## Abstract

A method to reduce coherent noise in digital holographic phase contrast microscopy is proposed. By slightly shifting the specimen, a series of digital holograms with different coherent noise patterns is recorded. Each hologram is reconstructed individually, while the different phase tilts of the reconstructed complex amplitudes due to the specimen shifts are corrected in the hologram plane by using numerical parametric lens method. Afterward, the lateral displacements of the phase maps from different holograms are compensated in the image plane by using digital image registration method. Thus, all phase images have same distribution, but uncorrelated coherent noise patterns. By a proper averaging procedure, the coherent noise of phase contrast image is reduced significantly. The experimental results are given to confirm the proposed method.

© 2011 OSA

## 1. Introduction

In digital holography, cameras like charge-coupled device (CCD) or complementary metal-oxide semiconductor (CMOS) are used to record holograms, and object wave fronts are reconstructed numerically by simulating the propagation of optical beams [1]. In combination with microscopy, digital holographic microscopy (DHM) permits nondestructive full-field, high-resolution quantitative phase contrast imaging and has been implemented in a number of applications, such as microstructure topography measurement [2,3] and living cell imaging [4,5]. However, because of the temporally and spatially coherent illumination, the coherent noise or speckles, which originates from rough surface, refractive inhomogeneities, multiple reflection, scratches, dust particle, etc., degrades the image quality and measurement accuracy. Hence, it is of particular importance to reduce the inherited coherent noise of DHM. Based on digital signal processing techniques, many approaches were proposed, i.e. classical filtering [6], Wiener filtering with an aperture function [7], discrete Fourier filtering [8], and wavelet filtering [9]. 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. The multiple holograms obtained by different ways, e.g., using multiple wavelengths [10], changing the incident angle of illumination [11,12], rotating linearly polarized reference beam with circularly polarized illumination [13], using multi-angle tilted illumination [14]. However, these methods mostly aimed at removing the speckle for intensity imaging of the rough and opaque object, and were inappropriate for phase imaging of the smooth and transparent object. Park et al. obtained multiple holograms by using independent speckle-fields illumination and achieve a significant reduction of coherent noise in phase contrast imaging by averaging procedure [15]. However, the disadvantage of this method is requirement to record and reconstruct two sets of digital holograms, one with the specimen and the other without it. 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 [16,17]. Dubois et al. demonstrated that the coherent noise can be eliminated by using a spatial partial coherent source [18,19]. Kemper et al. obtained high quality phase contrast images of living cells by using SLD and LED [20,21]. However, because of the limited coherence length, the specimen information within only limited depth can be recorded, and the system impulse response is broaden [22]. Moreover, for off-axis digital holography, the maximum number of spatial interference fringes is also restricted, and a stable optical structure is needed to ensure the equivalence of optical path in two beams for interference.

In this paper, we propose a method to reduce coherent noise for digital holographic phase contrast microscopy. By laterally shifting the specimen, a series of off-axis holograms are recorded. Due to the change of specimen position, the coherent noises of the phase images from different holograms are uncorrelated. In the reconstruction process, the differences between these phase images owing to specimen shift are corrected by using numerical phase compensation and image registration algorithms. Thus, the retrieved phase images have same distribution, but different coherent noise patterns. Consequently, by averaging the processed phase images, the coherent noise is reduced and the quality of the phase contrast image of the specimen is improved.

## 2. Coherent noise of reconstructed phase image in digital holography

In transmission configuration of DHM, a microscope objective (MO) collects the optical wave transmitted by the specimen, and produces a magnified image of the specimen near the camera. Therefore, in holography recording, the object wave can be considered as emerging directly from the magnified image. However, 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 hologram.

In this case, the magnified image can be regarded as the complex amplitude of specimen modulated by random complex amplitude, and expressed as

where $b\left(x,y\right)={A}_{b}\left(x,y\right)\mathrm{exp}\left(i{\phi}_{b}\right)$ is the complex profile which representing the transmission, thickness and refraction properties of the specimen, and $s\left(x,y\right)={A}_{s}\left(x,y\right)\mathrm{exp}\left(i{\phi}_{s}\right)$ is a random complex amplitude, which distorts the holographic fringe pattern, where*A*represents the amplitude and

*φ*represents the phase, respectivly.

