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Fourier transform profilometry based on a fringe pattern with two frequency components
ARTICLE IN PRESS
Optik
Optics
Optik 119 (2008) 57–62 www.elsevier.de/ijleo
Fourier transform profilometry based on a fringe pattern with two frequency components Wenjing Chen, Xianyu Su, Yipping Cao, Liqun Xiang, Qican Zhang Department of Opto-electronics, Sichuan University, Chengdu 610064, China Received 13 March 2006; accepted 12 May 2006
Abstract A modified Fourier transform profilometry (FTP) based on a fringe pattern with two frequency components is proposed, which provides a larger measuring range than that of the traditional FTP. We analyze the maximum measuring range and give an expression to describe the measurable slope of the height variation limitation of this method. The modified FTP provides us another approach to eliminate frequency overlapping. When the spectra distribution of a measured object is not spherical symmetry, we can avoid the frequency aliasing through projecting a fringe pattern with two frequency components, instead of increasing the density of the projected fringe and the resolution of a CCD camera. The theoretical analysis and primary experiments verified the method. r 2006 Elsevier GmbH. All rights reserved. Keywords: Fourier transform profilometry; Frequency aliasing; Three-dimensional shape measurement
1. Introduction The non-contract automatic three dimensional (3-D) shape measurement is very important in machine vision, solid modeling, industrial auto-measuring, etc. Fourier transform profilometry (FTP) proposed by Takeda et al. [1] is a popular 3-D sensing method, in which, a grating pattern is projected onto an object and a deformed fringe captured by a CCD camera is Fourier transformed and processed in its spatial frequency domain as well as in its space-signal domain. FTP has the following merits, only one (or two) fringe(s) needed, full field analysis, and high precision, so it has been extensively studied [2–8]. A sinusoidal fringe projecting combing with p phase shifting technique [4] can extend the measurable slope of height variation to nearly three times that of the unimproved FTP. Two-dimensional Corresponding author.
E-mail address: [email protected] (W. Chen). 0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.05.024
Fourier transform and frequency-weighted filtering are applied to provide a better separation of the useful information from noise when speckle-like structures and discontinuities exist in the fringe pattern [5]. The frequency-multiplex technology [6] permits the 3-D shape measurement of objects that have discontinues height step and /or spatially isolated surfaces. The phase error caused by sampling in FTP is discussed in detail [7]. Ref. [3] reviewed some of the developments in FTP over the past years. But up to now, the analysis about the maximum measuring range of FTP fits for only single frequency component fringe patterns. Although an oblique fringe pattern with two frequency components on the x and y directions is usually employed in FTP [9], the limitation of the measuring range based on this kind of fringe pattern projection has not been analyzed. In this paper, we analyze the maximum measuring range of FTP based on a fringe pattern with two frequency components, and give an expression
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describing the maximum range of the measurement. This kind of fringe pattern can easily obtain through programmable digital light projector (DLP) or changing the grating lines direction by rotating a single frequency component grating. When this kind of fringe pattern is projected onto the object, the location of the fundamental spectrum will move in the frequency domain. If there is frequency aliasing in the fx direction or fy direction, maybe there isn’t frequency aliasing after spectrum moving. If the spectra distribution of the measured object is not spherical symmetry, we can avoid the frequency aliasing through projecting a fringe pattern with two frequency components, instead of increasing the density of the projected fringe and the resolution of a CCD camera. The theoretical analysis and primary experiments verified the method.
The general crossed-optical-axis geometry of FTP is shown in Fig. 1, in which the optical axes P1P2 of a projector lens crosses that of a camera lens I1I2 at point O on a plane R, which is a fictitious plane normal to I1I2 and serves as a reference plane. d is the distance between P2 and I2, and L0 is the distance between I2 and plane R, A, C express points on R, D expresses a tested point on the measured object. Grating G has its lines normal to the plane of the figure. The grating image projected onto the object surface and observed through a CCD camera is a regular grating pattern. When a measured object is put on plane R and a sinusoidal grating pattern with one frequency component f0 in the x direction is projected onto it, the deformed fringe pattern observed through a CCD camera can be expressed by (1a)
On reference plane R ðhðx; yÞ ¼ 0Þ, the fringe pattern is still deformed, which can be expressed by g0 ðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cosð2pf 0 x þ f0 ðx; yÞÞ. Projector
CCD
Grating
(1b)
Camera
I1
P1
d
I2
P2
L0
D
Object h
C
A
x O
Fig. 1. The optical geometry.
