Improved Fourier-transform profilometry

Transcription

1 Impoved Fouie-tansfom pofilomety Xianfu Mao, Wenjing Chen, and Xianyu Su An impoved optical geomety of the pojected-finge pofilomety technique, in which the exit pupil of the pojecting lens and the entance pupil of the imaging lens ae neithe at the same height above the efeence plane no coplana, is discussed and used in Fouie-tansfom pofilomety. Futhemoe, an impoved finge-patten desciption and phase-height mapping fomula based on the impoved geometical genealization is deduced. Employing the new optical geomety, it is easie fo us to obtain the full-field finge by moving eithe the pojecto o the imaging device. Theefoe the new method offes a flexible way to obtain eliable height distibution of a measued object Optical Society of Ameica OCIS codes: , , Intoduction Fouie-tansfom pofilomety 1 (FTP) is one of the popula 3D sensing methods, 1 10 in which a gating patten is pojected onto an object, the defomed finge patten is Fouie tansfomed and pocessed in its spatial fequency domain as well as in its space-signal domain, and then the depth distibution of the test suface is econstucted. Howeve, in taditional FTP, the convesion fom phase to height is deduced depending on the supposition that not only ae the pojecto and the camea at the same height above the efeence plane, but also thei axes must coss at the same point on the efeence plane. When these conditions ae too stict to be satisfied, a lage measuement eo will be intoduced. In this pape what is believed to be a new phaseheight mapping fomula based on an impoved desciption of a efeence finge patten and a defomed finge patten in FTP is poposed in which the pojecto and the imaging system can be set feely as long as a full-field finge patten can be obtained. A stict theoetical analysis about the finge desciption as well as the convesion between the phase and the height ae given. This is a geneal fomula to obtain the eliable height distibution of a measued object, while the convesion fomula between the The authos ae with the Depatment of Optoelectonics, Sichuan Univesity, Chengdu, China X. Mao s addess is mx9971f@163.com. Received 30 May 2006; accepted 26 Septembe 2006; posted 5 Octobe 2006 (Doc. ID 71427); published 25 Januay /07/ $15.00/ Optical Society of Ameica phase and height 1 of the taditional FTP is just a special case. Employing the impoved FTP method, it is easie fo us to obtain the full-field finge by adjusting the pojecto o the imaging device. Both compute simulations and expeiments have veified ou analysis. 2. Pinciple of the Method A. Phase Calculation The impoved optical geomety of the FTP method is shown by the solid lines in Fig. 1. I 1 is the exit pupil of the pojecto, and I 20 is the entance of the CCD camea. The optical axis I 1 O cosses the efeence plane R at point O, and the optical axis I 20 O 1 is vetical to the efeence plane and cosses it at anothe point O 1. Note that the connecting line I 1 I 20 is not paallel to the efeence plane and that I 1 O and I 20 O 1 ae not coplana. D x, y and C 1 ae points on the 3D object and on the efeence plane, espectively, and they ae imaged at the same point on the CCD aay. I 1 A is a ay though the D point to the efeence plane. We add seveal dashed lines in the figue as guidelines fo ou analysis. I 1 H paallels the efeence plane and cosses I 20 O 1 at point H (I 1 H is not in the figue plane). I 1 K is vetical to the efeence plane R. We then daw a line I 2 O though point O, which paallels the optical axis of the CCD, and point I 20 is otated to point I 2 aound point D, and C 1 is otated to point C, which must be on line OA. BD is vetical to plane R, which denotes the height of point D on the measued object. I 1 F paallels the efeence plane R and cosses I 2 O and the extension of BD at points G and F, espectively. I 2 D cosses I 1 F at point P. 664 APPLIED OPTICS Vol. 46, No Febuay 2007

