1、1Study on phase-shifting profilometry of tilted measurement system（School of Mechanical Engineering, Quzhou College of Technology, Zhejiang 324000, China） Abstract: Based on research of the traditional phase-shifting profilometry (PSP) method, a tilted measurement system is proposed. The proposed te
2、chnology makes three constraints of the traditional phase-shifting profilometry measurement system less constrains. The exit pupil of the projecting system and the entrance pupil of the CCD imaging system are not horizontal and not placed at equal height. The optical axis in the CCD imaging system i
3、s not necessarily perpendicular with a specific tilting angle to the reference plane. The optical axises of the CCD imaging system and projecting system are not coplanar. The two axises do not intersect with each other. Compared to the traditional phase-shifting profilometry the proposed method offe
4、rs a flexible way to calculate height distribution. Key words: phase-shifting profilometry (PSP); measurement system; phase 1 Introduction 2Fringe pattern profilometry (FPP) is one of the most popular non-contact approaches to reconstructing three dimensional object surfaces. A Ronchi grating or sin
5、usoidal grating is projected onto a three-dimensional diffuse surface, the height distribution of which deforms the projected fringe patterns and modulates them in a phased domain. By retrieving the phase difference between the original and deformed fringe patterns three-dimensional profilometry can
6、 be achieved. In order to obtain phase maps from original and deformed fringes patterns researchers contributed various analysis methods, including Fourier transform profilometry (FTP) 1-5, and phase shifting profilometry (PSP) 6-8. PSP is one of the most popular approaches carrying out three diment
7、ional reconstructions. In recent years digital projection is widely used to generate sinusoidal fringe patterns9,10. The traditional phase-shifting profilometry cross-axis structure of the system requirements are relatively harsh, measurement system must meet the following three conditions of the co
8、nstraints 11. The exit pupil of the projecting system and entrance pupil of the CCD imaging system are horizontal and are planed at equal heights, the optical axis in the CCD imaging system is perpendicular 12, the optical axises of the 3CCD imaging system and projecting system are coplanar. The pro
9、posed technology makes the three constraints of the traditional phase-shifting profilometry measurement system less constraining. Through the strict theoretical analysis the mathematical relationship between the phase-height mapping formulae is given. 2 Principle theory 2.1 Principle theory of the t
10、raditional measurement system The traditional crossed-optical-axis geometry of the phase-shifting profilometry is shown in Fig.1. The distance l from the reference plane is same for the exit pupil of the projecting system and entrance pupil of the CCD imaging system. They are separated from each oth
11、er by the distance d in which the optical axes I1 of the projector lens crosses the camera lens I2 at point O on a plane R. Point C expresses points on R, and point D expresses a tested point on the measured object. The grating has its lines normal to the plane of the figure. Since the phase is modu
12、lated by the height of the object, the phase of point A is moved to point C. Therefore, the phase value of point C is the same as that of point A. The grating image is projected onto the object surface and observed through a CCD camera is a regular grating pattern. When a measured object is put on p
13、lane R, a sinusoidal grating pattern with one 4frequency component f0 along x direction is projected onto it. The deformed fringe pattern is observed through a CCD can be expressed as follow: Where A(x, y) is the background intensity, B(x, y) is the non-uniform distributions of reflectivity on the d
14、iffuse object. Given , Eq. (2) can be simplified as follow: The requirements of the traditional crossed-optical-axis structure of phase-shifting profilometry are relatively harsh. The system must meet the following three conditions of constraints. 2.2 Principle theory of the tilted measurement syste
15、m The phase-shifting profilometry of the tilted measurement system is shown in Fig. 2. In this system, the exit pupil of the projecting system is perpendicular to the reference plane R at point O. A three-dimensional coordinate system is defined with the origin O and axis lines X, Y and Z, as shown
16、in Fig 2. the deformed fringe pattern captured by CCD can be formulated as follow: First the traditional method for the phase calculation in four-step phase-shifting interferometry is introduced. After simplification, the reference fringe and deformed fringe can be 5rewritten as the simple form. The
17、 fringe pattern on the reference plane can be expressed as follow: For example, during the four-step phase-shifting process, the phase-shifts are usually to be set as 0, 90, 180, and 270. In general the intensity of an interferogram created by the interference of the reference and object can be form
