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Abstract
The application of digital holography in several fields is limited since speckle destroys the original information of the reconstructed image. This paper proposes a neighborhood filter based on multiple sub-reconstructed images according to the random distribution of speckle noise. In this method, the denoised value is equal to the weighted sum of neighboring pixel values, and the weight is calculated by the degree of correlation between different positions of multiple sub-holograms. The experimental results show that the method can greatly reduce the speckle noise, and its noise reduction performance is superior to traditional digital image processing algorithms.
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1. Introduction
Digital holography is a three-dimensional imaging technology [1], which uses CCD or CMOS to record the amplitude and phase of an object, and then realizes digital reconstruction through computer simulation. It is widely applied in the fields of microstructure measurement [2], holographic encryption [3], and optical metrology [4]. However, digital holography generally uses a laser as the light source. Due to the high coherence of the laser and the high roughness of the object, the scattered light will be coherently superimposed on the holographic plane to form speckle noise [5]. speckle will reduce the contrast and resolution of the reconstructed image [6], limiting the application of digital holography [2–4]. In order to suppress the speckle, numerous speckle reduction methods have been developed over the years.
Multi-look digital holography [7,8] has also been proven to be an effective optical method to reduce speckle noise. This method realizes speckle reduction by averaging multiple reconstructed images of the same object under different speckle distributions. It generally obtains multiple reconstructed images through polarization diversification [9], angle diversification [10], and wavelength diversification [7]. However, this kind of holography needs to rely on complex optical equipment. At the same time, the recording process will also consume a lot of time. Hincapie et al. proposed a single-shot speckle reduction method to overcome the shortcomings of the above-mentioned methods [11]. The method averages multiple sub-reconstructed images with uncorrelated speckle patterns obtained from a single hologram to suppress the speckle. Fukuoka et al. proposed a spatial mask method [12], which uses aperture overlap to obtain more sub-reconstructed images to enhance the noise reduction effect of the single-shot speckle reduction method. Although these two methods are simple and time-saving, they do not consider the distribution characteristics of speckle noise. Therefore, their speckle reduction performance is not excellent.
In addition, digital image processing methods have shown strong performance in the field of image noise reduction [13–15]. Non-local means filter [16–18] and BM3D [19] are classical noise reduction methods in digital image processing. They make full use of a large amount of redundant information in the image to reduce noise. However, these methods only have an excellent noise reduction effect for specific noise models. The speckle pattern of the reconstructed image of digital holography is randomly distributed [20], and it is difficult to estimate the noise model of the image. Therefore, these traditional digital image processing methods cannot achieve better results when dealing with coherent noise.
In this paper, combining digital image processing methods and the spatial mask method, we propose the neighborhood filter based on multiple sub-reconstructed images and the Pearson correlation coefficient [21]. Since the speckle is randomly distributed, the speckle patterns of the different sub-reconstructed images are different. According to this characteristic, the filter uses the correlation degree between different positions of multiple sub-reconstructed images to calculate the weight of the neighborhood pixels to the denoising pixels in the full reconstructed image. Experimental results show that this calculation method can effectively reduce the speckle in digital holography, and it is better than traditional digital image processing methods and the spatial mask method.
2. Principle and method
2.1 Off-axis lensless Fourier transform digital holography
In digital holography, the digital hologram recorded by CCD is obtained by the interference of the object light $O$ and the reference light $R$ [22]. The intensity distribution $I$ of the hologram can be described by Eq. (1):
where the superscript * denotes the complex conjugation.The hologram formed when the object light and the reference light are coaxial is called a coaxial hologram. The zero-order light and the twin image of the coaxial holography will overlap, causing the reconstructed image to become blurred. However, off-axis holography can solve this shortcoming [23]. The way is to form an angle between the object light and the reference light. As long as the deflection angle is large enough, the zero-order light and the twin image can be separated, and the image resolution will be improved. Next, we will do the mathematical proof. In off-axis holography, if the reference light wave vector is in the $y$-$z$ plane, $R$ will be equal to $a\cdot \exp [-j2 \pi \beta y]$. Therefore, Eq. (1) is rewritten as:
where $a$ is the amplitude of the reference light, and $\beta$ is the spatial carrier frequency. $\beta =sin\theta / \lambda$, where $\theta$ is the deflection angle between the object light and the reference light. Assuming that the illuminating light is C, the reconstructed light field $I_c$ can be described by Eq. (3):The lens has Fourier transform properties. If we use the lens to record the interference pattern of the object light spectrum and the reference light spectrum, the Fourier transform hologram can be obtained [22,23]. Fourier transform holography can complete the reconstruction process only by Fourier transforming the hologram. In addition, when the object light in off-axis holography is spherical, a lensless Fourier transform hologram can be obtained, as shown in Fig. 1. The lensless Fourier transform holography can make more effective use of the bandwidth of the image sensor, and make reconstruction steps simpler [24].
