Coherent noise suppression in digital holography ... - OSA Publishing

Coherent noise suppression in digital holography based on flat fielding with apodized apertures D. G. Abdelsalam1, 2 and Daesuk Kim 1,* 1

2

Division of Mechanical System Engineering, Chonbuk National University, Jeonju 561-756, Korea. Engineering and surface metrology lab., National Institute of Standards, Tersa St., El haram, El Giza, Egypt. *[email protected]

Abstract: Great number of approaches has been carried out in digital holography (DH) in order to overcome the problem of coherent noise in the reconstruction process. In this paper, we describe a new method that can be used to suppress the coherent noise in phase-contrast image. The proposed method is a combination of the flat fielding method and the apodized apertures technique. The proposed method is applied to a sample of 200 µ m step height. The quality of the phase-contrast image of the sample is refined and the coherent noise level is reduced drastically by the order of 65%. The proposed method can also applicable to noise reduction of intensity imaging. ©2011 Optical Society of America OCIS codes: (090.1995) Digital holography; (100.3010) Image reconstruction techniques; (100.2000) Digital image processing; (100.2980) Image enhancement; (100.5070) Phase retrieval.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

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Received 14 Jun 2011; revised 16 Aug 2011; accepted 18 Aug 2011; published 29 Aug 2011 12 September 2011 / Vol. 19, No. 19 / OPTICS EXPRESS 17951

17. T. Kozacki and R. Jo’z’wicki, “Digital reconstruction of a hologram recorded using partially coherent illumination,” Opt. Commun. 252(1–3), 188–201 (2005). 18. B. Steve, Howell., “Handbook of CCD Astronomy”, Cambridge, UK, (2006). 19. D. G. Abdelsalam, M. S. Shaalan, and M. M. Eloker, “Surface microtopography measurement of a standard flat surface by multiple-beam interference fringes at reflection,” Opt. Lasers Eng. 48(5), 543–547 (2010). 20. D. G. Abdelsalam, M. S. Shaalan, M. M. Eloker, and D. Kim, “Radius of curvature measurement of spherical smooth surfaces by multiple-beam interferometry in reflection,” Opt. Lasers Eng. 48(6), 643–649 (2010). 21. E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39(23), 4070–4075 (2000). 22. E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. 24(5), 291–293 (1999). 23. H. Lee, S. Kim, and D. Kim, “Two step on-axis digital holography using dual-channel Mach-Zehnder interferometer and matched filter algorithm,” J. Opt. Soc. Korea 14(4), 363–367 (2010). 24. D. G. Abdelsalam, B. J. Baek, Y. J. Cho, and D. Kim, “Surface form measurement using single-shot off-axis Fizeau interferometer,” J. Opt. Soc. Korea 14(4), 409–414 (2010). 25. E. Cuche, P. Marquet, and C. Depeursinge, “Aperture apodization using cubic spline interpolation: application in digital holographic microscopy,” Opt. Commun. 182(1-3), 59–69 (2000). 26. M. Born and E. Wolf, Principles of Optics (Cambridge: Cambridge University Press, UK), pp. 459–490 (1980). 27. V. G. Maximov, G. V. Simonova, and V. A. Tartakovskii, “The effect of the Gaussian inhomogeneity of laser beam intensity on the interferometric measurement uncertainty,” Russ. Phys. J. 48(5), 495–500 (2005).

