A multiple wavelength unwrapping algorithm for digital fringe profilometry based on spatial shift estimation

Transcription

1 University of Wollongong Research Online Faculty of Engineering an Information Sciences - Papers: Part A Faculty of Engineering an Information Sciences 214 A multiple wavelength unwrapping algorithm for igital fringe profilometry base on spatial shift estimation Pu Cao University of Wollongong, pc241@uowmail.eu.au Jiangtao Xi University of Wollongong, jiangtao@uow.eu.au Yanguang Yu University of Wollongong, yanguang@uow.eu.au Qinghua Guo University of Wollongong, qguo@uow.eu.au Publication Details P. Cao, J. Xi, Y. Yu & Q. Guo, "A multiple wavelength unwrapping algorithm for igital fringe profilometry base on spatial shift estimation," in Proceeings of SPIE 913: Three-Dimensional Image Processing, Measurement (3DIPM), an Applications 214, 214, pp. 913E-1-913E-1. Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library: research-pubs@uow.eu.au

2 A multiple wavelength unwrapping algorithm for igital fringe profilometry base on spatial shift estimation Abstract In this paper, a new approach is presente for solving the problem of spatial shift wrapping associate with Spatial Shift Estimation (SSE)-base Fringe Pattern Profilometry (FPP). The problem arises as the result of fringe reuse (that is, fringes perioic light intensity variance), an the spatial shift can only be ientifie without ambiguity with the range of a fringe with. It is emonstrate that the problem is similar to the phase unwrapping problem associate with the phase etection base FPP, an the propose metho is inspire by the existing ieas of using multiple images with ifferent wavelengths propose for phase unwrapping. The effectiveness of the propose metho is verifie by experimental results on an object with complex surface shape. 214 SPIE. Keywors fringe, igital, algorithm, unwrapping, profilometry, wavelength, estimation, multiple, spatial, shift Disciplines Engineering Science an Technology Stuies Publication Details P. Cao, J. Xi, Y. Yu & Q. Guo, "A multiple wavelength unwrapping algorithm for igital fringe profilometry base on spatial shift estimation," in Proceeings of SPIE 913: Three-Dimensional Image Processing, Measurement (3DIPM), an Applications 214, 214, pp. 913E-1-913E-1. This conference paper is available at Research Online:

3 A Multiple Wavelength Unwrapping Algorithm for Digital Fringe Profilometry base on Spatial Shift Estimation Pu Cao, Jiangtao Xi*, Yanguang Yu an Qinghua Guo School of Electrical, Computer an Telecommunications Engineering University of Wollongong, Wollongong, NSW2522, Australia ABSTRACT In this paper we present a review of the phase unwrapping problem in Fringe Pattern Profilometry (FPP), base on which we stuy the spatial shift wrapping problem in spatial shift estimation (SEE) base FPP. An approach for carrying out the spatial shift unwrapping is propose with its performance confirme by experiments. Keywors: fringe pattern profilometry, 3D measurement, phase unwrapping 1. Introuction: Fringe Pattern Profilometry (FPP) base on Digital Fringe Projection (DFP) is a promising optical noncontact threeimension (3D) profile measurement technologies ue to its accuracy an flexibility. Compare with the other methos, it has the avantages of simple system structure an high accuracy. Hence it provies a much more flexible an practical approach for 3D profile measurement. Figure 1 shows the system structure of a DFP base FPP, consisting of a igital vieo projector, a CCD camera an a reference plane. With the system, a frame of image with a particular fringe pattern is prouce by the igital projector an projecte onto the reference plane, an then onto the surface of the object when the reference plane is remove. The projecte images from the reference plane an the object surface are capture by the CCD camera, with the later being a eforme version of the former by the variance of the height of the object surface. As the eforme fringe pattern carries the information of surface shape, 3D profile of the object can be retrieve from these two fringe patterns. Figure 1. Schematic iagram of FPP system. *jiangtao@uow.eu.au; phone ; fax

