## Abstract

A new simultaneous three-dimensional (3D) displacement measurement technique based on the combination of digital holography (DH) and digital imaging correlation (DIC) is proposed. The current DH-based 3D displacement measurement technique needs three sets of DH setups, and only the phase images are utilized in measurements, with all the intensity images discarded. In contrast, the proposed new technique only adopts a single off-axis DH setup. In the proposed technique, the phase images are used to extract out-of-plane displacements, but the intensity images (instead of being discarded) are processed by an intensity correlation algorithm to retrieve in-plane displacement components. Because the proposed technique fully takes advantage of all the information obtained by an off-axis DH without additional optical arrangements, it is simpler and more practical than the existing DH-based 3D displacement measurement technique. Experiments performed on a United States Air Force (USAF) target demonstrate that both the in-plane and out-of-plane displacements can be accurately determined by the proposed technique.

© 2014 Optical Society of America

Holography is an important noncontact optical technique for profile and out-of-plane deformation measurements, with the advantages of high sensitivity and nonintrusive measurements. By using photoelectric recording devices such as CCD for hologram recording and computer programs for numerical reconstruction, the invention of digital holography (DH) [1,2] circumvents wet chemical processing of photosensitive plates and the complicated phase determination problems associated with conventional holography interferometry. Furthermore, it provides direct access to quantitative amplitude and phase images of the test object before and after deformation [2,3], based on which the quantitative phase difference between the two states and related out-of-plane deformation can be easily deduced. Due to the aforementioned advantages, DH has been employed as a practical and effective full-field out-of-plane displacement measurement technique with a high accuracy down to the nanometer scale [4].

Recently, simultaneous three-dimensional (3D) displacement measurement techniques have been in great demand and of great interest to researchers [5,6]. Although DH has advantages in out-of-plane displacement measurement, a single DH setup is not considered suitable for 3D displacement measurement. For simultaneous 3D displacement measurement, three DH setups have been utilized, with each in charge of one directional measurement [7]. Such a technique is complicated in configuration and not efficient in information utilization, as only the phase images are used, with all the intensity images of the three DH systems being discarded.

In the field of experimental mechanics, the two-dimensional digital image correlation (2D-DIC) technique [8] using a single fixed camera has been widely established as a simple but versatile tool for quantitative in-plane deformation measurement of nominal planar objects. By numerically comparing the two digital images with random intensity distributions recorded before and after deformation, 2D-DIC directly provides full-field in-plane displacements with subpixel accuracy [9]. By combining this with high-spatial-resolution microscopes, microscale or even nanoscale in-plane deformation can be realized. However, 2D-DIC is limited to only in-plane displacement measurement.

In this Letter, we propose a new 3D displacement measurement technique based on the combination of DH interferometry and 2D-DIC. In this technique, instead of using three DH setups, only a simple off-axis DH setup is adopted. Its phase differences are used for typical out-of-plane displacement. Intensity images in this technique, instead of being discarded, are processed by a DIC algorithm for determining the desired in-plane displacements. Therefore, all the information acquired by a DH setup is fully utilized and contributes to a complete 3D displacement measurement. This efficient information utilization further results in a significant reduction in system complexity and contributes to a simple setup. Furthermore, if integrated with microscope objectives, the proposed technique could provide nanometer accuracy in all three dimensions thanks to the nanometer accuracy of DH and 2D-DIC in out-of-plane and in-plane dimensions, respectively. These advantages could make the proposed methodology a quite promising technique for practical 3D displacement measurement of microelectromechanical systems (MEMS), bio samples, and so forth.

The methodology of the proposed technique is illustrated in Fig. 1. The setup is a simple off-axis DH setup working in reflection mode. In the recording process, the object wave first illuminates the object. Then, it is reflected by the object surface and carries the object surface information. At the CCD target plane, the object wave $O$ interferes with the reference wave $R$ to generate an optical hologram ${I}_{H}(x,y)$, which is sampled, digitalized by CCD, and transferred to a computer as a digital hologram ${I}_{H}(k,l)$ for numerical reconstruction.

