Further investigation on the phase stitching and system errors in digital holography Yongfu Wen,1,2,* Weijuan Qu,1 Haobo Cheng,2 Hao Yan,3 and A. Asundi4 1
Centre for Applied Photonics and Laser Technology, Ngee Ann Polytechnic, Clementi Road 535, Singapore 599489, Singapore 2 3
School of Optoelectronics, Beijing Institute of Technology, Beijing 100081, China
Department of Instrument Science and Engineering, Shanghai Jiaotong University, Shanghai 200242, China 4
Centre for Optical and Laser Engineering, School of Mechanical and Aerospace Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore *Corresponding author: [email protected] Received 10 September 2014; revised 6 November 2014; accepted 15 November 2014; posted 20 November 2014 (Doc. ID 222815); published 9 January 2015
In this work, an improved phase stitching algorithm is proposed in digital holography (DH) based on a deduced phase errors model and a global optimization algorithm. In addition, to correct the relative rotation error between the coordinate systems of a CCD and xy-motion stages, we presented a simple and reliable image-based correction method. The experimental results obtained from our proposed method are compared with those calculated from the existing phase stitching method to verify the performance of the presented method. It is shown that our new proposed methods are robust and valid for measurement of a large microstructure element. As far as we know, the improved phase stitching algorithm and imagebase correction method have not been discussed in DH, as we presented in this paper. © 2015 Optical Society of America OCIS codes: (090.1995) Digital holography; (120.5050) Phase measurement; (180.6900) Threedimensional microscopy. http://dx.doi.org/10.1364/AO.54.000266
1. Introduction
Microelectromechanical systems (MEMSs) and microstructure elements are very important in science and industrial applications [1]. Hence, for fabricating reliable MEMS and microstructure elements, a quick, noncontact, and noninvasive measurement tool is needed to provide accurate inspection and characterization of these structures. Recently, digital holography (DH) has been shown to offer new possibilities in noninvasive measurements because DH obtains both qualitative and quantitative intensity and phase information from a hologram with high resolution and has been developed into a prospect for being a new powerful tool for 1559-128X/15/020266-11$15.00/0 © 2015 Optical Society of America 266
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microdomain measurement [2–5]. Normally, the measurement of microstructure elements with both larger testing area and high lateral resolution is always desired. However, DH can only test a small area owing to the limitation of CCD size. To balance the problem, synthetic aperture technology digital holography (SADH) was proposed [6–8], which can effectively improve the lateral resolution and increase the field of view (FOV) [9,10]. The basic principle is that multiple subholograms are captured by moving the CCD (note that the object does not move) and are then stitched to a larger synthetic aperture digital hologram for reconstruction. However, the testing area of this technique is still limited. Especially for the measurement of a large microstructure array, SADH is not useful. To further expand the measurement area, phase stitching technology is one of the most effective methods. This technique
has been applied to interferometry for the measurement of a large optical plane mirror [11,12] and microsurface [13]. As far as we know, the phase stitching technique into DH was first introduced in [14]. The basic principle of this technique is that the large object is divided into many small subaperture areas, which can be tested by moving a DH system or object, and then these subaperture phase maps are stitched together by using a suitable stitching model with tilts, shift terms, and a least-squares (LS) algorithm. Thus, the testing area of DH can be enlarged by this technology. Later, a further experiment was done in [15,16] to verify the performance of the phase stitching technique. However, the existing phase stitching method used in DH only includes tilt and shift terms, which is not robust in practical measurements. This is because of the other phase errors that may be produced by the microscope objectives or the mismatch between the reference wave and the object wave [17,18]. In addition, in the phase stitching method, there is a precondition that the phase data in the overlapping area of two adjacent subapertures must be equal. However, the positioning error or alignment errors may mean that the phase data of the overlapping areas do not meet the precondition causing the stitching failure. To solve these problems, in this paper we further studied the phase stitching method in DH. By analyzing the mechanism of phase errors in DH, a reasonable mathematical model of phase errors is established. Based on this model, an improved stitching algorithm is explored for phase stitching. In addition, after analyzing the system errors, a simple and reliable image-based correction method is employed to correct the relative rotation error between the coordinate system of a CCD and motion stage. This technique has been developed for various experimental investigations to verify its performance. It is shown from calculated results that it is an effective, high-resolution technique for measurement of a large-size microstructure surface. 2. Basic Principle A.
Phase Errors Mathematical Model
A sketch map of a lenseless off-axis compact digital holoscope (CDH) [19] is described in Fig. 1(a). In Fig. 1(b), we assume that the reference wave is
generated by a point source located at xr ; yr ; zr , and the object wave is also generated by a point source located at xo ; yo ; zo , where zr is the distance between the point source of the reference wave and hologram plane; similarly, zo is the distance between the point source of the object wave and hologram plane. Thus, the reference wave RxH ; yH and object wave OxH ; yH in the hologram plane can be, respectively, expressed as follows [20]: jk xH − xr 2 yH − yr 2 ; RxH ; yH exp − 2zr jk 2 2 OxH ; yH A0 exp − x − xo yH − yo 2zo H × expjφobject xH ; yH ;
(2)
where j and k are the imaginary unit and the wavelength number, respectively, and φobject xH ; yH and A0 are the phase and amplitude of the tested object, respectively. The corresponding intensity distribution of the interference pattern I between the two waves can be written as IxH ; yH jOj2 jRj2 R O RO 1 A20 A0 expjφobject xH ; yH jk 1 1 x2H y2H − × exp 2 zo zr xo xr yo yr x − jk y − − − jk zo zr H zo zr H jk 2 jk 2 2 2 x yo − x yr 2zo o 2zr r A0 exp−jφobject x; y jk 1 1 − x2H y2H × exp − 2 zo zr xo xr yo yr − − x jk y jk zo zr H zo zr H jk 2 jk 2 2 2 − x yo x yr ; 2zo o 2zr r
(3)
where jOj2 jRj2 is the DC term, and R O and RO are the virtual term and real term, respectively. The asterisk denotes the complex conjugate. In the off-axis
Fig. 1. Sketch map: (a) off-axis reflection of a CDH; (b) location of the two point sources. 10 January 2015 / Vol. 54, No. 2 / APPLIED OPTICS
