experimental results of the amplitude and phase reconstruction indicate that the zero-order image is successfully suppressed and both the image contrast and signal-to-noise ratio are improved significantly. Besides the advantages of the simple calculation process and easy implementation, the novel nonlinear filtering method also has the advantage of no requirement either in the spectrum distribution or in the intensity ratio between the object wave and reference wave. Moreover, just one hologram which is recorded without using precision phase-shifting device is sufficient enough for reconstruction. Therefore, the approach is especially applicable for the off-axis digital holography including the requirement of a real-time dynamic detection.
Acknowledgements This work is supported by the Special Fund for Basic Research on Scientific Instruments of the National Natural Science Foundation of China (Grant No. 61227010) and the Natural Science Foundation of Tianjin (Grant No. 11JCYBJC01400).
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E.N. Leith, J. Upatnieks, K.A. Haines, J. Opt. Soc. Am. 55 (1965) 981. M. Kanka, R. Riesenberg, H.J. Kreuzer, Opt. Lett. 34 (2009) 1162. Y. Takaki, H. Kawai, H. Ohzu, Appl. Opt. 38 (1999) 4990. E. Cuche, P. Marquet, C. Depeursinge, Appl. Opt. 39 (2000) 4070. N. Pavillon, C. Arfire, I. Bergoënd, C. Depeursinge, Opt. Express 18 (2010) 15318. D. Kim, R. Magnusson, M. Jin, J. Lee, W. Chegal, Opt. Express 21 (2013) 3658. Y. Zhang, Q. Lü, B. Ge, Opt. Commun. 240 (2004) 261. B. Das, C.S. Yelleswarapu, D.V.G.L.N. Rao, Opt. Commun. 285 (2012) 4954. I. Yamaguchi, T. Zhang, Opt. Lett 22 (1997) 1268. I. Yamaguchi, J. Kato, S. Ohta, J. Mizuno, Appl. Opt. 40 (2001) 6177. C. Liu, Y. Li, X. Cheng, Z. Liu, F. Bo, J. Zhu, Opt. Eng. 41 (2002) 2434. X F Xu, L Z Cai, Y R Wang, R S Yan, J. Opt. 12 (2010) 015301. J. Vargas, J. Antonio Quiroga, C.O.S. Sorzano, J.C. Estrada, J.M. Carazo, Opt. Lett. 37 (2012) 443.
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[14] G. Chen, C. Lin, M. Kuo, C. Chang, Opt. Express 15 (2007) 8851. [15] J. Li, Z. Peng, P. Tankam, P. Picart, Opt. Commun. 283 (2010) 4166. [16] N. Pavillon, C.S. Seelamantula, J. Kühn, M. Unser, C. Depeursinge, Appl. Opt. 48 (2009) H186. [17] C.S. Seelamantula, N. Pavillon, C. Depeursinge, M. Unser, in: IEEE International Symposium on Biomedical Imaging from Nano to Macro 2009, 201–204. [18] B.K.P. Horn, Artif. Intell. 8 (1977) 201. [19] J.L. Juárez-Pérez, A. Olivares-Pérez, L.R. Berriel-Valdos, Appl. Opt. 36 (1997) 7437. [20] J.W. Goodman, Introduction to Fourier Optics, second ed., McGraw Hill, New York, 1996. [21] Y. Zhou, R. Chellappa, A. Vaid, B.K. Jenkins, IEEE Trans. Acoust. Speech Signal. Process. 36 (1988) 1141. [22] Q. Huynh-Thu, M. Ghanbari, Electr. Lett. 44 (2008) 800.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Nonlinear filtering method of zero-order term suppression for improving the image quality in off-axis holography Zhonghong Ma a, Yong Yang a,n, Qi Ge b, Lijun Deng a, Zhenxin Xu c, Xuna Sun c a Institute of Modern Optics, Nankai University, The Key Laboratory of Optical Information Science and Technology of the Education Ministry of China, Tianjin 300071, China b College of Zhonghuan Information, Tianjin University of Technology, Tianjin 300071, China c School of Electronical Engineering, Tianjin University of Technology, Tianjin 300072, China
art ic l e i nf o
a b s t r a c t
Article history: Received 20 September 2013 Received in revised form 31 October 2013 Accepted 12 November 2013 Available online 22 November 2013
In order to alleviate the crosstalk between the reconstructed image and the zero-order image in off-axis digital holography, this paper proposes a novel nonlinear filtering method to suppress the zero-order image. Its feasibility in eliminating the zero-order term is verified by the theoretical analyses, and its robustness is confirmed by applying the appropriately designed nonlinear filter to both a Fresnel hologram and an image plane hologram. Experimental results show that the image reconstructed from the proposed method has the characteristics of enhanced image contrast and improved signal-to-noise ratio. This method also accelerated the speed of processing via a simple calculation procedure. & 2013 Elsevier B.V. All rights reserved.
