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Phase aberration compensation of digital holographic microscopy with curve fitting preprocessing and automatic background segmentation for microstructure testing
Journal Pre-proof Phase aberration compensation of digital holographic microscopy with curve fitting preprocessing and automatic background segmentation for microstructure testing Liu Huang, Liping Yan, Benyong Chen, Yanjiang Zhou, Tao Yang
PII: DOI: Reference:
S0030-4018(20)30041-9 https://doi.org/10.1016/j.optcom.2020.125311 OPTICS 125311
To appear in:
Optics Communications
Received date : 27 October 2019 Revised date : 5 January 2020 Accepted date : 12 January 2020 Please cite this article as: L. Huang, L. Yan, B. Chen et al., Phase aberration compensation of digital holographic microscopy with curve fitting preprocessing and automatic background segmentation for microstructure testing, Optics Communications (2020), doi: https://doi.org/10.1016/j.optcom.2020.125311. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.
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Phase aberration compensation of digital holographic microscopy with curve fitting preprocessing and automatic background segmentation for microstructure testing
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Liu Huang, Liping Yan,* Benyong Chen, Yanjiang Zhou, and Tao Yang Precision Measurement Laboratory, Zhejiang Sci-Tech University, Hangzhou 310018, China
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*Corresponding author: [email protected] ABSTRACT
In the digital holographic microscopy, the phase aberration is a key problem limiting the quantitative phase measurement, especially for the microstructure whose phase distribution is
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dense with few and discontinuous background region. A numerical phase aberration compensation method with global curve fitting preprocessing and automatic extraction of the background region for microstructure testing is proposed. The global curve fitting is implemented to compensate most of the tilt, quadratic and partial high order aberrations and make the profile
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of the measured object significantly distinguishable. Then the background region is automatically extracted by directly segmenting the converted grayscale phase image. At last, the Zernike polynomial fitting is performed on the extracted background region, and the obtained coefficients are used to calculate the phase aberration of the whole phase image. Simulation and experimental results demonstrated that the proposed method is able to accurately extract background region and compensate the phase aberration for the complicated dense microstructure. Keywords: Digital holographic microscopy, Phase aberration compensation, Curve fitting, Microstructure
1. Introduction
Digital Holographic Microscopy (DHM) is a non-contact and full-field imaging technique
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for surface topography measurement of micro-object or microstructure. The DHM system records a single digital hologram to reconstruct the amplitude and the phase of an object wavefront by optical diffraction technique [1]. As having the advantages of non-invasive, high sensitivity and real-time quantitative phase imaging, DHM has been applied in many inspecting fields such as living cells [2-4], microstructures [5,6], microfluidics [7], and micro-electro-mechanical systems (MEMS) [8,9]. Generally, in DHM, due to the off-axis geometry, the mismatch of spherical
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curvature between the object and reference beams, and the optical aberration of the setup, the unwrapped phase distribution includes tilt, quadratic, as well as higher-order aberrations. These aberrations will result in an inaccurate recovering of the surface topography of the measured
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object. Especially for MEMS testing, the miniaturization trend and dense microstructures make the phase aberration compensation and quantitative phase measurement more difficult in DHM [6].
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In recent years, many methods have been proposed to compensate the phase aberration in DHM, which could be divided into the physical methods [10–16] and numerical methods. The physical methods need an additional hologram, or complex optical configuration, even other numerical methods to compensate the residual aberration. In contrast, the numerical methods
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[17–29] including the principal components analysis [20], Zernike polynomial fitting (ZPF) [21,22], least square fitting [23], spectral analysis [24-26] and so on can correct the phase aberration without needing additional hologram or precisely knowing the system parameters. Min et al. proposed a simple and effective numerical aberration compensation method for quantitative
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phase imaging of living cells with DHM that is based on the analysis of spatial frequency spectra of off-axis holograms [26]. These methods generally analyze or fit the whole phase distribution to obtain the aberration, which are applied for the detection of small thin samples in transmissive system, such as living cells, biological sample or thin lens. In these applications, the phases of the measured object can be regarded as a small perturbation to the overall reconstructed phase distribution. However, for MEMS testing, the phase distribution of functional structures is usually dense and continuous, so these above numerical compensation methods are inapplicable because the object phase is no longer a small perturbation. It is known that the reconstructed phase distribution contains the object region and the background region. In the background region, the phase distribution can be regarded as a flat. Hence, another compensation method of fitting the background region is proposed. By using the
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fitting coefficients of the background region, the phase aberration of the whole region can be calculated. Colomb et al. performed the ZPF in several background areas selected manually to compensate the phase aberration [27]. However, it is difficult to select enough background areas by hand when the structure of the object is complex. Recently, Nguyen et al. utilized the deep learning technique with convolutional neural network (CNN) to extract the background region [28]. This method is time-consuming for data collection and CNN model training. Liu et al.
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proposed an automatic and accurate phase aberration compensation method based on double fitting and background segmentation [29]. In this method, the compensation effect is mainly dependent on the first fitting result because the second fitting on the extracted background area
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only remove the residual aberrations of the first result. It works well in quantitative phase measurement of biological cells.
In this paper, we propose a phase aberration compensation method of DHM with curve
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fitting preprocessing and automatic background segmentation for microstructure testing. The global curve fitting is firstly implemented to remove most of the tilt, quadratic and cubic aberrations. Then, the obtained phase distribution is directly converted to grayscale image for background segmentation. Finally, the phase aberration of the whole phase distribution is
extracted background region.
2. Principle of the method
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calculated by using the coefficients obtained by performing the Zernike polynomial fitting on the
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In DHM, a digital hologram image is recorded by a digital camera and then numerically reconstructed in a computer. Usually, the angular spectrum and Fresnel transform algorithms are applied for the wavefront reconstruction and the arctangent operation is adopted to obtain the wrapped phase distribution of the object. Then, to reduce the influence of noises, necessary phase filtering is carried out before the phase unwrapping. After the phase unwrapping, the continuous phase distribution φ(x, y) can be expressed as [29]
( x, y) o ( x, y) a ( x, y) e ( x, y)
(1)
Where φo(x, y), φa(x, y) and φe(x, y) are the object phase, the phase aberration and the phase error related to residual noise, respectively. In Eq. (1), the aberration φa(x, y) containing the tilt, quadratic and higher-order aberrations can be expressed as [20]
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exp[ia ( x, y)] exp[i(lx x l y y)]exp[i( K x x 2 K y y 2 )]exp[ih ( x, y)]
(2)
Where lx, ly denote the tilt factors, Kx, Ky denote the quadratic factors, and φh(x, y) denotes the higher-order phase aberrations. Generally, this aberration φae(x, y) can be estimated by implementing Zernike polynomial fitting, written as [27]
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k 1
ae ( x, y) a j Z j ( x, y)
(3)
j 0
Where Zj(x, y) denotes the j-order Zernike polynomial in Cartesian coordinates, aj is the corresponding coefficient, k is the term number of Zernike polynomials. Then, the compensated
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phase distribution can be obtained by subtracting the estimated aberrations φae(x, y) from the original phase distribution φ(x, y). Finally, the phase error φe(x, y) caused by the residual noise
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can be eliminated by common filtering methods, for example, median filtering.