Based on the above description, this paper will use off-axis lensless Fourier transform digital holography to realize the recording process.
2.2 Proposed method
In general digital image processing, bilateral filter [25] and non-local means filter are proven to be effective methods to reduce image noise. Both of these filters are neighborhood filters, so they can be described by Eq. (4):
where $\overline {y}(i,j)$ is the denoised grayscale of the pixel $(i,j)$, $y(i',j')$ is the grayscale of the pixel $(i',j')$ in the neighborhood window $\Omega$ of the pixel $(i,j)$, $\omega$ is the weight of the pixel $(i, j)$ to the pixel $(i', j')$. The weight comes from the similarity between pixels, and the more similar the pixels, the greater the weight. In the bilateral filter, the similarity is measured by the grayscale and distance between different pixels. In the non-local means filter, the similarity is measured by the grayscale and distance between the neighborhoods of different pixels. However, when calculating the weights of these two filters, only grayscale and distance are used. It often leads to inaccurate estimation of pixels. In addition, laser speckle is randomly distributed, and there is no specific noise model, which will greatly increase the error rate of estimation. Therefore, inspired by neighborhood filters [17,18,25] and the spatial mask method, we propose a neighborhood filter based on Pearson correlation coefficient [21] and multiple sub-holograms, which takes into account the random distribution of laser speckles in digital holography. The proposed filter improves the weight calculation more reasonably. It should be noted that the denoising target of the filter is the reconstructed image of the entire hologram, and the target of weight calculation is the reconstructed image of multiple sub-holograms. Here we focus on the weight calculation. The weight calculation process is divided into two steps.In Step 1, $N$ sub-holograms are constructed by multiplying the original hologram and the moving binary mask window, as shown in Fig. 2. This method is mentioned in the spatial mask method [12]. The size of the entire hologram is $H_x$ x $H_y$, and the size of the mask window is $S_x$ x $S_y$, which is consistent with the size of the neighborhood window $\Omega$. Therefore, the number $N$ of sub-holograms can be calculated by Eq. (5):
where, $d_x$ and $d_y$ are the movement intervals of the mask window on the $x$-axis and $y$-axis, respectively.In Step 2, we take the grayscale of the same pixels of the $N$ sub-reconstructed images as a data set. The data set corresponds to the same pixel of the denoising target (The reconstructed image of the entire hologram). The similarity between different pixels in the denoising target is measured by calculating the Pearson correlation coefficient between the data sets. Assuming $(i, j)$ is a denoising pixel in the denoising targe, the corresponding data set is $X^{(i, j)}$, which is described by Eq. (6):
The data set $X^{(i+k_1, j+k_2)}$ corresponding to other pixels in the neighborhood of the pixel $(i, j)$ can be described by Eq. (7):
Fig. 3. Illustration of Pearson’s correlation coefficient calculation in a neighborhood window $\Omega$ of 3x3.
The above is the content of the weight calculation. To better describe our proposed neighborhood filtering method, we show the sequential steps for speckle reduction through Fig. 4. First, a reconstructed image with speckle noise is obtained by Fourier transforming the Fourier transform hologram. Second, determine the denoising pixel $(i,j)$. Third, determine the neighborhood $\Omega$ by the pixel $(i,j)$. Fourth, calculate the weight of pixels in $\Omega$ to the pixel $(i,j)$. The weight is calculated by the data set corresponding to the pixel $(i,j)$ and data sets corresponding to neighborhood pixels. Data sets are determined by the neighborhood $\Omega$ and multiple sub-reconstructed images. Finally, according to Eq. (10), we can estimate the new gray value of the pixel $(i,j)$ to achieve speckle noise reduction.