Keywords: Digital holography Digital image processing Image reconstruction techniques Nonlinear filtering method Zero-order term elimination
1. Introduction The off-axis holography [1] is used originally to separate the zero-order image and the two conjugate images called the virtual image and the real image. Confined by the pixel size [2] and pixel number [3] of the digital recording device, the zero-order image inevitably becomes a blurring to the recorded sample. It is necessary to eliminate the zero-order term for increasing the recorded bandwidth of the sample, and therefore improving the reconstructed image quality [4–7]. Several methods have been proposed to achieve this purpose in off-axis digital holography. By using experimental procedures, such as phase-shifting digital holography [8–12], the zero-order diffraction is effectively eliminated. However, it requires a precise vibration-free optical table during the hologram recording procedure. Since the multi-frame holograms are required [13], it also has some difficulties in dealing with the dynamic real-time detection of a moving target. Therefore the numerical suppression method [14] which needs neither multi-frame holograms nor high precision device is applied to digital holography. Many numerical methods have specific applicable conditions in coping with the zero-order term elimination. For example, the spatial filter window designed in Ref. [15] is only applicable to convolutional
n
Corresponding author. Tel.: þ 86 13072041093. E-mail address: [email protected] (Y. Yang).
0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.11.021
reconstruction of the hologram. A nonlinear reconstruction technique proposed by Pavillon et al. [16,17] is able to eliminate the zero-order term successfully, but it requires a larger reference amplitude than that of the object, and the image order needs to be confined in one quadrant of the Fourier spectrum. In this paper, a novel nonlinear filtering method for numerically eliminating the zero-order term in off-axis digital holography is proposed. In this method, an appropriate nonlinear filter is designed and applied to the spectra of the logarithmically processed hologram. Followed by the inverse Fourier transform and exponential operation, a new hologram free from zero-order term blurring is obtained. Since the major blurring caused by zero-order image is filtered, both the contrast and the quality of the image reconstructed from the new hologram are improved. Due to the simple calculation procedure and the requirement of no more than one hologram acquisition, the method has the potential capability of dynamic real-time detection. It's worth mentioning that the novel nonlinear filtering method requires neither the specific Fourier spectrum distribution nor the certain intensity ratio between the object and the reference waves. 2. Theoretical analyses The flowchart of a nonlinear filtering method is shown in Fig. 1, in which f (x, y) represents the input hologram and g (x, y) represents the processed hologram free from the zero-order term
Z. Ma et al. / Optics Communications 315 (2014) 232–237
233
Fig. 1. The flowchart of nonlinear filtering method.