4 Several FPP approaches have been evelope uring the past ecaes. The most wiely use are these base on phase ifference estimation (PDE). In these approaches, the projecte fringe patterns are sinusoial or perioic, an the eforme one reflecte from the object surface is consiere as the result of phase moulation of the original fringe pattern. The surface profile is obtaine by etecting the phase maps of the two fringe patterns. A number of fringe pattern analysis methos have been evelope, such as Fourier transform profilometry (FTP) [1], phase shifting profilometry (PSP), phase measuring profilometry (PMP) [2, 3, 4], moulation measurement profilometry (MMP) [5], spatial phase etection (SPD) [6,7], phase loc loop (PLL) profilometry [8], Moire technique (MT) [9], laser triangulation measurement [1], colour-coe fringe projection [11, 12] an other methos [13, 14]. Among existing approaches, FTP an PSP are most popular an wiely use. Although phase base approaches have been consiere as the most popular, they suffers from a number of weanesses. A major restriction is that fringe patterns must be either sinusoial or ieal perioic. However such a requirement is har to meet in practice ue to some factors, such as the nonlinear istortion on inherent to igital vieo projections. In orer to solve the problem, a profilometry approach was propose by Hu et al [15, 16, 17], which, instea of etecting the ifferences between the phase maps, is base on the estimation of spatial shift for corresponing pixels on the two fringe patterns. The approach is referre to as spatial shift estimation (SSE) profilometry approach. The spatial shift estimation approach is suitable for any fringe pattern, however, use of a perioic fringe patterns is still necessary in orer to have enough measurement resolution. In the spatial shift estimation-base approaches, spatial shift between corresponing pixels on the two fringe patterns is arbitrary, it can only be etecte without ambiguity within the range of [, λ], where λ is the wavelength, or the spatial with of the iniviual fringe, i.e., number of pixels per fringe stripe. Obviously, shift unwrapping is also require in orer to correctly restore the 3D shape of the object surface. However, spatial shift unwrapping for complex object using spatial shift estimation-base fringe pattern profilometry is still an outstaning issue, which motivate the wor presente in this paper. Since the spatial shift unwrapping problem exists in spatial shift estimation approach which is similar to the phase unwrapping problem in phase ifference estimation base fringe pattern profilometry, a review of the phase unwrapping problem is given in this paper. Phase unwrapping problem is a major problem associate with phase ifference estimation-base fringe pattern profilometry approaches. This problem arises because the phase ifference can only be etecte within the main value range of [-π, π], but the true phase ifference can be arbitrary. In orer to retrieve the actual surface shape of the object, phase unwrapping must be carrie out to obtain the actual phase maps. To solve the unwrapping problem in phase ifference estimation approach, Zhang et al [18] introuce a multiple wavelength phase unwrapping algorithm. In his metho, an image which only has a single fringe covers the whole measurement area is first projecte, an then a series of fringe images with a wavelength ecrease by a factor from its previous wavelength is use. Since the first image only contains one fringe, the unwrapping step is not require. Then the phase of secon image can be unwrappe by referring the first image. After the phase of secon image is obtaine, it can be use to correct the thir image. In general, the phase of each wavelength is unwrappe by referring to the unwrappe longer wavelength phase pixel by pixel. Base on Zhang s metho, we introuce a multiple wavelength unwrapping algorithm for spatial shift estimation approach. In our metho, a series of fringe images with a wavelength ecrease by a factor from its previous wavelength is also applie. The spatial shift of each wavelength is then unwrappe by referring to the unwrappe longer wavelength spatial shift pixel by pixel. Since the longest wavelength covers the whole measurement area, no spatial shift unwrapping step is necessary. The propose metho solves the spatial shift unwrapping problem in spatial shift estimation, hence it can enable the measurement of complex objects with significant step height or multiple separate objects using spatial shift estimation approach. This paper is organize as follows. In Section 2 we firstly give a brief introuction on the conventional phase ifference estimation base fringe pattern profilometry an the spatial shift estimation base technique, incluing their principles, system structures an relevant algorithms. Then in Section 3 we inicate that the unwrapping problem exists in spatial shift estimation approach which is similar to the phase unwrapping problem in phase ifference estimation base fringe pattern profilometry. The paper then gives a review of the multiple wavelength phase unwrapping algorithm introuce by Zhang, base on which we introuce a multiple wavelength unwrapping algorithm for spatial shift estimation approach. Finally in Section 4 experimental results are given to emonstrate the propose metho can be use to measure complex objects with significant step height or multiple separate objects using spatial shift estimation approach. 2. Principle of Fringe Pattern Profilometry