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DH configuration, the three terms are well-separated by filtering in the two-dimensional (2D) Fourier spectrum. Suppose that the spectrum ψ FH of the virtual image term R O is filtered out [as shown in Eq. (4)] by the frequency filtering method for numerical reconstruction. Throughout this paper, the reconstructed holographic image ψ I can be calculated from ψ FH by the angular spectrum method [20] as in Eq. (5) ψ FH nΔξ; mΔη FFT A0 exp j φobject xH ; yH k 1 1 xo xr 2 2 − − xH yH − k x zo zr H 2 zo zr y y −k o − r yH zo zr kz x2 y2o − kzo x2r y2r r o ; (4) 2zo zr
ϕI nΔxI ; mΔyI arctan
Imψ I nΔxI ; mΔyI : (7) Reψ I nΔxI ; mΔyI
According to Eqs. (4)–(7), if we can precisely adjust zo zr in the measurement process, the quadratic phase errors and off-axis tilts terms (in this case, the off-axis tilts are easily, completely eliminated by using the spectrum centering process) in the reconstructed phase can be suppressed, as shown in Fig. 2(a). When zo ≠ zr , the quadratic phase error and tilts terms (here, the off-axis tilts are difficult to completely eliminate by using the spectrum centering process, due to the fluctuating boundary [21]) appear in the reconstructed phase, as shown in Figs. 2(b) and 2(c). B. Improved Stitching Algorithm
In experiments, zo and zr are difficult to measure and quantify. In addition, human factors and the
8 < ψ I nΔx ; mΔy expjkd FFT−1 ψ F nΔξ; mΔη × hnΔξ; mΔη I I H ; jλd p : hnΔξ; mΔη exp j 2πd 1 − λnΔξ2 − λmΔη2
5
λ
where hnΔξ; mΔη is the optical transfer function in the spatial frequency domain, n and m are integers (−M∕2 ≤ n ≤ M∕2, −N∕2 ≤ m ≤ N∕2, and M × N is the number of pixels of the CCD), Δξ and Δη are the sampling intervals in the spatial frequency domain, ΔxH and ΔyH are the sampling intervals in the hologram plane, and ΔxI and ΔyI are the sampling intervals in the image plane. Then the intensity image and the phase map of the reconstruction image can be obtained as follows: I I nΔxI ; mΔyI jψ I nΔxI ; mΔyI j2 ;
(6)
positioning error of the motion stage can lead to a difference between zo and zr . In other words, the phase reconstructed directly from the hologram contains the phase of object, the two off-axis tilts terms, as well as a quadratic term. As far as we know, in existing DH phase stitching algorithms, the quadratic term is not considered in the stitching process. Here, we present an improved stitching algorithm. Without loss of generality, the ith subaperture phase reconstructed ϕi can be expressed ϕi ϕ0i ai bi x ci y di x2 y2 ;
(8)
Fig. 2. Reconstructed wrapped phase (top) and unwrapped phase (bottom) at different cases; (a) when zo zr , a quasi-flat phase is obtained; (b) when zo > zr , a converging quadratic wave and tilts remain; (c) when zo < zr , a diverging quadratic wave and tilts remain. 268
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where ϕ0i is the ith real subaperture phase of object, and ai , bi , ci , di are the corresponding coefficients of phase error terms. Thus, considering two subapertures ϕi and ϕj (note that in the overlapping area, ϕ0i ϕ0j , and the object can be considered as a rigid body), the difference between them in the common area fulfills the following equation
where n0 is the number of measured points at the overlapping area. If more than two subapertures are connected, a global optimization algorithm is utilized to keep the stitching errors from transmitting and accumulating. Let us assume that T subaperture phase maps need to be connected; thus, there are 4 × T − 1 parameters that have to be solved at once. Then a more general equation can be expressed as follows (for more details please refer to [22]): X T −1 T −1 X Qij − δij Ri i ; Pil Qil (12)
ϕi − ϕj Δϕij ϕ0i ai bi x ci y di x2 y2 − ϕ0j aj bj x cj y dj x2 y2
l
l
i
where i; j 0; 1; …; t − 1; t 1; …; T, l 0; 1; …; T
ai − aj bi − bj x ci − cj y
di − dj x2 y2 ζ1 ζ2 x ζ 3 y ζ 4 x2 y2 ;
δij
(9)
6 6 6 6 Qij 6 6 6 6P 4
i∩j
i∩j
P x i∩j P x2 i∩j P xy
3 ζ1;i 6 ζ2;i 7 7 Ri 6 4 ζ3;i 5; ζ4;i 2
3
i∩j
i∩j
0 P
x
y
i∩j P yx i∩j P y2
P y i∩j P yx i∩j P y2
P
i∩j
yx2 y2
i∩j
0
0
0
0
0
0
0
0
(14b)
0
3
7 07 7; 07 5 0
(14c)
3 P 2 x y2 i∩j 7 P 7 x2 y2 x 7 7 i∩j 7: P 2 2 x y y 7 7 i∩j 7 P 2 2 25 x y
(15)
i∩j
P
x2
y2
3−1 2
7 i∩j 7 P 7 2 2 x y x 7 7 i∩j 7 P 2 x y2 y 7 7 i∩j 7 P i∩j2 P 7 yx y2 x2 y2 2 5 i∩j
0
6 60 Qii 6 60 4
P i∩j 2 Z2 x y2
n0 6 6 P 6 x 6 ξ1 7 6 6 ξ2 7 6 i∩j 6 76 P 6 ξ3 7 6 y 4 5 6 6 P i∩j ξ4 P i∩j 2 6 Z2 x y2 4 x2 y2 2
(14a)
2
i∩j
2
3
i∩j
i∩j P x2 i∩j P xy
x2 y2
P Δϕ
(13)
6 7 P 6 7 6 7 Δϕx 6 7 i∩j 6 7 P Pij 6 7; 6 7 Δϕy 6 7 i∩j 6 7 4P 5 Δϕx2 y2
where the summation is taken in the overlapping area of two subaperture phases. Taking the partial derivatives of Eq. (10) with respect to each one of the coefficients ζ 1, ζ 2 , ζ3 , ζ 4 , the following matrix equation can be obtained P
if i j ; if i ≠ j
i∩j
(10)
n P x i∩j P y
1; 0;
2
where ζ1 , ζ2 , ζ3 , ζ4 are the corresponding coefficients of the relative phase error terms. The coefficients must be calculated to compensate the effect of relative phase errors between subaperture phases. For this purpose, the measurements are made so that the adjacent interferograms have common areas, and these phases are connected by minimizing the difference of the phase distributions in the common area. To reduce the influence of noise, the four coefficients can be obtained by using the LS method to minimize the differences Δϕij at the common area, as follows: X fΔϕij − ζ1 ζ 2 x ζ3 y ζ4 x2 y2 g2 → min;
2
ij
i∩j
P Δϕij i∩j P xΔϕij i∩j P yΔϕij
3
6 7 6 7 6 7 6 7 6 7 6 7; 6 7 6 7 6 P i∩j 7 6 x2 y2 Δϕ 7 4 ij 5
(11)
i∩j
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Summation in Eqs. (12) and (15) are taken in the indicated common area, and if there is no common area between subaperture ϕi and ϕj , the submatrices Pii and Qii are null matrices. After calculating these unknown coefficients R by solving Eq. (12), we can remove these relative phase errors of these subaperture phase maps to complete the phase stitching process. 3. Analysis and Correction of System Errors
In this section, the system inherent errors are analyzed. As mentioned previously, there is a precondition that the phase data in the overlapping area of two adjacent subapertures must be equal. However, in practical measurements, the positioning error produced by motion stages causes the data of the overlapping area to not meet the mentioned precondition. Thanks to the availability of high-precision computer controller stages and the software compensation method [23], the positioning error can be limited within a very small range. Here, we mainly focus on the rotation error caused by the misalignment between the coordinate system of a CCD and motion xy-stages. This error is often easy to ignore in the alignment process; however, it greatly affects the stitching result. Figure 3 shows the relationship between the coordinate system of a CCD and the coordinate system of xy-motion stage; the included angle between them is θ. Assume that Position A is the initial position where the first subaperture phase map is captured by the CCD, and Position B is the layout designing position where the second subaperture phase map
Fig. 3. Relationship between the coordinate system of a CCD and the xy-motion stage.
