Recovery of absolute phases for the fringe patterns of three selected wavelengths with improved antierror

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1 University of Wollongong Research Online Faculty of Engineering and Information Sciences - Papers: Part A Faculty of Engineering and Information Sciences 06 Recovery of asolute phases for the fringe patterns of three selected wavelengths with improved antierror capaility Jiale Long Wuyi University, Huazhong University of Science and Technology Jiangtao Xi University of Wollongong, jiangtao@uow.edu.au Jianmin Zhang Wuyi University, coolstarinfo@6.com Ming Zhu Huazhong University of Science and Technology, zhuming@mail.hust.edu.cn Wengqing Cheng Huazhong University of Science and Technology See next page for additional authors Pulication Details J. Long, J. Xi, J. Zhang, M. Zhu, W. Cheng, Z. Li & Y. Shi, "Recovery of asolute phases for the fringe patterns of three selected wavelengths with improved anti-error capaility," Journal of Modern Optics, vol. 6, (7) pp , 06. Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Lirary: research-pus@uow.edu.au

2 Recovery of asolute phases for the fringe patterns of three selected wavelengths with improved anti-error capaility Astract In a recent pulished work, we proposed a technique to recover the asolute phase maps of fringe patterns with two selected fringe wavelengths. To achieve higher anti-error capaility, the proposed method requires employing the fringe patterns with longer wavelengths; however, longer wavelength may lead to the degradation of the signal-to-noise ratio (SNR) in the surface measurement. In this paper, we propose a new approach to unwrap the phase maps from their wrapped versions ased on the use of fringes with three different wavelengths which is characterized y improved anti-error capaility and SNR. Therefore, while the previous method works on the two-phase maps otained from six-step phase-shifting profilometry (PSP) (thus fringe patterns are needed), the proposed technique performs very well on three-phase maps from three steps PSP, requiring only nine fringe patterns and hence more efficient. Moreover, the advantages of the two-wavelength method in simple implementation and flexiility in the use of fringe patterns are also reserved. Theoretical analysis and experiment results are presented to confirm the effectiveness of the proposed method. Keywords error, anti, improved, wavelengths, selected, three, capaility, patterns, recovery, fringe, phases, asolute Disciplines Engineering Science and Technology Studies Pulication Details J. Long, J. Xi, J. Zhang, M. Zhu, W. Cheng, Z. Li & Y. Shi, "Recovery of asolute phases for the fringe patterns of three selected wavelengths with improved anti-error capaility," Journal of Modern Optics, vol. 6, (7) pp , 06. Authors Jiale Long, Jiangtao Xi, Jianmin Zhang, Ming Zhu, Wengqing Cheng, Zhongwei Li, and Yusheng Shi This journal article is availale at Research Online:

3 Recovery of asolute phases for the fringe patterns of three selected wavelengths with improved anti-error capaility Jiale Long,, Jiangtao Xi *, Jianmin Zhang, Ming Zhu, Wenqing Cheng, Zhongwei Li 4 and Yusheng Shi 4 School of Information, Wuyi University, Jiangmen, Guangdong, 5900, China, School of Electronic Information and Communications, Huazhong University of Science and Technology, Wuhan, Huei, 40074, China School of Electrical Computer and Telecommunications Engineering, University of Wollongong, Wollongong, NSW, 5, Australia 4 State Key Laoratory of Material Processing and Die & Mould Technology, Huazhong University of Science and Technology, Wuhan, Huei, 40074, China Jiale Long: , longjiale_58@6.com *Corresponding author Jiangtao Xi: +6 44, jiangtao@uow.edu.au Jianmin Zhang: , zjm99_00@6.com Ming Zhu: , zhuming@mail.hust.edu.cn Wenqing Cheng: , chengwq@mail.hust.edu.cn Zhongwei Li: , zwli@hust.edu.cn Yusheng Shi: , shiyusheng@hust.edu.cn Acknowledgements: This work was partially supported y the Science Foundation for Young Teachers of Wuyi University under Grant numer 0zk06; Guangdong province innovation project of education department under Grant numer 0KJCX085; and the National Natural Science Foundation of China under Grant numer 679.

