A fringe period unwrapping technique for digital fringe profilometry based on spatial shift estimation

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1 University of Wollongong Research Online Faculty of Informatics - Papers (Archive Faculty of Engineering an Information Sciences 29 A fringe perio unrapping technique for igital fringe profilometry base on spatial shift estimation Pu Cao University of Wollongong Jiangtao Xi Faculty of Informatics, University of Wollongong, jiangtao@uo.eu.au Joe F. Chicharo University of Wollongong, chicharo@uo.eu.au Yanguang Yu University of Wollongong, yanguang@uo.eu.au Publication Details Cao, P., Xi, J., Chicharo, J. F. & Yu, Y. (29. A fringe perio unrapping technique for igital fringe profilometry base on spatial shift estimation. In P. Huang, T. Toshizaa & K. Haring (Es., Optical Inspection an Metrology for Non-Optics Inustries, Proceeing of SPIE: Vol.7432 (pp Washington, USA: SPIE. Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library: research-pubs@uo.eu.au

2 A fringe perio unrapping technique for igital fringe profilometry base on spatial shift estimation Abstract In this paper e present a revie of the phase unrapping problem in Fringe Pattern Profilometry (FPP, base on hich e stuy the spatial shift rapping problem in spatial shift estimation (SEE base FPP. An approach for carrying out the spatial shift unrapping is propose ith its performance confirme by experiments. Disciplines Physical Sciences an Mathematics Publication Details Cao, P., Xi, J., Chicharo, J. F. & Yu, Y. (29. A fringe perio unrapping technique for igital fringe profilometry base on spatial shift estimation. In P. Huang, T. Toshizaa & K. Haring (Es., Optical Inspection an Metrology for Non-Optics Inustries, Proceeing of SPIE: Vol.7432 (pp Washington, USA: SPIE. This conference paper is available at Research Online:

3 A Fringe Perio Unrapping Technique for Digital Fringe Profilometry base on Spatial Shift Estimation Pu Cao, Jiangtao Xi*, Joe F. Chicharo an Yanguang Yu School of Electrical, Computer an Telecommunications Engineering University of Wollongong, Wollongong, NSW2522, Australia ABSTRACT In this paper e present a revie of the phase unrapping problem in Fringe Pattern Profilometry (FPP, base on hich e stuy the spatial shift rapping problem in spatial shift estimation (SEE base FPP. An approach for carrying out the spatial shift unrapping is propose ith its performance confirme by experiments. Keyors: fringe pattern profilometry, 3D measurement, phase unrapping 1. Introuction: In recent years, fringe pattern profilometry (FPP has attracte increasing research effort as an enabling technology for non-contact measurement of three-imensional (3D object surfaces. Among various system implementation schemes for FPP, the one base on igital vieo projection (DVP is particularly attractive ue to the avantages of simple system structure an controllable fringe patterns. Figure 1 shos the system structure of a DVP base FPP, consisting of a igital vieo projector, a CCD camera an a reference plane. With the system, a frame of image ith a particular fringe pattern is prouce by the igital projector an projecte onto the reference plane, an then onto the surface of the object hen the reference plane is remove. The projecte images from the reference plane an the object surface are capture by the CCD camera, ith 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 to fringe patterns. Figure 1. Schematic iagram of FPP system. *jiangtao@uo.eu.au; phone ; fax Optical Inspection an Metrology for Non-Optics Inustries, eite by Peisen S. Huang, Toru Yoshizaa, Kevin G. Haring, Proc. of SPIE Vol. 7432, SPIE CCC coe: X/9/$18 oi: / Proc. of SPIE Vol