should be captured. Because of the relative rotation error between the two coordinate systems, Position C is the actual position of the second subaperture phase map, as shown in Fig. 4(a). According to the principle of stitching [14,15], the second subaperture phase map (at Position C) should be moved to its layout designing Position B and then can be stitched together with the first subaperture phase map (at Position A), as shown in Fig. 4(b). It is shown that the phase data of the overlapping area are mismatched. To overcome this issue, we can use the region-based search algorithm on the overlapping area to search for the best stitching position [13]. However, this takes much time, especially when the rotation error is large enough. In addition, when the features of objects at overlapping regions are neither obvious nor uniform, the search algorithm may be unreliable. In order to solve this problem, in this work we present an image-based error correction method, which is a simple and reliable method to obtain the rotation angle error for correcting the relative rotation error before measurement. The basic principle is described in Fig. 5(a), i.e., that the calibration target with a special feature pattern (here a character “+,” the center of the pattern, is considered as the feature point) is fixed on the motion stage and moved along with the x- or y-direction of the motion stage. The intensity information of the object for each position can be calculated from the corresponding hologram recorded by the CCD. After that, the feature point on an intensity scale can be extracted by a feature point extraction algorithm, which is a popular technique in image processing and achieves pixel-level accuracy. Then the linear fitting method with the LS criterion is used to fit the extracted feature points for getting the included angle θ. Figure 5(b) indicates real experimental data with the proposed correction method, and the included angle calculated is −3.1162°. Figures 6(a) and 6(b) display the stitching positions without correcting the relative rotation error. It is shown that the stitching positions are invalid for directly stitching due to the fact that the phase data of the overlapping area are all mismatched. In contrast, when the proposed correction method is adopted, the stitching position of two subapertures can be located at the right-hand position, where the two subapertures
Fig. 4. Positional relationship between two subaperture phase maps: (a) actual position of two subapertures captured by the CCD when there is relative rotation error; (b) stitching position of the two subapertures previously mentioned. 270
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Fig. 5. (a) Movement diagram of sample in the CCD plane at different positions; (b) experimental data.
Fig. 6. Stitching positions of two subapertures: (a) without correcting the rotation error, about 6°; (b) without correcting the rotation error, about 2°; (c) after correcting the rotation error.
can be directly stitched together through the phase stitching method, as shown in Fig. 6(c). Figure 7 shows the flowchart of the image-based correction method and stitching process for further understanding. 4. Experimental Implementation and Data Analysis A.
Experimental Setup
In the experiment, we have arranged a setup based on a CDH, as shown in Fig. 8. The digital hologram is
recorded by a CCD camera with pixel resolution 960 × 1280 and pixel size 4.65 μm × 4.65 μm. The light emitted from a 640 nm iFLEX-200 compact laser diode is coupled into a single-mode fiber as a point light source. The specimen is mounted on an xy-precision motor controlled stage, and the CDH system is fixed on a z axis motor controlled stage. The measurement area of the CDH system can be calculated, which is about 3 mm × 2.25 mm. Note that the following experiments were performed after correcting the rotation error by the proposed method.
Fig. 7. Flowchart of the image-based correction method and phase stitching process. 10 January 2015 / Vol. 54, No. 2 / APPLIED OPTICS
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Fig. 8. Sketch map of experiment setup.
Practically, especially in a reflection configuration CDH system, the quadratic term phase error is more or less produced due to the mismatch between the reference wave and the object wave. Based on this, two comparison experiments between the traditional stitching method and the proposed algorithm were presented to verify the validity of our proposed method. B. Experiment on USAF Test Target With Less Phase Errors
We applied the proposed stitching algorithm to measure the United States Air Force (USAF) target. In this configuration, the reconstructed subaperture phase data contain less unwanted phase errors. The object was moved along with the x-direction of the motion stages, while ensuring that there are overlapped common areas between adjacent subapertures. First, three holograms (1 × 3) for different parts of the USAF target were recorded, as shown in Figs. 9(a)–9(c). Then the holograms were transferred into the computer to reconstruct by the angular spectrum method. Figures 9(d)–9(f) show the unwrapped
phases calculated from corresponding holograms in Figs. 9(a)–9(c), respectively. Actually, the edge diffraction effect of the reconstructed subaperture phase data cannot be ignored in the phase stitching algorithm. To suppress the edge diffraction effect, throughout this paper, we waived the edge data (about 50 pixels) of each subaperture phase data before stitching processing. To verify the validity of the proposed stitching algorithm, in this work, we compared the proposed method with traditional phase stitching algorithms [14,15] on the same subaperture phase maps of object. Figures 10(a) and 10(b) show the stitched phase by the traditional method. Figures 10(c) and 10(d) show the stitched phase by the proposed method. It is shown that there are quadratic error and tilts in the stitched results by the traditional method. However, our new proposed algorithm can solve the problem. C. Experiment on Microstructure Objects with Large Phase Errors
The tested specimen was a multifunction test target with a 3.0 mm square grid and 0.25 mm spacing (Max Levy Autograph, Inc., DA030), as shown in Fig. 11(a). During the recording process, four subaperture holograms (2 × 2) were needed to cover the sample. Figure 12 shows the four holograms recorded and their reconstructed wrapped phases. These four subapertures were stitched together by the two stitching methods, as shown in Fig. 13. It is shown that the stitching phase error obtained from the traditional method is more obvious than that of our proposed method. For further study of the two stitching methods, the trademark of Edmund optics was used for a specimen, as shown in Fig. 11(b). For testing the mark
Fig. 9. Original data: (a)–(c) holograms recorded; (d)–(f) corresponding unwrapping phase data. 272
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Fig. 10. Phase stitching result: (a) 2D map obtained from the traditional method; (b) 3D map obtained from the traditional method; (c) 2D map obtained from our proposed algorithm; (d) 3D map obtained from our proposed algorithm.