4 Recovery of asolute phases for the fringe patterns of three selected wavelengths with improved anti-error capaility In a recent pulished work we proposed a technique to recover the asolute phase maps of fringe patterns with two selected fringe wavelengths. To achieve higher anti-error capaility, the proposed method requires employing the fringe patterns with longer wavelengths; however, longer wavelength may lead to the degradation of the signal-to-noise ratio (SNR) in the surface measurement. In this paper, we propose a new approach to unwrap the phase maps from their wrapped versions ased on the use of fringes with three different wavelengths which is characterized y improved anti-error capaility and SNR. Therefore, while the previous method works on the two phase maps otained from six-step PSP (thus fringe patterns are needed), the proposed technique performs very well on three phase maps from three steps PSP, requiring only 9 fringe patterns and hence more efficient. Moreover, the advantages of the two-wavelength method in simple implementation and flexiility in the use of fringe patterns are also reserved. Theoretical analysis and experiment results are presented to confirm the effectiveness of the proposed method. Keywords: phase shift; fringe analysis; phase unwrapping; projected fringes; surface measurement. Introduction With the development of digital technology, fringe projection profilometry (FPP) has ecome one of the most promising technologies for non-contact D shape measurement (-). To recover asolute phase maps from the wrapped ones, phase unwrapping is a necessary and important step in the implementation of FPP. Due to the complex nature of oject surface shape and various unwanted noises inherent to the projection and acquisition of fringe patterns, phase unwrapping is a challenging task. Many approaches have een studied and employed such as spatial (4-7) and temporal (8-). Spatial phase unwrapping approaches assume a continuity of scanned surfaces and likely fail in regions with depth discontinuities (step edges) (0). Temporal methods use multiple fringe patterns, which recovers the asolute phase on pixel-y-pixel asis, hence not suffering from error propagation etween pixels. In order to keep the efficiency of the temporal methods, it is always desirale to employ as less numer of images as possile. Zhao, et al. (8) proposed to use the fringe patterns of two wavelengths, one of which has a very long wavelength to cover the whole projection area, and hence can e used as a reference to unwrap the other fringe pattern. Li, et al. (9) also employ a projection image containing a single fringe as a reference to calculate the fringe order of the high spatial frequency fringe patterns. As indicated in () long spatial wavelength fringe patterns is disadvantageous y low phase-sensitivity to the oject surface shape, short spatial wavelength fringe patterns are always preferred in order to achieve accurate shape measurement. However, if the gap etween the frequencies of the two fringe patterns is larger than a certain value, the

