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Abstract
An adaptive filtering method for off-axis digital holographic reconstruction is presented. The spatial spectrum distribution of an off-axis digital hologram is clustered based on the K-means clustering algorithm of unsupervised machine learning . The spatial-spectrum filtering can be carried out by the quantitative comparison of different filtering interception windows. The first-order spectrum including the object information can be automatically located and intercepted after clustering the spatial spectrum distribution of the hologram. The experiment results demonstrate that the reconstruction images with better resolution and imaging quality can be achieved by the adaptive filtering algorithm. This adaptive filtering reconstruction based on K-means clustering provides a new way of automatic reconstruction imaging for off-axis digital holography.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Holography is an imaging technology to record and reconstruct the light field distribution of a three-dimensional sample based on optical interference and diffraction [1–3]. In the reconstruction imaging from holograms, there occurs the influence of the zero-order term and the conjugate term on the signal term [4]. The centers of the zero-order term, the signal term and the conjugate term of an off-axis hologram are apart from each other in its Fourier spectrum domain, while the three spectrum centers of an in-line hologram are located together. The object information included in the signal term of the off-axis hologram can be extracted by filtering in its Fourier spectrum domain. The Fourier spectrum of an off-axis digital hologram, i.e., hologram’s spatial-frequency spectrum, typically distributes around three center positions, corresponding to the centers of the zero-order, the positive and negative first-order spectrums. The center positions of the positive and negative first-order spectrums indicate the carrier frequency of the hologram’s spatial spectrum. The distance between the first-order spectrum center and the zero-order spectrum center completely depends on the off-axis interference angle in holographic recording. As the off-axis interference angle is taken as 0$^{\circ }$ , the centers of the positive and negative first-order spectrums are overlapped on the center of the zero-order spectrum, which is the case of an in-line digital hologram. In off-axis digital holographic imaging, the complex amplitude distribution of an object wave is typically reconstructed from the positive or negative first-order spectrum that includes the object spatial spectrum therein. The filtering interception processing of the first-order spectrum may affect the quality of the reconstructed map [5]. The spectrum interception range in off-axis digital holography is usually set in manual, via repeatedly comparing the reconstructed images in quality of due to the lack of quantitative criteria. The smaller the spectrum interception window is taken, the lower the utilization rate of the spatial bandwidth of a hologram is. The hologram’s spatial bandwidth determines the resolution of the reconstructed amplitude and phase images. If the interception window is taken too small, the resolution of the reconstruction image will be lower obviously. Conversely, for a too large interception window, the residue of the zero-order spectrum and the edge effect can cause some noise and distortion on the reconstructed amplitude and phase images. Particularly, in a batch data processing, setting filtering areas in manual is time-consuming and inefficient.
By consideration of the three separate centers in the spatial-spectrum intensity distribution of an off-axis hologram, the spectrum filtering in the imaging reconstruction can be expected more intelligent and faster by using statistical methods. In recent years, some adaptive spatial filtering methods have been presented. For instance, the filtering window in accord with the special shape of spectrum distribution can be obtained by adaptive spatial filtering such as via analyzing the spectrum histogram and based on region growing in digital holographic microscopy [6,7]. The automatic filtering based on the Butterworth filter in digital holographic reconstruction can be applied to the MEMS analysis of quantitative phase imaging [8], and the adaptive spatial filtering using weighting in digital holographic microscopy can reconstruct the 3D contours with sharp edges [9]. In addition, with the development of machine learning, the image learning-based recognition is increasingly combined into digital holography to improve the automatic processing in reconstruction imaging. Machine learning is generally categorized into supervised learning and unsupervised learning. Machine learning applied in digital holographic reconstruction is mostly as supervised learning, such as support vector machine (SVM) and convolutional neural network (CNN) [10,11]. The focus classification using deep convolutional neural networks in digital holographic microscopy can achieve automatic focusing [12]. The convolutional neural network combined with Zernike polynomial fitting can be used in the automatic compensation for the phase distortion in digital holographic imaging, which greatly improves the phase imaging quality of digital holography [13]. In addition, the convolutional neural network can be used for adaptive filtering in digital holographic reconstruction, because of its superior robustness and filtering effect [14]. Typically, the supervised learning method needs a great number of labeled data as the training sets [15]. This means that the training process typically requires a lot of extra computation. However, such extra amount of computation is unnecessary in some cases, especially for solving small-scale problems. On the contrary, the unsupervised learning methods without requiring pre-training have lower data redundancy and computational complexity [16]. The clustering algorithm is a typical unsupervised learning algorithm [17]. As one of the most commonly used algorithms in the field of data mining, the $K$-means clustering algorithm is popular in optical image recognition, medical identification, spectral analysis, fluid dynamics and morphological analysis, because of its characteristics of a small amount of computation and a strong convergence [18–23]. In the $K$-means clustering method, a set of data objects is divided into the specified number of clusters via an iteration process [24].
