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数字全息的过程就是利用相干光源对物体进行照射形成物光场EO,同时引入同样的一束无物体的参考光场与物光进行干涉,形成干涉条纹,通过CMOS/CCD进行记录并利用计算机进行重建的过程。记录的全息图为:
$I = {E_R}^2 + {E_O}^2 + {E_O}^*{E_R} + {E_R}^*$
(1) 式中, *代表共轭。而在数字全息的实验过程中,由于激光的高度相干特性以及照射物体表面的不规则,粗糙的表面会引起物体相位的骚乱,导致最后CMOS上接收的光信号携带散斑噪声。事实上,散斑噪声可以看成是许多随机相位点的集合,任一散斑点的强度可以表示为[16]:
$I_{\mathrm{SP}}=-2 \sigma^{2} \ln \left[2 \sigma^{2} P_{I}(I)\right]$
(2) 式中,PI(I)代表强度的概率密度,表达式为:
$P_{I}(I)=\frac{1}{2 \sigma^{2}} \exp \left(-\frac{I}{2 \sigma^{2}}\right)$
(3) 式中, σ2为方差,I≥0。
本文中重点研究对散斑噪声的去除,通过时间退偏以及空间退偏叠加的方式有效地抑制了相干噪声。接下来讨论具体的操作方法。
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图 1所示为时间退偏器的结构示意图。使用λ/4和λ/2波片调制物光,QWP(quarter wave-plate)和HWP(half wave-plate)分别为λ/4, λ/2波片。以MULLER矩阵的形式表示任意波片对光的调制[17]:
$\boldsymbol{M}=\left[\begin{array}{cccc}1 & 0 & 0 & 0 \\ 0 & \cos ^{2}(2 \alpha)+\sin ^{2}(2 \alpha) \cos \delta & (1-\cos \delta) \cos (2 \alpha) \sin (2 \alpha) & -\sin (2 \alpha) \sin \delta \\ 0 & (1-\cos \delta) \cos (2 \alpha) \sin (2 \alpha) & \sin ^{2}(2 a)+\cos ^{2}(2 \alpha) \cos \delta & \cos (2 \alpha) \sin \delta \\ 0 & \sin (2 \alpha) \sin \delta & -\cos (2 \alpha) \sin \delta & \cos \delta\end{array}\right]$
(4) 式中, α为波片的旋转角度,δ为相位调制量。λ/4波片的相位调制量为π/2;而λ/2波片的相位调制量为π。实际波片的相位调制量为[17]:
$\delta(\lambda)=\frac{2 \pi d\left(n_{\mathrm{e}}-n_{\mathrm{o}}\right)}{\lambda}$
(5) 式中, d为波片厚度,ne和no分别为波片的折射率及空气折射率。研究表明[3],当二元复合波片λ/4波片的角速度ω1与λ/2波片的角速度ω2之比为1:2时可达到理论上的理想退偏, 即ω1:ω2=1:2。此时,二元波片的复合Muller矩阵[3]为:
$\begin{aligned} \boldsymbol{M}_{t}=& \boldsymbol{M}_{\lambda / 4} \times \boldsymbol{M}_{\lambda / 2}=\boldsymbol{M}_{\delta=\pi / 2} \times \boldsymbol{M}_{\delta=\pi / 4}=\\ &\left[\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{array}\right] \end{aligned}$
(6) -
研究表明,毛玻璃可以减少光学系统的衍射噪声[18]。假设通过毛玻璃后的光场为G(x, y),毛玻璃引入的相位为φg,入射光的相位为φ0,则G(x, y)的表达式为:
$G(x, y)=G_{0} \exp \left[\mathrm{i} \varphi_{0}(x, y)\right] \exp \left[\mathrm{i} \varphi_{g}(x, y)\right]$
(7) 式中, G0为常数。
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本文中将时间退偏装置与空间退偏装置相结合,有效地降低了光源的相干性,并进而降低了全息图的散斑噪声。通过在物光侧将二元复合波片的装置与毛玻璃空间退偏装置相结合,对物光的相位进行了多重调制并抑制了散斑噪声。最终获得的相位结果为:
$\varphi=\delta(\lambda) \varphi_{\mathrm{g}}$
(8) -
本文中研究时空域退偏对于相干光的效果,采用邦加球进行仿真。邦加球由于其独特的偏振敏感特性, 一直以来用作分析偏振态的测试仪器。如图 2a所示,单色平面偏振光的斯托克斯矢量的邦加球表示为[19-20]:
$\boldsymbol{S}=\left[\begin{array}{cccc}S_{0} & S_{1} & S_{2} & S_{3}\end{array}\right]^{\mathrm{T}}= \\ \left[\begin{array}{cc}\mathrm{i} & \operatorname{icos}(2 \theta) \cos (2 \Phi) & \operatorname{icos}(2 \theta) \sin (2 \Phi) & \operatorname{isin}(2 \theta)]^{\mathrm{T}}\end{array}\right.$
(9) 式中, S1, S2, S3为球面上某点P的坐标,θ和Φ分别为对应球坐标的两个角度。邦加球的不同区域模拟不同的偏振态,坐标轴无量纲。当满足S02=S12+S22+S32时, 为全偏振光,代表球体表面的点; S02>S12+S22+S32, 为部分偏振光,代表在球体内部的点。如图 2b所示,邦加球上各点偏振态皆不同,邦加球水平最大圆周上的点表示线偏振光,从上极点到下极点右旋椭圆偏振光向左旋椭圆偏振光过度,球面为全偏振光,球心代表未经偏振调制的自然光,球体内为部分偏振光。采用二元波片旋转退偏与毛玻璃空间退偏时会产生任意偏振态的光,如图 2c所示,其中部分偏正光的坐标点未标出。