5 efore-mentioned methods would fail due to the existence of noise and disturance in the fringe patterns, and hence more intermediate fringe patterns should e employed (0). Recently, Servin, et al. () proposed a -step temporal phase unwrapping formula that uses so-called -sensitivity demodulated phases for measuring static surfaces. The first phase demodulation xy, has at most -wavelength sensitivity and the second one xy, is G-times (G>>) more sensitive. However, the phase error defined in the paper is ex, y [ x, y] G x, y] which should e limited within,. This condition implies a phase error tolerance ound /( G) for the method in (), eyond which the method may not work properly. In such cases, more fringe patterns should still e needed. Hence, minimization of the numer of fringe patterns while improving the anti-error capaility of phase unwrapping is a key prolem for application of temporal approaches in practice. Ding, et al. (4) proposed a method to unwrap asolute phase maps of fringe patterns with two selected frequencies. The two normalized spatial frequencies defined in (4) are the numers of fringes on the images which denoted y f and f, which are assumed to e integers. The performance of the method is limited y phase error tolerance ound, ( f f) and requirement of frequencies coprime (5). For purpose of increasing phase error tolerance ound, a new approach of asolute phase recovery is presented which ased on the use of fringe patterns with three selected spatial frequencies denoted y f, f and f, where the frequencies are also assumed to e integers (6). While the methods proposed in (4-6) provide efficient ways for phase unwrapping, they also suffer from a weakness, that is, the total numer of pixels perpendicular to the fringe must e an integer multiple of the numer of pixels within a fringe. Such a selection may not e convenient. Taking the experiment descried in (6) as an example, the resolution of the projector is 9 08 pixels. If the selected frequencies are ( f, f, f) (6,0,5) as in (6), the numers of pixels per fringe period will e, 9. and 9.8 respectively, which are not integers and thus are not implementale. Furthermore, only when f, f and f do not have a common factor larger than, there exists a unique solution for the phase unwrapping prolem as proved in (6). Such a selection may not e possile in some cases. For the cases of the horizontal resolution eing 04 pixels, which is common for many ordinary projectors, it is impossile to find three frequencies meeting the requirement. A possile solution to this prolem is to tailor the whole image to a smaller size, which may lead to the degradation of resolution in the D measurement which we have analyzed in (7). In order to solve the aove mentioned prolem, in the recent pulished work (7) we proposed a technique to recover the asolute phase maps of two sets of fringe patterns with flexile selection of fringe wavelengths (denoted y and ). It should e noted that wavelength as defined in this paper is different from that defined in physics. The wavelength here represents the total numer of pixels per fringe period. We have proved that when we get the wrapped phase maps y the six-step phase-shifting profilometry (PSP), the asolute phase maps can e recovered correctly. We also have proved that the method in (4) is a special case of the approach proposed in (7). That is, when and are co-prime,

6 and the numer of fringes on the images are integers, the conclusion in (4) tallies with the conclusion in (7). The allowed range of phase error in (7) is larger than that in (4) and the same with which in (6) when and are not co-prime. Furthermore, the computation in (7) is much less. However, we realized that the anti-error capaility as measured y phase error tolerance ound of the proposed technique depends on the greatest common divisor (g.c.m.) of the two fringe wavelengths (denoted y k ), that is the igger the k is, the etter the anti-error capaility can get. Let kg and kg where g and g are positive integers which are co-prime, the upper ound of the allowale phase error is k / / g g, if etter anti-error capaility is desired, g gshould e decreased. But at the same time, the validity of the method relies on R / k ggk, where R is the resolution of projector, which means once the projection is fixed, k should e increased as g and g decreased, and this will resulted in a multiple growth of the selected wavelengths. Taking the experiment descried in (7) as an example, the vertical resolution of the projector is 768 pixels, as the maximal phase error on the wrapped phase maps which get from six-step PSP is aout /00, two fringe wavelengths, meeting the criteria is easy to select, such as,47, 5,00. However, when the maximal phase error on the wrapped phase maps which get from three-step PSP increases to aout /0, the selected wavelengths of the fringes should e very long, such as 56,95, 59,65. As we know, for a phase-shifting technique, the longer the wavelength, the lower the phase sensitivity to the oject shape. In other words, short wavelength should e used to achieve high SNR associated with the fringe patterns for accurate phase unwrapping. Therefore, it is worthwhile to study if there is a room for further improvement in the antierror capaility. In this paper, we try to extend the method proposed in (7) to the cases of fringe patterns with three different wavelengths. The three wavelengths, and are still positive integers representing the total numer of pixels in a fringe period. We will demonstrate that the anti-error capaility can e consideraly improved without increasing the wavelengths of fringe patterns. In addition, the asolute phase maps can still e recovered correctly for the wrapped phase maps from the three-step PSP. Therefore, with suitale selections of fringe wavelengths, only 9 fringe patterns are needed, which is more efficient than the method proposed in (7). The paper is organized as follows. Section presents the phase unwrapping algorithm with three sets of fringe patterns. In Section, we give the principle for fringe wavelengths selection. The phase error ound is derived in Section 4. In Section 5, experiments and results are presented to validate the proposed method. Finally we conclude the paper in Section 6.. Asolute phase map recovery of fringe patterns with three difference wavelengths Consider that three sets of sinusoidal fringe patterns are projected onto the surface of an oject, which are reflected and captured y a camera. These fringe patterns of vertical resolution R pixels are with three different wavelengths, denoted y, and (in pixels) respectively. The projected image can e 4