In numerical reconstruction, propagation of the wave back to the object plane is calculated by a computer program, and the quantitative object wave field can be reconstructed as in Eq. (1) [2]:

In out-of-plane displacement measurement, the phase difference between the two states is calculated by Eq. (2), where ${\phi}_{1}$ and ${\phi}_{2}$ denote the phases before and after displacement [Figs. 1(b) and 1(c)], respectively [10]:

However, instead of being regarded as useless information and discarded as in the current DH-based 3D displacement measurement technique, the intensity images reconstructed at different states are processed by DIC to extract the desired in-plane displacements. The basic principle of the standard subset-based DIC is schematically illustrated in Figs. 1(e) and 1(f). First, a region of interest (ROI) is specified in the original intensity image (i.e., the reference image, denoted by $f(x,y)$), within which the regularly spaced pixel points are defined as points of interest. Then, to accurately determine the location of each measurement point of the ROI, a square reference subset of ($2M+1)\times (2M+1)$ pixels centered at the interrogated point $P(x,y)$—rather than a single pixel point—is tracked in the intensity image at different states (i.e., the deformed image, denoted by $g({x}^{\prime},{y}^{\prime})$ to find its most similar counterpart. Once the location and shape of the target subset with maximum similarity (or, equivalently, minimum difference) are found, the displacements of the reference subset center can be determined. The same tracking procedure is repeated on other points of interest to obtain full-field deformation of the ROI.

Although various correlation criteria have been defined for quantitatively evaluating the similarity between the two subsets, the following zero-mean normalized sum of squared differences (ZNSSD) criterion, which is robust against scale and offset changes in illumination lighting fluctuations, is highly recommended for practical use:

The ZNSSD criterion defined in Eq. (4) is a parametric objective function with respect to six unknown parameters of the deformed subset. Mathematically, this becomes a parametric optimization problem. If an initial guess of deformation ${\mathbf{p}}_{\mathbf{0}}$, which is sufficiently close to the true value, is provided, the objective function Eq. (4) can be iteratively optimized to the local minimum using the classic Newton–Rapshon (NR) iteration algorithm [9] or equivalent but more efficiently inverse compositional Gauss–Newton algorithm [11] to determine the desired displacement vector ${(u,v)}^{T}$. The same calculation can be extended automatically to the rest of the measurement points to acquire full-field displacements with sub-pixel accuracy.

To validate the effectiveness and accuracy of the proposed new 3D displacement technique, a series of experiments was performed to quantify the rigid body motion of a United States Air Force (USAF) target (a standard resolution chart). The optical setup used herein is a common lensless reflection DH setup. The wavelength, reconstruction distance, and pixel size used are 650 nm, 129 mm, and 4.65 μm, respectively. The field of view of the setup is $1280\times 960$ pixels. In the experiment, the USAF target first underwent in-plane displacements in the $x$ direction from 50 to 500 μm with an interval of 50 μm. The in-plane displacement was implemented by an $x\u2013y$ two-axis manual translation stage with a positioning sensitivity of 1 μm working in the way of a differential micrometer drive. After in-plane displacements, nine out-of-plane displacements are applied to the USAF target by rotations around the $x$ axis. The out-of-plane rotations are implemented by a manual rotation stage with one edge fixed and its parallel edge displaced in the $z$ direction (out-of-plane direction) with an interval of 100 μm by a differential micrometer drive. Considering the distance of 95 mm between the two edges, the rotation angles applied are from 3.62 to 32.57 min with an interval of 3.62 min.

At each state, a digital hologram is recorded and numerically reconstructed. From the reconstructed wavefront, the quantitative intensity and phase images are acquired. Examples of these are shown in Fig. 1. Figures 1(e) and 1(f) are the intensity images before and after deformation, respectively. Figures 1(b) and 1(c) are the phase images before and after deformation and the out-of-plane displacement deduced from the difference between the two phase images.