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图 3为所提方法的全息退偏装置。其中激光源为5mW,532nm绿光激光器,BS(beam spliter)为分光器,OBJ(objective)为20×倍率显微镜用于扩束,A为小孔,用于小孔滤波,QWP为λ/4波片,HWP为λ/2波片,两波片以ω1:ω2=1:2的角速度旋转, L为透镜,GG为毛玻璃,D为骰子,近似一个边长1cm的圆角立方体。CMOS用于记录全息图,其分辨率为1280pixel×1022pixel且像素大小为5.3μm×5.3μm,记录距离为45cm。将波片去除时为空间退偏装置,将毛玻璃去掉时为时间退偏装置。实验结果如图 4所示。
图 4中将上述提到的几种方法的实验结果进行了对比。图 4a为初始全息图重建,图 4b为空间退偏重建图,图 4c为时域退偏重建图,图 4d为时域空域退偏重建图。显然,图 3b空域退偏的效果最差,毛玻璃将散斑打乱,于是导致了整体质量的下降,同时将部分高频信息淹没,导致物体的边缘信息难以感知,但0级像的噪声显著降低,验证了毛玻璃降低相干度的作用。图 4c中采用时域退偏,拍摄的时间为10s,期间一共拍摄了70幅全息图,得出一个叠加平均的全息图。可以看到,对比初始全息图,时域退偏的重建图无论是±1级像还是0级像的噪声都得到了大幅度的降低。图 4c比起图 4b,其全局噪声也得到抑制,同时散斑噪声也远小于图 4a中所示。图 4d在图 4c的基础上添加毛玻璃进行时域空域叠加退偏,起到了进一步抑制噪声的作用,且从感知质量上来看,图 4d的去噪效果是最好的。用图像的信噪比来评价去噪的效果。信噪比定义为:
$R_{\mathrm{SNR}}=10 \lg \left\{\frac{\sum\limits_{x=1}^{N} \sum\limits_{y=1}^{N}\left[f_{0}(x, y)-\hat{f}_{O}(x, y)\right]^{2}}{\sum\limits_{x=1}^{N} \sum\limits_{y=1}^{N}\left[f_{\mathrm{p}}(x, y)-\hat{f}_{\mathrm{p}}(x, y)\right]^{2}}\right\}$
(10) 式中, fO为原始图像的像素值, $\hat{f}_{o}$为原始图像的像素平均值,fp为处理后的像素值, $\hat{f}_{\mathrm{p}}$为处理后的像素平均值,N为像素数。计算所得信噪比记录在表 1中。
Table 1. Signal-to-noise ratio of different methods
spatial-domain
depolarizationtime-domain
depolarizationspatiotemporal
depolarizationRSNR/dB 0.4459 1.5155 1.7162 为进一步验证上述方法的去噪效果,取各实验结果的频谱图进行了分析。图 5a~图 5d分别为图 4a~图 4d的频谱图,其中x, y轴为像素坐标,z轴为灰度值。
显然,对比图 5b~图 5d的频谱和图 5a中的频谱可以看出,上述方法都在一定程度上抑制了全息散斑噪声。以频谱的灰度值作为分析指标,从0级像可以直观地得出,空间退偏以及时间退偏对于0级像的散斑抑制效果比较明显,两种方法都将±1级像与0级像之间的噪声抑制了一定的程度; 同时,也将±1级像中的噪声也抑制了一部分,将时空退偏效果叠加之后可以看到,时空域退偏将噪声抑制了一部分也保留了许多+1级像中的物体信息。此外,实验过程中的噪声抑制与物体信息的丢失是同步的,因此无法精确地比较噪声抑制的同时物体信息的损失,对此,通过感知质量比较最后的去噪效果,并在之后采用信噪比来直观地比较时空退偏的效果。
为了直观地比较各种方法,进一步对频谱进行2维曲线的分析。图 6a~图 6d分别为图 5a~图 5d当x=0时的截面图,其中x轴为像素坐标,y轴为灰度值。
从图 6中可以直观地看出不同方法降噪的效果。频谱图降噪的评价指标为±1级像与0级像边缘之间的空隙质量。显然,从图 6a可以看出,初始全息图在记录过程中会受到多重噪声的影响,±1级像频谱与0级像频谱之间包含许多噪声,导致了如图 4a所示的模糊的重建像; 图 6b为空域退偏的效果,可以看到,±1级像与0级像大大分离,这是利用毛玻璃降低了激光的相干度, 从而抑制了散斑噪声,使得噪声下降,但是缺陷在于由于毛玻璃的不均匀,导致噪声去除的不理想,如图中品红色虚线框所示; 图 6c为时域退偏效果,经过多幅全息图的记录平均降低了噪声,但同时也会将少量高频细节叠加,会产生图中品红色虚线框所示的毛刺,观察±1级图像可以看出,此方法在去噪的同时也去除了许多物体信息; 图 6d为时空域退偏的效果,可以看出,比起初始全息图,此方法显著地去除了噪声,对比图 6b和图 6c, 此方法保留了许多物体信息。
如表 2所示,将图 6中的4种情况的参量全部都记录下来, 便于直观地比较不同方法的降噪效果。
Table 2. Parameters of noise suppression of different methods
number of pixels between-1 and 0 image edges in x and y directions number of pixels between+1 and 0 image edges in x and y directions original hologram spectrum Δx=59, Δy=140 Δx=56, Δy=143 spatial filtering spectrum Δx=180, Δy=172 Δx=182, Δy=176 temporal filtering spectrum Δx=163, Δy=188 Δx=162, Δy=187 spatiotemporal filtering spectrum Δx=85, Δy=164 Δx=112, Δy=168
基于时空域退偏的数字全息成像去噪研究
Research on digital holographic imaging denoising based on time-space domain depolarization