7 expressed as follows: I x, y a x, y x, y cos[ ( y)] i,,, y [0, R) () pi p p pi where xy, are the coordinates of a pixel in the projector and camera images, and pi ( y) y is the asolute phase of the projected image (or the carrier). The image reflected from the oject and captured y the camera (denoted as camera image) can e expressed as where ( y) ( y u( y)) ci pi, and I x, y a x, y x, y cos[ y ] i,,, y [0, R) () ci c c ci u y is spatial shift caused y the oject surface shape descried y triangular relationship among the positions of the projector, the camera and the target (). As ( y) is a liner function of y, we also have ci ( y) pi ( y) i ( y), where i y u( y). Hence, in order to reconstruct the D surface of the oject, we need y, which can e otained y ci ( y) pi ( y), thus requiring the asolute phase ( y). However, all existing methods can only yield ( y), which are the ci wrapped version of the asolute phases with their values falling within[, ). The asolute phases ( y) are related to the wrapped ones y the following: ci where mi i y m y y () ci i ci y are integers and are the fringe order index associated with pixel (x,y). As mentioned aove, ( y) are a spatially shifted version of the phase of the carrier fringe patterns ( y) ci and hence there is a unique corresponding mapping relationship etween ( y) and ( y) which purely depends on uyor ( ) the surface shape. With all the three fringe patterns eing projected on the same oject, uyshould ( ) e the same in relating ( y) and ( y), and the same relationship applies for the ci pi i ci ci pi i pi, ci pi wrapped versions of ( y) and ( y) as well. As the temporal approach unwrap the phase on pixel-ypixel asis, the same method can e applied to oth the carrier patterns as well as the patterns reflected from the target. In other words, we do not need to consider if these wrapped phases are from the carrier only patterns or these reflected from the oject surface. In the following, we will simply work on ( y). Without loss of generality, the ranges of y, y and y are shifted y, yielding the following : From Eq. (), we have: c c y y y c c c c 0,, (4) c ( y) m y c y c( y) m y c y c( y) m y c y With the help of assuming a pattern with only one fringe covering the whole image with its asolute phase ci (5) pi 5

8 ( y) 0 within (0, ), we have R ci ( y) 0 ( y) (for i=,,). Then we are ale to otain the following: i y y m y m y [ ]/ (6) y y m y m y [ ]/ (7) y y m y m y [ ]/ (8) Note that the left hand sides of Eqs.(6)~(8) can e otained y PSP, which must e integers as the right sides are integers. If there are one-to-one correspondences etween the right sides and the three integers m y, m y and m y, Eqs. (6)~(8) reveal a way to determine these three integers ased on the values c c c of the left hand sides. Comining Eqs.(6)~(8) and the expressions (4) we have: y y /, i.e., m y m y y y /, i.e., m y m y (9) (0) y y /, i.e., m y m y At the same time, from Eqs. (5) and the expressions (4), we have: 0 m y R / 0 m y R / 0 m y R / () () () (4) With inequalities (9)~(4) aove, an unique mapping from y y /, y y y y / to m y, m y and m y can e identified. /, Now let us to utilize an example to prove the effectiveness of this method, where we choose 54, 7, 90 and R 04. By varying m y, m y and m yover the range defined y Eqs. ()~(4), we are ale to otain m y m y, m y m y and m y m y. Then we check the values against the range given y Eqs. (9)~() and these meeting the conditions can e listed in Tale. From Eqs.(5), the whole range of y p (i.e.,[0, R ) ) can e separated ased on the value of m yas follows: m y, m y and m m 0, 0 y p, y p R /, R / y p R / R /, R / y p R 0, 0 y p, y p y R /, R / y p R / R /, R / y p R y (5) (6) 6