4 Several FPP approaches have been evelope uring the past ecaes. The most iely 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 to 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 iely use. Although phase base approaches have been consiere as the most popular, they suffers from a number of eanesses. A major restriction is that fringe patterns must be either sinusoial or ieal perioic. Hoever 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 as propose by Hu, et al [15, 16], hich, instea of etecting the ifferences beteen the phase maps, is base on the estimation of spatial shift for corresponing pixels on the to fringe patterns. The approach is referre to as spatial shift estimation (SSE profilometry approach, or Generalize Analysis Moel (GAM base approach [15][16]. Phase unrapping is a major problem associate ith FDE-base FPP approaches. This problem arises because the phase ifference can only be etecte ithin 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 unrapping must be carrie out to obtain the actual phase maps. Many methos have been propose to solve the unrapping problem in FDE approach [17][18]. In the SSE-base approaches, spatial shift beteen corresponing pixels on the to fringe patterns can also be arbitrary, it can only be etecte ithout ambiguity ithin the range of [, T], here T is the ith of the iniviual fringe. Obviously, shift unrapping is also require in orer to correctly restore the 3D shape of the object surface. Hoever, spatial shift unrapping for SSE-base FPP is still an outstaning issue, hich motivate the or presente in this paper. This paper is organize as follos. In Section 2 e firstly present a brief escription of conventional PDE an SSE base FPP, incluing oring principles, system structures an relevant algorithms. Then in Section 3 e revie the phase unrapping problem in PDE base approaches, base on hich e inicate that a similar problem exists in SSE approach. Section 4 presents a metho for solving the problem. Finally in Section 5 experimental results are presente to emonstrate the effectiveness of the propose metho. 2. Principle of Fringe Pattern Profilometry 2.1. Principle of Triangulation FPP is base on the triangulation principle escribe as follos. As the image prouce by the projector has a fringe structure, ithout loss of generality e can assume that light intensity varies perioically alone x irection, hile eeping constant along y irection, as shon 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 ell 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 hat happens hen a beam of light is projecte onto the point D on the object. When the object is remove, the same light beam (hence ith the same intensity shoul be projecte onto point H on the reference surface, hich is reflecte bac to the camera through point C. As the triangles E c E P H an CDH are similar, e have the folloing relationship: CD l h( x x enotes the coorination positions of point D. x Note that reference plane, given by: = (1 ( h enotes the istance beteen points C an the Proc. of SPIE Vol

5 h( x l CD The above relationship gives the founation for FPP. = ( PDE base approaches for FPP The FDE base FPP utilize fringe patterns that are perioic an can be expresse as [19, 2]: + s( x = b cos(2πf x + ψ = an the eforme fringe pattern can also be expresse as: ( x = + = b cos(2πf x + φ ( x + ψ 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. ψ is the initial phase of the -th orer harmonic component, an φ (x enotes the phase ifference beteen the -th orer harmonic components of these to fringe patterns. Equations (3 an (4 sho that sx ( an ( x are relate by the phase shift φ (x. Let us consier the light beam projecte at point D on the object an H on the reference plane hen the object is remove. The phase shift beteen C an D can be etermine by the spatial istance CD, an hence e have [17, 18]: φ( x = 2πf CD = 2 π fcd = φ( x (5 here φ( x = 2π fcd is the phase shift of the funamental component. Substituting Equation (5 to Equation (2 e have: h ( x l φ ( x = (6 2πf As points D an H are arbitrary, the erivations shoul apply to all the points on the projecte fringe pattern. Therefore e have: h( x l φ ( x = (7 2πf Equation (7 shos that as long as the gives φ (x can be etecte, e are able to calculate the height hx of the object surface. This is the founation of all PED base approaches. istribution ( 2.3. Spatial Shift Estimation base FPP The FDE base FFP 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. Hoever, 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, 16] introuce a metho hich is base on the spatial shift estimation (SSE rather than PSE. Proc. of SPIE Vol

