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Calibration of a fringe projection profilometry system using virtual phase calibrating model planes

This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2005 J. Opt. A: Pure Appl. Opt. 7 192 (http://iopscience.iop.org/1464-4258/7/4/007) View the table of contents for this issue, or go to the journal homepage for more

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INSTITUTE OF PHYSICS PUBLISHING

JOURNAL OF OPTICS A: PURE AND APPLIED OPTICS

J. Opt. A: Pure Appl. Opt. 7 (2005) 192–197

doi:10.1088/1464-4258/7/4/007

Calibration of a fringe projection profilometry system using virtual phase calibrating model planes Zhang Xiaoling, Lin Yuchi, Zhao Meirong, Niu Xiaobing and Huang Yinguo State Key Laboratory of Precision Measuring Technology and Instruments, College of Precision Instrument and Opto-Electronics Engineering, Tianjin University, Tianjin 300072, People’s Republic of China E-mail: shanxi [email protected] (X Zhang)

Received 6 September 2004, accepted for publication 25 January 2005 Published 9 March 2005 Online at stacks.iop.org/JOptA/7/192 Abstract A novel virtual phase calibrating model plane method for coordinate calibration in fringe projection profilometry is presented in this paper. Plane (X -axis and Y -axis) calibration and height (Z -axis) calibration are studied respectively. The virtual phase calibrating model plane method is designed to calibrate the coordinates. Experimental results show that the proposed calibration procedure using the virtual phase calibrating model plane method is more efficient in that it needs only to grab and process one series of images of the model plane, whereas the traditional calibration method requires at least two series of images. Keywords: 3D profile measurements, fringe projection profilometry, coordinate calibration, virtual phase calibrating model plane, image series, spatial phase unwrapping

(Some figures in this article are in colour only in the electronic version)

1. Introduction In fringe projection profilometry, the method based on phase shifting profilometry (PSP) [1, 2] can obtain the measured object’s 3D profile from the unwrapped phase of every pixel in measured object. The unwrapped phase is obtained by projecting a sine fringe to the measured object, grabbing the modulated fringe images by the measured object, processing the grabbing images to get the wrapped phase, and demodulating the wrapped phase using spatial phase unwrapping algorithms [3–5]. In the processing period from the unwrapped phase of the measured object to the 3D profile of the measured object, the measurement system must be calibrated correctly; this involves the height calibration (Z axis) and the plane calibration (X-axis and Y -axis). Many people have done research on these calibrations [6– 10] and presented many corresponding solutions, such as Srinivasan [11], Zhou and Su [12], Asundi [13], Xu [14], Hung [15], Sitnik [16], and Liu et al [17] on the height 1464-4258/05/040192+06$30.00 © 2005 IOP Publishing Ltd

calibration, and Li and Su [18], Sitnik [16], Lilley [19], and Huntley [20–23] on the X-axis and Y -axis calibration. Though all of these practical solutions are more accurate than before, there are many defects and limitations that must be modified and improved. All these plane calibration methods need to collect at least two series of images. One is a pure fringe image series of the model plane produced by projecting a sine fringe to the model plane, which provides the phase information of the model plane at a different location; the other is an image series of the model plane exposed to white light, which provides the relationship between the image coordinate of feature points and their 3D coordinates. A lot of time will be wasted on grabbing and processing the mass data of the two image series. In this paper, we present a novel virtual phase calibration model plane method, which only grabs and processes one image series, i.e. a pure fringe image series of the model plane, to obtain the coordinates of the X-axis, Y -axis and Z -axis. Section 2.1 describes how to extract the feature information of the virtual phase calibrating model plane by using phase shifting

Printed in the UK

192

Calibration of a fringe projection profilometry system using virtual phase calibrating model planes X Digital light projector(DLP)

Z

START Split screen

Place model plane at Z1=0

Obtain unwrapped phase φK at Zk

Grab modulated images series

Display card

Extract feature information

Image gathering card

k=k+1

CCD

computer display

Y

Calibrate the Xaxis and Y-axis

Model plane

Move the model plane to Zk

Figure 1. Measurement system.

technology. Section 2.2 discusses the relationship between the image coordinate and the object coordinate by using the least-square map function method [24, 25]. In section 3, we consider in detail the height calibration using piecewise linear interpolation [17]. Section 4 gives some experimental results.