Fig. 11. Two microstructure objects (the red rectangle areas are wanted measurement areas): (a) multifunction test target DA030; (b) trademark of Edmund optics.
Fig. 12. Reconstruction results of subapertures: (a)–(d) holograms; (e)–(h) corresponding wrapped phases. 10 January 2015 / Vol. 54, No. 2 / APPLIED OPTICS
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Fig. 13. Phase stitching result: (a) 2D map obtained from the traditional method; (b) 3D map obtained from the traditional method; (c) 2D map obtained from our proposed algorithm; (d) 3D map obtained from our proposed algorithm; (e) comparison of the results of the two methods in the cross section of the x-direction; (f) comparison of the results of the two methods in the cross section of the y-direction.
Fig. 14. Five pairs of subaperture holograms and intensity distributions. 274
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of “Optics,” five subaperture holograms (1 × 5) were captured by CDH, as shown in Fig. 14. Similarly, the comparison experiment results are given in Fig. 15. It is obvious that our proposed algorithm can provide more accurate results than that of the traditional stitching method. In this experiment, the processing time of the proposed algorithm and traditional method are 25.68 and 25.54 s, respectively, with a 2.67 GHz Core i7 CPU using MATLAB 7.11.
It must be pointed out that the traditional stitching method can be used to get a perfect stitching result when the reconstructed subaperture contains phase data without the quadratic term error. However, in the two presented comparison experiments, the traditional method always produces data of relatively lower quality, and one traditional result produces highly large errors (cf., Fig. 15) when the tested subaperture phase data have the quadratic term error.
Fig. 15. Phase stitching result: (a) 2D map obtained from the traditional method; (b) 3D map obtained from the traditional method; (c) 2D map obtained from our proposed algorithm; (d) 3D map obtained from our proposed algorithm; (e) comparison of the results of the two methods in the cross section of the 300th row; (f) comparison of the results of the two methods in the cross section of the 800th row. 10 January 2015 / Vol. 54, No. 2 / APPLIED OPTICS
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5. Conclusions and Discussion
The phase stitching technique can expand the measurement area of a DH system. It enables the possibility of using DH to measure a large MEMS or microstructure element without losing high resolution. In this work, we presented an improved phase stitching algorithm based on the analysis result of phase errors to overcome the shortcoming of the existing phase stitching algorithm in DH. In addition, as mentioned previously, the relative rotation error between the coordinate systems’ CCD and xy-motion stages is very important for the phase stitching algorithm. To correct the relative rotation error, a simple and reliable image-based method was employed by using the feature point extraction technique and the linear fitting method with the LS criterion. Finally, the comparison experiments based on a reflection system of DH were performed. The advantages of the proposed algorithm over the existing phase stitching method in DH were verified by the comparison experiments on different situations. It is shown that the proposed algorithms are available and reliable for the measurement of a large microstructure object. We think the proposed methods can also be applied to transmission systems of DH and digital holographic microscopy (DHM). This study is supported by Translational Research and Development grant MOE2012-TIF-1-T-003 from the Singapore National Research Foundation, and Innovation Fund grant MOE2013-TIF-2-G-012 from the Singapore Ministry of Education. References 1. T. Reissman, E. Garcia, N. Lobontiu, and Y. Nam, “Integrated electrostatic micro-sensors for the development of modeling techniques of defects in the actuation of large microelectromechanical systems (MEMS),” Proc. SPIE 6152, 615238 (2006). 2. G. Coppola, P. Ferraro, and M. Iodice, “A digital holographic microscope for complete characterization of microelectromechanical systems,” Meas. Sci. Technol. 15, 529–539 (2004). 3. F. Montfort, Y. Emery, and F. Marquet, “Process engineering and failure analysis of MEMS and MOEMS by digital holography microscopy (DHM),” Proc. SPIE 6463, 64630G (2007). 4. L. Xu, X. Peng, J. Miao, and A. K. Asundi, “Studies of digital microscopic holography with applications to microstructure testing,” Appl. Opt. 40, 5046–5051 (2001). 5. P. Ferraro, S. D. Nicola, and G. Coppola, “Testing silicon MEMS structures subjected to thermal loading by digital holography,” Proc. SPIE 5343, 235–243 (2004).