6 The SSE base approach is rather simple an straight forar. Let us consiercd, the istance beteen C an D again, hich 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 e have the folloing: u( x = (8 l h( x here ux ( = CD= xc x, hich is the spatial istance beteen x an x c. Note that x an x c are the points on ( x an sx ( having the same light intensity, that is ( x = s( xc. As the above erivation is vali for any x an x c, e can replace x by x, yieling the folloing: l u( x h ( x = (9 Note that ux ( is the spatial istance beteen a point x on ( x an the corresponing point on sx ( ith the same light intensity, that is: ( x = s( x u( x (1 Equations (9 an (1 provie a straight forar ay to obtain the 3D profile of the object surface. With ( x an sx ( available, if e are able to obtain ux ( to meet Equation (1, e then can utilize Equation (9 to yiel hx (, the height istribution of the object surface along x. 3D profile of the object surface. By repeating the proceure for all y e shoul be able to obtain the The spatial shift base approach has a particular avantage. The projecte fringe patterns are no longer require to be sinusoial, hich implies that even there are istortions ith the fringe patterns, sufficient three-imensional information on the object surface is containe in the variation beteen projecte an eforme fringe patterns. Thus the profilometry can be archive. This is the avantage of Generalize Moel. 3. The Unrapping Problem In this part, e first revie the unrapping problem in conventional FDE an SSE Phase Unrapping in conventional FDE: From Equation (7 e have h( x φ ( x = 2π (11 T l here T = 1/ fis the ith of an iniviual fringe. Obviously, φ (x can tae any value, epening on hx (,, l an T. Hoever, ith most PDE base approaches, φ ( x can only be ientifie ithin the range of[ π, π ]. In other ors, the phase is rappe into the main value range. In the folloing such a rappe phase is enote as φ ( x. Figure 2 shos an example hich emonstrates the ifference beteen φ ( x an φ ( x. Assuming e have an object ith its height istribution given in Figure 2 (a, the phase map φ ( x shoul be the one shon in Figure 2 (b base on Equation (11. Hoever, most PDE base approaches are only able to yiel a φ ( x shon in Figure 2 (c. If such a rappe φ ( x is use in Equation (7, e ill obtain a height istribution in Figure 2 (, hich obviously suffers from significant errors. Consequently, in orer to correct hx (, e must employ φ(x instea of φ ( x. Proc. of SPIE Vol

7 2 On' Hx go lou (a Original (x gun- - IOU Jr Jr ////z IOU (13 Wrappe (x N\\ ppe H lou ( Figure 2. Unrappe an rappe phase maps Phase unrapping refers to the process of restoring φ (x from φ ( x. As can be seen by Figure 2, the phase is rappe in the folloing ay. When φ (x increases an reaches points π,3π, 5π, (i.e., o number multiples of π, φ ( x rops from π to π. Similarly, hen φ (x varies ecreasingly an reaches the same points, φ ( x jumps from π to π. The phase unrapping shoul reverse the process. In other ors, hen e observe a phase rop from π to π, e shoul a 2π to the unrappe φ (x, an hen e notice a phase jump from π to π, e shoul a 2π Spatial Shift Unrapping in SSE base FPP: A rapping problem also exists in SEE approaches. From Equation (9 e have: h( x u ( x = (12 l Depening on hx (, anl, the shift function ux ( may tae any value as ell. Hoever, hen sx ( has a fringe structure ith a perioic fringe of itht, ux ( can only be etecte ithin the main value of [, T ]. In other ors, ux ( is rappe into[, T ], hich is enote as u (x an given as follos: ux ( u ( x = u( x T, here = Integer (13 T In orer to emonstrate the relationship, e utilize the same example in Figure 2. With hx ( shon in Figure 3 (a, e shoul have ux ( in Figure 3 (b. Hoever, hat e have is (x as shon by Figure 3 (c. Use of u (x in Equation (9 ill result in significant error in hx (, as shon by Figure 3 (. Therefore, e must or out a ay to restore ux (. The process is referre to as spatial shift unrapping. u Proc. of SPIE Vol