In this paper, we present an adaptive filtering method based on $K$-means clustering for the automatic filtering of spatial-frequency spectrum in off-axis digital holographic reconstruction. By clustering the position parameters in the hologram’s Fourier spectrum where the gray values over a certain threshold, the center position of the positive first-order spectrum and the filtering range can be obtained adaptively.
2. Method
2.1 Off-axis digital holographic reconstruction
The off-axis digital hologram is a fringe pattern formed via the optical interference of an object wave and a reference wave with a small angle between them. The intensity distribution of the hologram can be expressed as:
The intensity distribution of an off-axis digital hologram can be rewritten as:
The Fourier spectrum of the hologram can be expressed as:
By using the filter $H$ for filtering spatial spectrum, only the term $\mathcal {F}\left [O(x,y)R^\ast _{off}(x,y)\right ]$ in Eq. (4) can be retained. Next, this term is shifted into the center of the spectrum coordinate, and then it can be rewritten as $\mathcal {F}\left [O(x,y)R^\ast (x,y)\right ]$. Then, by operating inverse Fourier transform on $\mathcal {F}\left [O(x,y)R^\ast (x,y)\right ]$, the complex amplitude distribution on the recording plane can be obtained as [25]:
The complex amplitude distribution $U_z$ on the output plane can be achieved by using the angular spectrum propagation algorithm. For imaging reconstruction of an image plane hologram, the propagation distance is taken as z=0. The reconstructed intensity distribution on the output plane can be expressed as:
The phase distribution can be expressed as:
where $\textrm{Im}(U_z)$ is the imaginary part of $U_z$ and $\textrm{Re}(U_z)$ is the real part of $U_z$. Since the above $\varphi$ consists of wrapping and phase distortion, by using the least square estimation for phase unwrapping [26] and Zernike polynomial fitting for phase distortion compensation to the $\varphi$, the object actual phase can be obtained.2.2 Spatial spectrum domain $K$-means clustering and adaptive filtering interception
In the Fourier spectrum domain of an off-axis hologram, according to the Nyquist sampling theorem, the maximum spatial frequency of the hologram is expressed as:
where $\lambda$ is the wavelength, $f_{obj}$ is the maximum spatial frequency of the object, and $\theta$ is the angle between the object wave and the reference wave, named as off-axis interference angle. The first term can be represented as $f_c=\textrm{sin}\ \theta /\lambda$, which means the carrier frequency of the hologram.As shown in Fig. 1, the zero-order spectrum locates in the central region of the spatial spectrum domain, of which the center position is at the point $P_0(0,0)$, and the region radius is twice of the maximum spatial frequency, denoted as $2f_{obj}$. The centers of the positive and negative first-order spectrums are at the points $P_{+1}(f_x,f_y)$ and $P_{-1}(-f_x,{-f}_y)$, respectively, and their region radiuses are both equal to the maximum spatial frequency $f_{obj}$. In addition, $f_c$ is equal to the distance between the centers of the first-order spectrum and the zero-order spectrum, so that is $f_c=\sqrt {f_x^2+f_y^2}$.
The time complexity of the $K$-means algorithm is dependent on the category number $K$, the number of clustering elements $n$, the dimension of each clustering element and the iteration number $t$. We suppose that a set of points whose gray values are greater than the threshold in the spatial spectrum domain are denoted as $A=\left \{a_1,a_2{\ldots }a_n\right \}$, and each data object in the set has two dimensions, namely $a_i=\left \{u_i,v_i\right \}^\textrm{T}$. In the $K$-means clustering method, the $K$ points in the original data set $A=\left \{a_1,a_2{\ldots }a_n\right \}$ are selected as the initial center points according to the preset number of categories $K$. By considering that the centers of the zero-order spectrum and two first-order spectrums are typically separated from each other in the Fourier transform domain of off-axis holograms, herein, the number of categories $K$ is taken as 3 for clustering. Moreover, as the zero-order spectrum of the hologram is always in the center of the spectrum domain, we take the center of the spectrum distribution as one of the initial center points, and then the other two initial points are randomly selected.