18、ulated as follows Eq. (4). Four equations with respect to these phase-shifts can be derived as follows: In Eq. (4), (5) and (6), A(x, y) is the background intensity, B(x, y) is the non-uniform distributions of the reflectivity on the diffusion object. ?渍 0(x, y) is the phase distribution for H(x, y)
19、 =0. ?渍 1(x, y) is the phase modulations and the measured object height distribution. Then ?渍 0(x, y) and ?渍 1(x, y) can be calculated respectively. The phase can be solved using k-frame phase-shifting (k = 4) algorithm. Therefore the phase difference ?渍 1(x, y) can be further calculated as follow:
20、The phase ?驻?渍(x, y) directly originated from the height variation can be obtained as follow: The phase ?驻?渍(x, y) gives the principal value ranging from - to , and has discontinuities with 2 phase jumps. Continuous phase distribution can be obtained applying to phase 6?驻?渍(x, y) unwrapping algorith
21、m. The phase measurement profilometry diagram of the tilted measurement system is shown in Fig. 2. Projection grating frequency is f0, the phase-shift caused by the surface profile of the measured object is ?驻?渍(x, y), point P expresses a tested point on the measured object. Because the phase is mod
22、ulated by the surface height, the phase of point C actually is the same as B. T In order to further analyze the structure of the optical measurement system several lines are added into Fig. 2 to act as assisting lines for our analysis. herefore, the formula of the phase-shift value can be obtained a
23、ccording to the principle of similar triangles as follows: According to the principle of similar triangles, the following formula can be obtained: |AP| is the height of any point at the surface substituting |BC| into : Eq. (11) can be a simplified, conversion formula between the phase and height and
24、 can be expressed as follow: Where , Substituting ?驻?渍(x, y) into Eq. (12) the height variation can be obtained. Given and the right radix Eq. (12) can be expressed as follow: 7Eq. (14) is entirely consistent with the phase-height formula of the traditional phase-shifting profilometry. The optical a
25、xis of the CCD imaging system is necessarily perpendicular to the reference plane in the measurement system of the traditional TPMP. But the phase-shifting profilometry of the tilted measurement system is what we propose. The optical axis in the CCD imaging system is not necessarily perpendicular wi
26、th a specific tilting angle ?兹 2 to the reference plane. Because of this affects the calibration of object sizes along the X and Y directions. Therefore according to the calibration object sizes measured along the X and Y directions the value is between.-3 and +3. This parameter is not necessarily m
27、easured. The operation is simple and the corresponding result is more accurate. 3 Experimental results and discussion The projector model used is a NP200+ with the power of 180W, projection lens focal point is 21.8324 mm and a CCD imaging system model CM-140MCL was used in the experiment. The geomet
28、ric parameters were chosen as follows: projection system center coordinates I2 (0, 0, 1080 mm), the CCD imaging system center coordinates I2 (330 mm, 130 mm, 1030 mm), the projection system and the CCD imaging system as well as the angle (?兹 1 ,?兹 2) between reference plane. The normal direction ang
29、le is 812 and 3 respectively. The coordinates of any point on the measured object is expressed as P(x, y, H). The projection grating period is T0 = 4, the sinusoidal grating will be projected onto the reference plane by the projector system. Four different deformed fringe patterns can be obtained, s
30、hown in Fig. 3. After the deformed fringe pattern was obtained, the phase distribution of the object surface can also be obtained according to formula (9). The object height distribution is then calculated according to Eq. (12). The corresponding results are shown in Fig. 5 exhibiting three-dimensio
31、nal reconstruction mapping of the object (Fig. 4(a) and a vertical cross-section of the object (Fig. 4(b). To prove the correctness the measuring system we conducted an experiment, H1max is time domain phase measurement profilometry with the average height of ten equal intervals. H2max is a caliper
32、measurement of the average height in ten intervals. The corresponding results are shown in table 1. 4 Conclusions Based on the research of the traditional phase-shifting profilometry, phase-shifting profilometry of a tilted measurement system is proposed in this paper. Through the 9strict theoretica
33、l analysis, the mathematical relationship between the phase-height formulae is given. The measurement accuracy of the tilted measurement system was confirmed. The proposed technology makes the three constraints of the traditional phase-shifting profilometry measurement system less constrains. More i
34、mportantly the tilted system takes advantages of a simple operation process, measurement accuracy and is practical. Acknowledgements This work was supported by Natural Science Foundation of Zhejiang province, China under Contract Nos. X106874. The authors specially thank Dr. Lizai Pei for his valuab
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