Q*
Q f1max
fb −f0
0
f1min f0
x
Fig. 2. The spectrum distribution schematics.
In Eqs. (1a) and (1b), a(x, y) is the background intensity, b(x, y)is the non-uniform distributions of reflectivity on the diffuse object, f(x, y) is the phase modulations resulting from the object height distribution. f0(x, y) is the phase distribution for hðx; yÞ ¼ 0. Assuming qðx; yÞ ¼ 12bðx; yÞ expðifðx; yÞÞ, The Fourier spectrum of Eq. (1a) can be expressed as Gðz; ZÞ ¼ Aðz; ZÞ þ Qðz f 0 ; ZÞ þ Q ðz þ f 0 ; ZÞ.
2. Principle theory of the traditional FTP
gðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cosð2pf 0 x þ fðx; yÞÞ.
A
(2)
QðB; ZÞ is the spectrum of q(x, y). The spectrum distribution scheme on the x direction is shown in Fig. 2, where fb is the maximum of the zero order spectrum. f1max and f1min are the maximum and the minimum of the fundamental spectrum components, respectively. An instantaneous fundamental frequency f1 on the x direction is defined by [1] 1 qfðx; yÞ . (3) 2p qx In order to restore the object surface correctly, the fundamental component must be separated from all other spectra, that is
f1 ¼ f0 þ
f 1 min 4f b . By substituting Eq. (3) into Eq. (4), we obtain 1 qfðx; yÞ f0 4f b . 2p qx max
(4)
(5)
The phase variation caused by the height modulation must be limited in qfðx; yÞ (6) qx o2pðf 0 f b Þ, max then the measurable slope of the height variation extends the limitation, the fundamental components overlap the zero component. Aliasing will influence the precision of FTP or prevent obtaining a correct reconstruction. If jqfðx; yÞ=qxjmax satisfies Eq. (6), one of the 71 order fundamental components, for example QðB f 0 ; ZÞ, is filtered out through a suitable filter. The inverse Fourier transform is applied to the fundamental component, we obtain
R
^ yÞ ¼ gðx;
1 bðx; yÞ expði2pf 0 x þ ifðx; yÞÞ. 2
(7a)
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59
We do the same operation for Eq. (1b) to obtain
y
1 (7b) g^ 0 ðx; yÞ ¼ bðx; yÞ expði2pf 0 x þ if0 ðx; yÞÞ, 2 Dfðx; yÞ can be generated by Eqs. (7a) and (7b) [3]. Assuming the measuring system is perfect and the system parameters L0 and d have no error, the conversion formula between the phase and the height is [3]: L0 Dfðx; yÞ . hðx; yÞ ¼ Dfðx; yÞ 2pf 0 d
X
Y
fxy
f0 fx fb
θ
f0
(a)
(8)
y Y
3. Analysis on FTP with two frequency components grating pattern
fb
ð9bÞ
where fðx; yÞ is the phase modulation caused by the height distribution, Eq. (9b) can be rewritten as gðx; yÞ ¼ aðx; yÞ þ bðx; yÞ½expði2pðf 0 cosðyÞx þ f 0 sinðyÞyÞ þ ifðx; yÞÞ þ expði2pðf 0 cosðyÞx þ f 0 sinðyÞyÞ ifðx; yÞÞ=2.
ð10Þ
Let q ¼ 12 bðx; yÞ expðifðx; yÞÞ, the Fourier spectra of Eq. (10) are: Gðz; ZÞ ¼ Aðz; ZÞ þ Qðz f 0 cosðyÞ; Z f 0 sinðyÞÞ þ Q ðz þ f 0 cosðyÞ; Z þ f 0 sinðyÞÞ.