2. The X -axis direction and Y -axis direction calibration The measuring system is shown in figure 1. The model plane is overlapped by a standard white plane with a set of identical black circles (radius, r = 10 mm; distance between the centre of circles, d = 40 mm) and located on a special linear translation stage. The special linear translation stage can move along the Z -axis, and the model plane’s normal is parallel to the Z -axis. The detailed calibration scheme is shown in figure 2. At the first step, the model plane is placed at height Z 1 = 0, where it represents the XY plane in the space coordinates (X, Y, Z = 0). At the second step, we grab a sequence of nine fringecontoured π/4 phase-stepped images of the model plane. At the third step, by processing the grabbed images, the wrapped phases of the model plane are obtained, and then the unwrapped phase φ1 is calculated from sine patterns and grey codes for all pixels. At the fourth step, we accumulate the front eight frames of the nine frames of fringe images to get a mean grey-scale image. Then the positions of the black circles are determined. The centre coordinates of the black circles and phase φ1 of the model plane are regarded as the feature information of the model plane. Based on this feature information, the coordinates (x, y) for each pixel (i, j ) can be obtained. By now, we have finished the calibration of the X-axis and Y -axis at height Z 1 = 0. At the next step, we move the calibration plane to a new height Z k (k = 1, 2, 3, . . . , K ), and repeat the steps mentioned above. Thus we accomplish the calibration of X-axis and Y -axis at the height Z k (k = 1, 2, 3, . . . , K ), and obtain the unwrapped phase φk (k = 1, 2, . . . , K ). Finally, with the method of piecewise linear interpolation, the Z -axis calibration is obtained. 2.1. Extracting feature information of the model plane When the model plane is moved to height Z k (k = 1, 2, 3, . . . , K ) along the positive direction of the Z -axis, the light intensity Ik of the modulated image series is Ik (x, y) = rk (x, y)[Ak (x, y) + Bk (x, y) cos φk (x, y)], (1) where rk (x, y) is the reflectivity of point (x, y) in the model plane at height Z k , Ak (x, y) is the light intensity of the

END

YES

Process the next height Zk? Calibrate the Zaxis

Figure 2. Block scheme of the calibration process.

background, Bk (x, y) is the amplitude of sine fringe, and φk (x, y) is the wrapped phase. Using phase-shift technology of N + 1(N = 8) steps [7], namely the phase of sine fringe projected to the model plane shifts N + 1 times with π/4 phase step at height Z k (k = 1, 2, 3, . . . , K ), we can obtain K groups of fringe image series, and each group contains nine frames of fringe images. The light intensity Ik(n+1) of each frame image is
Ik(n+1) (x, y) = rk(n+1) (x, y) Ak(n+1) (x, y) + Bk(n+1) (x, y)   2πn × cos φk(n+1) (x, y) + −π , (2) N where rk(n+1) is the reflectivity of point (x, y) at each frame image at height Z k , Ak(n+1) the light intensity of background at each frame image, Bk(n+1) is the amplitude of sine fringe, φk(n+1) is the wrapped phase at each frame image, n = 0, 1, . . . , N (N = 8), and N is number of times of phase shifting. Then by accumulating and averaging the front eight frames of fringe images of each group of fringe image series, the mean light intensity Iksum is written as 7 Iksum =

n=0 Ik(n+1)

8

= rk(n+1) Ak(n+1) .

