Digital fringe profilometry based on triangular fringe patterns and 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 4 Digital fringe profilometry base on triangular fringe patterns an spatial shift estimation Pu Cao University of Wollongong, pc4@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, "Digital fringe profilometry base on triangular fringe patterns an spatial shift estimation," in Proceeings of SPIE: Dimensional Optical Metrology an Inspection for Practical Applications III, 4, pp. 9C--9C-5. 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 Digital fringe profilometry base on triangular fringe patterns an spatial shift estimation Abstract In this paper, we present a new approach for the 3D measurement using igital fringe projection. Instea of sinusoial fringe patterns an the traitional phase shift etection, the propose technique maes use of triangular patterns an the spatial shift estimation for extract the 3D shape. The propose technique is avantageous not only by improve immunization to nonlinear istortion associate with igital projections, but also reuce computational buren for its implementation. Theoretical analysis an experimental results are also presente to confirm the effectiveness of the propose technique. Keywors shift, spatial, patterns, triangular, profilometry, fringe, estimation, igital Publication Details P. Cao, J. Xi, Y. Yu & Q. Guo, "Digital fringe profilometry base on triangular fringe patterns an spatial shift estimation," in Proceeings of SPIE: Dimensional Optical Metrology an Inspection for Practical Applications III, 4, pp. 9C--9C-5. This conference paper is available at Research Online:

3 Digital Fringe Profilometry Base On Triangular Fringe Patterns an Spatial Shift Estimation Pu Cao, Jiangtao Xi*, Yanguang Yu an Qinghua Guo School of Electrical, Computer an Telecommunications Engineering University of Wollongong, Wollongong, NSW5, Australia ABSTRACT In this paper, we present a new approach for the 3D measurement using igital fringe projection. Instea of sinusoial fringe patterns an the traitional phase shift etection, the propose technique maes use of triangular patterns an the spatial shift estimation for extract the 3D shape. The propose technique is avantageous not only by improve immunization to nonlinear istortion associate with igital projections, but also reuce computational buren for its implementation. Theoretical analysis an experimental results are also presente to confirm the effectiveness of the propose technique. Keywors: fringe pattern profilometry, 3D measurement, igital fringe projection. Introuction: In recent years, optical noncontact three-imension (3D) profile measurement has attracte increasing research efforts ue to many potential applications. Among other approaches, the Fringe Pattern Profilometry (FPP) base on Digital Fringe Projection (DFP) has been proven to be one of the most promising techniques ue to the avantages of simple system structure, flexible fringe pattern generation an high accuracy. Figure shows the system structure of a DFP base FPP, consisting of a igital vieo projector, a CCD camera an a reference plane. A frame of image with a particular fringe pattern prouce by the igital projector is caste onto the reference plane, an then onto the surface of the object when the reference plane is remove. The light reflecte from the reference plane an the object surface are capture by the CCD camera, with the later being a eforme version of the former ue to the variance of the height of the object surface. The eforme fringe pattern carries the information of surface shape, an 3D profile of the object can be retrieve from the two fringe patterns. h(x,y) Reference Plane Figure. Schematic iagram of FPP system. *jiangtao@uow.eu.au; phone 6 434; fax Dimensional Optical Metrology an Inspection for Practical Applications III, eite by Kevin G. Haring, Toru Yoshizawa, Song Zhang, Proc. of SPIE Vol. 9, 9C 4 SPIE CCC coe: X/4/$8 oi:.7/.4979 Proc. of SPIE Vol. 9 9C-