According to the above analysis, accurate retrieval of Zernike coefficients is the key to phase compensation of the complex microstructure when the unwrapped phase distribution contains residual noise. Because the phase distribution of the flat background region can be
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regarded as the phase aberration of this region, here we select the background region to retrieve the exact Zernike coefficients. Therefore, the problem becomes how to automatically segment the background region from the phase distribution with overwhelming phase aberration. The framework of the proposed phase aberration compensation method is shown in Fig. 1. It mainly
fitting.
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includes the curve fitting preprocessing, background segmentation and Zernike polynomial
Background segmentation
Curve fitting preprocessing
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Fig. 1. Framework of the proposed phase aberrations compensation method.
2.1 Curve fitting preprocessing
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In curve fitting preprocessing, a cubic curve fitting is implemented. For a phase distribution φ(x, y) (M×N pixels), firstly, the cubic curve fitting is performed in each row, and the horizontally fitted phase distribution φh(i, j) can be obtained by
h (i, j ) ai j 3 bi j 2 ci j +di
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(4)
Where 1≤i ≤M, and 1≤ j ≤N, ai, bi, ci and di are the cubic fitting parameters of the i-th row. Then, the cubic curve fitting is performed on the obtained phase distribution φh in each
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column, and the estimated phase aberration φv can be obtained by
v (i, j ) a j i3 bj i 2 c j i d j
(5)
Where aj, bj, cj and dj are the cubic fitting parameters of the j-th column. Thus, φv(i, j) is the fitted surface distribution containing the horizontal and vertical phase aberrations. Then the phase
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distribution φp after the curve fitting preprocessing can be obtained by
p (i, j ) (i, j ) v (i, j )
(6)
After the curve fitting preprocessing, most of the tilt, quadratic and cubic phase aberrations
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are removed. Although there may be some residual aberration, the object profile can already identified, and the subsequent background segmentation can also be implemented. If the phase aberration is more complicated, fourth or higher-order curve fitting could be adopted to make the object profile distinguishable.
It should be noted that, here, we perform the global curve fitting row by row and column by column in sequence instead of performing the surface fitting on the whole phase distribution to roughly compensate the aberration. This is because the surface fitting characterizes the global common trend of the image. For some complicated measured objects whose phase distributions spread all over the image, the local features of different regions are different. In such case, the surface fitting result might be not good. However, the global curve fitting is able to effectively characterize the local details. For example, for an M×N image, only ten parameters are used to
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characterize the surface in the cubic surface fitting. But, the cubic curve fitting will obtain 4×(M+N) parameters to construct the surface. Therefore, the cubic curve fitting is able to retain more local details of the phase distribution, which is suitable for objects such as the dense microstructures. At last, the phase distribution φp is directly converted to 2D grayscale image for the background segmentation.
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2.2. Background segmentation In the background segmentation, the grayscale image is segmented and converted to the binary image by using the maximum between-class variance algorithm (OTSU) [30]. In real
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applications, the speckle noise from the coherence light source, the sampling noise and environmental vibrations [31] may result in some undesired holes and isolated points in the binary image, thus morphological closing and opening operations are adopted to improve the
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segmentation quality. Finally, the accurate background region can be extracted by collecting all the black pixels in the binary image. 2.3. Zernike polynomial fitting
The coordinate positions (xl, yl) and the phase values φ(xl, yl) of the pixels on the background
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region are adopted to construct the Zernike polynomial equations. The unknown Zernike polynomial coefficients can be obtained by solving the following linear equations Z1,1 Z 2,1
Z1, p Z 2, p
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( x1 , y1 ) Z1,0 (x , y ) Z 2 2 2,0 ( xl , yl ) Z l ,0 ( xm , ym ) Z m,0
Z l ,1
Zl , p
Z m,1
Z m, p
Z1, k 1 a0 Z 2, k 1 a1 Z l ,k 1 a p Z m,k 1 ak 1
(7)
Where 1≤ l ≤m, m is the number of pixels on the background region; (xl, yl) is the pixel coordinate of the background region; k is the number of Zernike polynomials, and Zl,p denotes the value of p-th Zernike polynomial for the pixel (xl, yl). Here we adopt Gram-Schmidt orthogonalization method [32] to solve the problem of the above equations being overdetermined.
After obtained the Zernike polynomial coefficients, the phase aberration of the whole phase distribution can be calculated by using Eq. (3). Then, the compensated phase distribution can be
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obtained by subtracting the estimated aberrations φae(x, y) from the original phase distribution φ(x, y).
3. Simulated and experimental results To validate the performance of the proposed method, a series of phase compensation experiments were conducted using the simulated and actual hologram images. To show the
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significance of the proposed method, the results using other common phase aberration compensation methods were compared, including: (1) performing the Zernike polynomial fitting in the whole phase distribution (ZPF) [21,22]; (2) performing double fitting and background
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segmentation (double fitting method) [29]. The hologram reconstruction and aberration compensation are carried out with MATLAB on a PC system that contains an Intel Core CPU running at 3.50GHz clock speed. The memory on this PC is 4GB RAM.
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3.1. Numerical simulation
Figure 2(a) shows a simulated ideal phase distribution (200×200 pixels) of a stepped microstructure, in which the height of the flat background is 0 rad. The step heights are 1 rad, 1.5 rad and 2 rad, respectively. In addition, Gaussian random noise with mean value of 0 and
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standard deviation of 0.05 rad is added into the phase distribution in Fig. 2(a) to simulate the disturbed phase distribution caused by the noise from DHM, as shown in Fig. 2(b). The simulated phase aberration is generated with 1~9 orders Zernike polynomials in ZEMAX classification. The
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Zernike polynomials and simulated coefficients are listed in the first and second columns of Table 1. Figure 2(c) shows the distorted phase distribution after adding the simulated aberration into the phase distribution in Fig. 2(b).
Figures 3(a)-3(b) show the phase aberration estimated by the cubic curve fitting, and the precompensated phase distribution, respectively. As it can be seen from Fig. 3(b), the phase distribution still exhibits a large distortion due to the residual aberration, but the profile of microstructure can be significantly distinguished. Figure 3(c) shows the converted grayscale image of Fig. 3(b). Figure 3(d) is the binary image of Fig. 3(c) by adopting OTSU, in which the black pixels constitute the background region. At last, we perform 9 Zernike polynomials fitting in the extracted background region and the fitted surface is shown in Fig. 3(e). (a)
(b)
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(c)
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Fig. 2. Simulated phase distributions: (a) ideal phase distribution of a stepped microstructure, (b) the disturbed phase distribution after adding Gaussian random noise into (a), and (c) the distorted phase distribution after adding the simulated aberration into (b).
Figure 3(f) is the compensation result after subtracting the phase aberration of Fig. 3(e) from the original distorted phase distribution of Fig. 2(c) and performing the median filtering. It is
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obvious that the compensation result is almost close to the ideal phase distribution in Fig. 2(a).