5 2.1. Principle of Triangulation FPP is base on the triangulation principle escribe as follows. As the image prouce by the projector has a fringe structure, without loss of generality we can assume that light intensity varies perioically alone x irection, while eeping constant along y irection, as shown in Figure 1. We can use s (x), (x) an h (x) to enote the variance of light intensity of the fringe pattern on the reference plane an object surface as well as the height istribution along x coorinate respectively. We also assume that the reference plane an the object surface have the same reflective characteristics. Let us consier what happens when a beam of light is projecte onto the point D on the object. When the object is remove, the same light beam (hence with the same intensity) shoul be projecte onto point H on the reference surface, which is reflecte bac to the camera through point C. As the triangles E c E P H an CDH are similar, we have the following relationship: CD (1) l h( x ) Note that x enotes the coorination positions of point D. h ( x ) enotes the istance between points C an the reference plane, given by: h( x ) l CD The above relationship gives the founation for FPP. (2) 2.2. PDE base approaches for FPP The PDE base FPP utilize fringe patterns that are perioic an can be expresse as [19, 2]: s( x) b cos(2f x ) an the eforme fringe pattern can also be expresse as: ( x) b cos(2f x ( x) ) In the above equations, f is the spatial frequency of the funamental component in the fringe patterns, an (3) (4) b is the amplitue of the -th orer harmonic component. is the initial phase of the -th orer harmonic component, an (x) enotes the phase ifference between the -th orer harmonic components of these two fringe patterns. Equations (3) an (4) show that s( x ) an ( x ) are relate by the phase shift (x). Let us consier the light beam projecte at point D on the object an H on the reference plane when the object is remove. The phase shift between C an D can be etermine by the spatial istance CD, an hence we have [21, 22]: ( x) 2fCD 2 fcd ( x) (5) where ( x ) 2 fcd is the phase shift of the funamental component. Substituting Equation (5) to Equation (2) we have: l ( x ) h ( x ) (6) 2f As points D an H are arbitrary, the erivations shoul apply to all the points on the projecte fringe pattern. Therefore we have:

6 l ( x) hx ( ) (7) 2 f Equation (7) shows that as long as the gives (x) can be etecte, we are able to calculate the height istribution hx ( ) of the object surface. This is the founation of all PDE base approaches Spatial Shift Estimation base FPP The PDE base FPP methos suffer from some limitations. In particular, the fringe pattern use to project is limite to be sinusoial or purely perioic in orer that the phase maps of s (x) an (x) exist an can be etecte. However, ue to many unesire factors inherent to igital projection, such as geometrical istortion an nonlinear intensity istortion, purely sinusoial fringe patterns are har to prouce. In orer to solve these problems, Hu et al [15] introuce a metho which is base on the spatial shift estimation (SSE) rather than PDE. The SSE base approach is rather simple an straight forwar. Let us consiercd, the istance between C an D again, which is obviously a function of the location of D (i.e. x,), the location of H (or C, i.e., x c ) an the height of the object at point H hx ( ). Therefore we have the following: u( x ) (8) l h( x ) where ux ( ) CD xc x, which is the spatial istance between x an x c. Note that x an x c are the points on ( x ) an s( x) having the same light intensity, that is ( x) s( xc). As the above erivation is vali for any x an x c, we can replace x by x, yieling the following: l u( x) h( x) (9) Note that ux ( ) is the spatial istance between a point x on ( x ) an the corresponing point on s( x ) with the same light intensity, that is: ( x) s( x u( x)) (1) Equations (9) an (1) provie a straight forwar way to obtain the 3D profile of the object surface. With ( x ) an s( x ) available, if we are able to obtain ux ( ) to meet Equation (1), we then can utilize Equation (9) to yiel hx ( ), the height istribution of the object surface along x. By repeating the proceure for all y we shoul be able to obtain the 3D profile of the object surface. A number of approaches were propose to retrieve the ux ( ) [15 16, 17]. Among these approaches, the one referre to as Inverse Function base Shift Estimation (IFSE) [16] is particularly interesting an briefe as follows. Wu et al [23] then improve this metho by combining IFSE with Jia s Multiple-step triangular-pattern phase shifting algorithm [24], which greatly improve the accuracy of measurement. The spatial shift base approach has a particular avantage. The projecte fringe patterns are no longer require to be sinusoial, which implies that even there are istortions with the fringe patterns, sufficient three-imensional information on the object surface is containe in the variation between projecte an eforme fringe patterns. Thus the profilometry can be archive. 3. The Unwrapping Problem As we now, with most PDE base approaches, ( x) can only be ientifie within the range of [-π, π]. In other wors, the phase is wrappe into the main value range. The real phase, ( x), shoul be continues as a function of ( x) : ( x) 2 mx ( ) ( x) (11)