3. Experimental result
Figure 5 shows the experimental setup of off-axis lensless Fourier transform digital holography. A He-Ne laser with a wavelength of 632.8nm is used as the light source. "BE" denotes the beam expander, and "BS" denotes the beam splitter. The beam splitting ratio of the beam splitter is 1:1. The resolution of the CCD is 960 x 1280, and the size of each pixel is 4.65${\mu }$m x 4.65${\mu }$m. The object of recording digital holograms is a coin. To meet the experimental requirements of lensless Fourier transform holography, the distance from the reference point light source P to the CCD should be equal to the distance from the object to the CCD.
We obtained a reconstructed image of a digital hologram using the experimental setup in Fig. 5, as shown in Fig. 6(a). As expected, the reconstructed image contains a lot of speckle noise. Figure 6(d) shows the effect of our method. At the same time, we selected the excellent non-local means method (NLM) in the neighborhood algorithm and spatial mask method (SDM) to compare with our method, as shown in Fig. 6(b)-6(c). The parameters for these three methods are: SDM ($S_x$ x $S_y$, 640 x 320; N, 320), NLM (the search window size, 9 x 9; the similarity window size, 7 x 7), and our method ($S_x$ x $S_y$, 640 x 320; $m$ x $n$, 9 x 9; N, 320; $h^{2}$, 0.5). As we can see, the three methods have different degrees of noise suppression. Compared to SDM and NLM, our method has more noise reduction. In addition, the right side of each image in Fig. 6 shows the magnification effect of the red box and the yellow box, which more clearly confirms that our algorithm has better noise reduction performance.
Fig. 6. (a) The reconstructed image of digital hologram with speckle. Denoised images with (b) SDM, (c) NLM and (d) our method.
According to $RD(i,j)=(I(i,j)-\overline I)/\overline I$, we can estimate the relative deviation of each pixel in the image [26]. Figure 7(a)-7(d) show the relative deviation of the green boxes (contains the letter "Z") in Fig. 6. The original image has a lot of burrs (speckle), which will cause the letter "z" not to be identified. After the original image is processed by SDM and NLM, although the letter "Z" is slightly visible, the image still has more burrs. In contrast, after processing the image with our method, not only the letter "Z" is easy to recognize, but also the burr is greatly reduced. Additionally, Fig. 6 shows that our method loses edge and sharpness more than other methods. Our method is a neighborhood filtering method. Like other neighborhood filters, its speckle reduction is realized by the weighted sum of neighborhood pixel values. Although this weighted sum can weaken the noise intensity, it will also destroy the original image information. Therefore, We conduct an experiment to test how well our method destroys the edge information of the original image. This way is to extract a straight contour line from the image, which is located on the area with high black and white contrast information, as shown in the green line of Fig. 6(a). The experimental results are shown in Fig. 7(e)-7(h). It can be seen that our method does not break the edge information very much, and the contrast between the black area and the white area is large. Compared with SDM and NLM, Our method has lower noise contrast.
Fig. 7. (a)-(d) denote the relative deviation of the green boxes in Fig. 6(a)-6(d), and (e)-(h) denote the evolution of intensity profiles for the green lines in Fig. 6(a)-6(d).
We further investigate the effect of the statistical properties of speckle on the performance of denoising methods. Figure 8 shows the gray histogram in the yellow boxes in Fig. 6. This histogram can reflect the statistical distribution of speckle. From Fig. 8, We can see that the noise distribution of the processed image has changed a lot. It is worth mentioning that the images processed by our method have a more concentrated distribution, which shows that our proposed method has a better speckle reduction effect.
Fig. 8. Gray histogram in the yellow boxes in Fig. 6. (a) Result of Fig. 6(a) (the reconstructed image); (b) result of Fig. 6(b) (NLM); (c) result of Fig. 6(c) (SDM); (d) result of Fig. 6(d) (our method).