8 2 Original H(n ID 2 4 BOO (a 1 12 Original U(a OB///// \ \ BOO (13 Wrappe U BOO 4 2- Wrappe H(n DC 2 4 BOO ( N 1 12 Figure 3. Unrappe an rappe shift maps 3.3. Spatial Shift Unrapping in SSE base FPP In orer to or out ho to unrap the spatial shift, e can see ho ux ( is rappe into u (x. From Figure 3 (b an (c, e observe the folloing: When ux ( varies increasingly an reaches points T, 2T, 3T, (that is, integer multiples of T, u (x exhibits a rop of T ith its value ropping from T to. T, (x ill When ux ( varies ecreasingly an reaches the same points, that is, the integer multiples of jump from to T. Spatial shift unrapping shoul reverse process from u (x to ux (. In other ors, hen e observe a rop from T to, e shoul a T to the rappe result, an hen e notice a jump from to T, e shoul a T. Assuming that the object has a continuous surface, an u (x is acquire in iscrete form, that is, u ( i =,2,3,..., N, spatial shift unrapping shoul be carrie out by the folloing proceure: u Proc. of SPIE Vol

9 Step 1: Initialization u( x = u ( x + T, here is etermine by the height of the object at x. hx ( Because u ( x << TO, from Equations (12 an (13 e have ux ( = an l ux ( = Integer T Step 2: Starting from u ( x an for u, herei=,2,3,..., N ; If ux ( i increases folloe by a rop of T, that is, u ( xi u ( xi 1 = T, e shoul increase by 1, that is, = + 1; else if ux ( i ecreases folloe by a jump of T, that is u ( xi u ( xi 1 = T, then ecrease by 1, that is, = 1; otherise, eep unchange, that is, = ; Compute the unrappe shift by u( x = u ( x + T The above proceure is straight forar, but a number of issues must be resolve for its implementation in practice. Firstly, e shoul etermine if u increases or ecreases. Seconly e shoul be able to etect the sharp rop an jump. In ieal cases hen surface is continuous, sx ( i an are free of noise, these can be carrie easily. Hoever, in practice both sx ( i an may contain noise, resulting in u corrupte by noise an isturbance. In orer to eliminate the influence of noise an isturbance, e shoul mae sure that u is as smooth as possible ith respect to x i. To this en e propose the folloing: Pre-processing of sx ( i an by means of a igital filter in orer to remove the noise an to smooth the aveform of sx ( i an. The challenges associate ith pre-processing is to eliminate the unante noise an isturbance hile eeping the original aveform of sx ( i an. Parameters of these filters shoul be selecte ith care. In orer to etermine the slop of u, e firstly evaluate the ifference of u over successive samples by Δu x = u ( x u ( x, an then tae the sign of the variance by ( i i i 1 δ i = sign{ Δu } (here i Δu If ( xi < respectively. We carry out the folloing: i 1 1 δi >, then u increases, or M j= i M i 1 δ equals to 1, an -1 for Δu ( x >, Δ ( x = i u an 1 If δi <, then u ecreases. M j= i M In practice, a sharp jump or rop may transverse a fe ata samples on u. Assuming that the jump or rop occurs ithin L samples, e shoul use the folloing to etect the rop: If u ( xi u ( xi L > T2, then u jumps, or If u ( xi u ( xi L < T2, then u rops i Proc. of SPIE Vol

10 here T 2 is the threshol hich shoul be chosen base on the quality of u. With the pre-processing of T sx ( i an, u is smooth enough an hence e choose T = Experiments an Results In orer to test the performance of the performance of the approach propose in Section III, experiments ere carrie out in our laboratory. The experimental setup is shon in Figure 5. The igital projector use is HITACHI CP-X26, an camera is Duncan Tech MS31. The igital camera is place on top of the projector ith a istance of 35 mm. The istance beteen the camera lens an the reference plan is 1295 mm. We use a ome set on a flat boar as the object, shon in Figure 6(a, here the maximum height is 22.8mm. The iameter of the bottom surface of the ome is 99mm, an the thicness of the base boar is 16mm. The resolution of the CCD camera is pixels, an the file of vision for CCD camera is 25mm 187mm. Hence, the equivalent spatial resolution is.1796 mm/pixel. Figure 5. The experimental system setup The capture fringe patterns on the reference plane an on the object surface are shon in Figures 6 (b an 6 (c respectively. We have 9 fringes, each covering 96 pixels. Figure 6. Fringe patterns observe Proc. of SPIE Vol