7 where mx ( ) are integers. This is calle the wrapping problem, an the retrieving of mx ( ) is calle phase unwrapping Spatial Shift Unwrapping in SSE base FPP: The wrapping problem also exists in SSE approaches. From Equation (9) we have: h( x) u( x) (12) l Depening on hx ( ), anl, the shift function ux ( ) may tae any value as well. However, when s( x ) has a fringe structure with a perioic fringe of with λ, ux ( ) can only be etecte within the main value of [, ]. In other wors, ux ( ) is wrappe into[, ]. Thus the real shift function, which is enote as U( x ), shoul be continues as a function of ux ( ): U( x) m( x) u( x), where mx ( ) are integers. (13) In orer to emonstrate the relationship, we utilize the example in Figure 2. With hx ( ) shown in Figure 2 (a), we shoul have U( x ) in Figure 2 (b). However, what we have is ux ( ) as shown by Figure 2 (c). Use of ux ( ) in Equation (9) will result in significant error in hx ( ), as shown by Figure 2 (). Therefore, we must wor out a way to restore U( x ). The process is referre to as spatial shift unwrapping. Figure 2. Unwrappe an wrappe shift maps

8 3.2. Multi-wavelength Phase Unwrapping Algorithm: To solve the phase unwrapping problem, Zhang et al [18] introuce a multiple wavelength phase unwrapping algorithm. In his metho, a single fringe covers the whole measurement area is first projecte. That is, suppose the measurement area has a resolution of W H an the projecte fringe images are vertical. The wavelength of first fringe is W. Since there is only one fringe covere the whole measurement area, no phase unwrapping is necessary, hence 1 1( x) 1( x) an m 1 ( x). After the first fringe is projecte, a series of fringe images with a wavelength ecrease by a factor N from its previous wavelength is use. In Zhang s research, the wavelength, 1,2,3, N 1 is use, where N 1, 2,3,. As we now 1 ( x) 1 ( x), since 1 N 2, it will have 2 N 1. Combining with Equation (11), N1( x) 2( x) m2 ( x) Integer (14) 2 2 Here the operator Integer[] is to obtain the closest integer value. The wrappe phase, ( x) 2, can be unwrappe pixel by pixel by referring to the retrieve m ( x ) 2 : 2( x) 2 m2( x) 2( x) (15) After ( x) 2 is obtaine, it can be use to correct ( x) 3. Hence in general, for / 1 N, N ( ) ( ) 1 x x m ( x) Integer (16) 2 2 an ( x) 2 m( x) ( x) (17) Since the unwrappe phase is obtaine pixel by pixel without accessing its neighbourhoo pixels in the same phase map, an the noise of the longer wavelength will not significantly affect that of the shorter ones. Therefore, this technique can measure surface profile with arbitrary step height, an the measurement noise is close to that using a single-wavelength phase-shifting technique with the shortest wavelength Multi-wavelength Spatial Shift Unwrapping Algorithm: In orer to wor out how to unwrap the spatial shift, a simple spatial shift unwrapping metho has been propose in [25], which unwraps the spatial shift using the neighbourhoo pixels. However, this metho has the limitation which cannot retrieve the correct shift istribution if the value ifference of neighbourhoo pixels is greater than λ/2. Hence it cannot solve the unwrapping problem for those complex objects which have arbitrary step height. To aress this problem, we introuce a multiple wavelength unwrapping algorithm for spatial shift estimation approach base on Zhang s metho. In this algorithm, we still assume the measurement area has a resolution of W H an the projecte fringe images are vertical. Then the wavelength of the fringe which projecte in first time is selecte to be W, that is 1 W. There is no nee for spatial shift unwrapping since the single fringe stripe covers the whole measurement area. Hence we have u1( x) U1( x) an m 1 ( x). We then choose 2 1 / N. From equation (12), it is easy to fin the real shift function U( x ) has no relationship with wavelength λ, that is, U2( x) U1( x) U( x). Thus combining with Equation (13), we can have: U1( x) u2( x) m2 ( x) Integer (18) 2 2 This means the wrappe spatial shift, u ( x ) 2, can be unwrappe by referring to the longer wavelength shift U ( x ) 1 pixel by pixel:

9 U2( x) 2m2( x) u2( x) (19) Because shift U ( x ) 1 is only use to obtain the integer m ( x ) 2, the noise of U ( ) 1 x oes not significantly affect the noise of U ( x ) 2. After U ( x ) 2 is retrieve, we can use it to get ( ) 3 / 1 N, we have U x, where / N 3 2. So in general, for U ( ) ( ) 1 x u x m ( x) Integer (2) an U ( x) m ( x) u ( x) (21) From Equation (2), we can fin that the unwrappe spatial shift map is retrieve by referring to the longer wavelength shift map. Compare to those algorithms which use the neighbourhoo pixels to o the unwrapping, this metho can measure the surface profile with arbitrary step height. Since the noise of longer wavelength will not significantly affect the noise of shorter one, the measurement noise is close to the noise of using the shortest wavelength. This algorithm also has the avantage that a ranom noise reuction filter can be use to smooth the retrieve shift maps for longer wavelength hence increasing the robustness of the calculation. 4. Experiments an Results In orer to test the performance of the performance of the approach propose in Section III, experiments were carrie out in our laboratory. The experimental setup is shown in Figure 3. The igital projector use is HITACHI CP-X26, an camera is Duncan Tech MS31. The igital camera is place on top of the projector with a istance of 35 mm. The istance between the camera lens an the reference plan is 1295 mm. The resolution of the CCD camera is pixels, an the fiel of vision for CCD camera is 25mm 187mm. Hence, the equivalent spatial resolution is.1796 mm/pixel. Figure 3. The experimental system setup To verify the propose metho can measure the surface profile with arbitrary step height, we choose a mas as the measure object. A four-wavelength spatial-shifting algorithm with 1 48 an N = 2 is use in our experiment. The unwrappe ux ( ) is retrieve using Wu s metho et al [23]. Figure 4 (a) shows the mas an Figures 4 (b) (e) show the triangular fringe images of object with ifferent wavelength.

10 (a) (b) (c) () (e) Figure 4. Capture triangular fringe images of objects with ifferent wavelength. (a) object; (b) fringes image ( λ = 48 ); (c) fringes image ( λ = 24 ); () fringes image ( λ = 12 ); (e) fringes image ( λ = 6 ); Figure 5 shows the reconstructe 3D surface shape of the object using the propose unwrapping metho. It can be seen from these figures, the mas is reconstructe successfully. The height information for those places with high step rop (mas ege, nose part) is retrieve correctly. Some etails, such as mouth an eyes part, are also recovere. This result emonstrate that the propose metho can successfully measure those complex objects with arbitrary step height. Figure 5. 3D reconstruct results

