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 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
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 . 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 , Wiener filtering with an aperture function , discrete Fourier filtering , and wavelet filtering . 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 . 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.
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
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 ; and the last two terms represent the interference terms with the object wave O (the virtual images), or its conjugate (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 , 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:
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 and the argument, 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 [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 can be represented as follows:Eq. (2) may be described mathematically. Assume that the coordinate system of the reconstruction plane is pq plane. The complex wave front Ocan be regarded as the complex amplitude of the sample modulated by complex amplitude of random coherent noise, and can be expressed as follows:
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 non-uniform 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. 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.
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 step height. Figure 3(a) shows an average of 50 off-axis interferograms captured when the sample is adjusted as an object in the Mach-Zender interferometer. In order to subtract the thermal noise, fifty dark frames 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 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.
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  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 .). The aperture is completely 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 . 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  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.
After the spatial filtering step, the digital reference wave is used for the centering process. Then, the final reconstructed object is obtained by adjusting the values of and. 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.
The one-dimensional (1D) surface profile height h can be calculated directly as follows: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 . They are non-contacting, 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 .
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 interferograms. The intensity variation value is calculated by, where is the mean and σ is the standard deviation. Taking 3σ 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 .
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.
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.
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].
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%.
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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