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(c)
(e)
(f)
Fig. 3. The compensation results of simulated stepped microstructure: (a) the phase aberration estimated by the cubic curve fitting, (b) the pre-compensated phase distribution, (c) the converted grayscale image of (b), (d) the binary image of (c) by adopting OTSU, (e) the fitted surface, and (f) the compensation result obtained with the proposed method.
For comparison, ZPF and the double fitting methods are also implemented and their
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compensation results are shown in Fig. 4. Figure 4(a) shows the phase compensation result obtained with ZPF method, which is also the result of first least squares fitting in the double fitting method. ZPF method removes a large amount of the phase aberrations without selecting specimen-free background. However, as Fig. 4(a) shown, the compensation result contains larger distortion. Figure 4(b) shows the phase gradient map calculated with wrapped phase distribution of Fig. 4(a). This phase gradient map can only characterize the edge information of the
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microstructure. Although the mathematical morphology operations are used, it is still difficult to obtain an accurate functional structure region or background region. Figure 4(c) shows the phase compensation result obtained with the double fitting method. It can be seen that the compensation
the second compensation does not show obvious effect. (a)
(b)
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result is also unsatisfactory because the first compensation result contains larger distortion and
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(c)
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Fig. 4. (a) The phase compensation result obtained with ZPF method [21,22], (b) the phase gradient map calculated with wrapped phase distribution of (a), and (c) the phase compensation result obtained with the double fitting method [29].
For a further observation, Figs. 5(a)-5(d) are generated by extracting the phase data from the 100th row of the phase distributions in Figs. 2(a), 3(f), 4(a) and 4(c), which all correspond to the
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row marked with the red dotted line in Fig. 3(d). It can be seen that the result of the proposed method (Fig 5(b)) is consistent with the ideal one (Fig 5(a)). The result of ZPF method (Fig 5(c)) is similar to the double fitting method (Fig 5(d)), and both exhibit a certain degree of bending. This indicates that the compensation effect of double fitting mainly depends on the first fitting result. (a)
(c)
(b)
(d)
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Fig. 5. The profile curves obtained with the different methods: (a) the ideal profile curve of Fig. 2(a), (b) the profile curve of Fig. 3(f) obtained with the proposed method, (c) the profile curve of Fig. 4(a) obtained with ZPF method [21,22], and (d) the profile curve of Fig. 4(c) obtained with the double fitting method [29].
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The Zernike polynomial coefficients retrieved with the three methods are presented in Table 1. It is clear that the coefficients obtained with the proposed method are much closer to the simulated coefficients. Moreover, the root mean square errors (RMSE) between the ideal phase
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distribution and the compensation results of three methods, and the time consumptions are summarized in Table 2. It can be seen that the result of the proposed method is more accurate. The total time consumption of the proposed method is 1.2693s, in which the running time of
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curve fitting preprocessing, background segmentation and Zernike polynomial fitting are 0.4485s, 0.6273s and 0.1935s, respectively. Although the time consumption of the proposed method is slightly longer than that of the other methods, it is acceptable. Table 1. Zernike Polynomials and Coefficients in Simulation Simulated
2x
0.5958
2y
-0.6220
ZPF [21,22]
Double fitting [29]
0.5958
0.6004
0.6003
-0.6173
-4
-7.3458×10
2
Proposed
-0.6220
-4
3(2 x 2 y 1) 2
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ai
Zi(x,y)
-0.6174
-7.3455×10
-4
-7.4845×10
-7.4658×10-4
-3.0864×10-6
-5.2596×10-6
-6.3167×10-6
-3.1115×10-6
6( x y )
2.3114×10-5
2.3146×10-5
2.3164×10-5
2.3159×10-5
8(3x y 3 y 2 y )
5.0431×10-9
4.9480×10-9
7.3081×10-9
4.9980×10-9
8(3x 3xy 2 x)
-7.9402×10-8
-7.9399×10-8
-7.7006×10-8
-7.9395×10-8
8(3x y y )
3.8103×10-10
3.3540×10-10
7.6269×10-10
4.6031×10-9
8( x 3 xy )
-7.1773×10-8
-7.1874×10-8
-7.2302×10-8
-7.6066×10-8
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6(2 xy ) 2
2
2
3
3
2
2
3
3
2
Table 2. Comparison Results of Three Methods Results comparison
Proposed
ZPF [21,22]
Double fitting [29]
RMSEs (rad)
0.0705
0.6333
0.6306
Time consumptions (s)
1.2693
0.1690
0.3783
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3.2. Experimental results To further evaluate the performance of the proposed method, a reflective off-axis DHM setup was constructed as shown in Fig. 6. The laser beam emitted from a 632.8nm He-Ne laser is coupled into a single mode fiber by a laser-to-fiber coupler (LFC), and then divided into the object and reference beams by a 2×2 fiber. The two beams are collimated by two fiber collimators (FCL) to produce the plane waves. Two adjustable attenuators are used to regulate the
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intensities of the object and reference beams. A 4X microscope objective (MO) with 0.10 NA is placed before the specimen mounted on a two dimensional linear translation stage. A CCD camera (1624×1228 pixels) with the pixel size of 4.4 μm×4.4 μm is used to record the hologram.
(a)
LFC
Laser
(b)
CCD
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Two experiments were carried out to verify the feasibility of the proposed method. LFC
Laser
Fiber
FC
BS2
CCD
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FCL
BS2
Attenuator
Specimen
Specimen Translation Stage
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Translation Stage
Adjustable Attenuator
MO
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MO
FCL
BS1
BS1
FCL
FCL
Fig. 6. The reflective off-axis DHM system: (a) optical schematic, and (b) experimental setup. LFC, laser-tofiber coupler; FC, fiber coupler; FCL, fiber collimator; BS1/BS2, nonpolarizing beam splitters; MO, microscope objective; CCD, CCD camera.
Firstly, the hologram of a USAF 1951 resolution target (group 6 element 5) was recorded with a distance of 53 mm. In spatial filtering processing, the cropped +1 order spectrum was moved to the center of the Fourier domain to eliminate most of the tilt aberration. After reconstructing the filtered hologram by the angular spectrum algorithm, the phase unwrapping algorithm based on Discrete Cosine Transform (DCT) least square [33] was adopted to obtain the continuous phase distribution.
The aberration compensation of USAF with the proposed method is shown in Fig. 7. As shown in Fig. 7(a), the unwrapped phase distribution (281×281 pixels) of the USAF target is a
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curved surface. In preprocessing, by performing the curve fitting of quintic polynomial, the precompensated phase distribution is shown in Fig. 7(b). Figure 7(c) shows the converted grayscale image with Fig. 7(b). From Figs. 7(b) and 7(c), it can be seen that, after correcting most of the phase aberration, the object can be clearly distinguished from the background in the grayscale image. Figure 7(d) shows the result of background segmentation by performing OTSU and morphology operation, where black pixels constitute the background region. Then, we calculated the phase aberration of the whole phase distribution by using the coefficients obtained by
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performing 14 Zernike polynomials fitting on the background region, as shown in Fig. 7(e). Figure 7(f) is the final compensation result by subtracting Fig. 7(e) from Fig. 7(a). (a)
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(c)
(f)
Fig. 7. The aberration compensation of USAF with the proposed method: (a) the unwrapped phase distribution, (b) the pre-compensated phase distribution, (c) the converted grayscale image with (b), (d) the binary image of background segmentation, (e) the estimated phase aberration, and (f) the final compensation result.
