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OPTICS LETTERS / Vol. 31, No. 24 / December 15, 2006
Least-squares fitting of carrier phase distribution by using a rational function in fringe projection profilometry Hongwei Guo, Mingyi Chen, and Peng Zheng Department of Precision Mechanical Engineering, Laboratory of Applied Optics and Metrology, Shanghai University, Shanghai 200072, China Received August 7, 2006; revised September 8, 2006; accepted September 22, 2006; posted October 2, 2006 (Doc. ID 73835); published November 22, 2006 A least-squares fitting technique for the carrier phase component in fringe projection profilometry is presented. The carrier phase distribution for an arbitrary measurement system can be perfectly described with a rational function, whose coefficients can be estimated by least-squares fitting to the measured reference phases, so that the restrictions and limitations in the existing techniques are eliminated. © 2006 Optical Society of America OCIS codes: 120.2830, 120.5050, 150.6910.
In fringe projection profilometry, the depth maps of object surfaces are calculated from the phase distributions from which the carrier phases have been removed. Therefore exactly determining the carrier phase component is important for guaranteeing measurement accuracies. The difficulty is that the carrier phase distribution generally contains nonlinearity in its profile due to the restrictions of system schemes. In the conventional techniques,1–4 this nonlinearity has been corrected on the bases of geometrical analyses of measurement systems. With these techniques, however, the system parameters must be known or adjusted carefully to satisfy some stringent conditions. The polynomial fitting methods5–8 can overcome this problem, but with them the accuracies cannot be guaranteed even if high-degree polynomials are used because of their poor asymptotic properties; moreover, the error will increase when the data are extrapolated outside the reference plane. Reference 7 suggests a rational function representation of the carrier phases, but it assumes the optical axis of the imaging lens to be perpendicular to the reference plane, which is not always valid in measurement practice. To the best of our knowledge, models that can exactly describe the carrier phases for an arbitrary measurement system are still not available. In this Letter, we demonstrate that the carrier phase distribution for an arbitrary system can be perfectly represented with a 2D rational function. Based on this fact, a least-squares fitting technique for the carrier phase component is proposed. Because it is a pixelwise operation to remove the carrier component from the object phases, we shall represent the carrier phase as a function of pixel coordinates. Figure 1(a) shows the position and orientation of the camera relative to the reference plane, where C is the center of the imaging lens, and R is the reference plane. The coordinate system 共OXYZ兲 is established, with the XOY plane being superposed on R. The coordinates of C are 共xC , yC , zC兲. The fictitious plane I passes through the origin O and is perpendicular to the optical axis of the imaging lens. In another system 共OX⬘Y⬘Z⬘兲, the X⬘OY⬘ plane is superposed on I, 0146-9592/06/243588-3/$15.00
and the X⬘ and Y⬘ axes are parallel to the image coordinate axes, i.e., i and j axes, respectively. For a general pixel 共i , j兲, which is the image of the point E on I, the mapping between the coordinates 共xE ⬘ , yE⬘ , zE⬘ 兲 and 共i , j兲 is linear, viz.,
⬘ 关xE
⬘ yE
⬘ 兴 = 关共i − io兲 zE
共j − jo兲
0兴,
共1兲
where 共io , jo兲 are pixel coordinates of the image of O, and is a scale factor. Assuming that the system 共OX⬘Y⬘Z⬘兲 can be transformed to 共OXYZ兲 by rotations around the X⬘, Y⬘, and Z⬘ axes in sequence,
Fig. 1. (Color online) Geometry for fringe projection profilometry. Positions and orientations of (a) the camera and (b) the projector relative to the reference plane, respectively. © 2006 Optical Society of America
December 15, 2006 / Vol. 31, No. 24 / OPTICS LETTERS
through the angles ␣, , and ␥ (which are basic system parameters for determining the orientation of the camera in 3D space), respectively, the coordinates of E in 共OXYZ兲 can be calculated with
冤冥冤 xE
cos ␥
yE = − sin ␥ zE
sin ␥
0
cos ␥
0
0
1
0
冤
1
⫻ 0 0
冥冤
cos 
0
− sin 
0
1
0
sin 
cos 
0
冥冤 冥
冥
0
cos ␣
sin ␣
− sin ␣
cos ␣
⬘ . yE
共2兲
0
冤冥冤 冤
共xCzE − xEzC兲/共zE − zC兲
yD = 共yCzE − yEzC兲/共zE − zC兲 zD
0
冥
⬘ + b2yE⬘ 兲/共1 + a1xE⬘ + a2yE⬘ 兲 共b1xE
冥
共3兲
where a1 = −sin  / zC, a2 = sin ␣ cos  / zC, b1 = cos  ⫻cos ␥ − xC sin  / zC, b2 = cos ␣ sin ␥ + sin ␣ sin  cos ␥ + xC sin ␣ cos  / zC, b3 = −cos  sin ␥ − yC sin  / zC, and b4 = cos ␣ cos ␥ − sin ␣ sin  sin ␥ + yC sin ␣ cos  / zC. Because D and E produce their images at the same pixel, the carrier phase at 共i , j兲 equals the reference phase at D. That is,
共i,j兲 = D .