9 m y 0, 0 y p, y p R /, R / y p R / R /, R / y R p (7) Where x denotes the largest integer not greater than x. As the vertical resolution of projection image is R 04,from Eqs.(5)~(7) we can see that the first, second and the third columns of Tale cover all the possile values of m y, m y and m y, and the last three columns meet the requirements of the desired range, that is, m y m y, m y m y and m y m y without repetition. The aove example shows that when the three wavelengths of fringe patterns have common factor, the phase maps can e unwrapped. In fact, no matter whether the three wavelengths of phase maps have or do not have common factors, phase unwrapping can always e done y the aove mentioned approach. Through aove analysis the asolute phase could e retrieved from the wrapped phase maps of fringe patterns with three selected fringe wavelengths y the following steps. () Select three fringe wavelengths,, using the criteria descried in Section to ensure the unique mapping from m y m y, m y m y, m y m y to m y, m y, m y and construct a lookup tale like Tale ; () Project the three sets of fringe patterns onto the oject and otain the unwrapped phase maps y, y and y ; () Calculate y y /, y y /, y y / and round them to the nearest integers. Find the row of the tale constructed in step whose value of m y m y m y m y,, m y m y is closest to the integers. Keep the records of m y, m y and m y in the same row; (4) Retrieve the asolute phase maps y Eqs.(5) using m y, m y and m y acquired in step. The process of creating the lookup tale does not need to take y which increases monotonically from π to π as the reference to analyze the interval distriution of the fringe orders like in (6), and hence does not need to do the interval partition at all. As only the simple inequations need to e checked, so, the computation is much less in compared to the amount required in (6). 0. Selection of the three fringe wavelengths From the analysis in appendix, there exists an unique mapping from,, m y m y m y to y y /, y y/ and y y / (i.e., m y m y, 7

10 m y m y, m y m y ) when R / k kkk, where R is the resolution of projector, and k is the g.c.m. of, and, kg, kg, kg, where k is the g.c.m. of g and g, k is the g.c.m. of g and g, k is the g.c.m. of g and g. That is, the constraint of the selection of three fringe wavelengths is: R / k k k k (8) 4. Phase error ound When we round the values of y y /, y y/ and y y / to the nearest integers and find the row of the tale constructed in step with the values of m y m y m y m y, and m y m y closest to the integers, three values are required to match, which should follow a proper order. For example, if y y / (i.e., the value of fourth column) is the one to match first, and y y / (i.e., the value of fifth column) follows, the anti-error capaility of the proposed technique first depends on the smallest gaps etween any two possile value of m y m y, and then m y m y. The larger the gaps, the less likely the error will happen during the rounding operations. In (7), we have proved that the minimal gap of m y m y must e equal or greater than k, where k is the g.c.m. of and. Then, we can deduce that the smallest gap etween the value of m y m y is k which is the g.c.m of and, and the smallest gap etween the value of m y m y is k which is the g.c.m of and, the smallest gap etween the value of m y m y is k which is the g.c.m of and. Hence, for achieving higher anti-error capaility, the first column to match should e the iggest one of k, k and k, and the last column to match should e the smallest one. From Tale, it is easy to discover that if two columns have een matched, the third one surely matched. Assuming phase errors in the phase maps y, y and y are y, yand y respectively, and max y, y, y, we have: max y y / max / k / y y / max / k / y y / max / k / Then we can find the upper ound of the allowale phase error, with which the asolute phase maps can e correctly recovered: 0 max k / 0 max k / (0) 0 k / max Note that, only two columns should e matched in this method, and so, the allowale phase error should (9) 8