2 Fig. 1. Geometical sketch map. At last we daw guidelines HG and OO 1. We define O 1 I 20 OI 2 L, I 1 G KO, I 1 I 20 s, HI 1 I 20 1, BCD, and BAD. The finge patten on the efeence plane may be expessed as 8 I 1 x, y I 0 1 cos 2 xf x, (1) whee I 0 is the illumination light intensity and f x is the spatial fequency of the efeence finge along the x axis, which can be expessed by x sin f x f cos 1 I 1 O, (2) whee f is the caie fequency of the gating. Compaed with the case in which the imaging axis cosses the pojecting axis at the same point on the efeence plane, if the influence on the CCD imaging quality is neglected, moving the CCD just causes the movement of the image s location on the CCD aay, wheeas the elationship among the adjusted pixel points is unchanged. Theefoe I 2 O can be egaded as a vitual optical axis. In OI 1 G, I 1 O sin, and consideing I 2 G I 20 H, we can obtain I 1 x, y I 0 1 cos 2 fx cos 1 2x sin2, (3) Fig. 2. whee x, y is the phase distibution caused by the height vaiation h x, y, which is expessed as 1 x, y 2 f CA cos, (6) whee CA can be calculated by the following pocess. Because ABD is simila to DI 1 F, and BCD is also simila to DPF, the following expession can be obtained: CA BD I 1P DF I 1 P L s sin 1 BD. (7) Analyzing simila tiangles PI 2 G, OI 2 C, and BCD, we obtain PG I 2 G I 1P OC s sin 1 L Simulated object. BC BD x L BD. (8) Substituting Eq. (8) into Eq. (7), we obtain BD CA L s sin 1 BD xs sin 1 L BD, (9) whee tan. (4) L s sin 1 When the measued object is placed, the defomed finge patten can be expessed as I 2 x, y I 0 1 cos 2 fx cos 1 2x sin2 x, y, (5) Fig. 3. Defomed finge distibution when Febuay 2007 Vol. 46, No. 5 APPLIED OPTICS 665

3 Fig. 4. (a) Calculated height eo when 1 0 ; (b) calculated height eo when 1 10 ; (c) calculated height eo when 1 30 ; (d) calculated height eo when 1 50 ; (e) calculated height eo when 1 10 ; (f) height eo distibution using the taditional method when so that x, y 2 f cos BD L s sin 1 BD xs sin 1 L BD. (10) Afte simplification, the efeence finge and the defomed finge can be ewitten as the following simple foms: I 1 x, y I 0 1 cos 2 f 0 x 0 x, y, (11) I 2 x, y I 0 1 cos 2 f 0 x x, y, (12) whee f 0 f cos, and 0 x, y and x, y ae the phase distibutions included in the efeence finge and the defomed finge. I 1 x, y and I 2 x, y ae Fouie tansfomed and pocessed in thei spatial fequency domain as well as in thei space-signal domain; then 0 x, y and x, y can be calculated, and the coe phase distibution x, y that diectly coesponds to the height distibution of the tested object can be obtained 11 : x, y x, y 0 x, y. (13) B. Phase-Height Mapping In this subsection the convesion fomula between the phase and the height will be deduced. Analyzing ABD and CBD we obtain CA BD cot cot, (14) whee cot OC L C 2 f 0 L ( C denotes the phase of finge patten at point C on the efeence plane). Because OA A 2 f 0 D 2 f 0 ( D and A epesent the phase of finge patten at point D on the object and the phase of finge patten at point A on the efeence plane, espectively), CA OA OC DC 2 f 0. In the optical geomety as shown in Fig. 1, DC 0, thus Analyzing AKI 1 we obtain CA DC 2 f 0. (15) OA 2 f 0 D cot L s sin 1 2 f 0 L s sin. (16) 1 Table 1. Mean-Squae Deviations and the Maximal Height Distibution Coesponding to Diffeent Angles Angle (degees) Mean-squae deviations Maximal height (mm) APPLIED OPTICS Vol. 46, No Febuay 2007

4 Substituting Eqs. (15) and (16) and cot OC L C 2 f 0 L into Eq. (14), afte simplification, we obtain BD DC L L s sin 1 2 f 0 L L DC C s sin 1, (17) whee L, s, 1, and can be measued diectly and f 0 f cos. In addition, the DC epesents the phase diffeence between the object and efeence planes that can be calculated by Eq. (13), and C is the phase of point C on the efeence plane. If 1 0 and d, Eq. (17) can be simplified as Fig. 6. Defomed finge of a 3D object. L DC BD, (18) 2 f 0 d DC which is just the taditional phase-height mapping fomula. 1 Theefoe Eq. (17) is a geneal fomula fo obtaining the eliable height distibution of a measued object, while the convesion fomula between the phase and height 1 of the taditional FTP is just a special case of ou fomula. 3. Compute Simulations In this section, some compute simulations ae used to veify ou method. The optical geomety is shown in Fig. 1. Assume that the angle 1 is positive if point I 20 is above the point H; othewise, it is negative. The simulation object is built by the function 4*peaks(512) in MATLAB language, as shown in Fig. 2, in which the maximal height is h mm. The simulated defomed gating with peiod T 16 pixels is as shown in Fig. 3 when 1 0, the size of the image is pixels, and the system paametes ae 800, s 400, and L 2000 mm. When we adjust the position of the CCD to fom 1 0, 10, 30, 50, the econstucted height eo distibutions using ou method ae shown in Figs. 4(a) 4(d) fo each case. Figue 4(e) is the econstucted height eo distibution coesponding to In addition, when 1 50, a compaison expeiment is caied out to show that thee is a lage eo when the conventional phase-height mapping algoithm is used, as shown in Fig. 4(f): The eo (mean-squae deviation) ises to , and the maximal height is mm; when the new method is used, the eo (mean-squae deviation) is , and the maximal height is mm, as shown in Fig. 4(d). In Table 1, the mean-squae deviation and maximal height distibutions coesponding to diffeent angles ae listed. A detailed analysis of the eo will be discussed in a futue wok. In addition, we should note that the new method can coectly econstuct a 3D height distibution even if 1 90, which is equivalent to the case in which the pojecto emains stationay but the CCD otates aound the pojecto on the coesponding spheical suface is the case whee the CCD is just above the pojecto i.e., the two axes ae coplana, but the pojecto and the camea ae not at the same height. 4. Expeiment A pimay expeiment is used to veify ou method in which a model with the maximal height h 32.5 mm is measued. A finge patten with sinusoidal intensities is pojected though a pojecto (PLUS U3-880) on the test object. The pojecto and the CCD (MTV-1881EX) ae not at the same height above the efeence, and the optical axes of the pojecto and that of the CCD ae not in a common plane. The system paametes ae and s mm, Fig. 5. Refeence finge. Fig. 7. Coect height econstuction. 10 Febuay 2007 Vol. 46, No. 5 APPLIED OPTICS 667