共4兲
Figure 1(b) shows the relationship between the projector and R, where P is the center of the projector lens. The fictitious plane G passes through O and is perpendicular to the optical axis of the projector lens. Therefore the fringes projected on G are kept parallel to each other and are equally spaced with pitch p. The X⬙OY⬙ plane of the system 共OX⬙Y⬙Z⬙兲 is superposed on G, and the Y⬙ axis is parallel to the fringes. Assuming that 共OX⬙Y⬙Z⬙兲 can be transformed to 共OXYZ兲 by rotations around the X⬙, Y⬙, and Z⬙ axes in sequence, through the angles , , and , respectively, and that the coordinates of P in 共OX⬙Y⬙Z⬙兲 are 共xP⬙ , yP⬙ , zP⬙ 兲, the coordinates of D in 共OX⬙Y⬙Z⬙兲 can be calculated with
冤冥冤 ⬙ xD
1
0
0
y⬙D = 0
cos
− sin
z⬙D
sin
cos
0
冤
cos
⫻ sin 0
− sin cos 0
冥冤 冥冤 冥
cos
0
sin
0
1
0
− sin
0
cos
0
xD
0
yD .
1
0
⬙ − xD ⬙ zP⬙ 兲/共zD ⬙ − zP⬙ 兲 共xP⬙ zD
冤
0
冥
共d1xD + d2yD兲/共1 + c1xD + c2yD兲
冥
= 共d3xD + d4yD兲/共1 + c1xD + c2yD兲 , 0
冥 共5兲
PD crosses G at F. By using the equation for PD, i.e.,
共6兲
c2 where c1 = 共 cos sin cos − sin sin 兲 / zP⬙ , = −共cos sin sin + sin cos 兲 / zP⬙ , d1 = cos cos + xP⬙ ⫻共cos sin cos − sin sin 兲 / zP⬙ , d2 = −cos sin − xP⬙ ⫻共cos sin sin + sin cos 兲 / zP⬙ , d3 = sin sin cos + cos sin + yP⬙ 共cos sin cos − sin sin 兲 / zP⬙ , and d4 = −sin sin sin + cos cos − yP⬙ 共cos sin sin + sin cos 兲 / zP⬙ . Noting that DF is an equal-phase line, we have
D = F = O + 2xF⬙ /p,
⬘ + b4yE⬘ 兲/共1 + a1xE⬘ + a2yE⬘ 兲 , = 共b3xE 0
冤冥冤 xF⬙ zF⬙
The line CE crosses R at D. By utilizing the equation of CE, i.e., 共x − xC兲 / 共xE − xC兲 = 共y − yC兲 / 共yE − yC兲 = 共z − zC兲 / 共zE − zC兲, the equation of R, i.e., z = 0, and Eq. (2), the coordinates of D, 共xD , yD , zD兲, are calculated with xD
共x⬙ − xP⬙ 兲 / 共x⬙D − xP⬙ 兲 = 共y⬙ − yP⬙ 兲 / 共y⬙D − yP⬙ 兲 = 共z⬙ − zP⬙ 兲 / 共z⬙D − zP⬙ 兲, the equation for G, i.e., z⬙ = 0, and Eq. (5), the coordinates of F are obtained with yF⬙ = 共yP⬙ z⬙D − y⬙DzP⬙ 兲/共z⬙D − zP⬙ 兲
⬘ xE
0
3589
共7兲
where O is the phase at O. Utilizing Eqs. (1), (3), (4), (6), and (7) yields
共i,j兲 = 共r + si + tj兲/共1 + ui + vj兲,
共8兲
where u = 共a1 + c1b1 + c2b3兲 / 关1 − 共 a1 + c1b1 + c2b3兲io − 共 a 2 + c 1b 2 + c 2b 4 兲 j o 兴 , v = 共a2 + c1b2 + c2b4兲 / 关1 − 共a1 + c1b1 + c2b3兲io − 共a2 + c1b2 + c2b4兲jo兴, r = 0 − 2 关 共 d1b1 + d2b3兲io + 共d2b4 + d1b2兲jo兴 / 兵p关1 − 共a1 + c1b1 + c2b3兲io − 共a2 + c1b2 + c2b4兲jo兴其, s = 关0共a1 + c1b1 + c2b3兲 + 2共d1b1 + d2b3兲 / p兴 / 关 1 − 共a1 + c1b1 + c2b3兲io − 共a2 + c 1b 2 + c 2b 4兲 j o 兴 t = 关 0 共 a 2 + c 1b 2 + c 2b 4 兲 + 2 共 d 2b 4 + d1b2兲 / p兴 / 关 1 − 共 a1 + c1b1 + c2b3兲io − 共a2 + c1b2 + c2b4兲 ⫻jo兴. Differing from the existing methods,5–8 Eq. (8) can exactly model the carrier phase map for an arbitrary system scheme. Derivation of Eq. (8) makes it possible in principle to determine the carrier phase distribution from the system parameters. However, the difficulty remains that these parameters are hard to know exactly. As an alternative approach, the coefficients of Eq. (8) can be estimated from the measured phase distribution of a standard plane (i.e., reference plane). For simplifying the computation, we recast Eq. (8) in the following linear form, r + si + tj − u共i,j兲i − v共i,j兲j = 共i,j兲,
共9兲
where the reference phase 共i , j兲 has been measured. When 共i , j兲 varies, a linear system based on Eq. (9) is established, from which r, s, t, u, and v can be solved. Using Eq. (9) involves a biased estimation, by which the expectations of the estimated coefficients in the presence of noise slightly deviate from their true values. Because the deviations are very small, the results are still satisfactory. If a higher accuracy is required, the nonlinear system based on Eq. (8) must be solved, and an iterative procedure has to be implemented.
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OPTICS LETTERS / Vol. 31, No. 24 / December 15, 2006