11 e change to: 0 median[ k /, k /, k / ] () max In another word, if k k k, we can choose the two columns whose minimal gap are k and k to match only, then the allowale phase error changes to 0 max k / ( ) 0 max k / ( ) If,i.e., max k k 0 min( / ( ), / ( )). max is given, the three fringe wavelengths should e selected to meet two of the following three conditions at least. k / k / k / max max max For example, suppose the resolution of the projection image is R 04, if we want to unwrap the wrapped phase maps otained from three-step PSP whose maximal phase error is aout /0, the fringe wavelengths could e chosen as,, (54,7,90) whose phase error tolerance ound is 0 max / 8. If the method proposed in (7) is employed, a pair of which have large g.c.m should e selected, e.g., 08,60 whose phase error tolerance ound is 0 max / 9. If the method proposed in (6) is used, the selected frequencies are f, f, f (8,,5) whose phase error tolerance ound is 0 / 9 max as descried in (6), and the corresponding wavelengths are,, (8,85., 68.). Hence, for the same level of phase error tolerance, we can choose much shorter wavelength for the proposed method than that in (6) and (7). Considering that the fringe patterns with long wavelengths suffer from low SNR in the phase maps, the method proposed is advantageous y high SNR in the phase maps and hence accurate D measurement. () 5. Experiments Experiments are carried out to test the proposed method with a system consisting of a LG HW00 projector and Daheng HV5 camera, with their resolutions eing and respectively. The oject to e measured is a toy model which is shown in Fig.. Firstly, three sets of sinusoidal fringe patterns with fringe wavelengths of 54, 7 and 90 are projected onto the toy model, as shown in Fig. (a), () and (c). Note that the vertical resolution of projector is 04, i.e., R 04. The wrapped phase maps otained from three-step PSP are shown in Fig. (d), (e) and (f). Since the ackground does not contain useful information, a shadow noise detector/filter (8, 9) was employed which is shown in Fig. (a), such that the shadow-noised regions were discarded from further processing (9). The maximal phase error on the wrapped phase maps is aout /0, which is smaller than median( k /, k /, k / ( )) / 8. Hence ased on Eq.() the asolute phase maps can e successfully recovered. Fig. () is the unwrapped phase map of 54 which is characterized y monotonic variance over the areas of smooth shape change on the model, and hence results are considered 9

12 as correct. The anti-error capaility is consideraly increased in contrast to (7) and the SNR can e improved due to the use of the fringe patterns with short wavelengths. Secondly, three sets of fringe patterns with different wavelengths of 5, 0 and 5 are projected onto the same toy model, as shown in Fig. 4(a), 4() and 4(c). Four-step PSP is used and thus leads to lower level of noise in the wrapped phase maps, as shown in Fig.4(d), 4(e) and 4(f). The shadow noise detector/filter is also employed in Fig. 5(a). The maximal phase error on the wrapped phase maps is aout /5, which is smaller than the error ound /. Hence ased on Eq.() the asolute phase maps can e successfully recovered. Fig.5() is the unwrapped phase map of 5 which is characterized y monotonic variance over the areas of smooth shape change on the model, and hence results are considered as correct. The anti-error capaility and the SNR here are oth improved incontrast to the performance of (7). In order to see the topographic details of the oject clearly, we remove the carrier-plane from Fig.5() and get Fig.5(c). The D view of the recovered oject is shown in Fig Conclusion This paper proposes a new approach to recover the asolute phase maps ased on wrapped ones of the fringe patterns with three selected wavelengths. Compared with our previous work in (7), the anti-error capaility is increased for reliale phase unwrapping implementation and the SNR can e improved for the use of shorter wavelengths. With suitale selections of fringe wavelengths, the asolute phase maps can e correctly recovered using only 9 fringe patterns, which is more efficient than the method proposed in (7). Moreover, in comparison with the existing method in (6), the proposed method is advantageous y less computation required for constructing the checking tales and more flexiility in the design of fringe patterns. The effectiveness of the proposed method has een verified y experimental results. References: () Christopher J., Waddington, Jonathan D. Kofman. Opt. Eng. 04, 5, () Haixia Wang, Qian Kemao, Seah Hock Soon. Opt. Exp. 05,, () Z.H.Zhang. Opt Lasers Eng. 0, 50, (4) S. Zhang, X. Li, S.-T.Yau. Appl. Opt. 007, 46, (5) S. Li, W. Chen, X. Su. Appl. Opt. 008, 47, (6) Y. Shi. Opt. Exp. 007, 5, (7) M. Costantini. IEEE Trans. Geoscience and Remote Sensing. 998, 6, 8-8. (8) H. Zhao, W. Chen, Y. Tan. Appl. Opt.994,, (9) J. Li, L. G. Hasserook, C. Guan. J. Opt. Soc. Am. A. 00, 0, (0) J. M. Huntley, H. O. Saldner. J. Opt. Soc. Am. A. 997, 4, () M. Servin, J.M. Padilla, A. Gonzalez, G. Garnica. Opt. Exp. 05,, () Y. Hu, J. Xi, J. Chicharo, E. Li, Z. Yang. Appl. Opt. 006, 45, () G. Pedrini, I. Alexeenko, W. Osten, H. J. Tiziani. Appl. Opt. 00, 4, () Liu Yong, Huang Dingfa, Jiang Yong. Appl. Opt. 0, 5,