11 In orer to emonstrate the propose approach, e loo at a cross section of the fringe patterns. Figure 7 shos sxy (, 1 an ( x, y 1 respectively here y 1 = 75, hich is the mile line of the ome. Obviously, both fringe patterns are istorte an corrupte by the noise. s(xyl Figure 7. Fringe ata acquire In orer to eliminate the influence of noise, e firstly employe a 128th orer FIR lo pass filter. Then e normalize the amplitue of each iniviual fringe perio of ( x, y 1 accoring to the corresponing fringe on sxy (, 1. The results of pre-processing are shon in Figure 8. s(xyl Figure 8. Fringe ata pre-processe The inverse function analysis approach [21] as employe to calculateuxy (, 1. With this metho, e firstly ivie sxy (, 1 an into segments, each covering a half fringe perio either monotonically increasing or ecreasing. For each iniviual segment, e foun the inverse function using polynomial fitting, an then applie the inverse function to the corresponing segment on ( x, y 1 to obtain the rappe u (, x y 1 in Figure 9 (a. With the propose unrapping approach e recovere uxy (, 1 in Figure 9 (b, ith hich e obtaine the height istribution hxy (, 1 Proc. of SPIE Vol

12 in Figure 9 (c. From Figure 9 (c, the max height of hxy (, 1 is 23.8mm an hence the measurement accuracy is 1.228%. Therefore e can say that shape of the object can be successfully retrieve ith the propose approach ID IOU (a Unrappe u(x ID IOU ( ' h(x Figure 9. Spatial shift unrapping an high istribution estimation Then e employe the propose for all ( x, y 1 ith y 1 = 1, 2,...,8 an the 3D shape surface shape of the object as constructe shon in Figure 1. Figure 1. 3D Reconstruct Results Proc. of SPIE Vol

13 5. Conclusion In this paper, e stuie the spatial shift rapping problem associate ith 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 ithout ambiguity ith the range of a fringe ith. We presente a technique to carry out spatial shift unrapping to remey the problem. In orer to test the performance, e also carrie out experiments on an object ith simple hemisphere surface shape. The results have shon the effectiveness of the propose unrapping 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 revie," Opt. Lasers Eng. 35, , 21. H. Zhang, M. J. Lalor, an D. R. Burton, Spatiotemporal phase unrapping 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," Opt. Lasers Eng. 11, pp , J. Li, H. Su, an X. Su, To-frequency grating use in phase-measuring profilometry, Applie Optics, vol. 36, pp , Janurary X. Su, L. Su, W. Li, an L. Xiang, Ne 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. Iaasa, 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. Meaos, 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 ithout 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, Z. Yang, E. Li, an J. Chicharo, Generalize analysis moel for fringe pattern profilometry, in IEEE Instrumentation an Measurement Conference (IMTC, Ottaa, Canaa, May 25, pp K. Itoh, Analysis of the phase unrapping algorithm, Applie Optics, Vol. 21, Issue 14, pp , July 1982 D. C. Ghiglia, M. D. Pritt, To-Dimensional Phase Unrapping: Theory, Algorithms, an Softare. John Wiley & Sons, 1998 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 , Y. Hu, J. Xi, E. Li, J. Chicharo, an Z. Yang, Fringe pattern profilometry base on inverse function analysis, Proceeings of 25 International Symposium on Intelligent Signal Processing an Communication Systems (ISPACS 25, December 25, pp Copyright IEEE 25. Proc. of SPIE Vol