4 A number of FPP approaches have been introuce. The most wiely use methos are on the basis of phase etection (PD) or phase ifference etection (PDD) [-6]. In these approaches, the eforme fringe pattern is consiere as the result of phase moulation of the original fringe pattern, an hence etection of phase maps from original an eforme fringe patterns enables the retrieval of the 3D shape. However, PDD approaches also suffer from a number of isavantages. A major problem is the influence of nonlinear istortions inherent to igital vieo projection [7, 8]. Such istortions mae it ifficult for the original fringe patterns to be either sinusoial or ieal perioic, which are require by PDD base approaches. As an effort to solve the problem, Hu, et al. [9] propose an approach referre to as Spatial Shift Estimation (SSE) profilometry, which instea of etecting the phase ifference between the phase maps, is base on the estimation of spatial shift for corresponing pixels on the two fringe patterns. Compare to PDD base approaches, SSE techniques are avantageous in that non-sinusoial fringe patterns can be employe, an that they o not suffer from the nonlinear istortion associate with the igital fringe projection. However, these avantages are not fully exploite, as the fringe patterns are usually sinusoial in the existing wor on SSE [9,, ]. Although the influence of nonlinear istortions can be remeie, computational buren associate with the reporte wor on SSE is still rather heavy. In this paper, we will investigate the use of non-sinusoial fringe patterns with the aim to improve the efficiency in terms of computation. In particular, we propose to employ triangular patterns which will lea to significant reuction in computation buren in contrast to other commonly use ones, such as sinusoial fringe patterns. This paper is organize as follows. In Section we firstly present a brief escription of conventional PDD an SSE base FPP, incluing woring principles, system structures an relevant algorithms. Shortcomings an limitation associate with the use of sinusoial fringe patterns are iscusse in Section 3. In Section 4 we introuce a triangular fringe pattern instea of sinusoial base on theoretic analysis. Experiment results are presente in Section 5 which to verify the effectiveness an avantages of the triangular fringe patterns. Section 6 conclues the paper.. Problem Statement.. Principle of Triangulation Operation of FPP is base on the triangulation principle escribe as follows. As the image projecte has a fringe structure, without loss of generality we assume that light intensity varies perioically alone x irection, while eeping constant along y irection, as shown in Figure. Hence we can use sx ( ) an ( x ) to enote the variance of light intensity of the fringe pattern on the reference plane an object surface respectively. We will use h (x) to enote the height istribution of the object surface along x-coorinate. 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. From Figure, we can see 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 are able to obtain the following relationship to etermine the height of object at x : Note that reference plane. h( x ) l CD x enotes the coorination of point D. ) = () ( x h enotes the istance between points C an the CD is the istance between points C an D. The above relationship is the founation for FPP... PDD-base Profilometry Phase Difference Detection (PDD) is a class of wiely use approaches for FPP. With PDD, the fringe patterns projecte are sinusoial or perioic, which can be expresse as follows [, ]: + π ψ () = sx ( ) = bcos( fx+ ) an the eforme fringe pattern reflecte from the object surface is: Proc. of SPIE Vol. 9 9C-

5 + π φ ψ (3) = ( x) = b cos( f x+ ( x) + ) In the above expressions, f is the spatial frequency of the funamental component in the fringe patterns, an b is the amplitue of the -th orer harmonic component. ψ is the initial phase of the -th orer harmonic component. φ ( x) enotes the phase ifference between the -th orer harmonic components of these two fringe patterns, that is, the phase shift between C an D can be etermine by the spatial istance CD, that is [, ]: ( x) φ ( ) = π = π = φ( ) (4) fcd x fcd fcd x where φ = π is the phase shift of the funamental component. Substituting Equation (4) to Equation () we have: lφ ( x) hx ( ) = (5) π f Equation (5) shows that as long as φ ( x) can be etecte, we are able to calculate the height istribution hx ( ) of the object surface. This is the founation of all PDD base approaches. A number of fringe pattern analysis methos have been evelope to etect φ ( x), such as Fourier Transform Profilometry (FTP) [3], Phase Shifting Profilometry (PSP), Phase Measuring Profilometry (PMP) [4, 5, 6], Moulation Measurement Profilometry (MMP) [], Spatial Phase Detection (SPD) [3, 4], Phase Loc Loop (PLL) profilometry [5], Moiré Technique (MT) [6], colour-coe fringe projection [7, 8] an other methos [9, ]..3. Nonlinear Distortions associate with Digital Projection As mentione above, all PDD base approaches require projection of sinusoial or perioic fringe patterns. However, in practice, it is har to meet this requirement ue to the unesire factors inherent to igital projection. A major problem is the nonlinear projection luminous response, referre to as Gamma istortion [7, 8], which is introuce by visual isplay systems in orer to enhance human perception of the sensation of lightness. The istortion can be moelle as follows [7, 8]: wx ( ) = vx ( ) γ for u [,] (6) where vx ( ) is the image intensity function elivere to projector, an wx ( ) is the actual output image intensity istribution. γ is a fractional number within < γ < 3. Obviously, even if a pure sinusoial fringe pattern is elivere to the projector, the resulting one will no longer be a sinusoial ue the influence of Gamma istortion. In orer to overcome the nonlinear istortion, a number of methos have been propose. Guo, et al. [] propose a metho which approximates the gamma value using iterative statistical analysis of igital fringe patterns. However, it oes not wor if the single parameter moel in Eq. (6) is not able to accurately escribe the istortion, which always happens in practice. Baer, et al. [8] introuce a efocus metho to eal the gamma istortion, where the high orer harmonic waveforms are filtere out by means of efocusing the projector thus no aitional computation for correction or compensation is require. However, the requirement to ajust parameter for gamma moeling an efocusing is complex. Zhang, et al. [] also propose a phase error compensation metho by using a looup table (LUT) to store an compensate the phase error. This metho oes not employ a mathematical moel, which is suitable for nonanalytical situations. However, the result of this compensation metho is unstable since it only computes the gamma at center of projecte image. The accuracy of the metho is epene by the length of LUT. Hence it is time-consuming for high accuracy compensation. Zhang an Yau [3] then presente another LUT-base metho which requires no pre-computing of the gamma. However, the accuracy of the results is still epens on the length of LUT. Pan, et al. [4] introuce an iterative phase compensation algorithm base on the theoretical analysis of the phase error. This metho wors well for gamma compensation, but it is vulnerable to the influence cause by other factors, such as bacgroun brightness an reflectivity of reference plane. Liu, et al. [5] evelope a complicate gamma moel to increase the accuracy of compensation, while the computing time is also increase. Proc. of SPIE Vol. 9 9C-3

