Absolute three-dimensional shape measurement with two-frequency square binary patterns

Transcription

1 871 Vo. 56, No. 31 / November / Appied Optics Research Artice Absoute three-dimensiona shape measurement with two-frequency square binary patterns CHUFAN JIANG AND SONG ZHANG* Schoo of Mechanica Engineering, Purdue University, West Lafayette, Indiana 4797, USA *Corresponding author: szhang15@purdue.edu Received 26 Juy 217; revised 3 September 217; accepted 1 October 217; posted 2 October 217 (Doc. ID 33418); pubished 3 October 217 This paper presents a nove method to achieve absoute three-dimensiona shape measurement soey using square binary patterns. This method uses six patterns: three ow-frequency phase-shifted patterns and three phase-shifted high-frequency patterns. The phase obtained from the ow-frequency phase temporay unwraps the phase obtained from high-frequency patterns. The projector is defocused such that the high-frequency patterns produce a high-quaity phase, but the phase retrieved from ow-frequency patterns has a arge harmonic error that fais the two-frequency tempora phase unwrapping process. In this paper, we deveop a computationa framework to address the chaenge. The proposed computationa framework incudes four major approaches to aeviate the harmonic error probem: (i) use more than one period of ow-frequency patterns enabed by a geometric constraint-based phase unwrapping method; (ii) artificiay appy a arge Gaussian fiter to ow-frequency patterns before phase computation; (iii) create an error ookup tabe to compensate for harmonic error; and (iv) deveop a boundary error correction method to aeviate probems associated with fitering. Both simuation and experimenta resuts demonstrated the success of the proposed method. 217 Optica Society of America OCIS codes: (12.12) Instrumentation, measurement, and metroogy; (1.588) Phase unwrapping; (11.586) Phase unwrapping; (1.57) Phase retrieva INTRODUCTION High-speed three-dimensiona (3D) shape measurement based on a digita fringe projection (DFP) method is one of the most popuar optica methods due to its fexibiity, accuracy, and speed especiay when a phase-shifting agorithm is empoyed [1]. However, the singe-frequency phase-shifting agorithm typicay uses an arctangent function to compute phase vaue whose range is from π to π [2]. This step is often regarded as phase wrapping, and the phase-unwrapping agorithm is usuay required to recover a smooth phase for 3D shape reconstruction. In genera, phase unwrapping can be broady cassified into spatia and tempora phase unwrapping categories. A spatia phase unwrapping agorithm determines the number of 2πs to be added to a point by referring to phase vaues of other points on the same phase map, assuming the surface geometry is smooth. Ghigia and Pritt [3] edited a book that coected numerous spatia phase unwrapping methods, and Su and Chen [4] reviewed the popuar quaity-guided phase unwrapping agorithms. The fundamenta imitation of the spatia phase unwrapping method is the entire surface smoothness assumption, i.e., if an object surface is not smooth, the spatia phase unwrapping method cannot be empoyed. Tempora phase unwrapping can unwrap the phase by referring to the additionay acquired information. Over the history, the DFP system can empoy two-waveength and mutiwaveength phase-shifting agorithms [5 7] or hybrid binary coding with phase-shifting methods [8,9] for tempora phase unwrapping. For high-speed appications, it is aways desirabe to use the ower number of fringe patterns; as a resut, the two-frequency phase unwrapping agorithm is preferabe. The phase obtained from the ow-frequency phase unwraps the phase obtained from high-frequency patterns pixe by pixe without referring to neighboring pixes. However, the capabiity of the two-frequency method is significanty imited by the noise impact [1]. To reduce the noise impact, Hyun and Zhang recenty proposed the enhanced two-frequency phase-shifting agorithm [11] by empoying the geometricconstraint-based phase unwrapping method [12]. A typica DFP system requires sinusoida fringe patterns for high-quaity 3D shape measurement. However, owing to the use of 8-bit representation of the sinusoida fringe patterns, the measurement speed is typicay imited to be up to 12 Hz if a digita-ight-processing (DLP) projector is used [1], abeit use of other projection techniques such as LED arrays [13], a aser specke [14], or a mechanica projector X/17/ Journa 217 Optica Society of America