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6. L. Martínez-León and B. Javidi, “Synthetic aperture singleexposure on-axis digital holography,” Opt. Express 16, 161–169 (2008). 7. G. Indebetouw, Y. Tada, J. Rosen, and G. Brooker, “Scanning holographic microscopy with resolution exceeding the Rayleigh limit of the objective by superposition of off-axis holograms,” Appl. Opt. 46, 993–1000 (2007). 8. X. Xu, L. Cai, and G. Lu, “Advanced aperture synthesis by wave-front combination in generalized phase-shifting interferometry,” Chin. Phys. Lett. 29, 084202 (2012). 9. Y. Hao and A. Asundi, “Studies on aperture synthesis in digital Fresnel holography,” Opt. Lasers Eng. 50, 556–562 (2012). 10. H. Jiang, J. Zhao, J. Di, and C. Qin, “Numerically correcting the joint misplacement of the subholograms in spatial synthetic aperture digital Fresnel holography,” Opt. Express 17, 18836–18842 (2009). 11. M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurement of large plane surface shape with interferometric aperture synthesis,” Proc. SPIE 1720, 444–447 (1992). 12. M. Bray, “Stitching interferometer for large plano optics using a standard interferometer,” Proc. SPIE 3134, 39–50 (1997). 13. S. Tang, “Stitching: high spatial resolution microsurface measurements over large areas,” Proc. SPIE 3479, 43–49 (1998). 14. C. Dai, Y. Yu, G. Chen, and A. Asundi, “Study of the holographic phase stitching technique,” Proc. SPIE 7000, 70001T (2008). 15. W. Zhou, G. Chen, Y. Yu, and W. Qu, “Study on phase stitching technique on digital holography,” in Advances in Imaging, OSA Technical Digest (CD) (Optical Society of America, 2009), paper DWB23. 16. X. Xu, G. Lu, and G. Han, “Phase stitching and error correction in aperture synthesis for generalized phase-shifting interferometry,” Appl. Opt. 52, 4864–4870 (2013). 17. E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt. 38, 6994–7001 (1999). 18. W. Qu, O. Chee, Y. Yu, and A. Asundi, “Microlens characterization by digital holographic microscopy with physical spherical phase compensation,” Appl. Opt. 49, 6448–6454 (2010). 19. V. R. Singh, L. Sui, and A. Asundi, “Compact handheld digital holographic microscopy system development,” Proc. SPIE 7522, 75224L (2010). 20. W. Qu, O. Chee, and V. R. Singh, “Quasi-physical phase compensation in digital holographic microscopy,” J. Opt. Soc. Am. A 26, 2005–2011 (2009). 21. C. Zuo, Q. Chen, W. J. Qu, and A. Asundi, “Phase aberration compensation in digital holographic microscopy based on principal component analysis,” Opt. Lett. 38, 1724–1726 (2013). 22. M. Otsubo, K. Okada, and J. Tsujiuchi, “Measurement of large plane surface shapes by connecting small aperture interferograms,” Opt. Eng. 33, 608–613 (1994). 23. Z. Lechniak, A. Werner, and K. Skalski, “Methodology of offline software compensation for errors in the machining process on the CNC machine tool,” J. Mater. Process. Technol. 76, 42–48 (1998).
Centre for Applied Photonics and Laser Technology, Ngee Ann Polytechnic, Clementi Road 535, Singapore 599489, Singapore 2 3
School of Optoelectronics, Beijing Institute of Technology, Beijing 100081, China
Department of Instrument Science and Engineering, Shanghai Jiaotong University, Shanghai 200242, China 4
Centre for Optical and Laser Engineering, School of Mechanical and Aerospace Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore *Corresponding author: [email protected] Received 10 September 2014; revised 6 November 2014; accepted 15 November 2014; posted 20 November 2014 (Doc. ID 222815); published 9 January 2015
In this work, an improved phase stitching algorithm is proposed in digital holography (DH) based on a deduced phase errors model and a global optimization algorithm. In addition, to correct the relative rotation error between the coordinate systems of a CCD and xy-motion stages, we presented a simple and reliable image-based correction method. The experimental results obtained from our proposed method are compared with those calculated from the existing phase stitching method to verify the performance of the presented method. It is shown that our new proposed methods are robust and valid for measurement of a large microstructure element. As far as we know, the improved phase stitching algorithm and imagebase correction method have not been discussed in DH, as we presented in this paper. © 2015 Optical Society of America OCIS codes: (090.1995) Digital holography; (120.5050) Phase measurement; (180.6900) Threedimensional microscopy. http://dx.doi.org/10.1364/AO.54.000266
1. Introduction
Microelectromechanical systems (MEMSs) and microstructure elements are very important in science and industrial applications [1]. Hence, for fabricating reliable MEMS and microstructure elements, a quick, noncontact, and noninvasive measurement tool is needed to provide accurate inspection and characterization of these structures. Recently, digital holography (DH) has been shown to offer new possibilities in noninvasive measurements because DH obtains both qualitative and quantitative intensity and phase information from a hologram with high resolution and has been developed into a prospect for being a new powerful tool for 1559-128X/15/020266-11$15.00/0 © 2015 Optical Society of America 266
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microdomain measurement [2–5]. Normally, the measurement of microstructure elements with both larger testing area and high lateral resolution is always desired. However, DH can only test a small area owing to the limitation of CCD size. To balance the problem, synthetic aperture technology digital holography (SADH) was proposed [6–8], which can effectively improve the lateral resolution and increase the field of view (FOV) [9,10]. The basic principle is that multiple subholograms are captured by moving the CCD (note that the object does not move) and are then stitched to a larger synthetic aperture digital hologram for reconstruction. However, the testing area of this technique is still limited. Especially for the measurement of a large microstructure array, SADH is not useful. To further expand the measurement area, phase stitching technology is one of the most effective methods. This technique
has been applied to interferometry for the measurement of a large optical plane mirror [11,12] and microsurface [13]. As far as we know, the phase stitching technique into DH was first introduced in [14]. The basic principle of this technique is that the large object is divided into many small subaperture areas, which can be tested by moving a DH system or object, and then these subaperture phase maps are stitched together by using a suitable stitching model with tilts, shift terms, and a least-squares (LS) algorithm. Thus, the testing area of DH can be enlarged by this technology. Later, a further experiment was done in [15,16] to verify the performance of the phase stitching technique. However, the existing phase stitching method used in DH only includes tilt and shift terms, which is not robust in practical measurements. This is because of the other phase errors that may be produced by the microscope objectives or the mismatch between the reference wave and the object wave [17,18]. In addition, in the phase stitching method, there is a precondition that the phase data in the overlapping area of two adjacent subapertures must be equal. However, the positioning error or alignment errors may mean that the phase data of the overlapping areas do not meet the precondition causing the stitching failure. To solve these problems, in this paper we further studied the phase stitching method in DH. By analyzing the mechanism of phase errors in DH, a reasonable mathematical model of phase errors is established. Based on this model, an improved stitching algorithm is explored for phase stitching. In addition, after analyzing the system errors, a simple and reliable image-based correction method is employed to correct the relative rotation error between the coordinate system of a CCD and motion stage. This technique has been developed for various experimental investigations to verify its performance. It is shown from calculated results that it is an effective, high-resolution technique for measurement of a large-size microstructure surface. 2. Basic Principle A.