The aberration compensations of the ZPF method and the double fitting method are shown
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in Fig. 8. Figure 8(a) shows both the aberration compensation result of ZPF method and the first fitting result of the double fitting method. Figures 8(b) and 8(c) show the phase gradient map and the binary image, respectively. And Fig. 8(d) is the compensation result of double fitting. It can be seen that because there is a large background area in the phase distribution, the measured object has a smaller impact on the overall ZPF, so the differences between Figures 7(f), 8(a), and 8(d) are not very obvious. However, by comparing Fig. 7(d) with Fig. 8(c), it can be seen that the
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background area extracted by the proposed method is a little more accurate than that by the double fitting method. (b)
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Fig. 8. The aberration compensation of USAF with the ZPF method and the double fitting method: (a) the compensation result of ZPF method [21,22], (b) the phase gradient map calculated with the wrapped phase distribution of (a), (c) the binary image after background segmentation with (b), and (d) the compensation result of the double fitting method [29].
For a more quantitative evaluation, Fig. 9 plots three profile curves extracted from the phase distributions of Figs. 7(f), 8(a), and 8(d), respectively. These profile curves are measured along the red dotted line in Fig. 7(d). It can be seen that the profile curve obtained with ZPF method (black curve) has an obvious larger deviation and fluctuation due to the step structures participating in the global fitting. After removing the residual aberration, the compensation result of double fitting (blue curve) improves a lot, but its fluctuation is still larger than that of the profile curve obtained with our method (red curve).
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Fig. 9. The profile curves of the compensation results obtained by three methods.
Secondly, a 5mm×5mm MEMS chip with dense microstructure was tested. Figure 10 shows the 2D structure of the chip observed by an optical microscope with 400X magnification.
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Fig. 10. 2D structure of the MEMS chip observed by an optical microscope.
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In this DHM experiment, the recording distance was 49.5 mm. The compensation results of our method, ZPF method and the double fitting method are shown in Figs. 11 and 12, respectively. Figure 11(a) is the pre-compensated phase distribution (442×442 pixels) obtained by performing a global quintic curve fitting, and Fig. 11(b) is the converted grayscale image of
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Fig. 11(a). Figure 11(c) shows the result of background segmentation by performing OTSU and morphology operation, where black pixels constitute the background region. And Fig. 11(d) is the final compensation result obtained with the proposed method. (a)
(b)
(c)
(d)
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Fig. 11. The aberration compensation result of the MEMS chip obtained with the proposed method: (a) the precompensated phase distribution, (b) the converted grayscale image of (a), (c) the binary image of background segmentation, and (d) the final compensation result obtained with the proposed method.
Figure 12(a) is the compensation result obtained with 14 Zernike polynomials fitting. Figure
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12(b) shows the phase gradient map calculated with the wrapped phase distribution of Fig. 12(a). Figure 12(c) shows the result of background segmentation of Fig. 12(b). Figure 12(d) shows the compensation result obtained with the double fitting method. In Figs. 11 and 12, the height values
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are calculated according to the relation of h(x,y)=λφo(x,y)/4π.
Comparing the compensation results of the three methods with the 2D MEMS image shown in Fig. 10, it seems that all the three methods can roughly retrieve the profile of the microstructure. However, from Fig. 12(c), it can be seen that, for the dense microstructure, it is
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difficult to distinguish the object area and the background area because the phase gradient only charactering edge information. Thus, some microstructure area is incorrectly identified as the background. In addition, as shown in Figs. 11(d), 12(a) and 12(d), the height values of retrieved
(a)
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results are different.
(b)
(d)
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Fig. 12. The aberration compensation results of the MEMS chip obtained with ZPF and the double fitting methods: (a) the result of ZPF method [21,22], (b) the phase gradient map of (a), (c) the binary image of background segmentation with (b), and (d) the result of the double fitting method [29].
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(b)
Fig. 13. Comparison of the profile curves obtained by the profilometer and three methods: (a) the profile curve obtained by the profilometer, and (b) the profile curves obtained by three methods.
For an accurate quantitative evaluation, the compensation results of three methods are compared with the measurement result of a stylus profilometer (S1910DX3; ACCRETECH).
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Figure 13(a) plots the profile curves measured along the red dotted line in Fig. 10 by the stylus profilometer, and Fig. 13(b) plots three profile curves obtained by the three methods. It can be seen that the result obtained by our method is more consistent with that obtained by the profilometer in terms of the microstructure height. Both the profile curves obtained by the ZPF
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method and the double fitting method have a larger deviation of about 20 nm. The above experimental results demonstrate that our method is able to effectively measure the profile of the MEMS chip even though its line width is narrow and dense. The essential reason why our method works well is that we are able to extract the background area automatically by performing high-order curve fitting to pre-compensate most of the phase aberration. Comparing with the global surface ZPF, the high-order curve fitting uses more parameters to construct more accurate aberration surfaces. More importantly, we calculate the global phase aberration by using the ZPF coefficients of the extracted background region, which avoids the influence of phase value of object on the fitting. Therefore, in our method, the precompensation is only used to accurately extract the background area, its aberration compensating error will not directly influence the final result. For the ZPF method and the double fitting
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method, both require that the phase distribution of the measured object is only a small perturbation to the overall phase distribution. This means that proportion of the object area must be very small, otherwise, large error will be introduced to final results. In addition, the phase gradient calculation is only able to detect the object edge information, it can not distinguish the object area and background area. Therefore, for the MEMS chip with dense microstructure, both the compensation results of ZPF method and the double fitting method are unsatisfactory. It
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should be noted that if the object is too dense so that the object area exceeds 50% of the entire phase distribution, or the phase value of object is comparable to the phase aberration, our method may not be effective.
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4. Conclusion
In summary, we have presented a numerical phase aberration compensation method of DHM
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with the curve fitting preprocessing and the automatic background segmentation. By using the global curve fitting, the proposed method has the prominent merit of effectively precompensating the phase aberrations for the dense microstructure object whose phase distribution spread all over the image. Simulation and experimental results show that the proposed method can obtain a more distinguishable object profile in curve fitting preprocessing, which makes the
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background segmentation and fitting more accurate. Compared with ZPF method and the double fitting method, the proposed method realizes a more precise phase aberration compensation by using the ZPF coefficients of the extracted background region, especially for the dense
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microstructures. Thus, the proposed phase aberration compensation method has significant application for the microstructure testing in DHM system.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (51875530); Natural Science Foundation of Zhejiang Province (LZ18E050003); Program for Changjiang Scholars and Innovative Research Team in University (IRT_17R98).