1. Introduction In digital holography, cameras like charge-coupled device (CCD) are used to record interferograms (or digital holograms), and object wave fronts are reconstructed numerically by simulating the propagation of optical beams [1]. Because of the temporally and spatially coherent illumination, the coherent noise, which estimated to come from shot noise (random arrival of photons) and scattering noise due to some dust particles or scratches on optical elements, etc., degrades the image quality and subsequently the measurement accuracy. Hence, it is of particular importance to reduce the inherited coherent noise of DH. Based on digital signal processing techniques, many approaches were proposed, i.e. classical filtering [2], Wiener filtering with an aperture function [3], discrete Fourier filtering [4], and wavelet filtering [5]. Nevertheless, these methods have the disadvantages of reducing the resolution of image. Other methods were presented by superposing several reconstructed images with different speckle patterns [6–10]. However, these methods mostly aimed at removing the coherent noise for intensity imaging of the rough object. Recently, partial coherent light sources were adopted for digital holographic phase contrast imaging. Pedrini et al. presented a lensless short coherence digital holography in biological samples microscopy [11,12]. Dubois et al. demonstrated that the coherent noise can be eliminated by using a spatial partial coherent source [13,14]. Kemper et al. obtained high quality phase contrast images of living cells by using SLD and LED [15,16]. However, because of the limited coherence length, the specimen information within only limited depth can be recorded, and the system impulse response is broaden [17]. In this paper, the flat fielding method with apodized apertures is used to reduce the coherent noise in phase-contrast image [18–20]. The coherent noise is further reduced when the flat fielding method with apodized apertures is applied. The experimental results show that the coherent noise is reduced by the order of 65% when the proposed method (flat fielding method with apodized apertures technique) is applied. We believe that the estimated value 65% noise reduction may be varied with the region of interest, surface property of a target sample, amount of averaging the digital holograms, etc. In this paper, the estimated value 65% noise reduction is calculated for only one region of interest. To our knowledge, this is the first time that a combination of the flat fielding method and the apodized apertures technique is used to suppress coherent noise in phase-contrast image. 2. Theory The interferogram (or digital hologram) intensity resulting from the interference of the object wave O and the reference wave R was recorded by a CCD camera is given by 2

2

I (k , l ) = O + R + R*O + RO* .

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(1)


Here, I is the interferogram (or digital hologram) intensity, k and l are integers. In Eq. (1), * stands for the complex conjugate: the first two intensity terms are of the zero order, which can be directly filtered in the Fourier domain [21]; and the last two terms represent the interference terms with the object wave O (the virtual images), or its conjugate O* (the real images), being modulated by the spatial carrier frequency in the spatial frequency domain. After filtering out the object and conjugate terms by a spatial filtering approach [21], the filtered spectrum data in the spatial frequency domain turn back to the spatial domain by using inverse 2D FFT. For off-axis digital holography, the propagation of the filtered complex wave is calculated numerically. While, such Fresnel transform based propagation step is not necessary for interferometry. Let us define the filtered complex wave Ψ as follows: Ψ = R *O . (2) The above filtered complex wave is an array of complex numbers. An amplitude-contrast image and a phase-contrast image can be obtained by using the following intensity

[Re(Ψ )2 + Im(Ψ )2 ] and the argument arctan[Re(Ψ ) / Im(Ψ )] , respectively. Finally, in order to obtain the object information, the Ψ needs to be multiplied by the original reference wave called a digital reference wave ( RD (m, n)) [22–24]. Here, m and n are integers. If we assume that a perfect plane wave is used as a reference for interferogram recording, the computed replica of the reference wave RD can be represented as follows:

RD (m, n) = AR exp[i (2π / λ )(k x m∆x + k y n∆y )] , where,

(3)

AR is the amplitude, λ is the wavelength of the laser source, ∆x = ∆y are the pixel

sizes, and k x and

k y are the two components of the wave vector that must be adjusted such

that the propagation direction of RD matches as closely as possible with that of the experimental reference wave. By using this digital reference wave concept, we can obtain an object wave which is reconstructed in the central region of the observation plane. As stated above, the reconstructed complex wave fronts O in the reconstruction plane contain both the amplitude and the phase information. With coherent illumination, undesired scattering occurs due to the refractive inhomogeneities, multiple reflection, rough surface, scratches, dust particle, etc., and introduce irregular diffraction pattern in interferogram (hologram) and consequently in the reconstructed object wave (amplitude and phase). To illustrate the principle of coherent noise more clearly, the filtered complex wave in Eq. (2) may be described mathematically. Assume that the coordinate system of the reconstruction plane is pq plane. The complex wave front O can be regarded as the complex amplitude of the sample modulated by complex amplitude of random coherent noise, and can be expressed as follows:

