Mod a l phase measuring deflectometry

: In this work, a model based method is applied to phase measuring deflectometry, 10 which is named as modal phase measuring deflectometry. The height and slopes of the 11 surface under test are represented by mathematical models and updated by optimizing the 12 model coefficients to minimize the discrepancy between the reprojection in ray tracing and 13 the actual measurement. The pose of the screen relative to the camera is pre-calibrated and 14 further optimized together with the shape coefficients of the surface under test. Simulations 15 and experiments are conducted to demonstrate the feasibility of the proposed approach.


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Another issue we will discuss is the system calibration, which is an everlasting topic in 118 metrology. The classical PMD calibration method can be split into three separate steps for the 119 calibration of screen, cameras, and the system geometry [3,4]. A flexible way to calibrate the has been applied to calibrate stereoscopic PMD system by Ren et al. [24]. Laser tracker was.

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employed to measure the system geometry [12,25]. In this work, we present a method to 123 simultaneously estimate the height and slopes of the SUT in PMD by using mathematical 124 models (e.g. Chebyshev polynomials, Zernike polynomials, or B-splines), which is called 125 modal phase measuring deflectometry (MPMD). Moreover, a post-optimization for the screen 126 geometry after the pre-calibration of the PMD system is also addressed in MPMD.

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Section 2 introduces the principle of the proposed MPMD with its post-optimization.

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configurations are carried out in section 3 to demonstrate the performance of MPMD. The

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The fundamental principle of deflectometry is the law of reflection. The angle of incidence 134 equals the angle of reflection when light coming from the screen goes into the camera after 135 the reflection on the specular surface as depicted in Figure 1(a). Conventionally, this scene is 136 considered in a reverse way illustrated in Figure 1(b). The probe ray P = (x n , y n , 1) T from the

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Unlike the classical PMD that calculates the slopes from the fringe phases with pre-known represent both height and slopes of the SUT with well-established mathematical models w 151 and to optimize the coefficient vector c to best explain the captured fringe patterns in measurement. In order to avoid recalculating basis function during iterations in optimization, where w is the vector of the known basis functions, which can be Chebyshev polynomials, The xand y-slopes in camera coordinates (x c , y c ) are represented as 168 Therefore, the normal of the mirror surface facing towards the camera is . For a probe vector P, its sampling point on the SUT is X and the vector of the reflected ray from X can be calculated as 171 2 ,   r p p n n, 5 where  N n N is the normalized normal and  P p P is the normalized probe vector.

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The reflected ray can be determined once its direction (r) and one point on it (X) are and reflected ray. Another purpose of pre-calibration is to initialize coefficient vector c by 181 fitting a camera calibration plane close to where the SUT is going to place. 188 In addition, the proposed MPMD method can easily adapt to multi-camera PMD system

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The noise on fringe phase is set as 2π/100 rad rms which is not difficult to achieve in 221 practical phase retrieval, especially in PMD. The fringe period is 16 screen pixels with the 222 pixel pitch of the screen is 187.5 μm. Therefore, the measurement uncertainty of a pattern 223 point in one direction on the screen is 30 μm rms.

Error evaluation
If we only concern on the shape of a SUT, instead of its absolute coordinates, the piston, tip 226 and tilt terms in the result can be removed from the error evaluation as illustrated in Figure 4.

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In MPMD, the screen pose can be adjusted with the model coefficient during optimization. It

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In most applications of the PMD, the shape of the specular surface is the goal, so we 234 evaluate the shape error with piston, tip, and tilt removed in this work.

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Reconstruction under perfect calibration 236 If the system calibration is perfect, there is no need to optimize the screen pose. The shape is  Figure 5(d-f). All these results indicate the MPMD can reconstruct the 248 SUT with a shape error less than 1 μm rms with the simulated system configuration and phase 249 noise condition, if the system is perfectly calibrated.

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Reconstruction under imperfect calibration 251 However, the practical system calibration is never perfect. In fact, the inaccurate system 252 calibration is one of the major error sources in PMD, which limits its practical application.

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The pre-calibration error is simulated and added into the true pose of the screen as shown in Chebyshev polynomials as the model for an instance, large shape error shown in Figure 7(e) will observed from the reconstruction result in Figure 7.

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The height and slopes represented together by coefficients are modified to "best" reflect 266 the probe rays onto the screen as illustrated in Figure 7(b-d). However, the screen in ray 267 tracing is "placed" at a wrong location owning to the incorrect geometry knowledge. The

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A similar observation is made in Figure 9 when the post-optimization is implemented to a 284 stereo-PMD system according to Eq. (9).  measurements. An LCD screen (Dell P2414H with 1920×1080 pixels and 0.2745 mm×0.2745 295 mm pixel pitch) and a CCD camera (Manta G-145 with 1388×1038 pixels and 12-bit pixel 296 depth) compose a mono-PMD system. Another camera (Manta G-145 with 1388×1038 pixels 297 and 12-bit pixel depth) is added for a stereoscopic configuration as shown Figure 10. The

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SUT is 200 mm long and 95.3 mm wide and the mirror is concave along one dimension.
299 Figure 10. Experiment setup mainly consists of two cameras, one LCD screen, and the SUT.
above, the purpose of the pre-calibration is to initialize the parameters for screen pose and 303 model coefficients.
The pre-calibration can be the same as the classical PMD calibration. An LCD screen displaying phase shifting fringe patterns is employed as the calibration target for camera 307 calibration. A flat mirror with markers is used in geometry calibration to determine the screen 308 pose in camera coordinates. An optimization is finally operated to refine the screen pose as 309 suggested by Xiao et al. [23]. The pre-calibrated system geometry is illustrated in Figure 11.     one m while the iteration updates. The resultant shape is finalized in Figure 13(c) when the 338 shape change or the variation of the reprojection error is negligible.

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delivers the shape result. Figure 14(a) shows both camera rays trace large errors at the start of the iteration. It is mainly because the shape coefficients are initialized from a plane pose used 343 in camera calibration, which makes the shape guess is so different from the SUT.    affordable geometry error on screen pose will affect the shape result little. However, it does 377 not mean the geometry calibration is not important anymore. In fact, the system pre-

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The proposed MPMD is not good at reconstructing local waviness on the specular SUT.

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In some applications, high frequency components are interested and significant. It requires a 409 large number of model coefficients to represent these high frequency terms. However, this 410 will highly increase the computing time and become it impractical. How to better deal with 411 the reprojection residuals and reconstruct the high frequency components in MPMD will be 412 addressed in our future work.

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We would like to thank Yuankun Liu in Sichuan University for the extensive discussion on