13 (4) Y. Ding, J. Xi, Y. Yu, J. Chicharo. Opt. Let. 0, 6, (5) Y. Ding, J. Xi, Y. Yu, W. Q. Cheng, S. Wang, J. Chicharo. Opt. Exp. 0, 0, 8-5. (6) Y. Ding, J. Xi, Y. Yu, F. Deng. Opt. Laser. Eng. 05, 70, 8-5. (7) J. Long, J. Xi, M. Zhu, W. Cheng, R. Cheng, Z. Li, Y. Shi. Appl. Opt. 04, 5, (8) X. Su, G. von Bally, D. Vukicevic. Opt. Commun. 99, 98, (9) K. Liu, Y. Wang, D. L. Lau, Q. Hao, L. G. Hasserook. Opt. Exp. 00, 8, Appendix: Analysis on selection of the three fringe wavelengths The validity of this method relies on the existence of unique mapping from m y m y m y m y,, m y m y to m y, m y, m y. This requires that oth sides of Eqs.(6)~(8) must not yield the same value for any two different pixel numer index such as ya and y ( a words, the following must hold: a a a a m y m y m y m y m y m y m y m y m ya m ya m y m y for ya ). In other y () From Tale we can see that if the first two equations hold, the last one surely holds. So, without loss of generality, Eqs.() can e simplified to Eqs.(4). Let us first discuss the simpler case where, and are coprime with each other. a a m y m y m y m y m ya m ya m y m y for ya The aove can e proved y reductio ad asurdum. Suppose there exist ya and y ( a y (4) ) making the two side of Eqs.(4) equal, and three possile situations may make the two sets of m y, m y, m y different: firstly, the three pairs of integers are all different with each other, i.e., m y m y a, and m y m y ; secondly, there are two pairs of integers different, such m y m y a a as m y m y, m y m y and m y m y ; thirdly, only one pair of them is different, such as a a a, m y m y and m y m y m y m y a a a. Without losing generosity, we take the first case into account where m y m y, m y m y. When the two sides of (4) equal, we have: m y m y a a a m y m y / m y m y / m ya m y / m ya m y / a As, and are coprime two y two, Eqs.(5) must e equivalent to the following: a, a, m y m y n m y m y n m ya m y n m ya m y n Where n, n are integers and n, n 0. From Eqs.(6) we know: a, (5) (6)