The procedure of $K$-means clustering is shown in Fig. 2. In the data allocation, the algorithm typically uses the standard Euclidean distance for clustering [27]. In the flowchart, $G_k^t$ denotes anyone of the three clusters, which is the cluster $G_k$ after the iteration of $t$ times, and $c_k^{t+1}=\left \{u_k^{t+1},v_k^{t+1}\right \}^T$ is a new center point determined by the expression as:
where ${\sum _{a_i\in G_k^t}a_i}$ is the sum of the elements in cluster $G_k^t$, and $\left |G_k^t\right |$ is the number of the elements in cluster $G_k^t$.The hologram generated by a computer is used to illustrate the procedure of $K$-means clustering. Figure 3 shows the synthetic process of a computer-generated hologram (CGH), which is designed as an image-plane hologram. The Monarch pattern from the standard test images in Fig. 3(a) and the pattern of USAF-1951 resolution test target in Fig. 3(b) are used as amplitude distribution and phase distribution, respectively, to produce a computer-generated hologram. The pixel number of the above-mentioned patterns is designed as 512$\times$512 pixels. The computer-generated hologram in Fig. 3(c) is produced from both patterns in Fig. 3(a) and Fig. 3(b). Figure 3(d) shows the zoomed-in picture of the red-framed area in Fig. 3(c). The interference fringes in the hologram can be seen more clearly in the zoomed picture.
Fig. 3. (a) Amplitude distribution of Monarch pattern; (b) phase distribution generated from the pattern of USAF-1951 resolution-test-target; (c) computer-generated digital hologram by using the patterns in (a) and (b); (d) zoomed-in picture of the red-framed area in (c).
For adaptive filtering, after obtaining the spatial-frequency spectrum of the hologram by Fourier transform, the operation of $K$-means clustering on this spatial-frequency spectrum is performed firstly. The procedure of $K$-means clustering on the spatial-frequency spectrum of the hologram is illustrated in Fig. 4.
Fig. 4. Clustering spatial spectrum based on $K$-means: (a) spatial-frequency spectrum distribution of the hologram via Fourier transform, (b) spectrum image after binarization processing on (a), and (c) spectrum image after $K$-means clustering.
The spatial-frequency spectrum in Fig. 4(a) is Fourier spectrum of the hologram shown in Fig. 3(c). Figure 4(b) shows its binarized spectrum image after performing image binarization on this spatial-frequency spectrum. The 1000-1500 discrete points in Fig. 4(a) where the spectrum intensity is greater than the threshold value are selected as initial data for $K$-means clustering calculation. The spectrum intensity values of the discrete points are binarized as 1, and their two-dimensional coordinate positions are $a_i=\left \{u_i,v_i\right \}^\textrm{T}$. Then, the three groups of data objects, each belonging to the zero-order spectrum, the positive and negative first-order spectrums, are clustered into their individual clusters according to the $K$-means clustering algorithm. Figure 4(c) shows the spectrum image after clustering, where the clustered distributions of the zero-order, the positive and negative first-order spectrums are indicated with the green, red and blue colors, respectively. The positions of the clustering centers after the clustering operation may slightly deviate from the three actual center positions in the spatial-frequency spectrum. The actual centers of the zero-order, the positive and negative first-order spectrums can be obtained by further searching for the maximum value in the small area.
The resolution of the reconstructed images from off-axis digital holograms is quite dependent on spatial-frequency filtering in the holographic reconstruction process. The larger filtering window means that much more high frequency components are intercepted from the Fourier spectrum of a hologram, which is benefit for achieving the higher resolution reconstruction. However, the size of the filtering window for the first-order spectrum is limited by consideration of the effect of the zero-order spectrum on it.
In the presented filtering, the size of filtering window is selected adaptively by evaluating the effect of the zero-order spectrum distribution on the first-order spectrum. The spectrum intensity is used as an evaluation parameter for the selection of filtering window radius. The curve of spectrum intensity at the line between the positive first-order and the zero-order centers on the spatial spectrum map is shown in Fig. 5, in which the longitudinal axis is with logarithmic values.
Fig. 5. (a) Spatial-frequency spectrum of a hologram; (b) curve of spectrum intensity at the line connecting the zero-order center and the positive first-order center in (a).
With the intensity of the positive first-order spectrum center as a basis value, the ratio of the intensity at the other positions of the positive first-order spectrum to the intensity basis value is referred to as an intensity decay rate. The intensity decay rate of the first-order spectrum is used to estimate the filtering window. Firstly, by calculating the spectrum intensity values of all the points along the red line in Fig. 5(a), the intensity decay rate of the positive first-order spectrum can be obtained. Then, the reconstruction imaging under different intensity decay rates is performed to find out a proper filtering interception range for the first-order spectrum. In the simulation, the positions with 3dB and 5dB intensity decay rates relative to the basis value, of which both are in the line between the zero-order center and the positive first-order center and closest to the zero-order center, are used as the border to form the interception windows, respectively. The distance from the selected point to the positive first-order center in the spectrum map is taken as a filtering window radius, while the positive first-order center is at the center of the filtering window. Figure 6 shows the spectrum filtering interceptions with 3dB and 5dB intensity decay rates, and the reconstructed amplitude and phase images from them.