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*Credit Author Statement
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Liu Huang:
Software, Validation, Investigation, Writing - Original Draft
Liping Yan:
Conceptualization, Methodology, Writing - Review & Editing Supervision, Project administration, Funding acquisition
Yanjiang Zhou:
Resources, Writing - Review & Editing
Data Curation
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Tao Yang:
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Benyong Chen:
PII: DOI: Reference:
S0030-4018(20)30041-9 https://doi.org/10.1016/j.optcom.2020.125311 OPTICS 125311
To appear in:
Optics Communications
Received date : 27 October 2019 Revised date : 5 January 2020 Accepted date : 12 January 2020 Please cite this article as: L. Huang, L. Yan, B. Chen et al., Phase aberration compensation of digital holographic microscopy with curve fitting preprocessing and automatic background segmentation for microstructure testing, Optics Communications (2020), doi: https://doi.org/10.1016/j.optcom.2020.125311. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.
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Phase aberration compensation of digital holographic microscopy with curve fitting preprocessing and automatic background segmentation for microstructure testing
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Liu Huang, Liping Yan,* Benyong Chen, Yanjiang Zhou, and Tao Yang Precision Measurement Laboratory, Zhejiang Sci-Tech University, Hangzhou 310018, China
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*Corresponding author: [email protected] ABSTRACT
In the digital holographic microscopy, the phase aberration is a key problem limiting the quantitative phase measurement, especially for the microstructure whose phase distribution is
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dense with few and discontinuous background region. A numerical phase aberration compensation method with global curve fitting preprocessing and automatic extraction of the background region for microstructure testing is proposed. The global curve fitting is implemented to compensate most of the tilt, quadratic and partial high order aberrations and make the profile
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of the measured object significantly distinguishable. Then the background region is automatically extracted by directly segmenting the converted grayscale phase image. At last, the Zernike polynomial fitting is performed on the extracted background region, and the obtained coefficients are used to calculate the phase aberration of the whole phase image. Simulation and experimental results demonstrated that the proposed method is able to accurately extract background region and compensate the phase aberration for the complicated dense microstructure. Keywords: Digital holographic microscopy, Phase aberration compensation, Curve fitting, Microstructure
1. Introduction
Digital Holographic Microscopy (DHM) is a non-contact and full-field imaging technique
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for surface topography measurement of micro-object or microstructure. The DHM system records a single digital hologram to reconstruct the amplitude and the phase of an object wavefront by optical diffraction technique [1]. As having the advantages of non-invasive, high sensitivity and real-time quantitative phase imaging, DHM has been applied in many inspecting fields such as living cells [2-4], microstructures [5,6], microfluidics [7], and micro-electro-mechanical systems (MEMS) [8,9]. Generally, in DHM, due to the off-axis geometry, the mismatch of spherical
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curvature between the object and reference beams, and the optical aberration of the setup, the unwrapped phase distribution includes tilt, quadratic, as well as higher-order aberrations. These aberrations will result in an inaccurate recovering of the surface topography of the measured
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object. Especially for MEMS testing, the miniaturization trend and dense microstructures make the phase aberration compensation and quantitative phase measurement more difficult in DHM [6].
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In recent years, many methods have been proposed to compensate the phase aberration in DHM, which could be divided into the physical methods [10–16] and numerical methods. The physical methods need an additional hologram, or complex optical configuration, even other numerical methods to compensate the residual aberration. In contrast, the numerical methods
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[17–29] including the principal components analysis [20], Zernike polynomial fitting (ZPF) [21,22], least square fitting [23], spectral analysis [24-26] and so on can correct the phase aberration without needing additional hologram or precisely knowing the system parameters. Min et al. proposed a simple and effective numerical aberration compensation method for quantitative
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phase imaging of living cells with DHM that is based on the analysis of spatial frequency spectra of off-axis holograms [26]. These methods generally analyze or fit the whole phase distribution to obtain the aberration, which are applied for the detection of small thin samples in transmissive system, such as living cells, biological sample or thin lens. In these applications, the phases of the measured object can be regarded as a small perturbation to the overall reconstructed phase distribution. However, for MEMS testing, the phase distribution of functional structures is usually dense and continuous, so these above numerical compensation methods are inapplicable because the object phase is no longer a small perturbation. It is known that the reconstructed phase distribution contains the object region and the background region. In the background region, the phase distribution can be regarded as a flat. Hence, another compensation method of fitting the background region is proposed. By using the
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fitting coefficients of the background region, the phase aberration of the whole region can be calculated. Colomb et al. performed the ZPF in several background areas selected manually to compensate the phase aberration [27]. However, it is difficult to select enough background areas by hand when the structure of the object is complex. Recently, Nguyen et al. utilized the deep learning technique with convolutional neural network (CNN) to extract the background region [28]. This method is time-consuming for data collection and CNN model training. Liu et al.
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proposed an automatic and accurate phase aberration compensation method based on double fitting and background segmentation [29]. In this method, the compensation effect is mainly dependent on the first fitting result because the second fitting on the extracted background area
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only remove the residual aberrations of the first result. It works well in quantitative phase measurement of biological cells.
In this paper, we propose a phase aberration compensation method of DHM with curve
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fitting preprocessing and automatic background segmentation for microstructure testing. The global curve fitting is firstly implemented to remove most of the tilt, quadratic and cubic aberrations. Then, the obtained phase distribution is directly converted to grayscale image for background segmentation. Finally, the phase aberration of the whole phase distribution is
extracted background region.
2. Principle of the method
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calculated by using the coefficients obtained by performing the Zernike polynomial fitting on the
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In DHM, a digital hologram image is recorded by a digital camera and then numerically reconstructed in a computer. Usually, the angular spectrum and Fresnel transform algorithms are applied for the wavefront reconstruction and the arctangent operation is adopted to obtain the wrapped phase distribution of the object. Then, to reduce the influence of noises, necessary phase filtering is carried out before the phase unwrapping. After the phase unwrapping, the continuous phase distribution φ(x, y) can be expressed as [29]
( x, y) o ( x, y) a ( x, y) e ( x, y)
(1)
Where φo(x, y), φa(x, y) and φe(x, y) are the object phase, the phase aberration and the phase error related to residual noise, respectively. In Eq. (1), the aberration φa(x, y) containing the tilt, quadratic and higher-order aberrations can be expressed as [20]
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exp[ia ( x, y)] exp[i(lx x l y y)]exp[i( K x x 2 K y y 2 )]exp[ih ( x, y)]
(2)
Where lx, ly denote the tilt factors, Kx, Ky denote the quadratic factors, and φh(x, y) denotes the higher-order phase aberrations. Generally, this aberration φae(x, y) can be estimated by implementing Zernike polynomial fitting, written as [27]
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k 1
ae ( x, y) a j Z j ( x, y)
(3)
j 0
Where Zj(x, y) denotes the j-order Zernike polynomial in Cartesian coordinates, aj is the corresponding coefficient, k is the term number of Zernike polynomials. Then, the compensated
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phase distribution can be obtained by subtracting the estimated aberrations φae(x, y) from the original phase distribution φ(x, y). Finally, the phase error φe(x, y) caused by the residual noise
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can be eliminated by common filtering methods, for example, median filtering.
