In phase measuring deflectometry (PMD), the fringe pattern deformed according to slope deviation of a specular surface is digitized employing a phase-shift technique. Without height-angle ambiguity, carrier-removal process is adopted to evaluate the variation of surface slope from phase distribution when a quasi-plane is measured. However, the difficulty lies in the fact that the nonlinearity is generally contained in the carrier frequency due to the restrictions of system geometries. This paper investigates nonlinear carrier components introduced by the generalized imaging process in PMD. Furthermore, the analytical expression of carrier components in PMD is presented for the first time. The presented analytical form of carrier components can be extended to analyze and describe various effects of system parameters on carrier distortion. Assuming a pinhole perspective model, carrier phase distribution of arbitrary geometric arrangement is modeled as a function of spatial variables by exploring ray tracing method. As shown by simulation and experimental results, the carrier distortion is greatly affected by non-telecentric camera operation. Experimental results on the basis of reference subtraction technique further demonstrate that restrictions on reflection system geometry can be eliminated when the carrier phase is removed elaborately.
© 2012 OSA
Topography-related phase components are extracted from a projected or reflected fringe pattern in phase analysis based optical sensing techniques [1,2], and hence height map of the object is retrieved on the basis of system parameters. It is essential to establish the phase-to-height conversion model in phase measuring related profilometry.
It is known that the height-angle ambiguity problem is one of the difficulties in phase measuring deflectometry (PMD) and that the surface slope is coupled with object height in the phase map. It is difficult to isolate absolute height or slope variation from phase map without any further geometric information or assumptions. Solutions that proposed in the literature can be classified into analytical fitting methods  and geometric constraints enhancing methods [2,4–7]. Analytical models express the phase-to-height relation in accordance with the system geometry and the models are fitted to the observed data iteratively. However, accuracy of calibration parameters limited the improvement of the overall accuracy. On the other hand, Knauer et al.  enhanced geometric constraints by employing stereo measurement with two cameras. Muhr et al.  introduced a reference wire to constrain the direction of some reflected rays. Guo et al.  shifted the monitor to determine the locus of incident ray for each pixel and further reconstruct the 3D shape in the least squares sense. Shifting the screen and camera respectively, Tang et al.  calculated the slope and reconstructed the shape by the phase maps of the recorded fringe patterns. Consequently, the request of mechanical shifting and geometrical restrictions of the time-consuming calibration reduced the flexibility of these techniques.
However, the ambiguity can be neglected if a quasi-plane with very small depth variation is measured. In this situation, the phase distribution is only slope dependent and conversion of the measured phase distribution to the object slope distribution is required. Topography of the specular object is reconstructed by 2D numerical integration. Nevertheless, the phase distribution is sensitive to the geometric parameters and the local slope. The overall accuracy is limited by the accuracy of geometric parameters. The geometric constraints enhancing methods with time-consuming calibration [2,3] is not necessary. Here, conventional carrier removal methods  can serve as easier solutions to the phase-to-slope translation accordingly.
Carrier removal methods are usually limited by specific geometric arrangements. In fringe projection methods, system arrangement is usually adjusted carefully to help remove nonlinear carrier components and guarantee measurement accuracy. In order to eliminate restrictions on system geometries, models that describe exactly carrier phase distribution and phase-to-height relationship have been developed for arbitrary fringe projection profilometry system [9,10]. Various carrier-removal techniques have been proposed to determine the carrier frequency subsequently [8,9,11,12] and which might be adopted in the PMD. However, carrier phase distribution in a generalized PMD should be clarified to choose and develop carrier removal methods accordingly. To the best of authors’ knowledge, analytical description of the carrier phase distribution in PMD with an arbitrary geometric arrangement is still not available.
In PMD, the slope-related phase distribution in two perpendicular directions are extracted from x and y direction fringe patterns respectively . A fringe pattern with an x direction carrier frequency can be expressed as Eq. (1). Where x and y are spatial variables; I is the overall intensity distribution; A and B are background and modulation intensity, respectively; f0 is the carrier frequency for phase modulation; and only is the phase to be measured. Usually, the system is arranged in simple setup modes and employs a telecentric imaging assumption to ensure that f0 performs as a spatially independent constant. However, the generalized imaging process with non-telecentric operation will introduce unequal fringe spacing and show the nonlinear distortion in x and y direction in a very different way. The carrier must be removed to obtain the unaffected phase map .
In section 2, we demonstrate the analytical description of carrier component in x and y direction for a generalized PMD. Section 3 presents computer simulations of carrier phase distortion in x and y direction. Reference subtraction technique is investigated by experimental work for carrier-removal in the PMD. Conclusions are drawn in section 4.