The intensity maps are processed by a recently proposed high-accuracy and high-efficiency DIC algorithm, called the inverse compositional Gauss–Newton algorithm [11], for the in-plane displacement determinations. The lines and numbers on the USAF target are taken as the surface texture to be tracked. Due to the relatively low image contrast and high noise in the reconstructed intensity images, DIC analyses were performed with a large subset size of $201\times 201$ pixels. In Fig. 2(a), the calculated in-plane displacement vector field at a translation of 500 μm in the $x$ direction is presented as an example. It can be seen that the measured displacement vectors are all along the $x$ direction. At the same time, the out-of-plane displacements are also calculated from the phase images to be less than 200 nm, as presented in the inset of Fig. 2(a). As the in-plane displacements are the main concern in this experiment, the out-of-plane displacements are not presented in detail. To quantitatively evaluate the in-plane displacement, the averaged values of the $x$-directional displacement field at each state are calculated and compared with the applied ones in Fig. 2(b). It can be seen that the measured and applied values are in good agreement. The corresponding mean bias error and standard deviation (std) are presented in Fig. 2(c). Considering the mechanical error of the manual translation stage, the accuracy achieved in 3D is acceptable.

On the other hand, the quantitative phase images carrying the 3D profile information are processed for out-of-plane displacement determinations. First, the phase difference at each state is calculated by Eq. (2). Second, the useful displacement information is extracted from the phase difference by filtering in the Fourier domain, followed by the phase unwrapping algorithm. Finally, the transformation from phase differences into out-of-plane displacements in Eq. (3) is performed. The 3D out-of-plane displacement at the rotation angle of 32.57 min is presented in Fig. 3(a) as an example. From the 3D out-of-plane displacement maps, the rotation angles are calculated and compared with the angle applied in Fig. 3(b). The measured and applied out-of-plane displacement values are in good agreement. At the same time, the rotation-induced in-plane displacements are determined by the DIC algorithm, as presented in the inset of Fig. 3(b). This shows that the in-plane displacement vectors are all along the $y$ direction because of the applied rotation around the $x$ axis. As the in-plane displacements are not a concern in this experiment, they are not presented in detail. The according bias and std errors of the measured rotation angle are presented in Fig. 3(c), which indicate a high-accuracy measurement with a maximum absolute error of less than 0.8 min.

In conclusion, a novel methodology for 3D displacement measurement based on the combination of DH and DIC techniques is proposed for the first time to the best of the authors’ knowledge. Compared to current DH techniques for 3D displacement, this technique is simpler and more practical. Instead of wasting the intensity images as in the current DH-based 3D displacement measurement technique, both the intensity and phase images are fully utilized for 3D displacement measurement with the help of the powerful DIC technique. Two real experiments prove that the proposed new 3D displacement measurement technique is feasible for simultaneous 3D displacement measurement. The efficiency in information utilization further provides the advantage of a much simpler system configuration with only one simple off-axis DH system for the whole 3D displacement measurement. Finally, although it is not presented in this work, the proposed technique possesses the capability of 3D displacement measurement with nanometer accuracy if a DH microscope configuration is adopted.

In the preliminary experiments conducted in this Letter, although the measured in-plane and out-of-plane displacements are satisfying, it was observed that the reconstructed intensity images are less ideal than the speckle patterns recorded using a regular optical system. Thus, future work should focus on the following aspects. First, a proper filtering process, which can effectively suppress the noise but preserve the original intensity distributions, should be developed for the optimal reconstruction of intensity images. Second, for the objects without obvious surface texture, techniques for fabricating appropriate speckle patterns on the test object surface should be employed. Finally, the possible nanometer accuracy in all 3D displacement measurements of the proposed technique needs to be demonstrated.

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11272032, 11322220, and 91216301); the Program for New Century Excellent Talents in University (Grant No. NCET-12-0023); the Science Fund of State Key Laboratory of Automotive Safety and Energy (Grant No. KF14032); and the Beijing Nova Program (xx2014B034).

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