According to the above analysis, accurate retrieval of Zernike coefficients is the key to phase compensation of the complex microstructure when the unwrapped phase distribution contains residual noise. Because the phase distribution of the flat background region can be
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regarded as the phase aberration of this region, here we select the background region to retrieve the exact Zernike coefficients. Therefore, the problem becomes how to automatically segment the background region from the phase distribution with overwhelming phase aberration. The framework of the proposed phase aberration compensation method is shown in Fig. 1. It mainly
fitting.
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includes the curve fitting preprocessing, background segmentation and Zernike polynomial
Background segmentation
Curve fitting preprocessing
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Fig. 1. Framework of the proposed phase aberrations compensation method.
2.1 Curve fitting preprocessing
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In curve fitting preprocessing, a cubic curve fitting is implemented. For a phase distribution φ(x, y) (M×N pixels), firstly, the cubic curve fitting is performed in each row, and the horizontally fitted phase distribution φh(i, j) can be obtained by
h (i, j ) ai j 3 bi j 2 ci j +di
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(4)
Where 1≤i ≤M, and 1≤ j ≤N, ai, bi, ci and di are the cubic fitting parameters of the i-th row. Then, the cubic curve fitting is performed on the obtained phase distribution φh in each
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column, and the estimated phase aberration φv can be obtained by
v (i, j ) a j i3 bj i 2 c j i d j
(5)
Where aj, bj, cj and dj are the cubic fitting parameters of the j-th column. Thus, φv(i, j) is the fitted surface distribution containing the horizontal and vertical phase aberrations. Then the phase
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distribution φp after the curve fitting preprocessing can be obtained by
p (i, j ) (i, j ) v (i, j )
(6)
After the curve fitting preprocessing, most of the tilt, quadratic and cubic phase aberrations
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are removed. Although there may be some residual aberration, the object profile can already identified, and the subsequent background segmentation can also be implemented. If the phase aberration is more complicated, fourth or higher-order curve fitting could be adopted to make the object profile distinguishable.
It should be noted that, here, we perform the global curve fitting row by row and column by column in sequence instead of performing the surface fitting on the whole phase distribution to roughly compensate the aberration. This is because the surface fitting characterizes the global common trend of the image. For some complicated measured objects whose phase distributions spread all over the image, the local features of different regions are different. In such case, the surface fitting result might be not good. However, the global curve fitting is able to effectively characterize the local details. For example, for an M×N image, only ten parameters are used to
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characterize the surface in the cubic surface fitting. But, the cubic curve fitting will obtain 4×(M+N) parameters to construct the surface. Therefore, the cubic curve fitting is able to retain more local details of the phase distribution, which is suitable for objects such as the dense microstructures. At last, the phase distribution φp is directly converted to 2D grayscale image for the background segmentation.
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2.2. Background segmentation In the background segmentation, the grayscale image is segmented and converted to the binary image by using the maximum between-class variance algorithm (OTSU) [30]. In real
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applications, the speckle noise from the coherence light source, the sampling noise and environmental vibrations [31] may result in some undesired holes and isolated points in the binary image, thus morphological closing and opening operations are adopted to improve the
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segmentation quality. Finally, the accurate background region can be extracted by collecting all the black pixels in the binary image. 2.3. Zernike polynomial fitting
The coordinate positions (xl, yl) and the phase values φ(xl, yl) of the pixels on the background
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region are adopted to construct the Zernike polynomial equations. The unknown Zernike polynomial coefficients can be obtained by solving the following linear equations Z1,1 Z 2,1
Z1, p Z 2, p
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( x1 , y1 ) Z1,0 (x , y ) Z 2 2 2,0 ( xl , yl ) Z l ,0 ( xm , ym ) Z m,0
Z l ,1
Zl , p
Z m,1
Z m, p
Z1, k 1 a0 Z 2, k 1 a1 Z l ,k 1 a p Z m,k 1 ak 1
(7)
Where 1≤ l ≤m, m is the number of pixels on the background region; (xl, yl) is the pixel coordinate of the background region; k is the number of Zernike polynomials, and Zl,p denotes the value of p-th Zernike polynomial for the pixel (xl, yl). Here we adopt Gram-Schmidt orthogonalization method [32] to solve the problem of the above equations being overdetermined.
After obtained the Zernike polynomial coefficients, the phase aberration of the whole phase distribution can be calculated by using Eq. (3). Then, the compensated phase distribution can be
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obtained by subtracting the estimated aberrations φae(x, y) from the original phase distribution φ(x, y).
3. Simulated and experimental results To validate the performance of the proposed method, a series of phase compensation experiments were conducted using the simulated and actual hologram images. To show the
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significance of the proposed method, the results using other common phase aberration compensation methods were compared, including: (1) performing the Zernike polynomial fitting in the whole phase distribution (ZPF) [21,22]; (2) performing double fitting and background
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segmentation (double fitting method) [29]. The hologram reconstruction and aberration compensation are carried out with MATLAB on a PC system that contains an Intel Core CPU running at 3.50GHz clock speed. The memory on this PC is 4GB RAM.
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3.1. Numerical simulation
Figure 2(a) shows a simulated ideal phase distribution (200×200 pixels) of a stepped microstructure, in which the height of the flat background is 0 rad. The step heights are 1 rad, 1.5 rad and 2 rad, respectively. In addition, Gaussian random noise with mean value of 0 and
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standard deviation of 0.05 rad is added into the phase distribution in Fig. 2(a) to simulate the disturbed phase distribution caused by the noise from DHM, as shown in Fig. 2(b). The simulated phase aberration is generated with 1~9 orders Zernike polynomials in ZEMAX classification. The
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Zernike polynomials and simulated coefficients are listed in the first and second columns of Table 1. Figure 2(c) shows the distorted phase distribution after adding the simulated aberration into the phase distribution in Fig. 2(b).
Figures 3(a)-3(b) show the phase aberration estimated by the cubic curve fitting, and the precompensated phase distribution, respectively. As it can be seen from Fig. 3(b), the phase distribution still exhibits a large distortion due to the residual aberration, but the profile of microstructure can be significantly distinguished. Figure 3(c) shows the converted grayscale image of Fig. 3(b). Figure 3(d) is the binary image of Fig. 3(c) by adopting OTSU, in which the black pixels constitute the background region. At last, we perform 9 Zernike polynomials fitting in the extracted background region and the fitted surface is shown in Fig. 3(e). (a)
(b)
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(c)
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Fig. 2. Simulated phase distributions: (a) ideal phase distribution of a stepped microstructure, (b) the disturbed phase distribution after adding Gaussian random noise into (a), and (c) the distorted phase distribution after adding the simulated aberration into (b).
Figure 3(f) is the compensation result after subtracting the phase aberration of Fig. 3(e) from the original distorted phase distribution of Fig. 2(c) and performing the median filtering. It is
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(a)
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obvious that the compensation result is almost close to the ideal phase distribution in Fig. 2(a).
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(c)
(e)
(f)
Fig. 3. The compensation results of simulated stepped microstructure: (a) the phase aberration estimated by the cubic curve fitting, (b) the pre-compensated phase distribution, (c) the converted grayscale image of (b), (d) the binary image of (c) by adopting OTSU, (e) the fitted surface, and (f) the compensation result obtained with the proposed method.