2. Analytical description of carrier phase distribution
Figure 1 shows arrangement of the PMD. Computer generated phase-shifting fringe patterns in x and y direction with an equal spacing Ps are displayed on a liquid crystal display (LCD) screen. The carriers in both horizontal and vertical directions are employed to record the slope-related phase variation . As shown on the right side, a CCD camera observes and records the distorted fringe patterns via the tested specular surface. The slope variation is estimated by relating the angular deviation of reflection with equivalent phase differences under the assumption that the height deviation of the quasi-plane surface is relatively small. 3D shape of the tested surface is reconstructed by further 2D integration .
Figure 2 shows the detailed geometrical arrangement of the PMD for further derivation of carrier phase distribution. Here, xyz and XYZ are defined as the world coordinate and local coordinate of CCD plane, respectively. Considering a generalized imaging system, the CCD is placed behind a lens having an optical center (Xf, Zf). The optical axis of the CCD camera crosses the imaging center (X0, Z0) perpendicularly. The LCD plane is vertical to xz-plane, making an angle θ with xy-plane. P(x0, z0) is the original point on LCD, where the phase is set to zero. The light reflected by any point C on the specular surface, and received by CCD passing through the optical center, can be traced back to its source location on Bx or By to calculate the phase distribution.
A. Reference phase distribution of carrier fringe in x direction
In Fig. 2(a), the phase axis Px can be expressed by a line representation,Eq. (2) and (3) according to the reflection law. The x coordinate of Bx is described as follow,
Assume that the uniform fringe pitch on the LCD is T0 and the corresponding carrier frequency is f0. According to the geometry and Eq. (2), the carrier phase can be represented as,Fig. 2(a) we can write,Eq. (6) into Eq. (5) we get the carrier phase distribution on CCD of x direction carrier fringe pattern,
B. Reference phase distribution of carrier fringe in y direction
The carrier phase in y direction is calculated according to the geometry shown in Fig. 2(b) and (c). From similar triangles we get,Fig. 2(b) we can write,
The side view of PMD is shown in Fig. 2(c). A light beam is traced back to intersect at By on the virtual image of LCD screen, which can be described according to the system parameters as follow,
We can write the carrier phase of By,
Consequently, Eq. (7) and (15) demonstrate the fact that the carrier phase distribution is spatially-varying. Comparing x and y direction carrier components, the x direction carrier phase is described as a function of x direction spatial variable, whereas the y direction phase distortion is modulated by two spatial variables.
It should be noted that the above conclusion is only valid when the camera imaging plane is perpendicular to xz-plane and Y-axis is parallel with y-axis. Furthermore, these restrictions on the system’s geometry can be further eliminated to describe arbitrary arrangements by employing our approach. Assume the CCD is rotated around Z-axis and the optical center of imaging lens (Xf, Zf) in YZ-plane by angles Φ and ρ, respectively. The x-direction fringe pattern is not imaged as pure vertical fringe pattern but tilted according to the rotation. The light beams originated from the LCD screen with phase variation passing through the optical center of the imaging lens are not disturbed. Only the image mapping transformation is modified according to the rotation and shifting of the camera coordinates XYZ in the world coordinates xyz. We can easily estimate the new phase distribution by coordinate transformation of the imaging plane. The rotation matrix R and shifting matrix t can be given byEqs. (7) and (15) by Eq. (19) according to the rotation angles ρ and Φ, the revised carrier distribution for arbitrary geometric arrangement is determined. Obviously, the x-direction and y-direction carrier component are both related to x and y variables. Therefore the results have been extended to describe carrier phase distribution under an arbitrary geometric arrangement. Fortunately, as shown by Eq. (1), the carrier fringes, cos[2πf0(x,y)], serve as an information carrier and would leave the topography-related phase unaffected if the carrier frequency is removed elaborately.
3. Computer simulation and experimental work
The PMD was simulated based on light tracing and topography of a specimen was measured to verify the theoretical analysis. Figure 3 shows the established simulation model of PMD which consists of a LCD screen displaying phase-shifting fringe patterns, a CCD camera with an imaging lens and a specular surface. According to the system arrangement shown in Fig. 2 and reflection law, each incident light on CCD camera was traced back to its original source location on the LCD screen for phase quantification.
Reference phase of a plane in x and y direction has been simulated by ray tracing method using the model shown in Fig. 3. According to Fig. 2, the plane object was placed 450.0mm away from the LCD screen and the CCD camera. The angle between CCD optical axis and normal direction of the LCD screen was 90°. Both CCD plane and the LCD screen were placed perpendicular to xz-plane, and Y-axis of CCD coordinate was parallel with y-axis in the world coordinate. The fringe pitch on the LCD screen was 10mm and the focal length of camera lens was 28mm. In order to demonstrate the distortion of carrier phase distribution, the linear components have been subtracted by the plane-fitting technique.