(3)

Comparing equation (1) with (3), we can see that the phase disturbance of the sine fringe is eliminated. So, when eight fringe images are accumulated and averaged, a new mean grey-scale image of no fringe information can be obtained. Figure 3(a) gives one frame of fringe images of the model plane, and figure 3(b) gives the mean grey-scale image. We can see that the fringes are eliminated in figure 3(b). Because it does not contain fringes, the mean grey-scale image does not contain modulated phase information. So when processing it to get the feature information of the black circles, we need not unwrap the modulated phase information, whereas the unwrapped phase is a puzzling subject with mass and complex computation. Thus it becomes simple and fast to get the feature information of the model plane. After we process the mean grey-scale image with thresholding and morphological technology, we can get the marginal points of these black circles with a circle radial scanning method, and 193

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(b)

(a)

Figure 3. Fringe pattern on calibrating model plane and corresponding mean grey-scale image. (a) One frame of fringe images. (b) Mean grey-scale image.

get the centre coordinates (i k , jk ) of these black circles with a least-square fitting method. Hence, we obtain the feature information of the model plane. With a least-square map function method, the map relationship between the object coordinate system and the image coordinate system can be obtained; this will be described in the next section. 2.2. The map relationship to implement the X-axis and Y -axis calibration The origin point (0, 0, 0) in the object coordinate system is set at the centre point of the left upper black circle, and this origin point remains fixed during the measurement. The map relationship between the object coordinate system and the image coordinate system can be obtained and written as Re al X k (i, j ) =

N N −n  

cmn i km jkn ,

(4)

dmn i km jkn ,

(5)

n=0 m=0

Re alYk (i, j ) =

N N −n   n=0 m=0

where Re al X k (i, j ) is the X coordinate of every pixel point (i k , jk ) at height Z k , Re alYk (i, j ) is the Y coordinate of the point, N is the number of pixels of the model plane, and cmn and dmn are coefficients which can be obtained by estimating the minimum error sum squares of the black circles’ centres. Because the radii of these black circles and the distances between the centres of circles in the model plane are known, the centres’ coordinates (i ok , jok ) and the 3D coordinates (xok , yok , z ok ) of every black circle can be obtained. We introduce the centres’ coordinates (i ok , jok ) into equations (4) and (5): Re al X ok (i ok , jok ) =

N N −n  

m n cmn i ok jok ,

(6)

n=0 m=0

Re alYok (i ok , jk ) =

N N −n  

m n dmn i ok jok ,

(7)

n=0 m=0

where Re al X ok (i, j ) is the X coordinate of every centre (i ok , jok ) of black circles at height Z k , and Re alYok (i, j ) is the Y coordinate of the point. The error square sum of equations (6) and (7) is respectively Error X o =

total  p=1

194

(Re al X ok − xok )2 ,

(8)

ErrorYo =

total 

(Re alYok − yok )2 ,

(9)

p=1

where Error X o is the error of the X coordinate of these black circles’ centre, ErrorYo is the error of the Y coordinate, and total is the number of black circles in the model plane. Concretely, when the partial differentiation of error square sum (equations (8) and (9)) is zero, we use simultaneous equations (equations (10) and (11)) to obtain the coefficients cmn and dmn . ∂Error X o = 0. ∂cmn

(10)

∂ErrorYo = 0. ∂dmn

(11)

When the coefficients cmn and dmn are obtained, the X-axis and Y -axis coordinate (xk , yk ) of every pixel point in the model plane at height Z k can be obtained by introducing their pixel coordinates (i k , jk ) into equations (4) and (5). This means that the calibration of the X-axis and Y -axis is realized.

3. The Z -axis (height) calibration The paper adopts a piecewise linear interpolation method to realize the Z -axis calibration. At first, the phases φk (k = 1, 2, . . . , K ) at height Z k (k = 1, 2, 3, . . . , K ) are regarded as reference phases. Based on these reference phases and their heights Z k , every pixel’s height (Z -axis coordinate) can be obtained by linear interpolation of φk and Z k . And then we analyse the measurement accuracy. If the precision does not meet the measurement requirements, additional fringe images of the model plane are grabbed at an intermediate location, such as at Z k−12+Z k . Then the method of Z -axis calibration mentioned above is repeated, until the measurement accuracy is satisfied. The details are described in the following. In the detection range of the fringe projection system, the height of the measured object is encoded in the phases φk and the heights Z k ; an abridged general view is shown in figure 4(a). E p is the centre of the exit pupil of the projection lens and E c is the centre of the entrance pupil of the imaging lens. The plane Z 1 is regarded as the zero reference plane, and plane Z K is the highest calibration plane. The distance between these two planes is H ; the measured object lies in this range. Point P in the image plane corresponds to point Ak (k = 1, 2, . . . , K ) in