6 .4. SSE-base Profilometry As an effort to combat the istortion problem, Hu, et al. propose a class of approaches calle Spatial Shift Estimation (SSE) profilometry [9]. The iea of SSE base approaches is rather simple an straight forwar. Let us loo at Figure again. As x an x c are the points on ( x ) an sx ( ) with the same light intensity, that is =, we have: ( x) s( xc) x ( ) = sx ( ux ( )) (7) where ux ( ) = CD= x x, which is the spatial shift between x c an c all values of x, x can be replace by x, yieling the following: x an c Hence Eq. () can be expresse as: x. As the above erivation is vali for ( x) = s( x u( x)) (8) l u( x) h ( x) = (9) Equations (8) an (9) provie a straight forwar way to measure the 3D profile of the object surface. For every pixel on ( x ), we will locate the corresponing pixel on sx ( ) with the same intensity (i.e. ( x) = s( x u( x)) ), an then we wor out the spatial istance between the two points (i.e., ux ( )). The height of the object at the location of x can be etermine by Equation (9). By repeating the proceure for all pixels on ( x ) (incluing all the y values), 3D profile of the object surface can be obtaine. At iscusse above, the ey to reconstruct object surface using SSE is to obtain the shift istribution ux ( ) from ( x ) an sx ( ). A number of approaches were propose to achieve this [9,, ]. Among these approaches, the one referre to as Inverse Function base Shift Estimation (IFSE) [] is particularly interesting an briefe as follows. For a monotonic segment of sx ( ), there exists a function which is unique an the inverse function of sx ( ), that is: s s x = x [ ( )] Applying this inverse function s () v to the eforme signal ( x ), we have: s ( ( x)) = s { s[ x u( x)]} = x u( x) Hence the shift istribution function ux ( ) can be retrieve by: ux ( ) = x s ( x ( )) Now the ey problem is to obtain the inverse function s () v f v to approximate s () v (). A simple way is to employ the polynomial curve fitting, that is, employing a polynomial function (). As s [ s( x) ] = x, f () v can be etermine by minimization of the following average square error (i.e., the curve fitting error): N e = [( f( s( xi)) xi) ] (3) N i = where N is the number of ata samples, f () v is the polynomial of orer, that is, f( v) = av + a v av + a. The accuracy of the above estimation epens on an the characteristic of the inverse function, which then epens on that of sx ( ). The above spatial shift base approach has a great avantage. The projecte fringe patterns are no longer require to be sinusoial or perioic, leaing to an increase freeom for the selection of the fringe patterns. Also, the approach oes not suffer from the influence of nonlinear istortion. () () Proc. of SPIE Vol. 9 9C-4