For comparison, ZPF and the double fitting methods are also implemented and their
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compensation results are shown in Fig. 4. Figure 4(a) shows the phase compensation result obtained with ZPF method, which is also the result of first least squares fitting in the double fitting method. ZPF method removes a large amount of the phase aberrations without selecting specimen-free background. However, as Fig. 4(a) shown, the compensation result contains larger distortion. Figure 4(b) shows the phase gradient map calculated with wrapped phase distribution of Fig. 4(a). This phase gradient map can only characterize the edge information of the
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microstructure. Although the mathematical morphology operations are used, it is still difficult to obtain an accurate functional structure region or background region. Figure 4(c) shows the phase compensation result obtained with the double fitting method. It can be seen that the compensation
the second compensation does not show obvious effect. (a)
(b)
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result is also unsatisfactory because the first compensation result contains larger distortion and
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(c)
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Fig. 4. (a) The phase compensation result obtained with ZPF method [21,22], (b) the phase gradient map calculated with wrapped phase distribution of (a), and (c) the phase compensation result obtained with the double fitting method [29].
For a further observation, Figs. 5(a)-5(d) are generated by extracting the phase data from the 100th row of the phase distributions in Figs. 2(a), 3(f), 4(a) and 4(c), which all correspond to the
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row marked with the red dotted line in Fig. 3(d). It can be seen that the result of the proposed method (Fig 5(b)) is consistent with the ideal one (Fig 5(a)). The result of ZPF method (Fig 5(c)) is similar to the double fitting method (Fig 5(d)), and both exhibit a certain degree of bending. This indicates that the compensation effect of double fitting mainly depends on the first fitting result. (a)
(c)
(b)
(d)
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Fig. 5. The profile curves obtained with the different methods: (a) the ideal profile curve of Fig. 2(a), (b) the profile curve of Fig. 3(f) obtained with the proposed method, (c) the profile curve of Fig. 4(a) obtained with ZPF method [21,22], and (d) the profile curve of Fig. 4(c) obtained with the double fitting method [29].
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The Zernike polynomial coefficients retrieved with the three methods are presented in Table 1. It is clear that the coefficients obtained with the proposed method are much closer to the simulated coefficients. Moreover, the root mean square errors (RMSE) between the ideal phase
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distribution and the compensation results of three methods, and the time consumptions are summarized in Table 2. It can be seen that the result of the proposed method is more accurate. The total time consumption of the proposed method is 1.2693s, in which the running time of
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curve fitting preprocessing, background segmentation and Zernike polynomial fitting are 0.4485s, 0.6273s and 0.1935s, respectively. Although the time consumption of the proposed method is slightly longer than that of the other methods, it is acceptable. Table 1. Zernike Polynomials and Coefficients in Simulation Simulated
2x
0.5958
2y
-0.6220
ZPF [21,22]
Double fitting [29]
0.5958
0.6004
0.6003
-0.6173
-4
-7.3458×10
2
Proposed
-0.6220
-4
3(2 x 2 y 1) 2
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Zi(x,y)
-0.6174
-7.3455×10
-4
-7.4845×10
-7.4658×10-4
-3.0864×10-6
-5.2596×10-6
-6.3167×10-6
-3.1115×10-6
6( x y )
2.3114×10-5
2.3146×10-5
2.3164×10-5
2.3159×10-5
8(3x y 3 y 2 y )
5.0431×10-9
4.9480×10-9
7.3081×10-9
4.9980×10-9
8(3x 3xy 2 x)
-7.9402×10-8
-7.9399×10-8
-7.7006×10-8
-7.9395×10-8
8(3x y y )
3.8103×10-10
3.3540×10-10
7.6269×10-10
4.6031×10-9
8( x 3 xy )
-7.1773×10-8
-7.1874×10-8
-7.2302×10-8
-7.6066×10-8
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6(2 xy ) 2
2
2
3
3
2
2
3
3
2
Table 2. Comparison Results of Three Methods Results comparison
Proposed
ZPF [21,22]
Double fitting [29]
RMSEs (rad)
0.0705
0.6333
0.6306
Time consumptions (s)
1.2693
0.1690
0.3783
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3.2. Experimental results To further evaluate the performance of the proposed method, a reflective off-axis DHM setup was constructed as shown in Fig. 6. The laser beam emitted from a 632.8nm He-Ne laser is coupled into a single mode fiber by a laser-to-fiber coupler (LFC), and then divided into the object and reference beams by a 2×2 fiber. The two beams are collimated by two fiber collimators (FCL) to produce the plane waves. Two adjustable attenuators are used to regulate the
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intensities of the object and reference beams. A 4X microscope objective (MO) with 0.10 NA is placed before the specimen mounted on a two dimensional linear translation stage. A CCD camera (1624×1228 pixels) with the pixel size of 4.4 μm×4.4 μm is used to record the hologram.
(a)
LFC
Laser
(b)
CCD
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Two experiments were carried out to verify the feasibility of the proposed method. LFC
Laser
Fiber
FC
BS2
CCD
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FCL
BS2
Attenuator
Specimen
Specimen Translation Stage
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Translation Stage
Adjustable Attenuator
MO
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MO
FCL
BS1
BS1
FCL
FCL
Fig. 6. The reflective off-axis DHM system: (a) optical schematic, and (b) experimental setup. LFC, laser-tofiber coupler; FC, fiber coupler; FCL, fiber collimator; BS1/BS2, nonpolarizing beam splitters; MO, microscope objective; CCD, CCD camera.
Firstly, the hologram of a USAF 1951 resolution target (group 6 element 5) was recorded with a distance of 53 mm. In spatial filtering processing, the cropped +1 order spectrum was moved to the center of the Fourier domain to eliminate most of the tilt aberration. After reconstructing the filtered hologram by the angular spectrum algorithm, the phase unwrapping algorithm based on Discrete Cosine Transform (DCT) least square [33] was adopted to obtain the continuous phase distribution.
The aberration compensation of USAF with the proposed method is shown in Fig. 7. As shown in Fig. 7(a), the unwrapped phase distribution (281×281 pixels) of the USAF target is a
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curved surface. In preprocessing, by performing the curve fitting of quintic polynomial, the precompensated phase distribution is shown in Fig. 7(b). Figure 7(c) shows the converted grayscale image with Fig. 7(b). From Figs. 7(b) and 7(c), it can be seen that, after correcting most of the phase aberration, the object can be clearly distinguished from the background in the grayscale image. Figure 7(d) shows the result of background segmentation by performing OTSU and morphology operation, where black pixels constitute the background region. Then, we calculated the phase aberration of the whole phase distribution by using the coefficients obtained by
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performing 14 Zernike polynomials fitting on the background region, as shown in Fig. 7(e). Figure 7(f) is the final compensation result by subtracting Fig. 7(e) from Fig. 7(a). (a)
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(c)
(f)
Fig. 7. The aberration compensation of USAF with the proposed method: (a) the unwrapped phase distribution, (b) the pre-compensated phase distribution, (c) the converted grayscale image with (b), (d) the binary image of background segmentation, (e) the estimated phase aberration, and (f) the final compensation result.