Calibration of a fringe projection profilometry system using virtual phase calibrating model planes

Ep

Projector

Y

CCD

Ec

z1

J zK-1

zK Z

AK H

…….

A2

Plane Zk

A1

Plane Z2

E

AK-1 AK

P(i,j) I

Plane Z1

A1

OP

O

X (a)

(b)

Figure 4. Relationship between heights and phases. (a) An abridged general view. (b) A detailed view. Y

(b)

worktable

Length (pixel)

Width (pixel)

Width (pixel)

X

(a)

worktable

Length (pixel)

Figure 5. Image of the measured plane at different heights Z . (a) At Z k = 60 mm. (b) At Z k = 100 mm.

plane Z k (k = 1, 2, . . . , K ). The phase of Ak is φk , and the phases φk are not equal to each other. For analysing in detail the relationship between the height and phase of points in the model plane shown in figure 4(a), the object coordinate system O XY Z and the image coordinate system O P I J are aligned as shown in figure 4(b). When the model plane is overlapping at the initial height Z 1 = 0, the unwrapped phase φ(i, j, 1) is obtained by spatial phase unwrapping algorithms, where (i, j ) is the pixel coordinate in the image coordinate system. In turn, we move the model plane equidistantly to the several heights Z k = (k − 1)Z , where Z is the movement interval, and the unwrapped phases φ(i, j, k) of model plane are obtained. This means that each pixel point (i, j ) in the image coordinate system at height Z k corresponds to phase series φ(i, j, k). Namely, when the height of the measured object lies in the range between Z 1 and Z K = (K − 1)Z , the unwrapped phase is definitely among φ(i, j, 1) and φ(i, j, K ). The phases φ(i, j, k) of the model plane at different heights are regarded as reference phases. Using these reference phases, the unwrapped phase φ M (m, n) of the each point M(m, n) in the image coordinate system of the measured object is obtained. Then, using piecewise linear interpolation, the height Z M (m, n) of the point M(m, n) is obtained: Zb − Za Z M (m, n) = Z a + φ(m, n, b) − φ(m, n, a) (12) × [φ M (m, n) − φ(m, n, a)], where φ M (m, n) requires φ(m, n, a)
φ M (m, n)
φ(m, n, b),

(13)

and a, b ∈ [1, K ], Z a and Z b are the heights of φ(m, n, a) and φ(m, n, b) respectively.

Figure 6. Error without calibration.

Using equations (12) and (13), we can calculate the height of the measured object. After confirming the phase of the model plane, we can use a look-up table to reduce the quantity of mathematical operations, from the four additions, single multiplication and single division found in equation (12). This then simplifies the computations required to two additions and a single multiplication, which improves the processing speed considerably.

4. Results and discussion 4.1. The experimental results of the X-axis and Y -axis calibration A measured plane, with dimensions length × width = 100 mm × 130 mm is placed vertically on the worktable. Figure 5 gives the image of the measured plane at different heights Z k . Figure 5(a) shows the plane at height Z k = 60 mm, and figure 5(b) shows the plane at height Z k = 100 mm. We can see that the length and width in the first image are about 204 and 265 pixels respectively, whereas the length and width in the second image are about 212 and 275 pixels. This means that a large error will be introduced if X-axis and Y -axis calibrations are not done. The error is shown in figure 6. When we adopt the virtual phase calibrating model plane method to calibrate the X-axis and Y -axis, the X coordinate and Y coordinate of every pixel of the measured plane is obtained. Thus the length and width of the measured plane 195