7 3. Limitations of sinusoial fringe patterns for SSE-base profilometry Although SSE approaches provie more egrees of freeom in fringe pattern selection, in the existing wor on SSE [9,, ], sinusoial fringe patterns are still employe. The issue of choosing fringe patterns with the aim to achieve best efficiency an accuracy performance has remaine an open issue. As a matter of fact, use of sinusoial fringe pattern is never the best choice for IFSE base SSE approach. As escribe above a straight forwar metho to obtain the inverse function of sx ( ) is ata fitting by the following polynomial function of -th orer. The coefficients a, a, L, a, by the least square principle, can be etermine by solving the equation below: N N N N xi L xi yi i= i= i= a N N N N + xi xi x i a L xiyi i= i= i= = i= M M M M M a N N N N + xi xi x i xi y i L i= i= i= i= where N is the number of ata samples. From Equation (4), it is clear that with the increase of, there will be a significant increase in computational buren. Hence in orer to improve the efficiency in terms of computational buren, the polynomial shoul be as simple as possible, that is, its orer as low as possible. In orer to show the how the orer of the polynomial is relate to the shape of the fringes, we stuie the sinusoial fringe, an Figure shows the monotonic part of a sinusoial function, where the length of this segment is pixels. We performe the ata fitting on its inverse function with ifferent egrees of polynomial. Table shows the relationship between the curve fitting error e an, the egrees of polynomial use to fitting, which clearly show that the egree of the polynomial must be high enough to approximate the inverse function. (4) O6 5 6 I m (pws) Figure. Selecte monotonic interval of sinusoial waveform e e Table : curving fitting error in ifferent polynomial egree Proc. of SPIE Vol. 9 9C-5

8 4. SSE-base profilometry using triangular fringe patterns As analyze above, sinusoial fringe patterns are not the optimal choice for IFSE base SSE approach. Generally speaing, for a specific value of e, the more linear the inverse function s () v, the lower the orer. Hence we shoul choose the fringe pattern sx ( ) in such a way that its inverse function is as linear as possible. Base on this scenario we choose a triangular fringe pattern which is perioic an without loss of generality the light intensity within the first fringe perio can be expresse as follows: a x, when x T T s() x = (5) x a ( ), when T x T T where a is the contrast of the projecte fringe, which is in the range of [,]. T is the perio of fringe. The whole image is a perioic extension of sx ( ). Figure 3 shows an example of the pattern of 4 fringes with T = pix xels. h a =. an op E Figure 3. Triangle waveform From Equation (5), it is seen the triangular function is linear an monotonic over every half of its perio nt x ( n + ) T an ( n + ) T x ( n + )T, where n is the fringe inex. Obviously, such a linear function is very goo as its inverse function is also a linear function. Consiering sx ( ) within the first fringe perio, its inverse function is given by the following: T v, when x T a s () v = (6) v T ( ), when T x T a The above relation implies that a linear function is sufficient to estimate the inverse function if there is no istortion associate with the projection. This is very avantageous as only two coefficients are require for a pure linear function. In the existencee of nonlinear istortions, such triangular patterns still have great avantages, as the projecte patterns are still close to linear, an so oes the inverse function. Let us consier that the projecte triangular fringe patterns are subject to Gamma istortion epicte by Equation (5). The actual projecte pattern in the first fringe perio as follows: γ a x γ, when x T T s() x = (7) x a γ ( ), γ when T x T T Proc. of SPIE Vol. 9 9C-6

9 As γ is typically a fractional value within the range < γ < 3, it is easy to evaluate the minimal egree of the polynomial for it to approximate the inverse function with a given error. Figure 4 gives the results of numerical computation for the case when γ =. The x-axis is variable x, an y-axis is the results of y= f [ s( x) ]. When a perfect inverse function is employe, we shoul have y= f [ s( x) ] = x, which is the soli line. Using a 3 r orer polynomial, the ashe line is obtaine, which is very close to the soli line with a very small curve fitting error e =.3. Hence a thir orer polynomial is enough to estimate the inverse function. In contrast, when sinusoial fringe patterns are utilize, a th orer polynomial is require to yiel the same curve fitting error. Hence the triangular patterns are much better than the sinusoial patterns. li 9 8 iea inverse function approximate inverse function %= =3,= Figure 4. Curve of inverse functions 5. Simulation In this section, simulation was employe to test the performance of the propose triangular fringe pattern. A flat boar with its height nown as mm is simulate as the object surface. The fringe images were than istorte using gamma correction which the gamma value is set to.6. Figure 5. The simulate fringe images Figure 5 gives the simulate fringe pattern images. Figures 5 (a) an (b) show the sinusoial fringe pattern on the reference plane an on the object surface respectively, an Figures 5 (c) an () are using the triangular fringe pattern. Figure 6 (a) shows the height istribution of cross section of the flat boar at the central line. In these figure, x label stans for the length of flat boar (in pixels), an h label is the height (mm). Note that the egree of the polynomial for Proc. of SPIE Vol. 9 9C-7