U o ( p, q ) = Eo ( p, q ).co ( p, q ),

(4)

where Eo ( p, q ) = AE ( p, q ) exp(iϕ E ) is the complex amplitude profile which representing the o o reflection and thickness properties of the sample, and co ( p, q) = Aco ( p, q) exp(iφco ) is a complex amplitude of random coherent noises, which distorts the holographic fringe pattern, where A represents the amplitude and φ represents the phase, respectively. Main sources of such noises are reduced drastically by application of the proposed method (flat fielding method with apodized apertures technique). Firstly, the flat fielding method is applied in order to correct the interferogram (or digital hologram) image, and then the apodized apertures is applied to reduce any further noise remained in the corrected interferogram (or digital hologram) image.

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3. Flat fielding with apodized apertures The use of digital detectors in digital holography for recording a series of off-axis interferograms (or digital holograms) is usually accompanied by dark current (thermal noise), shot noise, and scattering noise due to some dust particles, scratches, etc., on optical elements. Such coherent noise can affect the quality of the reconstruction of the object wave (amplitude and phase). Subtracting the dark current (thermal noise) clears the camera of any accumulated charge and reads out the cleared CCD. Figure 1(a) shows a three-dimensional (3D) thermal noise captured when there is no illumination through the optical system. Figure.1(b) shows a 3D influence of the non-uniformity of illumination which is mainly caused by the nonuniform Gaussian intensity distribution, instability of the laser source used, and the vignetting of lenses used in the system. Some factors which cause coherent noise have been reduced drastically by application of the flat fielding method.

Fig. 1. Three-dimensional (3D) intensity distribution shows the effect of (a) dark frame (thermal noise), (b) influence of the non-uniformity of illumination (flat frame).

A CCD camera is calibrated by a process known as “Flat fielding” or “Shading correction”. Flat fielding can be illustrated in the following formula [18–20], −

−

−

−

−

I C = [ M ( I R − I B )] / ( I F − I B ) ,

(5) −

−

where I C is the average of calibrated captured interferograms,

I R is the average of non-

−

calibrated captured interferograms, I B is the average of dark frames, M is the average pixel −

value of the corrected flat field frame, and I F is the average of flat field frames. Dark frames have been taken in advance and stored in the computer when there is no illumination through the optical system. Subtracting the dark frame clears the camera of any accumulated charge and reads out the cleared CCD. Flat field frames measure the response of each pixel in the CCD array to illumination and is used to correct any variation in illumination over the CCD sensor. The non-uniformity of illumination is mainly caused by the non-uniform Gaussian intensity distribution, as depicted in Fig. 2(a). And also, the vignetting of lenses used in the system, which decrease the light intensity towards the image periphery or the dust particles on optical components like a glass window in front of CCD as illustrated in Fig. 2(b), can be some inherent sources of errors in practical experiments. The flat fielding process corrects the uneven illumination.

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Fig. 2. Uneven illumination which produces darkness at the edges of the image (a) inhomogenity of the laser beam, (b) shadow detection of the dust particles hanging at the CCD camera aperture.

Flat field frames are also captured in advance and stored in the computer by blocking the wave from the object arm. Note that flat field frames are captured when laser source is switched on. The captured single off-axis interferograms have been calibrated by using the stored dark and flat field frames. The specimen is a sample of 200 µ m step height. Figure 3(a) −

shows an average of 50 off-axis interferograms I R captured when the sample is adjusted as an object in the Mach-Zender interferometer. In order to subtract the thermal noise, fifty dark − frames I B have been captured when there is no illumination through the optical system. Shot noise (random arrival of photons), fixed pattern noise (pixel to pixel sensitivity variation), and scattering noise due to some dust particles or scratches on optical elements have been − suppressed drastically when the flat fielding Eq. (5) is applied. Where I F is the average of 50 off-axis interferograms captured by blocking the wave from the object arm when laser source is switched on. Figure 3(b) shows the correction of Fig 3(a) after application of the flat fielding method.

Fig. 3. Average of 50 off-axis interferograms of a specimen of 200 µ m step height. (a) Before correction with flat fielding. (b) After correction with flat fielding.

