A Directional Space-scale Based Analysis Method for Three-dimensional Profile Detection by Fringe Projection Technique

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1 International Journal of Optics and Applications 213, 3(5): DOI: /j.optics A Directional Space-scale Based Analysis Method for Three-diensional Profile Detection by Fringe Projection Technique Mohaed Bailich 1,*, Mustapha Bahich 2, Benaissa Bellach 1, Mohaed Rahoun 1 1 Equipe d électronique et télécounications, École Nationale des Sciences Appliquées, UniversitéMohaed Preier, Oujda, Morocco 2 Equipe d optique appliquée et traiteent d iages, Facultédes sciences Ben Msik, UniversitéHassan II Mohaedia, Casablanca, Morocco Abstract The optical ethods of etrology are widely used in science and engineering to easure a wide range of paraeters (such as deforations) that are related to the optical phase inforation coded in the intensity iages. Two of ostly used techniques, naely a phase shifting algorith (PS) and a wavelet transfor algorith (WT), are applied in order to estiate optical phase inforation, after an initial set of fringe patterns acquireent using the fringe pattern technique. In this paper, the 2D wavelet approach used to the phase retrieval proble for surface shape and object s defect easureent is presented and coparatively evaluated. Ai of this paper is the in-depth investigation of the advantages of the 2D wavelet approach and the gain of a better insight on its use in the industrial context. Extensive work has been conducted for investigating the influence of several factors (such as the noises, the surface variation, the odulation rate, etc.) on the perforance of the wavelet phase retrieval technique, while a detailed analysis of the obtained results is also given. Keywords Coputer vision, Defect detection, Fringe projection technique, Phase retrieval, Wavelet transfor 1. Introduction The phase retrieval in optical etrology is very iportant process for any industrial applications[1]. Many phase retrieval algoriths have been conceived which differ in the adopted approach such as Fourier transfor ethod[2], phase shifting[3], direct phas e detection and wavelet transfor ethod[4]. It is well known in the digital signal processing theory that the wavelet transfor and tiefrequency analysis techniques are ore suitable to analyze non-stationary signals than stationary ones[5]. One of the tie-frequency analysis techniques that have been used successfully in fringe pattern deodulation is the wavelet transfor. This technique has been dilated to exploit its suitability for fringe pattern deodulation. Two-diensional continuous wavelet transfor (2D-CWT) techniques have been used successfully to deodulate fringe patterns and retrieve the phase even when the recording process is accopanied by the presence of noise[6]. In this paper we propose the use of the 2D-CWT algorith to deodulate fringe patterns and the study of soe environental * Corresponding author: bailich@gail.co (Mohaed Bailich) Published online at Copyright 213 Scientific & Acadeic Publishing. All Rights Reserved paraeters effects on the accuracy of results. In the following section, the fringe pattern technique is introduced. The background theory of the 2D-CWT fringe pattern deodulation algorith is discussed in section 3. Coputer-generated and real fringe patterns are used to test and discuss the effects of previously entioned paraeters on the phase retrieval algorith reliability, and we conclude in sections 4 and 5 respectively. 2. Fringe Projection Technique The fringe projection is an optical etrology technique that offers a topographic description of an illuinated surface (the object of test). This technique is based on the projection of a rectilinear fringes network fro a video projector on the studied surface (or the object) which its distortions appear as a odification of the fringes orientation in the x & y axis[7]. This odification results fro a variation of the phase distribution that is directly related to the object distortions: 2 x,y tan.z x,y (1) p Where is the phase, p is the interfringe, z is the distortion of interest and θ is the angle the CCD caera and the video projector.

distribution that codes the projected fringe pattern.

2 112 Mohaed Bailich et al.: A Directional Space-scale Based Analysis Method for Three-diensional Profile Detection by Fringe Projection Technique Therefore, to deterine the z distortion we should retrieve first the phase distribution that codes the projected fringe pattern. There are any algorith that can be used to deodulate fringe patterns, such as Fourier transfor based algorith, phase stepping, digital phase locked loop, direct phase detection and wavelet transfor based ethod[8]. Video projector Caera cos sin sin cos R (4) The coplex Morlet wavelet that will be used in this work is essentially a plane wave within a Gaussian window and is given by x, y expik xcos y sin exp x y 2 (5) With is the rotation angle. The Fourier transfor of this wavelet is s r r k s k ˆ, exp cos sin 2 (6) Figure 1. Fringe projection principle 3. Wavelet Fringe Pattern Analysis Proble Generally the intensity of a fringe pattern can be expressed as follows: I x,y I 1V cos y x,y (2) Where I, V and are the average intensity, the fringe visibility and the odulation rate respectively. x, y is the phase of fringe pattern and x&y are the siple indices for the x and y axis respectively. A fringe pattern tends to reseble to non-stationary signals. This otivates researchers to investigate wavelet transfor as a tie-frequency technique for fringe pattern deodulation proble. The wavelet transfor can detect the local characteristics of signals and obtain a tie frequency description of it[9]. A wavelet is an oscillating function with a copact width and a zero ean. A faily of 2D analyzing wavelets are generated by translating the other wavelet via the location paraeter R 2, dilating it with the scale paraeter s> and rotating it by the angle. It can be expressed as[1]: 1 R s x s (3) s,, R designates the rotation atrix by the angle α by: Figure 2. 2D Morlet wavelet, (a) Real part, (b) Iaginary part The 2D-CWT of a two-diensional signal f is given 1 x Wf s,, f 2 x R dx (7) R s s R 2,s,, 2 f x, y is projected onto Where In 2D-CWT, the fringe pattern the wavelets s,, basis and the resulted wavelet