Fig. 6. Comparison of reconstruction imaging under different filtering windows: (a) filtering window intercepted with 3dB decay rate, and (b) filtering window with 5dB intensity decay rate; (c) and (d) amplitude images reconstructed each from (a) and (b); (e)-(f) zoomed-in images of the red-framed areas in (c)-(d); (g) and (h) phase maps reconstructed each from (a) and (b); (i)-(j) zoomed-in images of the red-framed areas in (g)-(h), respectively; (k) and (l) curves of phase variation along the red lines in (i) and (j), respectively.
It can be seen in Fig. 6(a) that there exists a little zero-order spectrum on the lower right edge of the intercepted filtering window of the first-order spectrum. This residual zero-order spectrum causes some period fringes appearing in the reconstructed amplitude image in Fig. 6(c). Figures 6(e) and 6(f) are the zoomed patterns in the red-framed areas of the amplitude images in Figs. 6(c) and 6(d), respectively, which show almost similar clarity and resolution. The RMSE between the reconstructed amplitude with 3dB filtering interception in Fig. 6(c) and the amplitude of the original object is 0.179, while the RMSE between the reconstructed amplitude with 5dB filtering interception in Fig. 6(d) and the amplitude of the original object is 0.170. The smaller the RMSE indicates the smaller error value between the reconstructed image and the original image. Figures 6(i) and 6(j) are the zoomed patterns in the red-framed areas of the phase images in Figs. 6(g) and 6(h), respectively, of which the resolutions both are up to the sixth groups of the sixth line pairs. The curves in Figs. 6(k) and 6(l) depict the phase variation along the red lines in Figs. 6(i) and 6(j), respectively. The two curves further indicate that the reconstructed phase images with 5dB and 3dB interception have similar resolution and phase contrast. However, there has more obvious local phase deterioration in Fig. 6(i) than that in Fig. 6(j). As shown in the red-circled parts, the reconstructed phase image in Fig. 6(i) shows some dented distortion in the marked area because of 3dB filtering interception, but the quality of the phase image in Fig. 6(j) with 5dB filtering interception is slightly better than the former, which can be attributed to the more remnant of the zero-order spectrum with 3dB filtering interception.
The reconstruction of computer-generated digital holograms is in similar with that of ordinary holograms [28]. In the above simulation, the feasibility and practicability of the adaptive reconstruction in off-axis digital holography based on the $K$-means clustering algorithm are exhibited with the computer synthesized digital hologram.
3. Experiment and results
The off-axis digital holographic imaging for practical objects is performed by using the filtering algorithm based on $K$-means clustering. The optical setup of an off-axis digital holographic system for recording image plane holograms is shown in Fig. 7.
Fig. 7. Diagram of digital holographic recording system: HWP: half wave plate; BEC: beam expanding and collimating system; PBS: polarizing beam splitter; M: mirrors; BS: beam splitter, L: lens.
A laser beam of wavelength 633nm is expanded into a wide one by using a beam expanding and collimating system (BEC). After passing through a polarizing beam splitter (PBS), the beam is split into one reflected $s$-polarized beam as an object wave and another transmitted $p$-polarized beam as a reference wave. The half-wave-plate HWP$_1$ is placed in front of the BEC to adjust the intensity ratio of the object wave and the reference wave, and the half-wave-plate HWP$_2$ is put behind the polarizing beam splitter PBS to adjust the polarization of the reference wave into the $s$-polarized. After passing through the beam splitter BS, the object wave and the reference wave propagate forward at an angle between them. Then, the object wave and the reference wave interfere to form interference fringes on the CCD sensor area, for recording digital hologram. In this interference geometry, the angle between the object and reference waves are termed as off-axis interference angle. The CCD sensor area is of 960$\times$1280 pixels, with the pixel size of 6.45$\times$6.45 µm$^2$. The propagation distance in the reconstruction for image plane holograms is taken as the value of zero.