blurring; Ln and exp denote the logarithm and the exponential operation respectively; the DFT and DFT 1 indicate the Fourier transform and inverse Fourier transform respectively; F corresponds to the Fourier spectrum of the logarithmically processed hologram and H(u, v) corresponds to the properly designed nonlinear filter. According to the principle that an image intensity is jointly determined by the image illumination and the surface reflection properties [18], the nonlinear filtering method treats a hologram f (x,y) as a product of the incident component i (x, y) and the reflection component r (x, y), as is shown in Eq. (1). For a given hologram f (x, y), the incident component corresponding to the image illumination is found to have a slight fluctuation from place to place, therefore its Fourier spectrum is mainly concentrated on the low frequency domain. Different from the incident component, the reflection component corresponds to the detail of the image, thus the spectrum is mainly concentrated on the high frequency domain. f ðx; yÞ ¼ iðx; yÞ U rðx; yÞ;
ð1Þ
The two components i (x, y) and r (x, y) are separated and meanwhile the multiplication is converted into an addition after taking the logarithm of both sides of Eq. (1), as is shown in Eq. (2). Moreover, the logarithm operation is helpful for improving the gray scale of the hologram and narrowing the brightness scope of the hologram [19]. ln ½f ðx; yÞ ¼ ln ½rðx; yÞ þ ln ½iðx; yÞ:
ð2Þ
Applying Fourier transforms on both sides of Eq. (2), one obtains the Fourier spectrum of the logarithmically processed hologram: ℑf ln ½f ðx; yÞg ¼ ℑf ln ½rðx; yÞg þ ℑf ln ½iðx; yÞg:
ð3Þ
Eq. (3) can be further simplified as: Fðu; vÞ ¼ Rðu; vÞ þ Iðu; vÞ;
ð4Þ
where F(u,v), R(u,v) and I(u,v) are the Fourier transform of ln[f(x,y)], ln[r(x,y)] and ln[i(x,y)] respectively. The spectrum I(u,v) corresponding to the illumination intensity of the hologram mainly distributes in the low frequency domain, and the spectrum R(u,v) corresponding to the reflection property mainly distributes in the high frequency domain. Such spectral distribution provides the probability of both eliminating the low frequency spectrum and meanwhile enhancing the high frequency spectrum by employing an appropriately designed nonlinear filter. In designing an appropriate filter H, the problems are how to enhance the intensity of 7 1 order diffraction and how to suppress the zero-order diffraction disturbance as much as possible. In order to achieve such purpose, a filter is designed with totally different functions on the high and low frequency spectrum, as shown in Fig. 2.
Fig. 2. Schematic diagram of the nonlinear filter function.
The filter function can be mathematically written as Eq. (5): 2
Hðu; vÞ ¼ ðr H r L Þ½1 e cðD
ðu;vÞ=ÞD20
þ rL
ð5Þ
where rL and rH values determine what kinds of effects do the filter have on the low-frequency and high-frequency components; c is the parameter used to control the sharpness of the filter function and D0 is the cut-off frequency of the filter; D(u, v) is the frequency at the point of (u, v). Under the condition of rL o1 and rH 41, the filter function is certain to achieve the purpose of suppressing the low frequency component and enhancing the high-frequency component. Under the situation that the above parameters were taken the following specific values: rL ¼0, rH ¼ 1, c ¼0.5, the filter can be considered as an inverse Gaussian filter. The recorded digital hologram on the CCD plane can be written as [20]: 2 2 f ðx; yÞ ¼ I H ¼ ðψ O þ ψ R Þðψ O þ ψ R Þn ¼ ψ O þ ψ R þ ψ O ψ n R þψ n O ψ R ð6Þ where ΨO and ΨR are respectively the complex amplitude of the object wave and the reference wave. By simple transformation, the intensity distribution of the hologram can be seen as the product of the incident component In and the reflection component Re, as shown in Eq. (7). ψ ψ I H ¼ jψ R j2 ð1 þ O Þð1 þ O Þn ¼ I n Re ; ð7Þ ψR ψR According to the above mentioned analysis, the nonlinear filtering method is applicable to suppress the zero-order term and to enhance the high-order term. By taking the logarithm on both sides of Eq. (7), the hologram with compressed luminance ranges can be obtained, as shown in Eq. (8): ψ ψ lnðI H Þ ¼ lnðjψ R j2 Þ þ ln ½ð1 þ O Þð1 þ O Þn ð8Þ ψR ψR Then applying the Fourier transforms on both sides of Eq. (8) and the Fourier spectra of the logarithmically processed hologram can be obtained, as shown in Eq. (9): ψ ℑf ln ½I H gðu;vÞ ¼ ℑf ln ½jψ R j2 gðu;vÞ þ ℑ ln ½ð1 þ O Þ ψR ðu;vÞ ψO n þℑ ln ½ð1 þ Þ ð9Þ ψR ðu;vÞ The low frequency spectrum is suppressed and the high frequency spectrum is enhanced after applying the designed nonlinear filter to the input hologram. Following the subsequent inverse Fourier transform and the exponential operation, the new
234
Z. Ma et al. / Optics Communications 315 (2014) 232–237
hologram with eliminated zero-order term are obtained. gðx; yÞ ¼ I H New ¼ expfℑ
1
½ℑf ln ½I H gðu;vÞ Hðu; vÞg
ð10Þ
The reconstructed image from such hologram has the characteristics of improved image contrast and high signal-to-noise ratio, as confirmed by both the simulation and the experimental results.