International Journal of Optics and Applications 213, 3(5): 111-117 113 coefficients s,, W f reach their axial value when the two diensional analyzing wavelet and the fringe pattern are locally ore

Then, the ridge of the wavelet coefficients W s, for each pixel in the fringe pattern with the array coordinates is detected and its corresponding coefficient arguent will be taken as the pixel s

3 International Journal of Optics and Applications 213, 3(5): coefficients s,, W f reach their axial value when the two diensional analyzing wavelet and the fringe pattern are locally ore siilar and their frequencies are very close. To deodulate a fringe pattern, its wavelet transfor is first calculated. Then, the ridge of the wavelet coefficients W s, for each pixel in the fringe pattern with the array coordinates is detected and its corresponding coefficient arguent will be taken as the pixel s phase: ridge( )= W(s,, ) = ax( W(s,, ) ) (8) I W s,, arctg ReW s (9),, Where s and are the scale and rotation angle corresponding to ridge value. 4. Results and Discussions In the nuerical siulation, a coputer generated phase that is shown in Fig.3-a and is described by the following equation is generated. xy, x y 2 2 exp (1) The resultant odulated fringe pattern by the phase distribution is shown in Fig.3-b and given by the equation I x,y 1.5 cos y x,y (11) With the odulation rate is set to.6 radian/pixel. Figure 3. (a) A coputer-generated object, (b) its corresponding fringe pattern We suppose that the acquisition of the fringe pattern is ade by a CCD caera with resolution 512 x 512 and the pixel diension equals to 7.4 x 7.4. There are different noise types in real world. Multiplicative noise is coon beside additive noise. Multiplicative noise, that is naturally dependent on the iage data, is ultiplied by (not added to) the original iage. In this study, the ultiplicative noise that will be used is the known as speckle noise. To analyze the noise iunity of the wavelet analysis algorith, we apply it to several exaples of the previous fringe pattern with both additive and ultiplicative increasing Gaussian noise levels. The fig.4-a presents, in the case of additive noise, the plots of RMS (Root ean square) errors of the estiated phase distributions with the noise variance for the both frequency and phase based algoriths. The Fig.4-b presents the obtained RMSE results in the ultiplicative noise case. The analysis of these results shows clearly that the phase based wavelet analysis algorith gives the best values of the phase distribution with less error values. This is certainly due to the fact that the phase distribution is slowly variable and it presents sall spatial frequencies values that are very affected by noise and especially in the presence of ultiplicative one. Another issue that affects the perforance of the fringe pattern deodulation algorith is the high spatial frequencies variance of the surface object. Fig.5 shows the RMS error values of the obtained phase with the object spatial frequency increent β. With increasing values of β, as shown in Fig.5, the RMSE values of the estiated phase increase for both of algoriths. It can be seen that RMSE goes on reducing as the object surface is slowly variant and while β value keeps under 6 where can be observed that the algoriths lead to an accurate phase coputation. Notice that in this part, we fix the odulation rate with varying the axial spatial frequency that the phase distribution contains. The results analysis confirs the theoretical clai that for s all values of the axial s patial frequency that are less than the odulation rate the wavelet phase retrieval ethod exhibits a higher precision, whereas for the higher values of the axial spatial frequencies than the ethod couldn t estiate accurately the phase ap. Fig.6 shows effect of the variation of carrier frequency[11] (i.e odulation rate) on the accuracy of the estiated phase ap. In Fig.6, which is the case where the axiu spatial frequency containing in the phase distribution is 1.6 rad/pixel, the RMSE is considerably high when the odulation rate is less than the axiu high frequency. When the odulation rate is higher than 1.6 rad/pixel, the obtained RMS error is constant as can be seen fro Fig.6 and has an average of.1 rad. The phase recovery will reain accurate if a spatial carrier is added so as to reove the deforation sign abiguity, and at the sae tie, to reduce the noise effect.

4 RMS Error (rad) RMS Error (rad) 114 Mohaed Bailich et al.: A Directional Space-scale Based Analysis Method for Three-diensional Profile Detection by Fringe Projection Technique phase estiation frequency estiation Noise variabce (a) 1.5 phase estiation frequency estiation Noise variance (b) Figure 4. RMS Errors versus noise variance. (a) additive noise, (b) ultiplicative noise

5 RMS Error (rad) International Journal of Optics and Applications 213, 3(5): phase estiation frequency estiation Beta varying Figure 5. RMS Error versus axial spatial variation Figure 6. RMS Error versus odulation rate In the experiental work, a ain part of a doll body was used as test object to easure its height profile (Fig.7). To obtain the height variation of reference plane and the carrier frequency of the projected fringes in the iage plane, the CCD caera was focused to the center of the projected fringe pattern and then reference iage was stored to caera s eory as RGB iage. After this, the ain part of the doll s body was put on the reference plane and object

distribution of object is calculated by subtracting the carrier using the reference phase and the height distribution of the object that was calculated using experiental paraeters is shown in Fig.9.