10 the inverse function is selecte to be 3 (=3). Figures 6 (b) an (c) show the results using sinusoial fringe in the cases of =3 an respectively. h (mm) J x 6 Retrieve Height Distribution using Triangular Fringe, =3 h () 5 5 ar 4 5 Retrieve Height Distribution using Sinusoial Fringe, = Retrieve Height Distribution using Sinusoial Fringe, = Figure 6. Retrieve height istribution result (simulate boar) The reconstructe 3D surface shape of the object is shown in Figure 7. Figure 7 (a) is the result using triangular fringe in the cases of =3. The results which retrieve by using sinusoial fringe with ifferent polynomial egrees are also shown in Figures 7 (b) an (c) where the egrees are 3 an respectively. (a) 3D Reconstruct Result using Triangular Fringe, =3 Proc. of SPIE Vol. 9 9C-8

11 (b) 3D Reconstruct Result using Sinusoial Fringe, =3 (c) 3D Reconstruct Result using Sinusoial Fringe, = Figure 7. 3D reconstruct results (simulate boar) The error istribution of these reconstruct results is also calculate an shown in table : Triangular Fringe (=3) Sinusoial Fringe (=3) Sinusoial Fringe (=) Max Error (Positive) Max Error (Negative) Mean Square Error.494mm -.55mm.7mm.93mm -.33mm.96mm.3633mm -.394mm.84mm Table : error istribution of reconstruct results (simulate boar) From the results above we can see that, with same polynomial egree, the triangular fringe performs much better than sinusoial fringe. Also for the same level of accuracy, the egree of polynomial associate with triangular fringe patterns can be much lower than that with the sinusoial patterns. The egree of the polynomial has significant impact on the efficiency of the 3D measurement in terms of computational buren an hence the time require. For the examples stuie above where the egrees of the polynomials for sinusoial an triangular patterns are an 3 respectively, there will be a significant reuction in terms of computational buren with the propose approach. 6. Experiments an Results Experiments were also carrie out in our laboratory to verify the valiity of the propose metho. The experimental setup is shown in Figure 8. The igital projector use is HITACHI CP-X6, an camera is Duncan Tech MS3. 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 95 mm. Proc. of SPIE Vol. 9 9C-9

12 The first object we use is a flat boar where the height is 8mm. The resolution of the CCD camera is pixels, an the fiel of vision for CCD camera is 5mm 87mm. Hence, the equivalent spatial resolution is.796 mm/pixel. Figure 8. The experimental system setup The capture sinusoial fringe patterns on the object surface an on the reference plane are shown in Figures 9 (a) an (b) respectively, an Figures 9 (c) an () are capture triangular fringe patterns on the object surface an on the reference plane. In the figure, we have 3 fringes, each covering 3 pixels. la) Ib) (C) Figure 9. Fringe patterns observe (boar) II () We ivie the fringe into the monotonic intervals an applie IFSE approach to each of the intervals to obtain the height istribution of the boar. Figure (a) shows the height istribution of cross section of the boar at the central line. In these figure, x label stans for the length of boar (in pixels), an h label is the height (mm). Note that the egree of the polynomial for the inversee function is selecte to be 3 (=3). mm) 5 ' Retrieve Height Distribution using Triangular Fringe, = Retrieve Height Distribution using Sinusoial Fringe, =3 Proc. of SPIE Vol. 9 9C-

13 - - h (mm) o x Retrieve Height Distribution using Sinusoial Fringe, = Figure. Retrieve height istribution result (boar) We also give the results of height istribution which retrieve by using sinusoial fringe with ifferent polynomial egrees. Figures (b) an (c) show height istributions obtaine for the cross section of the boar, where the egree of the polynomials is 3 an respectively. (a) 3D Reconstruct Result using Triangular Fringe, =3 _i i-.i/! I (b) 3D Reconstruct Result using Sinusoial Fringe, =3 (c) 3D Reconstruct Result using Sinusoial Fringe, = Figure. 3D reconstruct results (boar) The 3D surface shape of the object is also reconstructe shown using triangular patterns as escribe above an the results are shown in Figure (a) where the egree of polynomial is 3. For comparison, use of the sinusoial patterns an polynomials of egree 3 an are also epicte in Figures (b) an (c). Since the height of this boar is alreay nown. The error istribution of these reconstruct results is calculate an shown in table 3: Proc. of SPIE Vol. 9 9C-