ð11Þ
QðB; ZÞ is the spectrum of q(x, y). Eq. (11) shows that the fundamental spectrum shape of the fringe pattern with two frequency component is as same as that of the fringe pattern with only one frequency component, but its center is put at point ðf 0 cosðyÞ; f 0 sinðyÞÞ. We define an instantaneous synthetical frequency fX on the X direction: fX ¼ f0 þ
1 qfðx; yÞ . 2p qX
x f0
(9a)
where fx is the frequency component on the x direction and fy is that on the y direction. The synthetical frequency vector fxy is on the X direction, which is normal to the grating lines, and its value is f0. So fx ¼ f0 cos(y) and fy ¼ f0 sin(y). y is the angle between the normal of the grating lines and the x-axis. The scheme is shown in Fig. 3(a). Projecting this kind of grating pattern onto the object, the deformed fringe pattern captured by the CCD camera is written by gðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cosð2pðf 0 cosðyÞx þ f 0 sinðyÞyÞ þ fðx; yÞÞ,
X
f0
In FTP with two frequency components, the grating pattern can be written as gðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cosð2pðf x x þ f y yÞÞ,
x
fy
(12)
(b) Fig. 3. (a) A schematics of coordination conversion. (b) A schematics of eliminating frequency by a fringe pattern with two frequency component.
The (X, Y) coordinates are obtained by rotating the (x, y) coordinates by an angle y, as shown in Fig. 3(a). The relationship between the (X, Y) coordinates and the (x, y) coordinates is " # " #" # x cos y sin y X ¼ , sin y cos y y Y " # " #" # x cos y sin y X ¼ . ð13Þ y sin y cos y Y So the following equation must be satisfied for avoiding frequency overlapping: 1 qfðx; yÞ 4f b . (14) f0 2p qX max Employing the partial differential equation: qfðx; yÞ qfðx; yÞ qx qfðx; yÞ qy ¼ þ , qX qx qX qy qY qfðx; yÞ qfðx; yÞ qfðx; yÞ ¼ cos y þ sin y. qX qx qy Eq. (14) can be rewritten as
1 qfðx; yÞ 2p qX ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2 2 1 qfðx; yÞ qfðx; yÞ ¼ f0 cos y þ sin y 4f b , 2p qx qy
f0
ð15Þ
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that is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 qfðx; yÞ qfðx; yÞ cos y þ sin y o2pðf 0 f b Þ, qx qy (16)
(17)
when y ¼ 01, Eq. (15) is simplified as 1 qfðx; yÞ f0 4f b . 2p qx max
(18)
It is obvious that Eq. (5) is a special case of Eq. (16). In Eq. (16), if ðqfðx; yÞ=qxÞ4ðqfðx; yÞ=qyÞ,
15 Height mm
when y ¼ 901, Eq. (15) is simplified as 1 qfðx; yÞ 4f b , f0 2p qy max
10 5 0 -5 300
-10 -15 300 250 200 150 100 Pixel x (a)
50
0 0
200 100 y el Pix
(a)
150
Pixel y
140 130 120
(b)
110 100 100
150
110
(b)
120
130 Pixel x
140
150
160
130 Height mm
15
120 110 100 100
(c)
110
120
130 140 Pixel x
150
160
Height mm
5 0 -5 300 0 0
200 100 ly Pixe
0.5
40 30 20 10 0
-10 300 250 200 150 100 (d) Pixel x
10
-10 -15 300 250 200 150 100 50 Pixel x (c)
50
0 0
300 250 200 150 y 100 ixel P 50
Fig. 4. FTP simulation based on a fringe pattern with single frequency component. (a) The simulated object. (b) The Deformed fringe pattern. (c) The contour distribution of frequency spectrum. (d) The restored object.
Height mm
Pixel y
140
0
-0.5 300
(d)
200 Pixel 100 x
0 0
200 100 ly Pixe
300
Fig. 5. FTP simulation based on a fringe pattern with two frequency components. (a) The deformed fringe pattern. (b) The contour distribution of frequency spectrum. (c) The restored object. (d) The error distribution.
ARTICLE IN PRESS W. Chen et al. / Optik 119 (2008) 57–62
then sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 qfðx; yÞ qfðx; yÞ qfðx; yÞ . cos y þ sin y o qx qy qx
4. Computer simulations and primary experiments Computer simulations verify the above analysis. The simulated object is shown in Fig. 4(a), the image size is 256 256 pixels. It has steep height variation on the x direction. When projecting sinusoidal fringe with 14 periods (26/1.41 pixels per period) onto the object, the deformed fringe pattern captured by a CCD camera has local shadow and under-sampling, as shown in Fig. 4(b). Its Fourier spectra are not symmetrical. On the x-axis in the frequency domain, the foundational spectrum overlaps the zero spectrum, Fig. 4(c) is the enlarged contour map of the spectrum distribution. The prefect foundational spectrum cannot be filtered out, so correct restored surface cannot be obtained, as shown in Fig. 4(d). When projecting a fringe pattern with two frequency components onto the object, Fig. 5(a) shows the deformed fringe pattern, which still has 14 periods (26 pixels per period) in the direction normal to grating
It shows that if the 71 order fundamental spectra are not spherical symmetry, the frequency aliasing can be avoided through projecting an oblique fringe with two frequency components as shown in Fig. 3(b). Monitor Object
290mm
790mm
Digital Camera
Frame memory
DLP projector
61
IBM_PC
Printer
Fig. 6. The experimental block diagram.
320 300 Pixel y
280 260 240 220 200
(b)
Height mm
(a)
20 10 0 -10 600 500 400 300 Pi xe 200 l y 100
(c)
500 300 400 x 200 l e 0 0 100 Pix
600
330 340 350 360 370 380 390 400 410 420 Pixel x
700
(d)
320 300 15 10 5 0 -5 600 500 400 300 200 100
Height mm
Pixel y
280 260 240
Pi
220
(e)
ly
xe
200
340
360
380 Pixel x
400
420
(f)
0
700 500 600 300 400 100 200 x Pixel
Fig. 7. (a) The deformed fringe with single frequency component. (b) The enlarged spectrum contour maps of (a). (c) The restored shape from (a). (d) The deformed fringe with two frequency components. (e) The enlarged spectrum contour maps of (d). (f) The restored shape from (d).
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lines. The shape of the foundational spectrum is not changed, but the position of the spectra move as an angle. As shown in Fig. 5(b), the foundational components separate from the zero component. Filtering out the foundational component and calculating its inverse Fourier transform, we can restore the surface of the measured object, as shown in Fig. 5(c). The difference between the original object and the restored surface is shown as Fig. 5(d). Some experiences are applied in our demonstration. The experimental block diagram is shown in Fig. 6, a TV camera (TM1881) with a 16 mm focal length lens, via a special video-frame grabber DT3152 digitizing the image in 576 768 picture elements. The system parameters are L0 ¼ 780 mm and d ¼ 290 mm. We measure an object with two heart-shape parts. First, we project a sinusoidal fringe pattern with one frequency component onto the object. The deformed fringe pattern captured by the CCD is cut to form a image with 512 680 pixels, as shown in Fig. 7(a). Fig. 7(b) shows the enlarged contour map of the Frequency spectra, in which the foundational spectrum overlaps the zero spectrum. So we cannot restore the correct shape of the measured object from the fundamental component, as shown in Fig. 7(c). Then the grating is rotated by an angle y, an oblique fringe pattern with two frequency components is projected onto the measured object and captured by the same camera, as shown in Fig. 7(d). The enlarged contour map of its Fourier spectra is shown in Fig. 7(e). It shows that the fundamental spectra separate from the zero component by moving a certain angle. The correct restored surface is as shown in Fig. 7(f).
5. Conclusions We analyze the maximum measuring range of FTP based on a fringe pattern with two frequency components and give an expression describing measuring range. The formula of maximum range of measurement of FTP given by Takeda is a special case of our expression. It shows that the position of the fundamental components of the deformed fringe pattern can be moved by rotating the direction of the grating lines. If
the fundamental spectra of the measured object are nonspherical symmetry, it gives us another approach to eliminate frequency overlapping, instead of just changing system parameters L0 and d or increasing the density of the projected fringe and the resolution of a CCD camera.
Acknowledgement This project was supported by National Natural Science Foundation of China.
References [1] M. Takeda, H. Ina, S. Koboyashi, Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry, J. Opt. Soc. Am. 72 (1) (1982) l56–l60. [2] M. Takeda, K. Motoh, Fourier transform profilometry for the automatic measurement of 3-D object shapes, Appl. Opt. 22 (24) (1983) 3977–3982. [3] X.-L. Su, W. Chen, Fourier transform profilometry: a review, Opt. Lasers Eng. 35 (5) (2001) 263–284. [4] J Li, X.-Y. Su, L.-R. Gou, An improved Fourier transform profilometry for automatic measurement of 3-D object shapes, Opt. Eng. 29 (12) (1990) 1439–1444. [5] J.F. Lin, X.-Y. Su, Two-dimensional Fourier transform profilometry for the automatic measurement of threedimensional object shapes, Opt. Eng. 34 (11) (1995) 3297–3302. [6] M. Takeda, Q. Gu, M. Kinoshita, et al., Frequencymultiplex Fourier-transform profilometry: A single-shot three-dimensional shape measurement of objects with large height discontinuities and/or surface isolations, Appl. Opt. 36 (22) (1997) 5347–5354. [7] W. Chen, Y. Hu, X. Su, Error caused by sampling in Fourier transform profilometry, Opt. Eng. 38 (6) (1999) 927–931. [8] J.L. Li, X. Su, H.J. Su, Removal of carrier frequency in phase-shifting techniques, Opt. Laser Eng. 30 (1) (1998) 107–115. [9] M.A. Herra´ez, D.R. Button, M.J. Lalor, Accelerating fast Fourier transform and filtering operations in Fourier fringe analysis for accurate measurement of three-dimensional surface, Opt. Laser Eng. 31 (1) (1999) 135–145.
Optik
Optics
Optik 119 (2008) 57–62 www.elsevier.de/ijleo
Fourier transform profilometry based on a fringe pattern with two frequency components Wenjing Chen, Xianyu Su, Yipping Cao, Liqun Xiang, Qican Zhang Department of Opto-electronics, Sichuan University, Chengdu 610064, China Received 13 March 2006; accepted 12 May 2006
Abstract A modified Fourier transform profilometry (FTP) based on a fringe pattern with two frequency components is proposed, which provides a larger measuring range than that of the traditional FTP. We analyze the maximum measuring range and give an expression to describe the measurable slope of the height variation limitation of this method. The modified FTP provides us another approach to eliminate frequency overlapping. When the spectra distribution of a measured object is not spherical symmetry, we can avoid the frequency aliasing through projecting a fringe pattern with two frequency components, instead of increasing the density of the projected fringe and the resolution of a CCD camera. The theoretical analysis and primary experiments verified the method. r 2006 Elsevier GmbH. All rights reserved. Keywords: Fourier transform profilometry; Frequency aliasing; Three-dimensional shape measurement
1. Introduction The non-contract automatic three dimensional (3-D) shape measurement is very important in machine vision, solid modeling, industrial auto-measuring, etc. Fourier transform profilometry (FTP) proposed by Takeda et al. [1] is a popular 3-D sensing method, in which, a grating pattern is projected onto an object and a deformed fringe captured by a CCD camera is Fourier transformed and processed in its spatial frequency domain as well as in its space-signal domain. FTP has the following merits, only one (or two) fringe(s) needed, full field analysis, and high precision, so it has been extensively studied [2–8]. A sinusoidal fringe projecting combing with p phase shifting technique [4] can extend the measurable slope of height variation to nearly three times that of the unimproved FTP. Two-dimensional Corresponding author.
E-mail address: [email protected] (W. Chen). 0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.05.024
Fourier transform and frequency-weighted filtering are applied to provide a better separation of the useful information from noise when speckle-like structures and discontinuities exist in the fringe pattern [5]. The frequency-multiplex technology [6] permits the 3-D shape measurement of objects that have discontinues height step and /or spatially isolated surfaces. The phase error caused by sampling in FTP is discussed in detail [7]. Ref. [3] reviewed some of the developments in FTP over the past years. But up to now, the analysis about the maximum measuring range of FTP fits for only single frequency component fringe patterns. Although an oblique fringe pattern with two frequency components on the x and y directions is usually employed in FTP [9], the limitation of the measuring range based on this kind of fringe pattern projection has not been analyzed. In this paper, we analyze the maximum measuring range of FTP based on a fringe pattern with two frequency components, and give an expression
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W. Chen et al. / Optik 119 (2008) 57–62
describing the maximum range of the measurement. This kind of fringe pattern can easily obtain through programmable digital light projector (DLP) or changing the grating lines direction by rotating a single frequency component grating. When this kind of fringe pattern is projected onto the object, the location of the fundamental spectrum will move in the frequency domain. If there is frequency aliasing in the fx direction or fy direction, maybe there isn’t frequency aliasing after spectrum moving. If the spectra distribution of the measured object is not spherical symmetry, we can avoid the frequency aliasing through projecting a fringe pattern with two frequency components, instead of increasing the density of the projected fringe and the resolution of a CCD camera. The theoretical analysis and primary experiments verified the method.
The general crossed-optical-axis geometry of FTP is shown in Fig. 1, in which the optical axes P1P2 of a projector lens crosses that of a camera lens I1I2 at point O on a plane R, which is a fictitious plane normal to I1I2 and serves as a reference plane. d is the distance between P2 and I2, and L0 is the distance between I2 and plane R, A, C express points on R, D expresses a tested point on the measured object. Grating G has its lines normal to the plane of the figure. The grating image projected onto the object surface and observed through a CCD camera is a regular grating pattern. When a measured object is put on plane R and a sinusoidal grating pattern with one frequency component f0 in the x direction is projected onto it, the deformed fringe pattern observed through a CCD camera can be expressed by (1a)
On reference plane R ðhðx; yÞ ¼ 0Þ, the fringe pattern is still deformed, which can be expressed by g0 ðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cosð2pf 0 x þ f0 ðx; yÞÞ. Projector
CCD
Grating
(1b)
Camera
I1
P1
d
I2
P2
L0
D
Object h
C
A
x O
Fig. 1. The optical geometry.
Q*
Q f1max
fb −f0
0
f1min f0
x
Fig. 2. The spectrum distribution schematics.
In Eqs. (1a) and (1b), a(x, y) is the background intensity, b(x, y)is the non-uniform distributions of reflectivity on the diffuse object, f(x, y) is the phase modulations resulting from the object height distribution. f0(x, y) is the phase distribution for hðx; yÞ ¼ 0. Assuming qðx; yÞ ¼ 12bðx; yÞ expðifðx; yÞÞ, The Fourier spectrum of Eq. (1a) can be expressed as Gðz; ZÞ ¼ Aðz; ZÞ þ Qðz f 0 ; ZÞ þ Q ðz þ f 0 ; ZÞ.
2. Principle theory of the traditional FTP
gðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cosð2pf 0 x þ fðx; yÞÞ.
A
(2)
QðB; ZÞ is the spectrum of q(x, y). The spectrum distribution scheme on the x direction is shown in Fig. 2, where fb is the maximum of the zero order spectrum. f1max and f1min are the maximum and the minimum of the fundamental spectrum components, respectively. An instantaneous fundamental frequency f1 on the x direction is defined by [1] 1 qfðx; yÞ . (3) 2p qx In order to restore the object surface correctly, the fundamental component must be separated from all other spectra, that is
f1 ¼ f0 þ
f 1 min 4f b . By substituting Eq. (3) into Eq. (4), we obtain 1 qfðx; yÞ f0 4f b . 2p qx max
(4)
(5)
The phase variation caused by the height modulation must be limited in qfðx; yÞ (6) qx o2pðf 0 f b Þ, max then the measurable slope of the height variation extends the limitation, the fundamental components overlap the zero component. Aliasing will influence the precision of FTP or prevent obtaining a correct reconstruction. If jqfðx; yÞ=qxjmax satisfies Eq. (6), one of the 71 order fundamental components, for example QðB f 0 ; ZÞ, is filtered out through a suitable filter. The inverse Fourier transform is applied to the fundamental component, we obtain
R
^ yÞ ¼ gðx;
1 bðx; yÞ expði2pf 0 x þ ifðx; yÞÞ. 2
(7a)
ARTICLE IN PRESS W. Chen et al. / Optik 119 (2008) 57–62
59
We do the same operation for Eq. (1b) to obtain
y
1 (7b) g^ 0 ðx; yÞ ¼ bðx; yÞ expði2pf 0 x þ if0 ðx; yÞÞ, 2 Dfðx; yÞ can be generated by Eqs. (7a) and (7b) [3]. Assuming the measuring system is perfect and the system parameters L0 and d have no error, the conversion formula between the phase and the height is [3]: L0 Dfðx; yÞ . hðx; yÞ ¼ Dfðx; yÞ 2pf 0 d
X
Y
fxy
f0 fx fb
θ
f0
(a)
(8)
y Y
3. Analysis on FTP with two frequency components grating pattern
fb
ð9bÞ
where fðx; yÞ is the phase modulation caused by the height distribution, Eq. (9b) can be rewritten as gðx; yÞ ¼ aðx; yÞ þ bðx; yÞ½expði2pðf 0 cosðyÞx þ f 0 sinðyÞyÞ þ ifðx; yÞÞ þ expði2pðf 0 cosðyÞx þ f 0 sinðyÞyÞ ifðx; yÞÞ=2.
ð10Þ
Let q ¼ 12 bðx; yÞ expðifðx; yÞÞ, the Fourier spectra of Eq. (10) are: Gðz; ZÞ ¼ Aðz; ZÞ þ Qðz f 0 cosðyÞ; Z f 0 sinðyÞÞ þ Q ðz þ f 0 cosðyÞ; Z þ f 0 sinðyÞÞ.
ð11Þ
QðB; ZÞ is the spectrum of q(x, y). Eq. (11) shows that the fundamental spectrum shape of the fringe pattern with two frequency component is as same as that of the fringe pattern with only one frequency component, but its center is put at point ðf 0 cosðyÞ; f 0 sinðyÞÞ. We define an instantaneous synthetical frequency fX on the X direction: fX ¼ f0 þ
1 qfðx; yÞ . 2p qX
x f0
(9a)
where fx is the frequency component on the x direction and fy is that on the y direction. The synthetical frequency vector fxy is on the X direction, which is normal to the grating lines, and its value is f0. So fx ¼ f0 cos(y) and fy ¼ f0 sin(y). y is the angle between the normal of the grating lines and the x-axis. The scheme is shown in Fig. 3(a). Projecting this kind of grating pattern onto the object, the deformed fringe pattern captured by the CCD camera is written by gðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cosð2pðf 0 cosðyÞx þ f 0 sinðyÞyÞ þ fðx; yÞÞ,
X
f0
In FTP with two frequency components, the grating pattern can be written as gðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cosð2pðf x x þ f y yÞÞ,
x
fy
(12)
(b) Fig. 3. (a) A schematics of coordination conversion. (b) A schematics of eliminating frequency by a fringe pattern with two frequency component.
The (X, Y) coordinates are obtained by rotating the (x, y) coordinates by an angle y, as shown in Fig. 3(a). The relationship between the (X, Y) coordinates and the (x, y) coordinates is " # " #" # x cos y sin y X ¼ , sin y cos y y Y " # " #" # x cos y sin y X ¼ . ð13Þ y sin y cos y Y So the following equation must be satisfied for avoiding frequency overlapping: 1 qfðx; yÞ 4f b . (14) f0 2p qX max Employing the partial differential equation: qfðx; yÞ qfðx; yÞ qx qfðx; yÞ qy ¼ þ , qX qx qX qy qY qfðx; yÞ qfðx; yÞ qfðx; yÞ ¼ cos y þ sin y. qX qx qy Eq. (14) can be rewritten as
1 qfðx; yÞ 2p qX ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2 2 1 qfðx; yÞ qfðx; yÞ ¼ f0 cos y þ sin y 4f b , 2p qx qy
f0
ð15Þ
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that is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 qfðx; yÞ qfðx; yÞ cos y þ sin y o2pðf 0 f b Þ, qx qy (16)
(17)
when y ¼ 01, Eq. (15) is simplified as 1 qfðx; yÞ f0 4f b . 2p qx max
(18)
It is obvious that Eq. (5) is a special case of Eq. (16). In Eq. (16), if ðqfðx; yÞ=qxÞ4ðqfðx; yÞ=qyÞ,
15 Height mm
when y ¼ 901, Eq. (15) is simplified as 1 qfðx; yÞ 4f b , f0 2p qy max
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0 0
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Fig. 4. FTP simulation based on a fringe pattern with single frequency component. (a) The simulated object. (b) The Deformed fringe pattern. (c) The contour distribution of frequency spectrum. (d) The restored object.