2 Research Artice Vo. 56, No. 31 / November / Appied Optics 8711 [15] coud achieve higher speeds. The binary defocusing method [16], in contrast, can achieve a much higher frame rate especiay when it is reaized on the advanced DLP patforms [17]. Therefore, to achieve higher-speed 3D shape measurement, a 1-bit binary pattern in ieu of an 8-bit is required, and the desired high-quaity sinusoida patterns is reaized through projector defocusing. However, to reaize a twofrequency phase unwrapping method, it is necessary to simutaneousy generate high-quaity sinusoida fringe patterns for two different frequencies, something not easiy reaized. This paper presents a nove method to reaize two-frequency phase unwrapping with square binary patterns through defocusing. The projector is defocused such that the high-frequency patterns produce a high-quaity phase, but the phase retrieved from ow-frequency patterns has a arge error introduced by the obvious binary structures. In particuar, we deveoped a computationa framework that incudes four major approaches to aeviate the arge error probem: (i) use more than one period of ow-frequency patterns enabed by a geometric constraintbased phase unwrapping method; (ii) artificiay appy a arge Gaussian fiter to ow-frequency patterns before phase computation; (iii) create an error ookup tabe (LUT) to compensate for harmonic error; and (iv) deveop a boundary errorcorrection agorithm to address the artifacts caused by fitering. Both simuation and experimenta resuts demonstrated the success of the proposed method. Section 2 expains the principes of the proposed method. Section 3 presents some simuation resuts. Section 4 shows experimenta vaidation, and, finay, Section 5 summarizes this paper. 2. PRINCIPLES A. Two-frequency Phase-shifting Agorithm In optica metroogy, various phase-shifting agorithms have been extensivey empoyed due to their accuracy and robustness [2]. Typicay, it achieves higher measurement accuracy with more fringe images. Despite its sensitivity to noise, a three-step phase-shifting agorithm is sti preferabe for high-speed appications since it requires the east number of patterns for pixewise phase computation. Mathematicay, three phase-shifted fringe images with equa phase shifts can be described as I 1 x;y I t x;y I x;y cos ϕ 2π 3 ; (1) I 2 x;y I t x;y I x;y cos ϕ ; (2) I 3 x;y I t x;y I x;y cos ϕ 2π 3 ; (3) where I t x;y is the average intensity or texture, I x;y is the intensity moduation, and ϕ is the phase. Simutaneousy soving these three equations eads to the texture I t x;y I 1 I 2 I 3 3; (4) as we as the phase pffiffiffi ϕ x;y tan 1 3 I 1 I 3 : (5) 2I 2 I 1 I 3 Mathematicay, Eq. (5) yieds phase vaues ranging from ony π to π with moduus of 2π; and this phase is often referred to as a wrapped phase with 2π discontinuities. A spatia or tempora phase unwrapping is typicay needed to remove 2π discontinuities for 3D shape measurement. Phase unwrapping essentiay determines the integer number k x;y for each pixe x;y to unwrap phase ϕ x;y. In other words, the reationship between the unwrapped phase Φ x;y and the wrapped phase ϕ x;y is Φ x;y k x;y 2π ϕ x;y : (6) Here k x;y is often regarded as fringe order. If fringe order k x;y can be uniquey determined, the unwrapped phase Φ x;y is regarded as absoute phase; and if k x;y has an unknown integer number of the overa offset, the unwrapped phase is regarded as reative phase. Spatia phase unwrapping typicay gives ony reative phase whie tempora phase unwrapping tends to recover absoute phase. Unike conventiona aser interferometry systems, DFP systems can project sufficienty ow-frequency fringe patterns that can recover phase uniquey without phase unwrapping (i.e., Φ x;y ϕ x;y, where Φ x;y refers to the unwrapped phase); and the ow-frequency phase can be directy used to determine fringe order for the high-frequency phase ϕ h x;y as Φ x;y T T h ϕ h k x;y Round 2π : (7) Here, Round() is the round operator that gives the cosest integer number, T is the ow-frequency fringe period in pixes, and T h is the high-frequency fringe period in pixes. It shoud be noted that since the ow-frequency phase is scaed up by a factor of T T h its noise coud introduce error to fringe order determination: the arger the number is, the more incorrect points it coud introduce. Therefore, this two-frequency phase unwrapping agorithm is not extensivey empoyed due to its sensitivity to noise. The next section introduces the enhanced two-frequency phase unwrapping agorithm that can increase the robustness of the two-frequency approach. B. Enhanced Two-frequency Phase Unwrapping Agorithm An et a. [12] deveoped the absoute fringe order determination method using the inherent geometric constraints of a standard DFP system (i.e., a singe camera and a singe projector). Briefy, such a phase unwrapping method works if the projection matrices P c and P p for the camera and the projector are known. These projection matrices represent the mathematica reationship between a point in a 3D word coordinate system x w ;y w ;z w and their projection point on the camera or projector sensor pane u; v coordinates. Mathematicay, given a virtua pane at z z w, each camera pixe u c ;v c can be mapped to projector pixe coordinates u p ;v p, defining a phase vaue Φ; and for a z z min pane, the mapped phase map, or Φ min, is caed minimum phase map. Φ min is a function of depth pane z min, fringe period T, and the projection matrices P c and P p, Φ min u c ;v c f z min ; T;P c ; P p : (8) An et a. [12] proved that for a given depth range, the fringe order k x;y can be determined by