It is shown apparently from Fig. 3(b) that most of random coherent noise (i.e. circulated noise which may come from dust particles and scratches) which distorts the holographic fringe pattern is reduced drastically by application of the flat fielding method. The noise is further reduced when the apodization with cubic spline interpolation technique [25] is applied to the calibrated interferogram (or digital hologram). Apodization of the interferogram (or digital hologram) could be achieved experimentally by inserting an apodized aperture in front of the CCD. However, as a digital image of the interferogram (or digital hologram) is acquired, it is more practical to perform this operation digitally by multiplying the digitized interferogram (or digital hologram) with a 2D function representing the transmission of the apodized aperture (this is explained in more detail in Ref [25].). The aperture is completely

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transparent (transmission equal to unity) in the large central part of the profile. At the edges, the transmission varies from zero to unity following a curve defined by a cubic spline interpolation [25]. After application of the apodized aperture technique to the corrected interferogram with flat fielding (Fig. 3(b)), the obtained off-axis interferogram is numerically processed [24] to obtain the object wave (amplitude and phase). Figure 4 shows the flow chart of the algorithm that was used to analyze the off-axis interferogram.

Fig. 4. Flowchart of the algorithm that was used to analyze the off-axis interferogram (or digital hologram).

After the spatial filtering step, the digital reference wave R D is used for the centering process. Then, the final reconstructed object is obtained by adjusting the values of k x and k y . A profile of the 2D function representing the transmission of the apodized aperture is presented in Fig. 5(a). The digital reference wave used in the calculation process should match as close as possible to the experimental reference wave. This has been done in this paper by selecting the appropriate values of the two components of the wave vector kx = 0.00279 mm−1 and ky = −0.002965 mm−1. Figure 5(b) shows the reconstructed amplitude-contrast image after application of the flat fielding method with the apodized aperture technique. In this study, the suppression noise value in phase-contrast image has been calculated in terms of height.

Fig. 5. Transmission profile of the apodized aperture. (a) The transmission from 0 to 1 at the edge of the aperture follows a cubic spline interpolation; the dashed line indicates the transmission of the unapodized aperture. (b) Numerically reconstructed amplitude-contrast image of Fig. 3(b).

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The one-dimensional (1D) surface profile height h can be calculated directly as follows:

h=

Φ λ. 4π

(6)

where Φ is the reconstructed phase for wavelength λ . In this paper, the measured height is not the real height whereas single wavelength is used. The discrepancy can be from the 2π ambiguity [24]. The main goal of the paper is the noise reduction not the real height measurement. 4. Experimental results Interferometry is a well-established technique for surface profiling [26]. They are noncontacting, nondestructive and highly accurate. In combination with computers and other electronic devices, they have become faster, more reliable, more convenient and more robust. Information about the surface under test can be obtained from interference fringes which characterize the surface. Figure 6 shows the schematic diagram of the proposed optical setup based on the Mach-Zehnder interferometric configuration at reflection. A laser diode beam passes through a collimating lens and expands. This expansion is necessary to illuminate a greater area of the surface to be imaged and to reduce the error measurement due to the inhomogeneity in the Gaussian beam. The collimated beam of the laser light falls upon the beam splitter BS 1, which transmits one half and reflects the other half of the incident light. The reflected beam acts as the object beam and it travels to the BS 2 and then to the specimen by way of the mirror M2. Meanwhile, the transmitted portion of the beam which is called the reference beam reflects from the mirror M1 to the BS 2. The interference fringes are generated at the BS 2 and recorded by the CCD camera via an imaging lens. The off-axis interferograms at perfect collimation have been captured and corrected with the proposed method. The corrected interferogram with the proposed method has been numerically processed to obtain the object wave [24].

Fig. 6. Schematic diagram of the optical setup based on the Mach-Zehnder interferometric configuration at reflection. O, object wave; R, reference wave; BS 1, BS 2, beam-splitters; M1, M2 mirrors.