Phase Errors Mathematical Model
A sketch map of a lenseless off-axis compact digital holoscope (CDH) [19] is described in Fig. 1(a). In Fig. 1(b), we assume that the reference wave is
generated by a point source located at xr ; yr ; zr , and the object wave is also generated by a point source located at xo ; yo ; zo , where zr is the distance between the point source of the reference wave and hologram plane; similarly, zo is the distance between the point source of the object wave and hologram plane. Thus, the reference wave RxH ; yH and object wave OxH ; yH in the hologram plane can be, respectively, expressed as follows [20]: jk xH − xr 2 yH − yr 2 ; RxH ; yH exp − 2zr jk 2 2 OxH ; yH A0 exp − x − xo yH − yo 2zo H × expjφobject xH ; yH ;
(2)
where j and k are the imaginary unit and the wavelength number, respectively, and φobject xH ; yH and A0 are the phase and amplitude of the tested object, respectively. The corresponding intensity distribution of the interference pattern I between the two waves can be written as IxH ; yH jOj2 jRj2 R O RO 1 A20 A0 expjφobject xH ; yH jk 1 1 x2H y2H − × exp 2 zo zr xo xr yo yr x − jk y − − − jk zo zr H zo zr H jk 2 jk 2 2 2 x yo − x yr 2zo o 2zr r A0 exp−jφobject x; y jk 1 1 − x2H y2H × exp − 2 zo zr xo xr yo yr − − x jk y jk zo zr H zo zr H jk 2 jk 2 2 2 − x yo x yr ; 2zo o 2zr r
(3)
where jOj2 jRj2 is the DC term, and R O and RO are the virtual term and real term, respectively. The asterisk denotes the complex conjugate. In the off-axis
Fig. 1. Sketch map: (a) off-axis reflection of a CDH; (b) location of the two point sources. 10 January 2015 / Vol. 54, No. 2 / APPLIED OPTICS
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DH configuration, the three terms are well-separated by filtering in the two-dimensional (2D) Fourier spectrum. Suppose that the spectrum ψ FH of the virtual image term R O is filtered out [as shown in Eq. (4)] by the frequency filtering method for numerical reconstruction. Throughout this paper, the reconstructed holographic image ψ I can be calculated from ψ FH by the angular spectrum method [20] as in Eq. (5) ψ FH nΔξ; mΔη FFT A0 exp j φobject xH ; yH k 1 1 xo xr 2 2 − − xH yH − k x zo zr H 2 zo zr y y −k o − r yH zo zr kz x2 y2o − kzo x2r y2r r o ; (4) 2zo zr
ϕI nΔxI ; mΔyI arctan
Imψ I nΔxI ; mΔyI : (7) Reψ I nΔxI ; mΔyI
According to Eqs. (4)–(7), if we can precisely adjust zo zr in the measurement process, the quadratic phase errors and off-axis tilts terms (in this case, the off-axis tilts are easily, completely eliminated by using the spectrum centering process) in the reconstructed phase can be suppressed, as shown in Fig. 2(a). When zo ≠ zr , the quadratic phase error and tilts terms (here, the off-axis tilts are difficult to completely eliminate by using the spectrum centering process, due to the fluctuating boundary [21]) appear in the reconstructed phase, as shown in Figs. 2(b) and 2(c). B. Improved Stitching Algorithm
In experiments, zo and zr are difficult to measure and quantify. In addition, human factors and the
8 < ψ I nΔx ; mΔy expjkd FFT−1 ψ F nΔξ; mΔη × hnΔξ; mΔη I I H ; jλd p : hnΔξ; mΔη exp j 2πd 1 − λnΔξ2 − λmΔη2
5
λ
where hnΔξ; mΔη is the optical transfer function in the spatial frequency domain, n and m are integers (−M∕2 ≤ n ≤ M∕2, −N∕2 ≤ m ≤ N∕2, and M × N is the number of pixels of the CCD), Δξ and Δη are the sampling intervals in the spatial frequency domain, ΔxH and ΔyH are the sampling intervals in the hologram plane, and ΔxI and ΔyI are the sampling intervals in the image plane. Then the intensity image and the phase map of the reconstruction image can be obtained as follows: I I nΔxI ; mΔyI jψ I nΔxI ; mΔyI j2 ;
(6)
positioning error of the motion stage can lead to a difference between zo and zr . In other words, the phase reconstructed directly from the hologram contains the phase of object, the two off-axis tilts terms, as well as a quadratic term. As far as we know, in existing DH phase stitching algorithms, the quadratic term is not considered in the stitching process. Here, we present an improved stitching algorithm. Without loss of generality, the ith subaperture phase reconstructed ϕi can be expressed ϕi ϕ0i ai bi x ci y di x2 y2 ;
(8)
Fig. 2. Reconstructed wrapped phase (top) and unwrapped phase (bottom) at different cases; (a) when zo zr , a quasi-flat phase is obtained; (b) when zo > zr , a converging quadratic wave and tilts remain; (c) when zo < zr , a diverging quadratic wave and tilts remain. 268
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where ϕ0i is the ith real subaperture phase of object, and ai , bi , ci , di are the corresponding coefficients of phase error terms. Thus, considering two subapertures ϕi and ϕj (note that in the overlapping area, ϕ0i ϕ0j , and the object can be considered as a rigid body), the difference between them in the common area fulfills the following equation
where n0 is the number of measured points at the overlapping area. If more than two subapertures are connected, a global optimization algorithm is utilized to keep the stitching errors from transmitting and accumulating. Let us assume that T subaperture phase maps need to be connected; thus, there are 4 × T − 1 parameters that have to be solved at once. Then a more general equation can be expressed as follows (for more details please refer to [22]): X T −1 T −1 X Qij − δij Ri i ; Pil Qil (12)
ϕi − ϕj Δϕij ϕ0i ai bi x ci y di x2 y2 − ϕ0j aj bj x cj y dj x2 y2
l
l
i
where i; j 0; 1; …; t − 1; t 1; …; T, l 0; 1; …; T
ai − aj bi − bj x ci − cj y
di − dj x2 y2 ζ1 ζ2 x ζ 3 y ζ 4 x2 y2 ;
δij
(9)
6 6 6 6 Qij 6 6 6 6P 4
i∩j
i∩j
P x i∩j P x2 i∩j P xy
3 ζ1;i 6 ζ2;i 7 7 Ri 6 4 ζ3;i 5; ζ4;i 2
3
i∩j
i∩j
0 P
x
y
i∩j P yx i∩j P y2
P y i∩j P yx i∩j P y2
P
i∩j
yx2 y2
i∩j
0
0
0
0
0
0
0
0
(14b)
0
3
7 07 7; 07 5 0
(14c)
3 P 2 x y2 i∩j 7 P 7 x2 y2 x 7 7 i∩j 7: P 2 2 x y y 7 7 i∩j 7 P 2 2 25 x y
(15)
i∩j
P
x2
y2
3−1 2
7 i∩j 7 P 7 2 2 x y x 7 7 i∩j 7 P 2 x y2 y 7 7 i∩j 7 P i∩j2 P 7 yx y2 x2 y2 2 5 i∩j
0
6 60 Qii 6 60 4
P i∩j 2 Z2 x y2
n0 6 6 P 6 x 6 ξ1 7 6 6 ξ2 7 6 i∩j 6 76 P 6 ξ3 7 6 y 4 5 6 6 P i∩j ξ4 P i∩j 2 6 Z2 x y2 4 x2 y2 2
(14a)
2
i∩j
2
3
i∩j
i∩j P x2 i∩j P xy
x2 y2
P Δϕ
(13)
6 7 P 6 7 6 7 Δϕx 6 7 i∩j 6 7 P Pij 6 7; 6 7 Δϕy 6 7 i∩j 6 7 4P 5 Δϕx2 y2
where the summation is taken in the overlapping area of two subaperture phases. Taking the partial derivatives of Eq. (10) with respect to each one of the coefficients ζ 1, ζ 2 , ζ3 , ζ 4 , the following matrix equation can be obtained P
if i j ; if i ≠ j
i∩j
(10)
n P x i∩j P y
1; 0;
2