When introducing a slight shift to the specimen in object plane, the magnified image also shifts correspondently and the complex amplitude of the object wave in the recording plane (*η*, *ξ*) is given by

*x*and Δ

*y*represent horizontal and vertical shifts, respectively,

*λ*denotes the wavelength,

*d*denotes the distance between the recording plane and the magnified image, the notation

**F**denotes the Fourier transform operation. When the object wave superposed with an auxiliary reference wave, a hologram is generated in recording plane.

In the recording step, the hologram is obtained by using a camera with a finite extent, which can be represented by a two-dimensional **rect** function. Subsequently, in reconstruction step, the complex amplitude of object wave in the image plane (*x'*, *y'*) is calculated by

*a*and

*b*are the length and width of the sensor array, respectively, the

*****denotes the convolution operation. Thus, after digital holographic reconstruction with convolution algorithm [23], the phase map which contained the specimen information and coherent noise can be calculated by

On the basis of above analysis, the specimen information and coherent noise are interdependent. Meanwhile, the finite hologram aperture causes the reconstructed complex amplitude to be convolved with a sinc function, which would make the coherent noise randomly vary with the specimen position. Thus, when recording a series of holograms by slightly shifting specimen, the reconstructed complex amplitudes would exhibit different coherent noise patterns. Then, by superposition process, the coherent noise of phase image would be reduced. However, because of a slight shift of the specimen, a phase tilt and a lateral shift occur in reconstruction image plane. First, the phase tilt of each reconstructed complex amplitude is corrected in the hologram plane by using numerical parametric lens method [24], where a digital phase mask is constructed by linear fitting along selected profiles extracted from a flat reference area in the reconstructed phase distribution, by which the reconstructed complex amplitude is multiplied before numerical propagation to the image plane. Hence, the propagation directions of all reconstructed wavefronts parallel to the optical axis and the differences of phase tilt are corrected. Afterward, the lateral displacements of the phase maps retrieved from the different holograms are compensated in the image plane by using digital image registration method [25]. The phase image reconstructed from the first recording digital hologram is designated as the reference image, and the other phase images reconstructed from the digital holograms recorded by small specimen shifts are seen as the shifted images. Subsequently, the relative displacements between these shifted images and the reference image are estimated with sub-pixel precision by motion estimation calculation, and then the shifted images are shifted in the x and y direction, respectively, by geometric image manipulations with the estimated parameters. So all phase images have same distribution, but uncorrelated coherent noise patterns. Finally, by a proper averaging procedure, the effect of coherent noise is suppressed so that the phase image quality is improved.

## 3. Experiments and results

The experimental setup is illustrated in Fig. 1
. The optical configuration is a modified Mach-Zender interferometer for transmission imaging. The emitted light from a frequency-doubled Nd:YAG laser source (532nm, 50mW) is divided into illumination beam and reference beam by a polarized beam splitter (PBS). A half-wave plate (λ/2) is placed in front of PBS to adjust the intensity ratio between the two beams. Another half-wave plate (λ/2) in the reference beam ensures the consistency of the polarization directions of the two beams. Both the beams are collimated as plane waves by beam expanders (BE_{1} and BE_{2} composed by spatial filters and collimating lenses). The transmitted wave of the specimen is collected by an infinity-corrected microscope objective (MO, the magnification is 20, numerical aperture is 0.4) combined with a tube lens (TL). The object wave of magnified image (*O*) is reflected by a beam splitter (BS) and superimposed with the reference wave (*R*). Off-axis holographic geometry is realized by tilting the mirror (M_{2}) with a small angle (*θ*). A CMOS camera (the interest region is 1024 × 1024 pixels, pixel pitch is 6.7μm × 6.7μm) is utilized to record the digital hologram. The observed specimen is onion epidermal cells, which is placed on a positioning stage for lateral shifting in a plane. Thus, a sequence of holograms can be obtained by slightly shifting the specimen. It is necessary to point out that the specimen displacement of each holography recording is exactly inputted and does not depend on the precision or accuracy of the mechanical stage.