14 n n (7) Because, and are coprime two y two, we have n n, n n, where n is an integer and n 0,then: a, a m ya m y n. Comining inequations ()~(4), m y m y n m y m y n, we have : R / m ya m y R /, R / m ya m y R / and a which means R / n R / R / m y m y R /, R / n R / and R / n R /. Hence, Eqs.() will hold when n. For the other two cases, it is ovious that the same conclusions will get. R n,where n is the asolute value of When, and are not coprime, there are two possile situations. Firstly, let k e the greatest common divisor (g.c.m.) of, and, we have kg, kg, kgwhere g, g and g are positive integers which are coprime with each other. Equations (4) can e reproduced as follow: a a kgm y kgm y kgm y kgm y i.e., kgm ya kgm ya kgm y kgm y a a gm y gm y gm y gm y gm ya gm ya gm y gm y Oviously Eqs.(8) is the same as Eqs.(4), and hence can e proved using the same approach. That is Eqs.(8) will hold when R gg g, where R R / k, which is equivalent to R / k. Secondly, when the g.c.m. of, and are not exists, ut they have g.c.m. etween any two of them. Let k e the g.c.m. of and, kg, kg, where g, g are positive integers which are coprime; k e the g.c.m. of and, kh, kh, where h, h are positive integers which are coprime; k e the g.c.m. of and. Equations (5) can e reproduced as follow: a a kg m y kgm y kg m y kgm y i.e., khm ya kh m ya khm y khm y Eqs. (9) can e rewritten as: a a m y m y / g m y m y / g m ya m y / h m ya m y / h a a (8) gm y gm y gm y gm y (9) hm ya h m ya hm y h m y As g, g are coprime and h, h are coprime too, Eqs.(0) must e equivalent to the following: Where n, n are integers and n, n 0 a, a, m y m y n g m y m y n g m y m y n h m y m y n h a a. From Eqs.() we know: n g (0) () n h () Note that k e the g.c.m. of and, and the g.c.m. of, and not exists, so, n g n k q, n h n k q, where q, q are positive integers which are coprime. Eq.() can e rewritten as follow: We have n q, n q, then: n q n q ()

15 m ya m y n g qg h / k g / kkk In the similar way, m ya m y n g / kkk, m ya m y nh / kkk. Comining inequations ()~(4), we have: R / m ya m y R /, R / m ya m y R / and, which means R / / kkk R /, R / m y m y R / R / / k k k R /. Hence, Eqs.() will hold when R / k k k. From the analysis aove, there exists an unique mapping from,, R / / k k k R / and m y m y m y to y y /, y y/ and y y / (i.e., m y m y m y m y, m y m y, ) when R / k kkk, where R is the resolution of projector, and k is the g.c.m. of, and, kg, kg, kg, where k is the g.c.m. of g and g, k is the g.c.m. of g and g, k is the g.c.m. of g and g.

16 Tale. Mapping from m y m y, m y m y and m y m y to m y, m y, m y when 54, 7, 90, R 04 m y m y m y m y m y m y m y m y m y

17 Figure. The photograph of the toy model Figure. Experiment results when,, 54,7,90. (a), () and (c) are the deformed fringe patterns; (d), (e) and (f) are the wrapped phase maps get y three-step PSP Figure. Experiment results when,, 54,7,90. (a) is the designed shadow noise filter; () is the recovered asolute phase map of 54. Figure 4. Experiment results when,, 5,0,5. (a), () and (c) are the deformed fringe patterns; (d), (e) and (f) are the wrapped phase maps get y four-step PSP. Figure 5. Experiment results when,, 5,0,5. (a) is the designed shadow noise filter; () is the recovered asolute phase map of 5.(c) is the asolute phase map of 5 which has een removed the carrier-plane. Figure 6. The D view of the recovered oject.

18 Fig Fig (a)

19 Fig.() Fig.(c)

20 Fig.(d) Fig.(e)

21 Fig.(f) Fig.(a)

22 Fig.() Fig.4(a)

23 Fig.4() Fig.4(c)

24 Fig.4(d) Fig.4(e)

25 Fig.4(f) Fig.5(a)

26 Fig.5() Fig.5(c)

27 Fig.6