3. Experimental results and parameter optimization
is 3.75 μm 3.75 μm) is used to capture the hologram which is formed by the interference between the diffracted light of the sample and the reference light. The Fresnel hologram of a letter A with a height of 0.8 cm is shown in Fig. 4(a). The corresponding Fourier spectra and the directly reconstructed image are shown in Fig. 4(b and c), respectively. The nonlinear filter with optimized parameters of D0 ¼367, rL ¼0.001, rH ¼ 1.3, c¼2 is applied to the logarithmically operated hologram of Fig. 4(a), with the final processed hologram
The schematic of the experimental set-up is shown in Fig. 3, in which a He–Ne laser (wavelength: 633 nm, power: 5 mW) is used as the light source. A commercially available CCD camera (chameleon CMLN-13S2M, 1280 960 pixels, and the individual pixel size
Fig. 3. The schematic diagram of the experimental setup. (BE: beam expander, BS: beam splitter, M: reflection mirror)
Fig. 5. Comparison of the image noise in different methods.
Fig. 4. A comparison of reconstruction is made between the traditional and the proposed method. (a) The experimentally obtained Fresnel hologram; (b) the Fourier spectrum of (a); (c) the directly reconstructed results of the original hologram (a); (d) the Fresnel hologram processed by the proposed method; (e) the Fourier spectrum of (d); (f) the reconstructed results of the processed hologram (d); (g) the hologram processed by a Laplacian operator; (h): the Fourier spectrum of (g); (i) the reconstructed result of the hologram (g).
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Fig. 6. Optimization of the parameters. (a) The directly reconstructed image of a dice hologram; (b) reconstructed image with zero-order term eliminated by the proposed method; (c) the curve of RMS values of the first quadrant of (a) versus the cutoff frequency of D0; (d) the curve of RMS values of the first quadrant of (a) versus the parameter rH.
Fig. 7. The reconstructed image obtained in different filter parameters of D0. (a) The reconstructed image obtained in an optimized value of D0 ¼ 367; (b) the reconstructed image in the case of D0 ¼ 267; (c) the reconstructed image in the case of D0 ¼ 467.
is shown in Fig. 4(d). The nonlinear filter is proved to be a powerful tool in eliminating the zero-order term, as is confirmed by both the Fourier spectrum and the reconstructed image in Fig. 4 (e and f) respectively. Here a hologram (Fig. 4(g)) processed by Laplacian operator [21] is provided as a comparison. The zeroorder term in Fig. 4(h) can be eliminated effectively, but a reduction in the signal-to-noise ratio is exposed as well. Fig. 4 (i) is reconstructed from the Laplace method with a factorized thickness of the letter A. The calculated root-mean-square (rms) value of the framed part of Fig. 4(i) is 0.4141, while the corresponding rms value of Fig. 4(f) is only 0.1810. The corresponding gray values of the solid lines in Fig. 4(c, f and i) are plotted in Fig. 5. Compared with the directly reconstructed image, the reconstructed image processed by the nonlinear filtering method greatly reduces the zero-order noise. As can be seen from Fig. 5, the image noise processed by nonlinear filtering method is lower than that of the Laplace method.