In experiment, a standard plane is used as a reference, on which the checkerboard pattern is printed for implementing the lateral calibration simultaneously. In the scheme, the optical axes of both projector and camera incline to the reference plane, a case that cannot be well handled by the existing techniques.1–8 The phase-shifting technique is used for phase recovery, and the sinusoid fringe patterns are projected with a DLP (digital light processing) projector, first on the reference plane and then on the tested object. The deformed patterns are captured by a CCD camera. Figure 2(a) shows a pattern on the reference plane, from which the oblique perspective of the reference plane induced by the incline of the camera is easily observed. Figure 2(b) shows the recovered phases. In the conventional techniques (see Ref. 9, for example) the carrier phases are usually removed by subtracting the measured reference phases from the object phases. However, such a method is inoperable here, because the phase values are not available for the pixels inside the black boxes or outside the reference plane region. By fitting Fig. 2(b) with the rational function, the carrier phase distribution can be reconstructed as shown in Fig. 2(c), where
the lost data in Fig. 2(b) have been padded by interpolation or extrapolation. [The maximum difference between the phase maps reconstructed with Eqs. (8) and (9) is 0.009 rad, which is neglectable]. Figure 2(d) shows the residual of (b) subtracting (c), which consists mainly of the noises and the local deformations of the reference plane. With the proposed technique, such error factors are effectively eliminated. As indicated in Ref. 5, inaccurate carrier phases will introduce additional bending (i.e., curvature errors) in the measurement results. This fact allows us to verify the validity of this technique by measuring a cylindrical object and then checking the resulting radius. For this purpose, a cookie jar is measured. In single-view measurements, we first recover the object phases from the captured fringe patterns [one of which is shown in Fig. 2(e)] and then subtract the carrier phases in Fig. 2(c) from the object phases. Finally, the 3D shape of each view is calculated by using the mapping relationship between the phases and depths.9 By the registration of six views,10 the 360° shape is obtained and shown in Fig. 2(f). Although additional errors may arise during the registration procedure, the final achieved accuracy depends mainly on single-view measurements. In Fig. 2(f), the least-squares radius of the cylinder is calculated as 95.335 mm. As a comparison, the object is also measured by using a coordinate measuring machine, and the least-squares radius is obtained as 95.294 mm. From the results, the accuracy of this technique is evident. In conclusion, we have proposed a least-squares fitting technique for determining the carrier phase component. This technique offers some important advantages over others. First, the model perfectly describes the carrier phase distribution for an arbitrary measurement system. Second, it allows for exactly determining the carrier phase map in the absence of system parameters. Third, the influence of noise and local deformations of the reference plane on the measurement can be effectively restrained. This work was sponsored by the National Natural Science Foundation of China (60678036), and in part by Shanghai Leading Academic Discipline Project (Y0102). H. Guo’s e-mail address is
[email protected]. References
Fig. 2. (Color online) Experimental results. (a) Fringe pattern on the reference plane. (b) Phase map (in radians) reconstructed from (a). (c) Least-squares fitting result (in radians) of (b) using the rational function. (d) Residual (in radians) obtained by subtracting (c) from (b). (e) Deformed fringe pattern on the object. (f) Reconstructed 3D shape.
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