In addition to a qualitative evaluation of the images, we also performed quantitative analysis using three image quality evaluation metrics. These metrics are:
(a) Normalized Root Mean Square Error(NRMSE) [18], defined by
(b) Equivalent Number of Looks (ENL) [18], defined by
where $\mu$ and $\sigma$ are the mean gray level of the image and standard deviation, respectively. The ENL can reflect the speckle-noise reduction ability of different methods. The larger the ENL value, the smoother the image.(c) Speckle Index (SI), defined by
where $\sigma (i,j)$ and $\mu (i,j)$ are the standard deviation and the mean gray level in the neighborhood of the pixel $(i,j)$, respectively. SI decreases with speckle noise reduction.Figure 9 shows the comparison of the three metrics values for the three methods (NLM, SDM, and our method). Without loss of generality, we choose the red box, yellow box, and green box of Fig. 6 as the calculation area. It can be seen that in the three boxes, the NRMSE and ENL of our method are the largest, and the SI of our method is the smallest, which shows that the performance of our method in metrics evaluation is also superior to NLM and SDM. Therefore, the quantitative and qualitative analysis has confirmed our method to be reliable. In other words, this method, which combines the spatial mask method and the neighborhood filter, is better than the spatial mask method and the original neighborhood filtering algorithm in speckle reduction.
Fig. 9. Evaluation metrics comparison between the SDM, NLM, and our method. (a) NRMSE, (b) ENL, and (c) SI.
Our proposed method is a neighborhood filtering method whose denoising performance is affected by the size of the neighborhood window $\Omega$. In addition, the control parameter $h^{2}$ is another important factor that affects the speckle reduction performance. To this end, we designed an experiment to analyze the effect of $h^{2}$ and the neighborhood window size on filter performance. Figure 10 shows the change curves of the three image evaluation metrics and $h^{2}$ in the red box when the neighborhood window size is $'5 x 5'$, $'7 x 7'$, $'9 x 9'$. With the increase of $h^{2}$, SI has been decreasing, NRMSE and ENL have been increasing. When $h^{2}>0.5$, the three metrics values gradually tend to be stable. With the larger size of the neighborhood window, NRMSE and ENL keep increasing, and SI keeps decreasing. Comparison in NRMSE and ENL: $'5$ x $5'$ < $'7$ x $7'$ < $'9$ x $9'$, and Comparison in SI: $'5$ x $5'$ > $'7$ x $7'$ > $'9$ x $9'$. A reasonable selection of parameters can optimize the noise reduction performance of our method.
Fig. 10. Change curves of the three image evaluation metrics and $h^{2}$ in the red box. (a) the Change curves of NRMSE, (b) the Change curves of ENL, and (c) the Change curves of SI.
The denoising performances of general digital image processing methods such as the Median filter, Wiener filter, Lee filter, Wavelet filter and BM3D have been compared with our method, as shown in Table 1. As before, our computational regions are still the red box, yellow box, and green box. The parameters for these methods are: Median filter (the filter window size, 3 x 3), Wiener filter (the filter window size, 3 x 3), Lee filter (the filter window size, 3 x 3), Wavelet filter (the mother wavelet, ’db’; the decomposition scale, 3), BM3D (the search window size, 9 x 9; the similarity window size, 7 x 7), and our method ($S_x$ x $S_y$, 640 x 320; $m$ x $n$, 9 x 9; N, 320; $h^{2}$, 0.5). According to the data of the three regions in Table 1, it can be concluded that the advantages of our method in NRMSE, ENL, and SI are obvious. It needs to be further emphasized that the comparison in ENL reflects the better smoothing effect of our method, and the comparison in SI reflects the lower noise level of our method. Therefore, in the field of coherent noise reduction, this digital image processing method that can take into account noise distribution characteristics has better performance than traditional digital image processing methods.
Table 1. Performance comparison between the Median filter, Wiener filter, Wavelet filter, Lee filter, BM3D, and our method.
4. Conclusion
Speckle noise is inevitable in digital holography. To eliminate speckle noise as much as possible, many denoising methods have been developed. Due to the random distribution of speckles, it is difficult for general digital image processing methods to obtain a good denoising effect. This paper proposes an innovative method, which combines the spatial masking method and neighborhood filters. This method analyzes the distribution characteristics of speckles according to the multiple sub-reconstructed images obtained by using the mask. Experiment results show that this method is more effective than the general method in reducing the speckle noise of digital holography. In addition, this method is specially designed for digital holographic speckle noise, which can provide a new idea in the field of digital holographic speckle denoising.
Funding
National Natural Science Foundation of China (62175039).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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