X Zhang et al

Table 1. The result of the measured plane after X Y correction (unit: mm). Height Z k of measured plane

Parameter

60

X -axis coordinate Y -axis coordinate

100

X -axis coordinate Y -axis coordinate

Feature of the measured plane

Corrected feature position

Measured parameter after XY correction

Up edge Worktable Left edge Right edge Up edge Worktable Left edge Right edge

69.5 200.3 131.2 231.4 69.7 200.4 131.1 231.5

130.8 100.2 130.7 100.4

Table 2. The height with 11 reference planes (unit: mm). Location

Ideal value (z)

Measured value (mz)

1 2 3

5.500 80.301 140.500

5.407 80.265 140.624

Figure 7. Error after correction.

can be obtained, as shown in table 1. We can see that the error of the length and width between these two images at Z k = 60 and 100 mm is 0.2 and 0.1 mm respectively, which is very small. Figure 7 gives the length error and width error of the measured plane at different height Z k after calibration. We can see the error is small and flat. 4.2. The experimental results of the Z -axis calibration Within the detection range of height, about 150 mm, we grab the fringe images of the model plane at 11 heights, which are Z = 0.001, 15.001, 30.001, 44.998, 60.000, 75.001, 90.000, 105.001, 120.001, 135.002 and 150.004 mm. And through getting the wrapped phase from the fringe images, unwrapping phase and mapping phase, the unwrapped phase series of the whole height are obtained. And then through the piecewise linear interpolation, the height of the measured object is obtained. To verify the error of the Z -axis direction calibration, we regard the model plane at three heights, Z = 5.500, 80.301 and 140.500 mm as measured planes; there are several solutions for doing this, as presented below. 196

Error (z = mz − z) −0.093 −0.036 0.124

Standard deviation 0.008 0.006 0.04

Figure 8. Error when using different reference planes.

4.2.1. Case 1. When all the reference planes (11 planes) participate in the linear interpolation, the heights of the measured planes are obtained as shown in table 2. 4.2.2. Case 2. When reference planes which are selected every two planes from the 11 planes participate in the linear interpolation, the heights of the measured planes are obtained as shown in table 3. 4.2.3. Case 3. When only two reference planes, Z 1 = 0.001 mm and Z k = 150.004 mm, participate in the linear interpolation, the heights of measured planes are obtained as shown in table 4. To analyse conveniently, we give the error scheme, shown in figure 8, when using different reference planes from the experimental results in tables 2, 3 and 4. From above experimental results, it can be seen the Z -axis do well and the reference planes participated in the interpolation distribute more uniform in the whole measurement field, the results of interpolation are more perfect.

Calibration of a fringe projection profilometry system using virtual phase calibrating model planes

Table 3. The height with six reference planes (unit: mm) Location

Ideal value (z)

Measured value (mz)

1 2 3

5.500 80.301 140.500

5.336 79.976 140.326

Error (z = mz − z) −0.164 −0.325 −0.174

Standard deviation 0.007 0.007 0.015

Table 4. The height with only two reference planes (unit: mm). Location

Ideal value (z)

Measured value (mz)

1 2 3

5.500 80.301 140.500

4.568 73.242 138.796

5. Conclusions A virtual phase calibrating model plane method for coordinate calibration in fringe projection profilometry is developed. The method only grabs one image series of the model plane to extract the feature information of the model plane, which is the phase information and the centre coordinate of the black circles of model plane. The method uses a least-square map function method to do the X-axis and Y -axis calibration, and piecewise linear interpolation to do the Z -axis calibration. Experimental results are given, which prove that the proposed method can realize plane calibration and height calibration economically. Compared with the traditional ways of calibration, the virtual phase calibrating model plane method reduces the hardware requirement of the system, and simplifies the process of grabbing and processing data.

Acknowledgments The work was supported by the Ministry of Education of the Chinese government (No. 99140).

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Error (z = mz − z) −0.932 −7.059 −1.704

Standard deviation 0.01 0.07 0.02

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