3 8712 Vo. 56, No. 31 / November / Appied Optics Research Artice Fig. 1. Proposed computationa framework for tempora phase unwrapping using phase-shifted square binary patterns. Φmin ϕ k x;y cei : (9) 2π Here, cei() is an operator yieding the nearest upper integer vaue. The geometric constraint-based phase unwrapping method permits the use of more than one period of ow-frequency fringe patterns for pixe-wise tempora phase unwrapping. Therefore, the enhanced two-frequency phase unwrapping method [11] essentiay uses the geometric constraint phase unwrapping method unwrap ow-frequency phase, and the unwrapped ow-frequency phase is then used to unwrap the high-frequency phase. Since the period of ow-frequency fringe patterns is reduced, the noise impact is aeviated. C. Proposed Tempora Phase Unwrapping Method Figure 1 shows the overa computationa framework of the proposed tempora phase unwrapping method. A arge Gaussian fiter is appied to the ow-frequency binary fringe patterns that are further used to compute the wrapped phase ϕ.anlutis then used to correct the phase error in the wrapped ow-frequency phase. The minimum phase Φ min is then appied to unwrap the ow-frequency phase to generate the unwrapped phase map Φ. The unwrapped ow-frequency phase then temporay unwraps the high-frequency phase pixe by pixe. This subsection wi eucidate the detais of the proposed computationa framework. As discussed in the previous subsection, the enhanced twofrequency phase-unwrapping agorithm improves the robustness of the two-frequency phase unwrapping approach. As discussed by An et a. [12], the geometric constraint-based tempora phase unwrapping agorithm has imited depth range to be approximatey T o tan α. Here T o is the one period of fringe span on the object space, and α is the ange between the projector and the camera. To aeviate this depth range imitation, one possibe soution is to reduce the ange α between the projector and the camera. However, it is not desirabe since the measurement quaity wi be compromised. Therefore, due to the imited depth range of the geometric constraint-based phase unwrapping agorithm, the ow-frequency fringe period sti has to be a rather arge number for practica appications. If the high-frequency binary patterns are defocused to be high-quaity sinusoida fringe patterns, the binary structure remains obvious for the ow-frequency fringe patterns. Therefore, the corresponding phase obtained from ow-frequency fringe patterns sti has a arge error. Figure 2 shows an exampe of binary patterns with two different frequencies (T h 18 pixes and T 54 pixes). Appying a 9 9 Gaussian fiter (standard deviation σ 3 pixes) to the high-frequency binary pattern shown in Fig. 2 generates a good-quaity sinusoida fringe pattern shown in Fig. 2(c). However, if the same Gaussian fiter is appied to the ow ow-frequency binary pattern, as shown in Fig. 2, the binary structures remain sharp and cear. A three-step phase-shifting agorithm is then appied to those burred binary patterns to compute phase maps. Figure 3 shows the wrapped phase ϕ h from the highfrequency burred binary patterns. The phase error is then computed by subtracting unwrapped phase Φ h by the idea phase Φ i, i.e., ΔΦ h Φ h Φ i. Figure 3(c) shows the cross-section of the phase error. Ceary, the phase error is very sma: rootmean-square (rms) error is approximatey.5 rad. Simiary, Figs. 3 and 3(d), respectivey, show the wrapped phase and the phase error for the ow-frequency fringe patterns. Obviousy, the phase error is very arge (rms error:.278 rad), as expected. Figure 4 shows the unwrapped high-frequency phase with a ot of incorrecty unwrapped points. This is easy to understand since it is not sufficient for a conventiona agorithm to temporay unwrap the high-frequency phase pixe by pixe due to the arge binary structura errors shown in Fig. 3(d). Fig. 2. Same amount of defocusing can make ony a onefrequency binary pattern a high-quaity sinusoida pattern. High-frequency binary pattern with period of T h 18 pixes; ow-frequency binary pattern with period of T 54 pixes; (c), (d) resutant pattern after appying 9 9 Gaussian fiter to the pattern shown in and, respectivey.