The fringe projection setup The fringe projection syste sensitivity increases with the projection angle; however, a large angle between the projection and viewing directions ay produce an unwanted

6 116 Mohaed Bailich et al.: A Directional Space-scale Based Analysis Method for Three-diensional Profile Detection by Fringe Projection Technique iage stored into the caera as shown in Fig.8. These stored two iages later downloaded to coputer for analysis. distribution of object is calculated by subtracting the carrier using the reference phase and the height distribution of the object that was calculated using experiental paraeters is shown in Fig.9. Figure 7. The fringe projection setup The fringe projection syste sensitivity increases with the projection angle; however, a large angle between the projection and viewing directions ay produce an unwanted shadow on the object. This shadow will lead to areas with issing data where the object cannot be contoured. To overcoe this proble and provide the best results, an angle no larger than the largest slope on the surface should be chosen, and also by using an appropriate phase unwrapping algorith that can identify the low quality shadow areas in the wrapped phase ap and produce an alost optial result for the. The use of ultiple projectors in fringe projection profiloetry has also been reported very useful to solve soe probles like the accurate 3D estiation in presence of local shadows[12]. Figure 8. Projected sinusoidal fringes on ain part of the doll The procedure outlined above was used to extract the phase fro the real iage that is shown in Fig.8. The 2D continuous wavelet transfor of reference iage and object iage were first calculated in the Matlab environent. The wrapped phase of the reference fringe iage and object fringe iage were calculated fro their coplex values and an appropriate unwrapping procedure was used to correct discontinuities in the wrapped phases. Finally phase Figure 9. 3D height profile of ain part of the doll 5. Conclusions This paper has studied the wavelet phase retrieval technique and has evaluated and copared its capabilities in ters of deodulating projected fringe patterns. Two-diensional continuous wavelet transfor has been used to retrieve the phase distribution of a fringe pattern. We conclude that the wavelet phase based algorith should be used when the signal to noise ratio of fringe pattern is high and also when the rapid variations (i.e high spatial frequencies) present. We has also seen that the accuracy of the estiated phase is satisfactory for the cases where the signal to noise ratio of the projected fringe pattern is low, even without fringe pattern filtering. The detailed investigation of the technique presented here has revealed that the Morlet wavelet, in conjunction with the high frequency carrier (i.e high odulation rate) perfors better in ters of noise iunity and accuracy in deodulating fringe patterns. In conclusion, both siulation and physical experients show that the directional wavelet based fringe analysis algorith is able to provide surface height easureents with a better reliability using one single fringe pattern that can easily be recorded by the detection syste. As a future work, we tend to investigate ore directional transfors for fringe pattern processing and etrological probles solutions. ACKNOWLEDGEMENTS The authors want to show gratitude to the reviewers for their helpful coents and suggestions.

7 International Journal of Optics and Applications 213, 3(5): REFERENCES [1] A. Patil and P. Rastogi, Moving ahead with phase, Opt and Lasers in Eng. 45 (27) [2] M. Takeda, H. Ina and S. Kobayashi, Fourier-transfor ethod of fringe-pattern analysis for coputer-based topography and interferoetry, JOSA. 72 (1982) [3] Song Zhang, Shing-Tung Yau, High-resolution, real tie 3D absolute cordinate easureent based on phase-shifting ethod, Opt. Express 14 (26) [4] M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassi, M. Sidki, S. Rachafi, Paul wavelet-based algorith for optical phase distribution evaluation, Optics Co 211 (22) [5] C. Torrence and G. P. Copo, A practical guide to wavelet analysis, Bulletin. Aerican Meteo. Soc. 79 (1998) [6] M Bahich, M Bailich, A Iloul, M Afifi, EM Barj, A coparative study of one and two-diensional wavelet-based techniques for noisy fringe patterns analysis, Opt. Co. 29 (213) [7] C. Quan, W. Chen, C.J. Tay, Phase-retrieval techniques in fringe-projectionprofiloetry, Opt. Lasers. Eng. 48 (21) [8] T. Yoshizawa, Handbook of Optical Metrology: Principles and Applications, CRC Press, 29. [9] Q. Keao, H. S. Seah and A. Asundi, Wavelets in optical etrology, Persp. Eng. Optics. (23) [1] J.-P. Antoine, R. Murenzi, Two-diensional directional wavelets and the scale-angle representation, Signal Proc. 52 (1996) [11] M Bahich, M Afifi, EM Barj, A nuerical spatial carrier for single fringe pattern analysis algorith, Optik. 122 (211) [12] Y. Cai, X. Su, Inverse projected-fringe technique based on ulti projectors, Opt. Laser Eng. 45 (27)

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