The digital hologram designated as the reference state is recorded firstly and the reconstructed phase image is shown in Fig. 2(a) . Figure 2(b) shows another phase image which is reconstructed from a digital hologram recorded by slightly shifting specimen relative to the reference state. The displacement between the phase images is 6.91 pixels by calculating with sub-pixel image registration algorithms [25]. The magnified portions of the same region are shown in Figs. 2(c) and 2(d) respectively, which are cut out from Figs. 2(a) and 2(b) and indicated by the white rectangle. Obviously, the coherent noise distribution has changed due to the specimen position shifting.

Further, in order to show the cross correlations between the coherent noise distributions, a sequence of holograms are recorded by shifting specimen. The phase image from the first hologram represents the reference state at the position zero, and so the phase images from other hologram have a certain distance to the reference state. The normalized cross correlation between the reference state and other state can be calculated by

*δ*, ${\overline{\varphi}}_{0}$ and ${\overline{\varphi}}_{\Delta \delta}$ denote the mean values of the phase distributions. The phase distribution used to calculate is selected from a uniform region, which indicated by the white rectangle in Fig. 2. The results of cross correlation coefficient for different displacement distances of the specimen are shown in Fig. 3 . As expected, the decorrelation of the coherent noise is considerable. The cross correlation coefficient decreases rapidly with the distance increasing, and when the displacement increases to 3.5 pixels which corresponds to the specimen shift of about 1.7μm in object plane, the cross correlation coefficient reduces to 0.17. Moreover, the displacement distance increases to more than 10 pixels, the cross correlation coefficient varies in the range of −0.07 ~0.07.

The results of averaging the processed phase images are shown in Fig. 4 , where the phase image from a single hologram is in Fig. 4(a) and the phase image after averaging 15 images is in Fig. 4(b). To avoid a smearing of the 2π phase, the sine and cosine values of the phase maps are calculated separately in the superposition process. It can be noticed that the concentric diffraction rings appeared in Fig. 4(a) are eliminated in Fig. 4(b). Furthermore, Fig. 5 shows the phase distributions along the straight line indicated by a white line in Fig. 4(a), with different number of phase images. Above results verify the parasitic coherent noise can be reduced by using the proposed method.

To further evaluate the effect of the averaging process quantitatively, Fig. 6 shows the relation between the variance value and the number of phase images used in the averaging process. The standard deviation can be expressed as

*ϕ*denotes the phase distribution after averaging process, $\overline{\varphi}$ denotes the mean values,

*n*and

*m*represent the pixel number of row and column of calculation region. The phase distribution used to calculate is also selected from a uniform region, which is indicated by the white rectangle in Fig. 4(a). As predicted, the standard deviation of the phase distribution decreases with the growing number of the phase image.

## 4. Conclusion

In this paper, a new approach to reduce coherent noise in digital holography phase microscopy is presented. It is accomplished by averaging several reconstructed phase images of the same specimen but with different coherent noise patterns, which are generated by slightly shifting specimen in recording hologram. For each specimen shifting, the displacements are not externally inputted and independent on the precision or accuracy of the mechanical stage. In the reconstruction process, the phase images from different holograms have same map, by correcting the differences due to the specimen shifting with numerical phase compensation and image registration algorithms. Consequently, by averaging these phase images, the coherent noise is reduced. Experimental results demonstrated the ability of the proposed method on improving the quality of the phase contrast image. Due to employing a set of holograms recorded by slightly shifting the specimen, the method is not suitable for real-time monitoring a specimen which may perform fast external or internal changes during holographic recording. However, the reconstruction rate depends on the performances of the computer and camera. If using proper automatic scanning scheme, specialized hardware and optimized software, the method allows for taking dynamic phase contrast microscopy. Furthermore, all acquired holograms can be post-processed for reconstruction after the acquisition sequence, provided that no real-time observation is needed. In such case, video frequency rate should be achievable with the method for phase contrast microscopy. Meanwhile, the method can also be applicable to noise reduction of intensity imaging.

## Acknowledgment

This work was supported by the National Science Foundation of China (No. 31000387), the Fundamental Research Funds for Central Universities in China (No. YWF-10-02-087), and the Science Foundation of Education Commission of Beijing (No. KZ200910005001).

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