Fig. 8. The variation of PSNR verses the intensity ratio between reference wave and object wave.
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Fig. 9. Nonlinear filtering method used to reconstruct the hologram with phase information. (a) The original image plane hologram of normal human hepatic cell; (b) the Fourier transform of the logarithmically processed hologram (a) A and B denote the first order and the high order spectra respectively; (c) the reconstructed phase information of original hologram; (d) the processed hologram by nonlinear filtering method; (e) the Fourier transform of the logarithmically processed hologram (d) A′ and B ′ denote the first order and the high order spectra respectively; (f) the reconstructed phase information of processed hologram.
In order to study what kinds of effects the filter parameters have on the quality of the reconstructed image, a dice hologram with the reconstructed image located in a particular quadrant are used for studying the performance of the nonlinear filters with different values of D0 and rH conveniently. The root-mean-square (RMS) value, which should have minimum value when the image has the best uniformity, is used as a quantitative evaluation of the image quality of the reconstructed image. The RMS value of the first quadrant of Fig. 6(b) which does not contain the reconstructed object should have the minimum value when the most appropriate filtering parameters are chosen. From the curves shown in Fig. 6(c and d), we can see that the reconstructed image has the lowest noise level when the filtering parameters are chosen as D0 ¼367 and rH ¼1.3. For a properly designed D0 of 367 pixels, the reconstructed image has the highest image quality, as shown in Fig. 7(a). Fig. 7 (b) shows that a filter with smaller D0 of 267 pixels is not able to eliminate the zero-order term completely. Although a filter with larger D0 of 467 pixels eliminates the zero-order term completely, as shown in Fig. 7(c), the overall image noise is higher than that of the image in Fig. 7(a). In the references by Pavillon et al. [16,17] demonstrated that the reconstructed image quality is highly depended on the intensity ratio between the reference wave and the object wave. The peak-signal-to-noise ratio (PSNR) [22] between the reconstructed image and the original image in our method is also calculated as the intensity ratio between the reference wave and the object wave changes. As is shown in Fig. 8, the PSNR has relatively constant values which would be a great advantage when there is a difficulty in determining the relative intensity of the reference wave. The robustness is also confirmed by applying the proposed method to sample with phase information. An image plane hologram of a normal human hepatic cell is recorded, as shown in Fig. 9(a). Its Fourier spectrum, as shown in Fig. 9(b), comprises the zero-order term, first order term and high order term. The high order term is caused by a slight saturation of CCD camera. For comparison, Fig. 9(d) shows the hologram without the zero-order
term, which is processed by the proposed method, and its Fourier spectrum is shown in Fig. 9(e), where only the first order spectrum and the high order spectrum can be distinguished. Results indicate that the zero-order term can be eliminated effectively, and both the first order spectrum and the high-order spectrum can be enhanced through comparison of patterns between region A and B (in Fig. 9(b)) and its counterpart region A′ and B′ (in Fig. 9(e)). Comparison between the reconstructed phase information of original hologram and the processed hologram (Fig. 9(c and f)) indicates that the proposed method reduces the noise level, as can be seen obviously from the marked elliptical region.
4. Conclusions In this paper, a properly designed nonlinear filter manifests its superiority in eliminating the zero-order term during the reconstruction of a Fresnel digital hologram and an image plane hologram. Experimental results of the amplitude and phase reconstruction indicate that the zero-order image is successfully suppressed and both the image contrast and signal-to-noise ratio are improved significantly. Besides the advantages of the simple calculation process and easy implementation, the novel nonlinear filtering method also has the advantage of no requirement either in the spectrum distribution or in the intensity ratio between the object wave and reference wave. Moreover, just one hologram which is recorded without using precision phase-shifting device is sufficient enough for reconstruction. Therefore, the approach is especially applicable for the off-axis digital holography including the requirement of a real-time dynamic detection.
Acknowledgements This work is supported by the Special Fund for Basic Research on Scientific Instruments of the National Natural Science Foundation of China (Grant No. 61227010) and the Natural Science Foundation of Tianjin (Grant No. 11JCYBJC01400).
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