11 5. Conclusion In this paper, we stuie the spatial shift wrapping problem associate with SSE-base FPP. The problem arises as the result of fringe reuse (that is, fringes perioic light intensity variance), an the spatial shift can only be ientifie without ambiguity with the range of a fringe with. We presente a technique to carry out spatial shift unwrapping to remey the problem. In orer to test the performance, we also carrie out experiments on an object with a mas which has complex surface shape. The results have shown the effectiveness of the propose unwrapping technique. [1] [2] [3] [4] [5] [6] [7] [8] [9] [1] [11] [12] [13] [14] [15] [16] [17] [18] [19] [2] [21] REFERENCES X. Su an W. Chen, Fourier transform profilometry: a review, Optics an Lasers in Engineering, vol. 35, pp , 21. H. Zhang, M. J. Lalor, an D. R. Burton, Spatiotemporal phase unwrapping for the measurement of iscontinuous objects in ynamic fringe-projection phase-shifting profilometry, Applie Optics, vol. 38, pp , June M. Halioua an H. C. Liu, Optical three-imensional sensing by phase measuring profilometry, Optics an Lasers in Engineering, vol. 11, pp , J. Li, H. Su, an X. Su, Two-frequency grating use in phase-measuring profilometry, Applie Optics, vol. 36, pp , Janurary X. Su, L. Su, W. Li, an L. Xiang, New 3D profilometry base on moulation measurement, Proceeings of SPIE, Vol. 3853, pp. 1 7, S. Toyooa an M. Tominga, Spatial fringe scanning for optical phase measurement, Optics Communications, Vol. 51, pp. 68 7, S. Toyooa an Y. Iwaasa, Automatic profilometry of 3-D iffuse objects by spatial phase etection, Applie Optics, vol. 25, no. 1, pp , May R. Roriguez-Vera an M. Servin, Phase loce loop profilometry, Optics an Lasers Technology, vol. 26, no. 6, pp , D. M. Meaows, W.. Johnson, an J. B. Allen, Generation of surface contours by moiré patterns, Applie Optics, vol. 9, no. 4, pp , April 197. A. Asuni an Z. Wensen, Unifie calibration technique an its applications in optical triangular profilometry, Applie Optics, vol. 38, no. 16, pp , June C. Wust an D. W. Capson, Surface profile measurement using color fringe projection, MVA, vol. 4, pp , P. Huang, Q. Ho, F. Jin, an F. Chiang, Colour-enhance igitial fringe projection technique for high-spee 3-D surface contouring, Optics Engineering, vol. 38, pp , A. J. Moore an F. Menoza-Santoyo, Phase emoulation in the space omain without a fringe carrier, Optics an Lasers in Engineering, vol. 23, pp , J. Villa, M. Servin, an L. Castillo, Profilometry for the measurement of 3-D object shapes base on regularize filters, Optics Communication, vol. 161, pp , Y. Hu, J. Xi, E. Li, J. Chicharo, an Z. Yang, Three-imensional profilometry base on shift estimation of projecte fringe patterns, Applie Optics, vol. 45, no. 4, pp , February 26. Y. Hu, J. Xi, J. Chicharo, W. Cheng, Z. Yang, Inverse Function Analysis Metho for Fringe Pattern Profilometry, IEEE Transactions on Instrumentation an Measurement, vol. 58, pp , 29. Y. Hu, J. Xi, J. Chicharo, an Z. Yang, Optimal filtering base shift estimation for fringe pattern profilometry by generalize analysis moel, in IEEE International Symposium on Intelligent Signal Processing an Communication Systems (ISPACS), Japan, December 26. S. Zhang, Phase unwrapping error reuction framewor for a multiple-wavelength phase-shifting algorithm, Optics Engineering, vol. 48, pp , 29. M. Taea, H. Ina, an S. Kobayashi, Fourier-transform metho of fringe-pattern analysis for computer-base topography an interferometry, Journal of the Optical Society of America A, vol. 72, pp , M. Taea an K. Mutoh, Fourier transform profilometry for the automatic measurement of 3-D object shapes, Applie Optics, vol. 22, pp , K. Itoh, Analysis of the phase unwrapping algorithm, Applie Optics, Vol. 21, Issue 14, pp , July 1982.

12 [22] [23] [24] [25] D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, an Software. John Wiley & Sons, K. Wu, J. Xi, Y. Yu an Z. Yang, 3D profile measurement base on estimation of spatial shifts between intensity ratios from multiple-step triangular patterns, Optics an Lasers in Engineering, vol.51, pp , 213. P. Jia, J. Kofman, an C. English, Multiple-step triangular-pattern phase-shifting an the influence of number of steps an pitch on measurement accuracy, Applie Optics, Vol. 46, pp , 27. P. Cao, J. Xi, J. Chicharo an Y. Yu, A Fringe Perio Unwrapping Technique for Digital Fringe Profilometry base on Spatial Shift Estimation, in Optical Inspection an Metrology for Non-Optics Inustries, Proceeings of SPIE Vol (SPIE, Bellingham, WA 29)