where ζ1 , ζ2 , ζ3 , ζ4 are the corresponding coefficients of the relative phase error terms. The coefficients must be calculated to compensate the effect of relative phase errors between subaperture phases. For this purpose, the measurements are made so that the adjacent interferograms have common areas, and these phases are connected by minimizing the difference of the phase distributions in the common area. To reduce the influence of noise, the four coefficients can be obtained by using the LS method to minimize the differences Δϕij at the common area, as follows: X fΔϕij − ζ1 ζ 2 x ζ3 y ζ4 x2 y2 g2 → min;
2
ij
i∩j
P Δϕij i∩j P xΔϕij i∩j P yΔϕij
3
6 7 6 7 6 7 6 7 6 7 6 7; 6 7 6 7 6 P i∩j 7 6 x2 y2 Δϕ 7 4 ij 5
(11)
i∩j
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Summation in Eqs. (12) and (15) are taken in the indicated common area, and if there is no common area between subaperture ϕi and ϕj , the submatrices Pii and Qii are null matrices. After calculating these unknown coefficients R by solving Eq. (12), we can remove these relative phase errors of these subaperture phase maps to complete the phase stitching process. 3. Analysis and Correction of System Errors
In this section, the system inherent errors are analyzed. As mentioned previously, there is a precondition that the phase data in the overlapping area of two adjacent subapertures must be equal. However, in practical measurements, the positioning error produced by motion stages causes the data of the overlapping area to not meet the mentioned precondition. Thanks to the availability of high-precision computer controller stages and the software compensation method [23], the positioning error can be limited within a very small range. Here, we mainly focus on the rotation error caused by the misalignment between the coordinate system of a CCD and motion xy-stages. This error is often easy to ignore in the alignment process; however, it greatly affects the stitching result. Figure 3 shows the relationship between the coordinate system of a CCD and the coordinate system of xy-motion stage; the included angle between them is θ. Assume that Position A is the initial position where the first subaperture phase map is captured by the CCD, and Position B is the layout designing position where the second subaperture phase map
Fig. 3. Relationship between the coordinate system of a CCD and the xy-motion stage.
should be captured. Because of the relative rotation error between the two coordinate systems, Position C is the actual position of the second subaperture phase map, as shown in Fig. 4(a). According to the principle of stitching [14,15], the second subaperture phase map (at Position C) should be moved to its layout designing Position B and then can be stitched together with the first subaperture phase map (at Position A), as shown in Fig. 4(b). It is shown that the phase data of the overlapping area are mismatched. To overcome this issue, we can use the region-based search algorithm on the overlapping area to search for the best stitching position [13]. However, this takes much time, especially when the rotation error is large enough. In addition, when the features of objects at overlapping regions are neither obvious nor uniform, the search algorithm may be unreliable. In order to solve this problem, in this work we present an image-based error correction method, which is a simple and reliable method to obtain the rotation angle error for correcting the relative rotation error before measurement. The basic principle is described in Fig. 5(a), i.e., that the calibration target with a special feature pattern (here a character “+,” the center of the pattern, is considered as the feature point) is fixed on the motion stage and moved along with the x- or y-direction of the motion stage. The intensity information of the object for each position can be calculated from the corresponding hologram recorded by the CCD. After that, the feature point on an intensity scale can be extracted by a feature point extraction algorithm, which is a popular technique in image processing and achieves pixel-level accuracy. Then the linear fitting method with the LS criterion is used to fit the extracted feature points for getting the included angle θ. Figure 5(b) indicates real experimental data with the proposed correction method, and the included angle calculated is −3.1162°. Figures 6(a) and 6(b) display the stitching positions without correcting the relative rotation error. It is shown that the stitching positions are invalid for directly stitching due to the fact that the phase data of the overlapping area are all mismatched. In contrast, when the proposed correction method is adopted, the stitching position of two subapertures can be located at the right-hand position, where the two subapertures
Fig. 4. Positional relationship between two subaperture phase maps: (a) actual position of two subapertures captured by the CCD when there is relative rotation error; (b) stitching position of the two subapertures previously mentioned. 270
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Fig. 5. (a) Movement diagram of sample in the CCD plane at different positions; (b) experimental data.
Fig. 6. Stitching positions of two subapertures: (a) without correcting the rotation error, about 6°; (b) without correcting the rotation error, about 2°; (c) after correcting the rotation error.
can be directly stitched together through the phase stitching method, as shown in Fig. 6(c). Figure 7 shows the flowchart of the image-based correction method and stitching process for further understanding. 4. Experimental Implementation and Data Analysis A.
Experimental Setup
In the experiment, we have arranged a setup based on a CDH, as shown in Fig. 8. The digital hologram is
recorded by a CCD camera with pixel resolution 960 × 1280 and pixel size 4.65 μm × 4.65 μm. The light emitted from a 640 nm iFLEX-200 compact laser diode is coupled into a single-mode fiber as a point light source. The specimen is mounted on an xy-precision motor controlled stage, and the CDH system is fixed on a z axis motor controlled stage. The measurement area of the CDH system can be calculated, which is about 3 mm × 2.25 mm. Note that the following experiments were performed after correcting the rotation error by the proposed method.
Fig. 7. Flowchart of the image-based correction method and phase stitching process. 10 January 2015 / Vol. 54, No. 2 / APPLIED OPTICS
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Fig. 8. Sketch map of experiment setup.