The aberration compensations of the ZPF method and the double fitting method are shown
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in Fig. 8. Figure 8(a) shows both the aberration compensation result of ZPF method and the first fitting result of the double fitting method. Figures 8(b) and 8(c) show the phase gradient map and the binary image, respectively. And Fig. 8(d) is the compensation result of double fitting. It can be seen that because there is a large background area in the phase distribution, the measured object has a smaller impact on the overall ZPF, so the differences between Figures 7(f), 8(a), and 8(d) are not very obvious. However, by comparing Fig. 7(d) with Fig. 8(c), it can be seen that the
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background area extracted by the proposed method is a little more accurate than that by the double fitting method. (b)
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Fig. 8. The aberration compensation of USAF with the ZPF method and the double fitting method: (a) the compensation result of ZPF method [21,22], (b) the phase gradient map calculated with the wrapped phase distribution of (a), (c) the binary image after background segmentation with (b), and (d) the compensation result of the double fitting method [29].
For a more quantitative evaluation, Fig. 9 plots three profile curves extracted from the phase distributions of Figs. 7(f), 8(a), and 8(d), respectively. These profile curves are measured along the red dotted line in Fig. 7(d). It can be seen that the profile curve obtained with ZPF method (black curve) has an obvious larger deviation and fluctuation due to the step structures participating in the global fitting. After removing the residual aberration, the compensation result of double fitting (blue curve) improves a lot, but its fluctuation is still larger than that of the profile curve obtained with our method (red curve).
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Fig. 9. The profile curves of the compensation results obtained by three methods.
Secondly, a 5mm×5mm MEMS chip with dense microstructure was tested. Figure 10 shows the 2D structure of the chip observed by an optical microscope with 400X magnification.
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400 X
Fig. 10. 2D structure of the MEMS chip observed by an optical microscope.
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In this DHM experiment, the recording distance was 49.5 mm. The compensation results of our method, ZPF method and the double fitting method are shown in Figs. 11 and 12, respectively. Figure 11(a) is the pre-compensated phase distribution (442×442 pixels) obtained by performing a global quintic curve fitting, and Fig. 11(b) is the converted grayscale image of
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Fig. 11(a). Figure 11(c) shows the result of background segmentation by performing OTSU and morphology operation, where black pixels constitute the background region. And Fig. 11(d) is the final compensation result obtained with the proposed method. (a)
(b)
(c)
(d)
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Fig. 11. The aberration compensation result of the MEMS chip obtained with the proposed method: (a) the precompensated phase distribution, (b) the converted grayscale image of (a), (c) the binary image of background segmentation, and (d) the final compensation result obtained with the proposed method.
Figure 12(a) is the compensation result obtained with 14 Zernike polynomials fitting. Figure
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12(b) shows the phase gradient map calculated with the wrapped phase distribution of Fig. 12(a). Figure 12(c) shows the result of background segmentation of Fig. 12(b). Figure 12(d) shows the compensation result obtained with the double fitting method. In Figs. 11 and 12, the height values
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are calculated according to the relation of h(x,y)=λφo(x,y)/4π.
Comparing the compensation results of the three methods with the 2D MEMS image shown in Fig. 10, it seems that all the three methods can roughly retrieve the profile of the microstructure. However, from Fig. 12(c), it can be seen that, for the dense microstructure, it is
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difficult to distinguish the object area and the background area because the phase gradient only charactering edge information. Thus, some microstructure area is incorrectly identified as the background. In addition, as shown in Figs. 11(d), 12(a) and 12(d), the height values of retrieved
(a)
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results are different.
(b)
(d)
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Fig. 12. The aberration compensation results of the MEMS chip obtained with ZPF and the double fitting methods: (a) the result of ZPF method [21,22], (b) the phase gradient map of (a), (c) the binary image of background segmentation with (b), and (d) the result of the double fitting method [29].
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(b)
Fig. 13. Comparison of the profile curves obtained by the profilometer and three methods: (a) the profile curve obtained by the profilometer, and (b) the profile curves obtained by three methods.
For an accurate quantitative evaluation, the compensation results of three methods are compared with the measurement result of a stylus profilometer (S1910DX3; ACCRETECH).
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Figure 13(a) plots the profile curves measured along the red dotted line in Fig. 10 by the stylus profilometer, and Fig. 13(b) plots three profile curves obtained by the three methods. It can be seen that the result obtained by our method is more consistent with that obtained by the profilometer in terms of the microstructure height. Both the profile curves obtained by the ZPF
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method and the double fitting method have a larger deviation of about 20 nm. The above experimental results demonstrate that our method is able to effectively measure the profile of the MEMS chip even though its line width is narrow and dense. The essential reason why our method works well is that we are able to extract the background area automatically by performing high-order curve fitting to pre-compensate most of the phase aberration. Comparing with the global surface ZPF, the high-order curve fitting uses more parameters to construct more accurate aberration surfaces. More importantly, we calculate the global phase aberration by using the ZPF coefficients of the extracted background region, which avoids the influence of phase value of object on the fitting. Therefore, in our method, the precompensation is only used to accurately extract the background area, its aberration compensating error will not directly influence the final result. For the ZPF method and the double fitting
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method, both require that the phase distribution of the measured object is only a small perturbation to the overall phase distribution. This means that proportion of the object area must be very small, otherwise, large error will be introduced to final results. In addition, the phase gradient calculation is only able to detect the object edge information, it can not distinguish the object area and background area. Therefore, for the MEMS chip with dense microstructure, both the compensation results of ZPF method and the double fitting method are unsatisfactory. It
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should be noted that if the object is too dense so that the object area exceeds 50% of the entire phase distribution, or the phase value of object is comparable to the phase aberration, our method may not be effective.
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4. Conclusion
In summary, we have presented a numerical phase aberration compensation method of DHM
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with the curve fitting preprocessing and the automatic background segmentation. By using the global curve fitting, the proposed method has the prominent merit of effectively precompensating the phase aberrations for the dense microstructure object whose phase distribution spread all over the image. Simulation and experimental results show that the proposed method can obtain a more distinguishable object profile in curve fitting preprocessing, which makes the
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background segmentation and fitting more accurate. Compared with ZPF method and the double fitting method, the proposed method realizes a more precise phase aberration compensation by using the ZPF coefficients of the extracted background region, especially for the dense
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microstructures. Thus, the proposed phase aberration compensation method has significant application for the microstructure testing in DHM system.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (51875530); Natural Science Foundation of Zhejiang Province (LZ18E050003); Program for Changjiang Scholars and Innovative Research Team in University (IRT_17R98).
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*Credit Author Statement
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Liu Huang:
Software, Validation, Investigation, Writing - Original Draft
Liping Yan:
Conceptualization, Methodology, Writing - Review & Editing Supervision, Project administration, Funding acquisition
Yanjiang Zhou:
Resources, Writing - Review & Editing
Data Curation
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Tao Yang:
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Benyong Chen: