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Phase-unwrapping algorithm in the presence of discontinuities using a system with crossed grating
Optics and ¸asers in Engineering 29 (1998) 49—59 ( 1998 Published by Elsevier Science Limited All rights reserved. Printed in Northern Ireland 0143—8166/98/$19.00 PII: S0143–8166(97)00050–X
Phase-unwrapping Algorithm in the Presence of Discontinuities Using a System with Crossed Grating Xinjun Xie,a Michael J. Lalor,b David R. Burtonb & Michael M. Shawa aCoherent and Electro-Optics Research Group, School of Electrical Engineering, Electronics and Physics, Liverpool John Moores University, Byrom Street, Liverpool, UK, L3 3AF bCoherent and Electro-Optics Research Group, School of Engineering and Technology Management, Liverpool John Moores University, Byrom Street, Liverpool, UK, L3 3AF (Received 26 February 1997; received in revised form 17 July 1997)
ABS¹RAC¹ A new phase-unwrapping algorithm for the phase map containing discontinuities by the use of a system with crossed grating is described in this paper. A crossed grating is projected onto the object in the usual way, the deformed grating image acquired is Fourier transformed and the frequency spectra for the individual gratings are separated. ºsing both phase distributions which have different sensitivities, the correct phase values in the presence of discontinuities, especially those caused by the object with height steps, can be obtained. ¹his algorithm is fast and accurate. ¹he results of the measurement of a three-dimensional object with height steps are presented. ( 1998 Published by Elsevier Science ¸td.
1 INTRODUCTION The techniques of phase-measuring profilometry are very important in many applications and have been extensively studied.1,2 The intensity recorded in an interferogram is a cyclical function of the phase. The phase is calculated by the inverse trigonometric function, but the phase values obtained by the method are in the range of n to !n radians. Early phase unwrapping,3~5 which is based on distinguishing the phase change between sample points, is the determination of the true phase from module 2n data, and these techniques were not tolerant of the discontinuities and noise in the phase distribution. Some attempts have been made to deal with the more difficult problem of the true phase discontinuities in the phase map. Bone6 has suggested identification of these discontinuities by search for the region in which the phase 49
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curvature exceeds some threshold. However, this method fails when the first derivative of the phase is continuous across a boundary. More recently, Huntley and Saldner7 described a method that could be applied successfully to this constrained problem domain, but this technique would require only that the phase could be changed slowly from an unambiguously unwrappable phase to the final desired phase. For the phase unwrapping in the presence of discontinuities, a algorithm called multichannel Fourier fringe analysis is proposed by Burton and Lalor.8 The algorithm requires the presence of more than one fringe pattern on the surface. Bhat9 studied a method based on the computation of the phase directly from the wrapped phase maps, by which the phase discontinuities are eliminated by converting the wrapped phase map into a continuous fringe pattern and the phase derivative is obtained using a differentiator in the Fourier space. A phase unwrapping algorithm using two phase images with different precision is proposed by Zhao et al. and can produce an approximately correct unwrapping in the presence of discontinuities.10 The method is based on the phase shifting technique and total of eight images are required for one measurement. Recently, a method has been proposed which is based on the identification of discontinuity sources that mark the start or end of a 2n phase discontinuity and a stable-marriages algorithm for preprocessing phase maps with discontinuity sources, which is implemented as a recursive procedure.11,12 None of these methods seems to deal completely with the problem. The main interest in this work is to solve the problem of discontinuities, especially those caused by the object with height steps. These generally take the form of lines along which the phase changes very rapidly form a steplike behavior in the true phase. The algorithm developed in this paper is based on the use of a crossed grating, which is equivalent to two individual gratings projected with an angle between two sets of grating lines. The captured deformed grating image is separated in the frequency space and inverse transformed. Two independent phase maps with different sensitivities are obtained, from which the correct phase values in the presence of discontinuities can be obtained. This paper discusses the process of this phase unwrapping algorithm in detail, and some results obtained through simulation and experiment for the measurement of a three-dimensional object with height step are presented.
2 FOURIER TRANSFORM In the fringe projection system, when a sinusoidal intensity grating is projected onto an object surface, the deformed grating by the surface can be
A system with crossed grating
51
expressed in the general form as follows I(x, y)"a(x, y)#b(x, y) cos[2nf x#'(x, y)] (1) 0 where a(x, y) and b(x, y) represent unwanted irradiance variations arising from the non-uniform light reflection or transmission by a test object; in most cases a(x, y), b(x, y) and '(x, y) vary slowly compared with the variation introduced by the spatial-carrier frequency, f . The phase function '(x, y) characterizes 0 the fringe deformation and is related to the shape of the object z"h(x, y). Equation (1) is Fourier transformed with respect to x by the use of a fast Fourier transform (FFT) algorithm, giving I( f, y)"A( f, y)#C( f!f , y)#C*( f#f , y) (2) 0 0 where A( f, y) is the transform of a(x, y), and C( f!f , y) and C*( f#f , y) are 0 0 the positive and negative frequency spectra of the modulated carrier fringes. After this is filtered and inverse-transformed, the phase distribution, '(x, y), can be obtained. If the phase of the reference plane is ' (x, y), then the phase distribution, 3 ' (x, y), produced by the object’s shape, h(x, y), can be expressed as 0 ' (x, y)"'(x, y)!' (x, y) (3) 0 3 3 PHASE UNWRAPPING The phase obtained from eqn (3) is indeterminate to an additive constant of 2np because the arctangent is defined over the range of !n to n. The true phase is given by /(x, y)"' (x, y)#2n(x, y)n (4) 0 where n(x, y), the number of turns, is an integer which may be positive or negative. The process of phase unwrapping determines the value of n(x, y) by comparing the phase values between adjacent pixels. This may result in incorrect results if there are discontinuities in the phase distribution. Figure 1(a) and (b) shows examples where this occurs. In Fig. 1(a), the phase jump across the height step is less than n, therefore the phase unwrapping is not required and the correct result can be estimated (see Fig. 1(c)). However, the phase jump in Fig. 1(b) is larger than n, and errors may occur if normal phase unwrapping is used. For example, if the phase jump is from 0 to 1)2n, the result obtained using the arctangent function is 0 and !0)8n (Fig. 1(d)). No additive information is gained by phase unwrapping in this case. Another
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Fig. 1. Fringe pattems on the object with height step and the calculated phase distributions along one vertical line.
problem is that we don’t know how many fringes have been moved between the top part of the image and the bottom part. To solve this problem, another phase distribution, / (x, y), is obtained by 1 the use of a different carrier frequency, f . If we choose a suitable value of 1 f such that it is less than f and that / (x, y) has a range of less than n, the 1 0 1 object height can be expressed as h(x, y)"k/(x, y)"k / (x, y) (5) 1 1 where k and k are constants and both are related to the setup of the optical 1 system. Figure 2 shows the optical geometry for the most general projection system. From the figure, the relationship between k and k is obtained as 1 ¸ k" (6) 2ndf 0 ¸ (7) k " 1 2ndf 1
A system with crossed grating
53
Fig. 2. The optical geometry for general projection system.
Substituting eqns (6) and (7) into eqn (5), we have /(x, y) f "0 (8) / (x, y) f 1 1 As it is assumed that f (f , so /(x, y)'/ (x, y). Using a carrier frequency 1 0 1 of f is more sensitive than using a carrier frequency of f . So the correct phase 0 1 jump can be obtained by comparing phase differences between adjacent pixels for the two phase distributions. The procedures are described below. 3.1 For phase discontinuities less than 2p Assuming that the phase jump between adjacent sample points, (i, j) to (i#l, j), is */ and */ for two phase distributions and the spatial frequencies 1 of two sets of gratings lines satisfy the condition of 2f (f , respectively, then 1 0 for the true phase jump, if n(D*/D(2n, gives: f D*/ D" 1D*/D(n 1 f 0
(9)
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Xinjun Xie et al.
When D*/ D)( f / f ) and (*/ /*/)(0, $2n should be added to the 1 1 0 1 phase values of /(x, y) for all sample points which have an index number bigger than i as that expressed by eqn (10). /(x , y )"/(x , y )$2n (10) i`1 j i`1 j The sign of the added 2n depends on the sign of */ and */ , and in this 1 way the error is automatically corrected. 3.2 For phase discontinuities greater than 2p When the phase jump, */, between two adjacent sample points is greater than 2n, the method to obtain the correct phase value is first to go through the procedure in Section 3.1, and then the absolute phase value of */ is exami1 ned to see whether it is greater than 2( f /f )n or not. If this is so, then the 1 0 following equation is applied:
A
B
*/ f 1 0 (11) 2n f 1 where INT is an operator to get the integer number which is equal to the integer part of its argument. After the integer number, n(x,y), is obtained, 2nn is added to the phase /(x, y) for all the sample points with an index number greater than i. n"INT
/(x , y )"/(x , y )#2nn i`1 j i`1 j This is valid for the phase discontinuities not greater than ( f /f )n. 0 1
(12)
3.3 Use of the crossed grating system If two fringe patterns at an angle, a, to each other are added together, a crossed fringe pattern is formed. Figure 3 shows the two separate fringe patterns; the corresponding crossed flange pattern is in Fig. 3(c). If the two flange patterns have the same period, P , and in the case of Fig. 3(a), the 0 period along the x direction is P , then for Fig. 3(b), the equivalent period in 0 the x direction is given by eqn (13) and shown in Fig. 4. P P " 0 1 cos a
(13)
If the crossed grating is projected onto the object, the captured image can be Fourier transformed and separated in the frequency space as two images. This has the same effect as if two separate fringe patterns had been projected. The frequencies of two sets of fringes in the x direction are given by the following
A system with crossed grating
Fig. 3. Uni-directional fringe patterns (a) and (b) and crossed fringe pattern (c).
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Fig. 4. The relationship between a grating’s period and the equivalent period in the x direction.
Fig. 5. (a) Simulated fringe pattem on an object with discontinuities; (b) real part of the frequency spectra; (c) imaginary part of the frequency spectra; (d) the phase distribution along one vertical line obtained by two separate fringe projections (1 and 2), and the correct phase values (3).
A system with crossed grating
57
two equations 1 f " (14) 0 P 0 cos a f " (15) 1 P 0 An example is shown in Fig. 5, where f /f "8. Figure 5(a) shows the 0 1 simulated projected crossed grating on the object which has large height step equivalent to 7)2n, in the vertical direction. Figure 5(b) and (c) show the real and imaginary parts of the frequency spectra, and Fig. 5(d) shows the unwrapped phase along one vertical line. Lines 1—3 in this diagram are the phase values obtained by projections with fringe frequencies of f and f , and the 0 1 unwrapped results, respectively. In this case the correct phase jump is 7)2n, but they are 0)8n and 0)9n, respectively, in lines 1 and 2.
4 EXPERIMENTS In the experimental measurements, a cross grating is used, which has a period of P "0)25 mm, with the sets of fringes oriented at an angle of about 80° to 0 each other. A standard 35 mm slide projector is used to project the grating lines onto the object. The image is captured by a CCD camera and the signal is processed by a personal computer system equipped with frame grabber. The object used in the experiment has a height step of about 10 mm. The distance from the camera to the object is 900 mm, and it is 250 mm from the camera to the projector. In the horizontal direction, f /f "5)5. Figure 6(a) shows the 0 1 image captured by the camera, whilst Fig. 6(b) shows the results obtained along one vertical line, where lines 1 and 2 are the results obtained with the fringe frequency of f in the horizontal direction, and the correct phase results, 0 respectively. The phase discontinuities in this case are 2)88 and 9)17 rad, respectively. Figure 6(c) shows the three-dimensional plots of the correct phase results. From Fig. 6(b), it can be seen that there is some noise in the results. This is due to the fact that the projected fringes are not the perfect sum of two sets of fringes and that the numbers of fringes in the horizontal direction in each set are not integers at exactly the same time.
5 CONCLUSIONS A new phase unwrapping algorithm for a phase map containing discontinuities through the use of a crossed grating has been proposed. The correct phase
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Fig. 6. (a) The image captured from the object surface with projected crossed grating. (b) 1—Results obtained by the fringe pattern with frequency of f ; and 2—corrected results. (c) The 0 results. three-dimensional plots of the corrected
values can be obtained by comparing the two phase distributions with different sensitivities in the presence of the discontinuities, especially those caused by the height steps. The theoretical analysis of the proposed method is described. Only one image is required for each measurement. The simulation
A system with crossed grating
59
and experiment on a three-dimensional object with height step has verified the theoretical analysis, and it is shown to be efficient and accurate.
REFERENCES 1. Reld, G. T., Automatic fringe pattern analysis: a review. Optics and ¸asers in Engineering, 7 (1986/87) 37—68. 2. Judge, T. R. & Bryanston-Gross, P. J., A review of phase unwrapping techniques in fringe analysis. Optics and ¸asers in Engineering 21 (1994) 199—239. 3. Takeda, M., Ina, H. & Kobayashi, S., Fourier transform method of fringe-pattern analysis for computer-based topography and interferometry. Journal of the Optical Society of America 72 (1982) 156—60. 4. Macy, W. W., Two-dimensional fringe-pattern analysis. Applied Optics 22 (1983) 3898—901. 5. Ghiglla, D. C., Mastin, G. A. & Romero, L. A., Cellular automata method for phase unwrapping. Journal of the Optical Society of America A 4 (1987) 267—80. 6. Bone, D. J., Fourier fringe analysis: the two-dimensional phase unwrapping problem. Applied Optics 30 (1991) 3627—32. 7. Huntley, J. M. & Saldner, H., Temporal phase unwrapping algorithm for automated interferogram analysis. Applied Optics 32 (1993) 3047—52. 8. Burton, D. R. & Lalor, M. J., Multichannel Fourier fringe analysis as an aid to automatic phase unwrapping. Applied Optics 33 (1994) 2939—48. 9. Bhat, G. K., A Fourier transform technique to obtain phase derivatives in interferometry. Optical Communications 110 (1994) 279—86. 10. Zhao, H., Chen, W. & Tan, Y., Phase-unwrapping algorithm for the measurement of three-dimensional object shapes. Applied Optics 33 (1994) 4497—500. 11. Cusack, R., Huntley, J. M. & Golderin, H. T., Improved noise-immune phase unwrapping algorithm. Applied Optics 34 (1995) 781—9. 12. Quiroga, J. A., Gonzalez-Cano, A. & Bemabeu, E., Stable-marriages algorithm for preprocessing phase maps with discontinuity sources. Applied Optics 34 (1995) 5029—38.
Phase-unwrapping Algorithm in the Presence of Discontinuities Using a System with Crossed Grating Xinjun Xie,a Michael J. Lalor,b David R. Burtonb & Michael M. Shawa aCoherent and Electro-Optics Research Group, School of Electrical Engineering, Electronics and Physics, Liverpool John Moores University, Byrom Street, Liverpool, UK, L3 3AF bCoherent and Electro-Optics Research Group, School of Engineering and Technology Management, Liverpool John Moores University, Byrom Street, Liverpool, UK, L3 3AF (Received 26 February 1997; received in revised form 17 July 1997)
ABS¹RAC¹ A new phase-unwrapping algorithm for the phase map containing discontinuities by the use of a system with crossed grating is described in this paper. A crossed grating is projected onto the object in the usual way, the deformed grating image acquired is Fourier transformed and the frequency spectra for the individual gratings are separated. ºsing both phase distributions which have different sensitivities, the correct phase values in the presence of discontinuities, especially those caused by the object with height steps, can be obtained. ¹his algorithm is fast and accurate. ¹he results of the measurement of a three-dimensional object with height steps are presented. ( 1998 Published by Elsevier Science ¸td.
1 INTRODUCTION The techniques of phase-measuring profilometry are very important in many applications and have been extensively studied.1,2 The intensity recorded in an interferogram is a cyclical function of the phase. The phase is calculated by the inverse trigonometric function, but the phase values obtained by the method are in the range of n to !n radians. Early phase unwrapping,3~5 which is based on distinguishing the phase change between sample points, is the determination of the true phase from module 2n data, and these techniques were not tolerant of the discontinuities and noise in the phase distribution. Some attempts have been made to deal with the more difficult problem of the true phase discontinuities in the phase map. Bone6 has suggested identification of these discontinuities by search for the region in which the phase 49
50
Xinjun Xie et al.
curvature exceeds some threshold. However, this method fails when the first derivative of the phase is continuous across a boundary. More recently, Huntley and Saldner7 described a method that could be applied successfully to this constrained problem domain, but this technique would require only that the phase could be changed slowly from an unambiguously unwrappable phase to the final desired phase. For the phase unwrapping in the presence of discontinuities, a algorithm called multichannel Fourier fringe analysis is proposed by Burton and Lalor.8 The algorithm requires the presence of more than one fringe pattern on the surface. Bhat9 studied a method based on the computation of the phase directly from the wrapped phase maps, by which the phase discontinuities are eliminated by converting the wrapped phase map into a continuous fringe pattern and the phase derivative is obtained using a differentiator in the Fourier space. A phase unwrapping algorithm using two phase images with different precision is proposed by Zhao et al. and can produce an approximately correct unwrapping in the presence of discontinuities.10 The method is based on the phase shifting technique and total of eight images are required for one measurement. Recently, a method has been proposed which is based on the identification of discontinuity sources that mark the start or end of a 2n phase discontinuity and a stable-marriages algorithm for preprocessing phase maps with discontinuity sources, which is implemented as a recursive procedure.11,12 None of these methods seems to deal completely with the problem. The main interest in this work is to solve the problem of discontinuities, especially those caused by the object with height steps. These generally take the form of lines along which the phase changes very rapidly form a steplike behavior in the true phase. The algorithm developed in this paper is based on the use of a crossed grating, which is equivalent to two individual gratings projected with an angle between two sets of grating lines. The captured deformed grating image is separated in the frequency space and inverse transformed. Two independent phase maps with different sensitivities are obtained, from which the correct phase values in the presence of discontinuities can be obtained. This paper discusses the process of this phase unwrapping algorithm in detail, and some results obtained through simulation and experiment for the measurement of a three-dimensional object with height step are presented.
2 FOURIER TRANSFORM In the fringe projection system, when a sinusoidal intensity grating is projected onto an object surface, the deformed grating by the surface can be
A system with crossed grating
51
expressed in the general form as follows I(x, y)"a(x, y)#b(x, y) cos[2nf x#'(x, y)] (1) 0 where a(x, y) and b(x, y) represent unwanted irradiance variations arising from the non-uniform light reflection or transmission by a test object; in most cases a(x, y), b(x, y) and '(x, y) vary slowly compared with the variation introduced by the spatial-carrier frequency, f . The phase function '(x, y) characterizes 0 the fringe deformation and is related to the shape of the object z"h(x, y). Equation (1) is Fourier transformed with respect to x by the use of a fast Fourier transform (FFT) algorithm, giving I( f, y)"A( f, y)#C( f!f , y)#C*( f#f , y) (2) 0 0 where A( f, y) is the transform of a(x, y), and C( f!f , y) and C*( f#f , y) are 0 0 the positive and negative frequency spectra of the modulated carrier fringes. After this is filtered and inverse-transformed, the phase distribution, '(x, y), can be obtained. If the phase of the reference plane is ' (x, y), then the phase distribution, 3 ' (x, y), produced by the object’s shape, h(x, y), can be expressed as 0 ' (x, y)"'(x, y)!' (x, y) (3) 0 3 3 PHASE UNWRAPPING The phase obtained from eqn (3) is indeterminate to an additive constant of 2np because the arctangent is defined over the range of !n to n. The true phase is given by /(x, y)"' (x, y)#2n(x, y)n (4) 0 where n(x, y), the number of turns, is an integer which may be positive or negative. The process of phase unwrapping determines the value of n(x, y) by comparing the phase values between adjacent pixels. This may result in incorrect results if there are discontinuities in the phase distribution. Figure 1(a) and (b) shows examples where this occurs. In Fig. 1(a), the phase jump across the height step is less than n, therefore the phase unwrapping is not required and the correct result can be estimated (see Fig. 1(c)). However, the phase jump in Fig. 1(b) is larger than n, and errors may occur if normal phase unwrapping is used. For example, if the phase jump is from 0 to 1)2n, the result obtained using the arctangent function is 0 and !0)8n (Fig. 1(d)). No additive information is gained by phase unwrapping in this case. Another
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Fig. 1. Fringe pattems on the object with height step and the calculated phase distributions along one vertical line.
problem is that we don’t know how many fringes have been moved between the top part of the image and the bottom part. To solve this problem, another phase distribution, / (x, y), is obtained by 1 the use of a different carrier frequency, f . If we choose a suitable value of 1 f such that it is less than f and that / (x, y) has a range of less than n, the 1 0 1 object height can be expressed as h(x, y)"k/(x, y)"k / (x, y) (5) 1 1 where k and k are constants and both are related to the setup of the optical 1 system. Figure 2 shows the optical geometry for the most general projection system. From the figure, the relationship between k and k is obtained as 1 ¸ k" (6) 2ndf 0 ¸ (7) k " 1 2ndf 1
A system with crossed grating
53
Fig. 2. The optical geometry for general projection system.
Substituting eqns (6) and (7) into eqn (5), we have /(x, y) f "0 (8) / (x, y) f 1 1 As it is assumed that f (f , so /(x, y)'/ (x, y). Using a carrier frequency 1 0 1 of f is more sensitive than using a carrier frequency of f . So the correct phase 0 1 jump can be obtained by comparing phase differences between adjacent pixels for the two phase distributions. The procedures are described below. 3.1 For phase discontinuities less than 2p Assuming that the phase jump between adjacent sample points, (i, j) to (i#l, j), is */ and */ for two phase distributions and the spatial frequencies 1 of two sets of gratings lines satisfy the condition of 2f (f , respectively, then 1 0 for the true phase jump, if n(D*/D(2n, gives: f D*/ D" 1D*/D(n 1 f 0
(9)
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Xinjun Xie et al.
When D*/ D)( f / f ) and (*/ /*/)(0, $2n should be added to the 1 1 0 1 phase values of /(x, y) for all sample points which have an index number bigger than i as that expressed by eqn (10). /(x , y )"/(x , y )$2n (10) i`1 j i`1 j The sign of the added 2n depends on the sign of */ and */ , and in this 1 way the error is automatically corrected. 3.2 For phase discontinuities greater than 2p When the phase jump, */, between two adjacent sample points is greater than 2n, the method to obtain the correct phase value is first to go through the procedure in Section 3.1, and then the absolute phase value of */ is exami1 ned to see whether it is greater than 2( f /f )n or not. If this is so, then the 1 0 following equation is applied:
A
B
*/ f 1 0 (11) 2n f 1 where INT is an operator to get the integer number which is equal to the integer part of its argument. After the integer number, n(x,y), is obtained, 2nn is added to the phase /(x, y) for all the sample points with an index number greater than i. n"INT
/(x , y )"/(x , y )#2nn i`1 j i`1 j This is valid for the phase discontinuities not greater than ( f /f )n. 0 1
(12)
3.3 Use of the crossed grating system If two fringe patterns at an angle, a, to each other are added together, a crossed fringe pattern is formed. Figure 3 shows the two separate fringe patterns; the corresponding crossed flange pattern is in Fig. 3(c). If the two flange patterns have the same period, P , and in the case of Fig. 3(a), the 0 period along the x direction is P , then for Fig. 3(b), the equivalent period in 0 the x direction is given by eqn (13) and shown in Fig. 4. P P " 0 1 cos a
(13)
If the crossed grating is projected onto the object, the captured image can be Fourier transformed and separated in the frequency space as two images. This has the same effect as if two separate fringe patterns had been projected. The frequencies of two sets of fringes in the x direction are given by the following
A system with crossed grating
Fig. 3. Uni-directional fringe patterns (a) and (b) and crossed fringe pattern (c).
55
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Fig. 4. The relationship between a grating’s period and the equivalent period in the x direction.
Fig. 5. (a) Simulated fringe pattem on an object with discontinuities; (b) real part of the frequency spectra; (c) imaginary part of the frequency spectra; (d) the phase distribution along one vertical line obtained by two separate fringe projections (1 and 2), and the correct phase values (3).
A system with crossed grating
57
two equations 1 f " (14) 0 P 0 cos a f " (15) 1 P 0 An example is shown in Fig. 5, where f /f "8. Figure 5(a) shows the 0 1 simulated projected crossed grating on the object which has large height step equivalent to 7)2n, in the vertical direction. Figure 5(b) and (c) show the real and imaginary parts of the frequency spectra, and Fig. 5(d) shows the unwrapped phase along one vertical line. Lines 1—3 in this diagram are the phase values obtained by projections with fringe frequencies of f and f , and the 0 1 unwrapped results, respectively. In this case the correct phase jump is 7)2n, but they are 0)8n and 0)9n, respectively, in lines 1 and 2.
4 EXPERIMENTS In the experimental measurements, a cross grating is used, which has a period of P "0)25 mm, with the sets of fringes oriented at an angle of about 80° to 0 each other. A standard 35 mm slide projector is used to project the grating lines onto the object. The image is captured by a CCD camera and the signal is processed by a personal computer system equipped with frame grabber. The object used in the experiment has a height step of about 10 mm. The distance from the camera to the object is 900 mm, and it is 250 mm from the camera to the projector. In the horizontal direction, f /f "5)5. Figure 6(a) shows the 0 1 image captured by the camera, whilst Fig. 6(b) shows the results obtained along one vertical line, where lines 1 and 2 are the results obtained with the fringe frequency of f in the horizontal direction, and the correct phase results, 0 respectively. The phase discontinuities in this case are 2)88 and 9)17 rad, respectively. Figure 6(c) shows the three-dimensional plots of the correct phase results. From Fig. 6(b), it can be seen that there is some noise in the results. This is due to the fact that the projected fringes are not the perfect sum of two sets of fringes and that the numbers of fringes in the horizontal direction in each set are not integers at exactly the same time.
5 CONCLUSIONS A new phase unwrapping algorithm for a phase map containing discontinuities through the use of a crossed grating has been proposed. The correct phase
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Xinjun Xie et al.
Fig. 6. (a) The image captured from the object surface with projected crossed grating. (b) 1—Results obtained by the fringe pattern with frequency of f ; and 2—corrected results. (c) The 0 results. three-dimensional plots of the corrected
values can be obtained by comparing the two phase distributions with different sensitivities in the presence of the discontinuities, especially those caused by the height steps. The theoretical analysis of the proposed method is described. Only one image is required for each measurement. The simulation
A system with crossed grating
59
and experiment on a three-dimensional object with height step has verified the theoretical analysis, and it is shown to be efficient and accurate.
REFERENCES 1. Reld, G. T., Automatic fringe pattern analysis: a review. Optics and ¸asers in Engineering, 7 (1986/87) 37—68. 2. Judge, T. R. & Bryanston-Gross, P. J., A review of phase unwrapping techniques in fringe analysis. Optics and ¸asers in Engineering 21 (1994) 199—239. 3. Takeda, M., Ina, H. & Kobayashi, S., Fourier transform method of fringe-pattern analysis for computer-based topography and interferometry. Journal of the Optical Society of America 72 (1982) 156—60. 4. Macy, W. W., Two-dimensional fringe-pattern analysis. Applied Optics 22 (1983) 3898—901. 5. Ghiglla, D. C., Mastin, G. A. & Romero, L. A., Cellular automata method for phase unwrapping. Journal of the Optical Society of America A 4 (1987) 267—80. 6. Bone, D. J., Fourier fringe analysis: the two-dimensional phase unwrapping problem. Applied Optics 30 (1991) 3627—32. 7. Huntley, J. M. & Saldner, H., Temporal phase unwrapping algorithm for automated interferogram analysis. Applied Optics 32 (1993) 3047—52. 8. Burton, D. R. & Lalor, M. J., Multichannel Fourier fringe analysis as an aid to automatic phase unwrapping. Applied Optics 33 (1994) 2939—48. 9. Bhat, G. K., A Fourier transform technique to obtain phase derivatives in interferometry. Optical Communications 110 (1994) 279—86. 10. Zhao, H., Chen, W. & Tan, Y., Phase-unwrapping algorithm for the measurement of three-dimensional object shapes. Applied Optics 33 (1994) 4497—500. 11. Cusack, R., Huntley, J. M. & Golderin, H. T., Improved noise-immune phase unwrapping algorithm. Applied Optics 34 (1995) 781—9. 12. Quiroga, J. A., Gonzalez-Cano, A. & Bemabeu, E., Stable-marriages algorithm for preprocessing phase maps with discontinuity sources. Applied Optics 34 (1995) 5029—38.