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摘要: 为了抑制全息记录过程中的散斑噪声,采用一种通过降低相干光源相干性的方法,并进行了理论分析和实验验证。利用旋转二元波片进行退偏,在10s内记录70幅全息图进行平均化降噪; 同时利用毛玻璃对相干光进行空间退相干,降低了光源的相干性从而降低了散斑噪声。将单独采用时间退偏操作的重建图像与单独采用空间退偏操作的重建图像以及两者相结合的重建图像进行了对比。通过实验验证了所提方法的可行性,通过分析频谱比较证实了几种方法,利用信噪比直观地证实了几种方法的有效性。结果表明,时间、空间退偏结合对散斑噪声的抑制有着明显的优势,空间退偏的重建像的信噪比为0.4459,时间退偏的重建像信噪比为1.5155,而两者结合后的信噪比为1.7162,采用时间退偏以及空间退偏叠加的方式有效地抑制了相干噪声。该实验为研究数字全息成像的过程中由于光源相干度过高带来的散斑噪声干扰的问题提供了一个有效的抑制方法。Abstract: In order to suppress speckle noise during holographic recording, a coherence method by reducing the coherent light source was used, and the theoretical analysis and experimental verification were performed. A rotating binary wave plate was used for depolarization, and 70 holograms were recorded in 10s for average noise reduction. At the same time, ground glass was used for spatial decoherence of the coherent light, which reduces the coherence of the light source and thus the speckle noise. The reconstructed image with the temporal depolarization operation alone was compared with the reconstructed image with the spatial depolarization operation alone, and the reconstructed image combining the two. The feasibility of the proposed method was verified by experiments, and several methods were confirmed by analyzing the frequency spectrum comparison. The signal-to-noise ratio was used to directly confirm the effectiveness of several methods. The experimental results show that the combination of time and space depolarization has obvious advantages in suppressing speckle noise. The signal-to-noise ratio of the reconstructed image with spatial depolarization is 0.4459, the reconstructed image with time depolarization is 1.5155, and the combined image with time depolarization is 1.7162. The method of time decrement and space decrement stack is used to suppress the coherent noise effectively. This experiment provides an effective method for the study of speckle noise caused by the over-high coherence of light source in the process of digital holographic imaging.
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Table 1. Signal-to-noise ratio of different methods
spatial-domain
depolarizationtime-domain
depolarizationspatiotemporal
depolarizationRSNR/dB 0.4459 1.5155 1.7162 Table 2. Parameters of noise suppression of different methods
number of pixels between-1 and 0 image edges in x and y directions number of pixels between+1 and 0 image edges in x and y directions original hologram spectrum Δx=59, Δy=140 Δx=56, Δy=143 spatial filtering spectrum Δx=180, Δy=172 Δx=182, Δy=176 temporal filtering spectrum Δx=163, Δy=188 Δx=162, Δy=187 spatiotemporal filtering spectrum Δx=85, Δy=164 Δx=112, Δy=168 -
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