5 5. Conclusion An impoved desciption about a efeence finge and a defomed finge and a phase-height mapping fomula have been deduced in the pape. Not only ae they suitable fo dealing with the case in which the connecting line between the exit pupil of the pojecto and the entance pupil of the imaging device ae nonpaallel to the efeence plane, but also fo the case in which the pojecting optical axis and the imaging axis ae not in a common plane. Employing the new theoy it is easie fo us to obtain the full-field finge by adjusting eithe the pojecto o the imaging device. Ou analysis shows that the taditional FTP aithmetic is just a special example of ou method. This poject was suppoted by the National Natue Funds of China ( , ). Fig. 8. Height econstuction by the taditional method. 1 17, and L mm. The gating peiod T 16 pixels, and the size of the captued image is pixels. Figues 5 and 6 ae the efeence finge and the defomed finge, espectively. Figue 7 is the econstuction height distibution by the new method whee the maximal height is 33.1 mm, wheeas the econstuction maximal height by the taditional method is 27.4 mm, as shown in Fig. 8. Although they have a simila shape, the impoved phase-height algoithm gives the coect height econstuction. Refeences 1. M. Takeda and K. Mutoh, Fouie-tansfom pofilomety fo the automatic measuement of 3D object shapes, Appl. Opt. 22, (1983). 2. J. Vanhezeele, P. Guillaume, and S. Vanlanduit, Fouie finge pocessing using a egessive Fouie-tansfom technique, Opt. Lases Eng. 43, (2005). 3. J. Li, X.-Y. Su, and L.-R. Guo, Impoved Fouie-tansfom pofilomety of the automatic measuement of theedimensional object shapes, Opt. Eng. 29, (1990). 4. D. Ganota, J. Joseph, and K. Singh, Object econstuction in multilaye neual netwok based pofilomety using gating stuctue compising two egions with diffeent spatial peiods, Opt. Lases Eng. 42, (2004). 5. J. Zhong and J. Weng, Phase etieval of optical finge pattens fom the idge of a wavelet tansfom, Opt. Lett. 30, (2005). 6. X. Su and W. Chen, Fouie tansfom pofilomety: a eview, Opt. Lases Eng. 35, (2001). 7. W.-S. Zhou and X.-Y. Su, A diect mapping algoithm fo phase-measuing pofilomety, J. Mod. Opt. 41, (1994). 8. G. Schiipa Spagnolo, G. Guattai, C. Sapia, D. Ambosini, D. Paoletti, and G. Accado, Contouing of atwok suface by finge pojection and FFT analysis, Opt. Lases Eng. 33, (2000). 9. K. J. Gåsvik, Optical techniques, in Intefeogams Analysis, D. Robinson and G. T. Reid, eds. (IOP, 1993). pp T. Yatagai, Intensity based analysis methods, in Intefeogams Analysis, D. Robinson and G. T. Reid, eds. (IOP, 1993) pp R.-H. Zheng, Y.-X. Wang, X.-R. Zhang, and Y.-L. Song, Twodimensional phase-measuing pofilomety, Appl. Opt. 44, (2005). 668 APPLIED OPTICS Vol. 46, No Febuay 2007