Practically, especially in a reflection configuration CDH system, the quadratic term phase error is more or less produced due to the mismatch between the reference wave and the object wave. Based on this, two comparison experiments between the traditional stitching method and the proposed algorithm were presented to verify the validity of our proposed method. B. Experiment on USAF Test Target With Less Phase Errors
We applied the proposed stitching algorithm to measure the United States Air Force (USAF) target. In this configuration, the reconstructed subaperture phase data contain less unwanted phase errors. The object was moved along with the x-direction of the motion stages, while ensuring that there are overlapped common areas between adjacent subapertures. First, three holograms (1 × 3) for different parts of the USAF target were recorded, as shown in Figs. 9(a)–9(c). Then the holograms were transferred into the computer to reconstruct by the angular spectrum method. Figures 9(d)–9(f) show the unwrapped
phases calculated from corresponding holograms in Figs. 9(a)–9(c), respectively. Actually, the edge diffraction effect of the reconstructed subaperture phase data cannot be ignored in the phase stitching algorithm. To suppress the edge diffraction effect, throughout this paper, we waived the edge data (about 50 pixels) of each subaperture phase data before stitching processing. To verify the validity of the proposed stitching algorithm, in this work, we compared the proposed method with traditional phase stitching algorithms [14,15] on the same subaperture phase maps of object. Figures 10(a) and 10(b) show the stitched phase by the traditional method. Figures 10(c) and 10(d) show the stitched phase by the proposed method. It is shown that there are quadratic error and tilts in the stitched results by the traditional method. However, our new proposed algorithm can solve the problem. C. Experiment on Microstructure Objects with Large Phase Errors
The tested specimen was a multifunction test target with a 3.0 mm square grid and 0.25 mm spacing (Max Levy Autograph, Inc., DA030), as shown in Fig. 11(a). During the recording process, four subaperture holograms (2 × 2) were needed to cover the sample. Figure 12 shows the four holograms recorded and their reconstructed wrapped phases. These four subapertures were stitched together by the two stitching methods, as shown in Fig. 13. It is shown that the stitching phase error obtained from the traditional method is more obvious than that of our proposed method. For further study of the two stitching methods, the trademark of Edmund optics was used for a specimen, as shown in Fig. 11(b). For testing the mark
Fig. 9. Original data: (a)–(c) holograms recorded; (d)–(f) corresponding unwrapping phase data. 272
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Fig. 10. Phase stitching result: (a) 2D map obtained from the traditional method; (b) 3D map obtained from the traditional method; (c) 2D map obtained from our proposed algorithm; (d) 3D map obtained from our proposed algorithm.
Fig. 11. Two microstructure objects (the red rectangle areas are wanted measurement areas): (a) multifunction test target DA030; (b) trademark of Edmund optics.
Fig. 12. Reconstruction results of subapertures: (a)–(d) holograms; (e)–(h) corresponding wrapped phases. 10 January 2015 / Vol. 54, No. 2 / APPLIED OPTICS
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Fig. 13. Phase stitching result: (a) 2D map obtained from the traditional method; (b) 3D map obtained from the traditional method; (c) 2D map obtained from our proposed algorithm; (d) 3D map obtained from our proposed algorithm; (e) comparison of the results of the two methods in the cross section of the x-direction; (f) comparison of the results of the two methods in the cross section of the y-direction.
Fig. 14. Five pairs of subaperture holograms and intensity distributions. 274
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of “Optics,” five subaperture holograms (1 × 5) were captured by CDH, as shown in Fig. 14. Similarly, the comparison experiment results are given in Fig. 15. It is obvious that our proposed algorithm can provide more accurate results than that of the traditional stitching method. In this experiment, the processing time of the proposed algorithm and traditional method are 25.68 and 25.54 s, respectively, with a 2.67 GHz Core i7 CPU using MATLAB 7.11.
It must be pointed out that the traditional stitching method can be used to get a perfect stitching result when the reconstructed subaperture contains phase data without the quadratic term error. However, in the two presented comparison experiments, the traditional method always produces data of relatively lower quality, and one traditional result produces highly large errors (cf., Fig. 15) when the tested subaperture phase data have the quadratic term error.
Fig. 15. Phase stitching result: (a) 2D map obtained from the traditional method; (b) 3D map obtained from the traditional method; (c) 2D map obtained from our proposed algorithm; (d) 3D map obtained from our proposed algorithm; (e) comparison of the results of the two methods in the cross section of the 300th row; (f) comparison of the results of the two methods in the cross section of the 800th row. 10 January 2015 / Vol. 54, No. 2 / APPLIED OPTICS
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5. Conclusions and Discussion
The phase stitching technique can expand the measurement area of a DH system. It enables the possibility of using DH to measure a large MEMS or microstructure element without losing high resolution. In this work, we presented an improved phase stitching algorithm based on the analysis result of phase errors to overcome the shortcoming of the existing phase stitching algorithm in DH. In addition, as mentioned previously, the relative rotation error between the coordinate systems’ CCD and xy-motion stages is very important for the phase stitching algorithm. To correct the relative rotation error, a simple and reliable image-based method was employed by using the feature point extraction technique and the linear fitting method with the LS criterion. Finally, the comparison experiments based on a reflection system of DH were performed. The advantages of the proposed algorithm over the existing phase stitching method in DH were verified by the comparison experiments on different situations. It is shown that the proposed algorithms are available and reliable for the measurement of a large microstructure object. We think the proposed methods can also be applied to transmission systems of DH and digital holographic microscopy (DHM). This study is supported by Translational Research and Development grant MOE2012-TIF-1-T-003 from the Singapore National Research Foundation, and Innovation Fund grant MOE2013-TIF-2-G-012 from the Singapore Ministry of Education. References 1. T. Reissman, E. Garcia, N. Lobontiu, and Y. Nam, “Integrated electrostatic micro-sensors for the development of modeling techniques of defects in the actuation of large microelectromechanical systems (MEMS),” Proc. SPIE 6152, 615238 (2006). 2. G. Coppola, P. Ferraro, and M. Iodice, “A digital holographic microscope for complete characterization of microelectromechanical systems,” Meas. Sci. Technol. 15, 529–539 (2004). 3. F. Montfort, Y. Emery, and F. Marquet, “Process engineering and failure analysis of MEMS and MOEMS by digital holography microscopy (DHM),” Proc. SPIE 6463, 64630G (2007). 4. L. Xu, X. Peng, J. Miao, and A. K. Asundi, “Studies of digital microscopic holography with applications to microstructure testing,” Appl. Opt. 40, 5046–5051 (2001). 5. P. Ferraro, S. D. Nicola, and G. Coppola, “Testing silicon MEMS structures subjected to thermal loading by digital holography,” Proc. SPIE 5343, 235–243 (2004).
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