14 Triangular Fringe (=3) Sinusoial Fringe (=3) Sinusoial Fringe (=) Max Error (Positive) Max Error (Negative) Mean Square Error.89mm -.794mm.47mm.4787mm -.44mm.6493mm.5456mm mm.67mm Table 3: error istribution of reconstruct results (boar) It can be seen that using polynomials with the same egree 3, the propose triangular patterns yiel much higher accuracy, while in orer to achieve the same accuracy, the egree of the polynomial must be much higher (i.e. ) in contrast to that use for triangular patterns (i.e., 3). We also teste the propose technique using another object which is a ome where the max height is.8mm. In this experiment, the igital camera is still place on top of the projector with a istance of 35 mm. The istance between the camera lens an the reference plan is change to 95 mm. Figure shows the capture fringe patterns using sinusoial fringe pattern (a) an triangular fringe pattern (b). We have about 8 fringes, each covering 4 pixels. ENNEMINalis Figure. Fringe patterns observe (ome) Figure 3 shows the height istribution obtaine for the mile line in this ome. Figure 3 (a) is the result using triangular fringe pattern with polynomial egree =3, an Figures 3 (b) an (c) gives the reconstructe object shape using sinusoial fringe with polynomial egree =3 an = respectively. From these results, it can be seen that only the egree of polynomial equals to or above, the employment of sinusoial pattern can achieve the same accuracy as using triangular pattern with polynomial egree =3. h ( ) h( oc Retrieve Height Distribution using Triangular Fringe, =3 Proc. of SPIE Vol. 9 9C-

15 Retrieve Height Distribution using Sinusoial Fringe, =3 (mm) h(x) Retrieve Height Distribution using Sinusoial Fringe, = Figure 3. Retrieve height istribution result (ome) Figure 4 gives the reconstructe 3D surface shape of the object. Figure 4 (a) is the result using triangular fringe in the cases of =3. Figures 4 (b) an (c) show the results using sinusoial fringe in the cases of =3 an respectively. (a) 3D Reconstruct Result using Triangular Fringe, =3 (b) 3D Reconstruct Result using Sinusoial Fringe, =3 Proc. of SPIE Vol. 9 9C-3

16 (c) 3D Reconstruct Result using Sinusoial Fringe, = Figure 4. 3D reconstruct results (ome) From the results above we can still see the triangular fringe performs much better than sinusoial fringe with same polynomial egree an for the same level of accuracy, the egree of polynomial associate with triangular fringe patterns can be much lower than that with the sinusoial patterns. These experimental results successfully verifie the efficiency of the propose metho. 7. Conclusion In this paper, we emonstrate a new approach for implementing FPP, where the fringe patterns are triangular an the height istribution is calculate base on spatial shift estimation using inverse functions. In contrast to the phase etection base technique using sinusoial fringe patterns, the propose technique is avantageous by () better immunization to nonlinear istortions associate with igital projection, an () improve efficiency in terms of computational buren require for 3D measurement. The performance of the propose technique has been verifie by experiments. It shoul be pointe out that the propose approach is base on projection of a single fringe pattern, an the measurement accuracy is not as high as the approaches using multiple image patterns, such as phase shift profilometry (PSP). Also, the spatial shift is calculate on fringe-by-fringe basis, leaving some measurement errors at the joint points between ajacent fringes. As a future wor we will investigate the use of multiple triangular fringe patterns with the aim to improve the measurement accuracy. [] [] [3] [4] [5] [6] [7] [8] REFERENCES Taea, M., Ina, H. an Kobayashi, S., "Fourier-transform metho of fringe-pattern analysis for computer-base topography an interferometry," J. Opt. Soc. Am. 7 (), 56 6 (98). Taea, M. an Mutoh, K., "Fourier transform profilometry for the automatic measurement of 3-D object shapes," Appl. Opt. (4), (983). Su, X. an Chen, W., "Fourier transform profilometry: a review," Opt. Lasers Eng. 35, 63-84(). Zhang, H., Lalor, M.J. an Burton, D.R., "Spatiotemporal phase unwrapping for the measurement of iscontinuous objects in ynamic fringe-projection phase-shifting profilometry," Appl. Opt. 38 (6), (999). Halioua, M. an Liu, H.C., "Optical three-imensional sensing by phase measuring profilometry," Opt. Lasers Eng., 85-5(989). Li, J., Su, H. an Su, X., "Two-frequency grating use in phase-measuring profilometry," Appl. Opt. 36 (), 77 8(997). Baer, M.J., Chicharo, J.F. an Xi, J., "An Investigation into Temporal Gamma Luminance for Digital Fringe Fourier Transform Profilometers," in IEEE International Symposium on Intelligent Signal Processing, (7). Baer, M.J., Xi, J. an Chicharo, J.F., "Elimination of γ Non-linear Luminance Effects for Digital Vieo Projection Phase Measuring Profilometers," in 4th IEEE International Symposium on Electronic Design, Test & Applications, 496-5(8). Proc. of SPIE Vol. 9 9C-4