In this paper, the effect of the inhomogeneity profile of the collimated beam or the intensity variation of the collimated beam from the laser source used has been presented. The intensity variation which may come from the instability of the laser source used is considered as one of the factors which produce coherent noise. The inhomogeneity profile of the collimated beam or the intensity variation can be estimated by calculating the mean and standard deviation of each interferogram image (50 interferograms have been captured). Figure 7(a) shows the normalized mean intensity of each interferogram against the number of

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−

−

interferograms. The intensity variation value is calculated by 3σ / x , where x is the mean and σ is the standard deviation. Taking 3 σ means 99.7% of the points are taken. Figure 7(a) shows that the calculated intensity variation value has been found to be of the order of 1.08%. This variation may come from instability of the laser source used. Figure 7(b) shows the intensity variation values of a mean of each successive 5 interferograms (50 interferograms have been captured) against the number of interferograms images. Normalization to compensate for the non-homogeneity of wavefronts intensities (reducing the intensity variation) is achieved by the flat fielding method [27].

Fig. 7. Intensity variation of the laser diode used (a) Normalized mean intensity of each interferogram image against number of images. (b) Intensity variation of a mean of each successive interferograms images against number of images for original and corrected interferograms.

Figure 8(a) and Fig. 8(b) show the reconstructed phase (after converting to height by using Eq. (6) of the original off-axis interferogram (before correction) and after application of the proposed method (a combination of the flat fielding method with the apodized aperture technique), respectively. It can be noticed that the quality of the reconstructed phase-contrast image is improved as shown clearly in Fig. 8(b) and Fig. 8(d). The reconstructed phase images shown in Fig. 8(a) and Fig. 8(b) are exactly in the same size inside the white rectangles in Fig. 3(a) and Fig. 3(b), respectively.

Fig. 8. Reconstructed phase images after converting to height from the off-axis interferograms. (a) Original (before correction with the proposed method). (b) After correction with the proposed method. (c) 3D of the selected rectangle of (a). (d) 3D of the selected rectangle of (b).

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The one-dimensional (1D) surface profile along the selected line of Fig. 8(a) and Fig. 8(b) is shown in Fig. 9(a) and Fig. 9(b), respectively. Figure 9(c) and Fig. 9(b) show the corresponding zoomed profile inside the black rectangle in Fig. 9(a) and Fig. 9(b), respectively.

Fig. 9. One-dimensional (1D) surface profile along the selected line of Fig. 8(a). (a) Original against flat fielding. (b) Flat fielding against flat fielding with apodized aperture. (c) The corresponding zoomed profile inside the black rectangle in (a). (d) The corresponding zoomed profile inside the black rectangle in (b).

Corresponding to the zoomed profile inside the black rectangle in Fig. 9(c), the height error reconstructed from the original off-axis interferogram is estimated to be of the order of 28 nm. While the height error reconstructed from the corrected off-axis interferogram with flat fielding is estimated to be of the order of 14 nm (i.e. the coherent noise is reduced by nearly 50%). Based on the zoomed profile inside the black rectangle in Fig. 9(d), the height error reconstructed from the apodized aperture has been estimated to be of the order of 12 nm (i.e. the coherent noise is reduced by nearly 15% compared to the flat fielding method). Above results verify the parasitic coherent noise can be reduced using the proposed method (a combination of the flat fielding method with the apodized aperture technique) by nearly 65%. The remained 12 nm is inevitable noise which may come from the non-perfect spatial filtering process in the spatial frequency domain since usually it is impossible to filter out only the object field in the spatial frequency domain. Note that the estimated height is calculated by the difference between the maximum point and the minimum point over the specific curve [19,20]. 5. Conclusion In this paper, a new approach to reduce coherent noise in digital holography phase-contrast image is presented. It is accomplished by combining the flat fielding method with the apodized apertures technique. Experimental results demonstrated the ability of the proposed method on improving the quality of the phase contrast image. The method can also applicable to noise reduction of intensity imaging. The proposed method is suitable for real-time monitoring a specimen. We claim that the proposed method (flat fielding method with apodized apertures) can suppress the coherent noise by the order of 65%. Acknowledgments This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MEST) (2011-0002487).

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