Height mm
Pixel y
140
0
-0.5 300
(d)
200 Pixel 100 x
0 0
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Fig. 5. FTP simulation based on a fringe pattern with two frequency components. (a) The deformed fringe pattern. (b) The contour distribution of frequency spectrum. (c) The restored object. (d) The error distribution.
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then sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 qfðx; yÞ qfðx; yÞ qfðx; yÞ . cos y þ sin y o qx qy qx
4. Computer simulations and primary experiments Computer simulations verify the above analysis. The simulated object is shown in Fig. 4(a), the image size is 256 256 pixels. It has steep height variation on the x direction. When projecting sinusoidal fringe with 14 periods (26/1.41 pixels per period) onto the object, the deformed fringe pattern captured by a CCD camera has local shadow and under-sampling, as shown in Fig. 4(b). Its Fourier spectra are not symmetrical. On the x-axis in the frequency domain, the foundational spectrum overlaps the zero spectrum, Fig. 4(c) is the enlarged contour map of the spectrum distribution. The prefect foundational spectrum cannot be filtered out, so correct restored surface cannot be obtained, as shown in Fig. 4(d). When projecting a fringe pattern with two frequency components onto the object, Fig. 5(a) shows the deformed fringe pattern, which still has 14 periods (26 pixels per period) in the direction normal to grating
It shows that if the 71 order fundamental spectra are not spherical symmetry, the frequency aliasing can be avoided through projecting an oblique fringe with two frequency components as shown in Fig. 3(b). Monitor Object
290mm
790mm
Digital Camera
Frame memory
DLP projector
61
IBM_PC
Printer
Fig. 6. The experimental block diagram.
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280 260 240 220 200
(b)
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ly
xe
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Fig. 7. (a) The deformed fringe with single frequency component. (b) The enlarged spectrum contour maps of (a). (c) The restored shape from (a). (d) The deformed fringe with two frequency components. (e) The enlarged spectrum contour maps of (d). (f) The restored shape from (d).
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lines. The shape of the foundational spectrum is not changed, but the position of the spectra move as an angle. As shown in Fig. 5(b), the foundational components separate from the zero component. Filtering out the foundational component and calculating its inverse Fourier transform, we can restore the surface of the measured object, as shown in Fig. 5(c). The difference between the original object and the restored surface is shown as Fig. 5(d). Some experiences are applied in our demonstration. The experimental block diagram is shown in Fig. 6, a TV camera (TM1881) with a 16 mm focal length lens, via a special video-frame grabber DT3152 digitizing the image in 576 768 picture elements. The system parameters are L0 ¼ 780 mm and d ¼ 290 mm. We measure an object with two heart-shape parts. First, we project a sinusoidal fringe pattern with one frequency component onto the object. The deformed fringe pattern captured by the CCD is cut to form a image with 512 680 pixels, as shown in Fig. 7(a). Fig. 7(b) shows the enlarged contour map of the Frequency spectra, in which the foundational spectrum overlaps the zero spectrum. So we cannot restore the correct shape of the measured object from the fundamental component, as shown in Fig. 7(c). Then the grating is rotated by an angle y, an oblique fringe pattern with two frequency components is projected onto the measured object and captured by the same camera, as shown in Fig. 7(d). The enlarged contour map of its Fourier spectra is shown in Fig. 7(e). It shows that the fundamental spectra separate from the zero component by moving a certain angle. The correct restored surface is as shown in Fig. 7(f).
5. Conclusions We analyze the maximum measuring range of FTP based on a fringe pattern with two frequency components and give an expression describing measuring range. The formula of maximum range of measurement of FTP given by Takeda is a special case of our expression. It shows that the position of the fundamental components of the deformed fringe pattern can be moved by rotating the direction of the grating lines. If
the fundamental spectra of the measured object are nonspherical symmetry, it gives us another approach to eliminate frequency overlapping, instead of just changing system parameters L0 and d or increasing the density of the projected fringe and the resolution of a CCD camera.
Acknowledgement This project was supported by National Natural Science Foundation of China.
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