Figure 4(a) and (c) show the 3D plot of the carrier residual in carrier phase in x and y direction by simulation. Figure 4(b) and (d) are the experimentally measured phase distribution of a plane mirror after subtracting the linear components in x and y direction, respectively. The experimental results are consistent with simulations in Fig. 4(a) and (c). Here, the system parameters are same as simulation values. Note that the carrier phase increases nonlinearly with increasing spatial variables x and y linearly. As expected by Eq. (7) and (15), the x direction phase distribution was only related to x direction spatial variable, whereas the y direction phase was two dimensional dependent. Consequently, this fact invalidates conventional linear carrier-removal technique  in fringe projection profilometry, e.g. plane-fitting technique, carrier-removal in the frequency domain by Fourier transform, etc.
The proposed carrier description in PMD with generalized imaging process was emphasized to give us guidance on how to choose or develop adequate carrier-removal techniques. Furthermore, experiments were conducted to study the elimination of restrictions on PMD schemes by a robust carrier-removal approach. The well developed carrier-removal techniques  in fringe projection profilometry might serve as alternative methods in this case, such as reference subtraction technique, least-squares approach, etc.
In the experimental work, we consider a well known reference subtraction technique for nonlinear carrier-removal. The undesirable side-effect of this approach is the increase of phase measurement uncertainty. In order to evaluate the increase of phase uncertainty, the carrier of an unwrapped phase map was removed by the series-expansion method  and reference subtraction method, respectively. A series function was fitted to the phase map on a plane object to estimate the carrier distribution. Subtraction of a series function will leave the phase uncertainty unaffected. On the other hand, the plane object was measured twice at the same position and the two unwrapped phase maps were subtracted to estimate the phase error. Phase uncertainty calculated from the carrier removal results of the two methods is plotted in Fig. 5 . It is shown that instead of being eliminated in the subtraction, the phase uncertainty was magnified by reference subtraction method. However, the reference subtraction method is straight forward and no matter whatever features of a carrier can be removed automatically.
Furthermore, a standard spherical convex mirror with a 15.360m nominal radius was measured to show the effectiveness of reference subtraction method in PMD. According to Fig. 2, the mirror is placed 650.0mm away from the LCD screen and 670.0mm away from the CCD camera. Angle between CCD optical axis and normal direction of the LCD screen is about 23.5°. The fringe pitch on the LCD screen is about 18.048mm and the focal length of camera lens is 28mm. Figure 6(a) shows a deformed fringe pattern reflected by a spherical convex mirror. Figure 6(b) shows the unwrapped phase map with linear carrier removed. By noting the nonlinear components contained in the slope-related phase map, conventional linear carrier-removal approaches would introduce additional shape in the reconstructed topography by 2D integration.
In Fig. 6(c), phase map of an extra reference plane was measured as the carrier components. By subtracting the reference phase map from the object phase map, the carrier frequency can be completely removed in the regardless of whether the carrier is linear or nonlinear. As shown in Fig. 6(c), the nonlinear carrier in slope-related phase in y direction was completely eliminated. Figure 6(d) shows the 3D plot of the reconstructed shape by 2D integration . Numerical integration performed like a low pass filter and here the least squares method was employed to recover the smooth surface iteratively. As depicted in Fig. 6(e), a spherical surface is fitted to the reconstructed shape and the measured radius is 15.818m which is close to the nominal value 15.360m. Hence reference subtraction technique is still valid for carrier-removal in PMD and elimination of the carrier distortion would leave the final reconstruction results unaffected.
In a simple situation that a quasi-plane is measured by PMD, the carrier-removal can be employed as a simply solution to translate phase distribution into local slope variation according to geometric parameters. In this paper, analytical carrier phase description in x and y direction for the PMD with generalized imaging process was proposed. In the derivation and computer simulations, the pinhole model of a CCD camera was used to analyze non-telecentric imaging operation. The carrier phase distribution was related to spatial variables by analytical ray tracing. Demonstrated by computer simulations and experimental work, the nonlinear carrier distortion in x and y direction that consistent with the derivation invalidated conventional linear carrier-removal techniques for topography-related phase retrieval. Experimental results of reference subtraction approach demonstrated the possibility of precisely eliminating of carrier components and leaving the slope-related phase unaffected. Significance of the presented carrier description lies in the fact that it can help us analyze various effects of system parameters in PMD and develop appropriate carrier-removal techniques. Studies focusing on these issues will benefit the community to borrow the existing carrier removal methods from fringe projection profilometry accordingly and further improve carrier removal techniques accordingly.
The authors wish to acknowledge the support by the Fundamental Research Funds for the Central Universities (No. ZYGX2011J053) and the National Nature Science Foundation of China (No. 60925019).
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