17 [9] [] [] [] [3] [4] [5] [6] [7] [8] [9] [] [] [] [3] [4] [5] Hu, Y., Xi, J., Li, E., Chicharo, J.F. an Yang, Z., "Three-imensional profilometry base on shift estimation of projecte fringe patterns," Appl. Opt. 45 (4), (6). Hu, Y., Xi, J., Chicharo, J.F., Li, E. an Yang, Z., "Discrete cosine transform base shift estimation for fringe pattern profilometry using generalize analysis moel," Appl. Opt. 45 (5), (6). Hu, Y., Xi, J., Chicharo, J.F., Cheng, W. an Yang, Z., "Inverse Function Analysis Metho for Fringe Pattern Profilometry," IEEE Trans. Instrum. Meas. 58 (9), (9). Su, X., Su, L., Li, W. an Xiang, L., "New 3D profilometry base on moulation measurement," Proc. SPIE 3853, 7(998). Toyooa, S. an Tominga, M., "Spatial fringe scanning for optical phase measurement," Opt. Commun. 5, 68 7(984). Toyooa, S. an Iwaasa, Y., "Automatic profilometry of 3-D iffuse objects by spatial phase etection," Appl. Opt. 5 (), (986). Roriguez-Vera, R. an Servin, M., "Phase loce loop profilometry," Opt. Laser Technol. 6, (994). Meaows, D.M., Johnson, W.O. an Allen, J.B., "Generation of surface contours by moiré patterns," Appl. Opt. 9 (4), (97). Wust, C. an Capson, D.W., "Surface profile measurement using color fringe projection," Mach. Vision Appl. 4, 93 3(99). Huang, P., Ho, Q., Jin, F. an Chiang, F., "Colour-enhance igitial fringe projection technique for high-spee 3-D surface contouring," Opt. Eng. 38 (6), 65 7(999). Moore, A.J., Menoza-Santoyo, F., "Phase emoulation in the space omain without a fringe carrier," Opt. Lasers Eng. 3, 39 33(995). Villa, J., Servin, M. an Castillo, L., "Profilometry for the measurement of 3-D object shapes base on regularize filters," Opt. Commun. 6, 3 8(999). Guo, H., He, H. an Chen, M., "Gamma correction for igital fringe projection profilometry," Appl. Opt. 43(4), 96-94(4). Zhang, S. an Huang, P.S., "Phase Error Compensation for a 3-D Shape Measurement System Base on the Phase Shifting Metho," Opt. Eng. 46 (6), 636(7). Zhang, S. an Yau, S., "Generic nonsinusoial phase error correction for three-imensional shape measurement using a igital vieo projector," Appl. Opt. 46 (), 36-43(7). Pan, B., Kemao, Q., Huang, L. an Asuni, A., "Phase error analysis an compensation for nonsinusoial waveforms in phase-shifting igital fringe projection profilometry," Opt. Lett. 34(4), 46 48(9). Liu, K., Wang, Y., Lau, D.L., Hao, Q. an Hassebroo, L.G., "Gamma moel an its analysis for phase measuring profilometry," J. Opt. Soc. Am. 7(3), (). Proc. of SPIE Vol. 9 9C-5