Firstly, the reconstruction imaging by adaptive clustering filtering is compared with that by typical manual filtering. The digital hologram of an amplitude-type resolution test target USAF-1951 is recorded in the experiment, of which the pattern of 850$\times$850 pixels used for amplitude reconstruction is shown in Fig. 8(a). The off-axis interference angle in this experiment is about 2$^{\circ }$. The reconstructed amplitude image with adaptive filtering interception based on $K$-means clustering is shown in Fig. 8(b). As a comparison, the amplitude image reconstructed by manual filtering interception is shown in Fig. 8(c). Figures 8(d) and 8(e) show the local zoomed-in parts of the amplitude images in Figs. 8(b) and 8(c). As shown in Fig. 8(d), the element 5-5 in the amplitude image can be distinguished, of which the resolution is 50.80 lp/mm. In Fig. 8(e), the element 5-3 with the resolution 40.32 lp/mm is distinguishable. As a result, the imaging resolution of amplitude reconstruction with the adaptive-interception filtering is higher than that with manual-selection filtering. On the other hand, the two curves in Fig. 8(f) are the amplitude distributions at the red lines of the 5-3 line pairs in Figs. 8(d) and 8(e). The solid-line amplitude shows higher contrast in its three peak positions, but the peak contrast in the dashed-line curve is much lower. This result shows that the amplitude image reconstructed by the adaptive filtering has better image contrast than that by manual-selection filtering. In addition, in the experiment, the time spent in $K$-means clustering is about 0.026 sec, and the time for spectrum filtering is about 3.666 sec, which demonstrates that the digital holographic reconstruction by adaptive filtering based on $K$-mean clustering is less time-consuming in the reconstruction process.
Fig. 8. (a) Hologram of amplitude-type resolution test target, and its amplitude images each reconstructed (b) by adaptive reconstruction based on $K$-means clustering and (c) by manual filtering interception; (d) and (e) zoomed images of the red-framed areas in (b) and (c), respectively; (f) intensity curves of amplitude at the mark lines.
Further, the phase imaging for biological samples is performed with the adaptive reconstruction method. The experimental results are shown in Fig. 9. The off-axis digital holograms of a cicada wing and a dragonfly wing both have the size of 512$\times$512 pixels. The off-axis interference angle in holographic recording is about 1$^{\circ }$. The dark parts of the holograms in Figs. 9(a) and 9(f) are wing veins, which are thicker than the other parts of the wings.
Fig. 9. (a) Hologram of cicada wing; (b) and (c) amplitude and phase images with adaptive filtering; (d) and (e) amplitude and phase images with manual-selection filtering; (f) Hologram of dragonfly wing; (g) and (h) amplitude and phase images with adaptive filtering; (i) and (j) amplitude and phase images with manual-selection filtering.
The amplitude and phase images reconstructed by adaptive filtering based on the $K$-means clustering algorithm and by manual filtering are also shown in Fig. 9. By using the adaptive filtering, the amplitude and phase images of the cicada wing are reconstructed as shown in Figs. 9(b) and 9(c), and the reconstructed amplitude and phase images of the dragonfly wing are shown in Figs. 9(g) and 9(h). For comparison, the reconstructed amplitude and phase images of the cicada wing and the dragonfly wing by manual spectrum filtering are given in Figs. 9(d) and 9(e), and in Figs. 9(i) and 9(j), respectively. It takes 0.003 seconds for $K$-means clustering in the holographic imaging process of cicada wing or dragonfly wing. It spends 0.598 seconds and 0.620 seconds to operate spectrum filtering in the reconstructions imaging of the cicada wing and the dragonfly wing, respectively. The imaging quality with the adaptive filtering is similar to that with a manual filtering, but the time cost for reconstruction imaging by the adaptive filtering decreases significantly without reducing the image quality.
4. Conclusions
An adaptive filtering reconstruction based on $K$-means clustering for off-axis digital holographic imaging is presented. By clustering the hologram’s spatial spectrum based on $K$-means clustering algorithm, the center of the positive first-order spectrum carrying object information can be positioned automatically. The filtering interception of the first-order spectrum can be performed adaptively with respect to the selected threshold. The results demonstrate that the adaptive filtering method based on $K$-means clustering can reconstruct the amplitude and phase images for off-axis digital holograms. The quality of the reconstructed amplitude image for a resolution test target USAF-1951 by using the adaptive filtering is slight better than that by manual filtering. For biological samples, the quality of the reconstructed phase image by the adaptive filtering is not inferior to that by manual filtering. Moreover, this adaptive reconstruction imaging can be completed without the requirement of manual intervention and preset parameters. The adaptive process is not affected by human eye judgment, which can provide with much more accuracy and stability.
Funding
National Natural Science Foundation of China (61575009).
Disclosures
The authors declare no conflicts of interest.
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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