4 h Research Artice Vo. 56, No. 31 / November / Appied Optics h Fig. 5. Phase error after appying a arger artificia Gaussian fiter (fiter size of 87 pixes) to the ow-frequency patterns. Cross-section of the phase error map (rms error:.61 rad); error LUT (c) (d) Fig. 3. Phase error after defocusing., Wrapped phase map obtained from three phase-shifted fringe patterns shown in Figs. 2(c) and 2(d), respectivey. (c) Cross-section of the high-frequency phase error map (rms error:.5 rad); (d) cross-section of the owfrequency phase error map (rms error:.278 rad) Fig. 4. Unwrapped phase using the two-frequency square binary patterns. Cross-section of the unwrapped phase map using a conventiona tempora phase unwrapping and three-step phase-shifting agorithm; cross-section of the unwrapped phase map using our method with the same fringe patterns. Therefore, it is often assumed that it is not possibe to reaize two-frequency tempora phase unwrapping using the three-step phase-shifting agorithm and the square binary patterns. This paper proposes to appy a arge Gaussian fiter to the ow-frequency binary patterns before appying the phaseshifting agorithm. In particuar, we suggest using the Gaussian fiter size of approximatey 1/6 of the fringe period and the standard deviation of approximatey 1/18 the fringe period. In this exampe case, the Gaussian fiter we used is pixes with a standard deviation of 29. The phase error is then computed after appying such a arge fiter. Figure 5 shows the phase error. Apparenty, the phase error is reasonaby sma now (approximatey rms error of.61 rad). We then created an LUT, ΔΦ, as a function of wrapped phase ϕ. Figure 5 shows the created LUT with 256 eements. The ow-frequency phase is compensated by the error LUT before being used for tempora phase unwrapping. The rationa of appying such a arge Gaussian fiter is that the defocusing effort of the projector wi not significanty change the phase error distribution for the ow-frequency binary patterns. Figure 6 shows the phase rms error changes with different fiter size. Obviousy, when fiter sizes increase, the phase rms error decreases initiay and then becomes amost 2 1 rmse FS (pixe) constant. The reason is that when the fiter size is arge enough, the burred pattern is very cose to idea sinusoida patterns. Figure 6 iustrates that when the fiter size is changed by 1 pixes, the maximum error difference is.3 rad; and such a sma change wi not affect the fringe order determination for tempora phase unwrapping. Figure 4 shows one cross-section of the unwrapped high-frequency phase from our method; ceary the entire phase is propery unwrapped. However, appying a arge Gaussian fiter coud introduce unwrapping artifacts near the boundary because some windowed pixes do not present when the Gaussian fiter is appied to a pixe cose to the image boundary or near the background pixes. The unwrapping artifacts near the boundary are caused by the inherent artifacts of Gaussian fitering. For these regions, the fitered phase cannot truthfuy represent the desired phase and thus phase error occurs on a ow-frequency phase map. When the ow-frequency phase is used to determine fringe order for the high-frequency phase, the fringe order information coud be incorrecty determined, eading to an incorrecty unwrapped phase. To aeviate the probem, we deveoped a boundary correction agorithm that is appied to both row and coumn directions of the unwrapped high-frequency phase map. Taking one row of pixes as an exampe, we first removed the background data and extracted a the indices of remained pixes as f i 1 ; Φ 1 ; i 2 ; Φ 2 ; ; i n ; Φ n g, where i j is the index of the pixe position in a row and Φ j denotes the phase vaue of pixe i j. We defined the first and ast r pixes on this raw data as unreiabe pixes to be corrected and assume that surface geometry on these pixes is smooth (i.e., without abrupt changes causing more than π changes for the phase from one pixe to its neighborhood pixes)..1 FS = 77 FS = 87 FS = 97 Fig. 6. Phase error variations with different fiter sizes (FS) for the binary patterns with a fringe period of 54 pixes. Phase rms error with different fiter sizes; cross-section of the phase error map for FS 77 pixes, 87 pixes, and 97 pixes (rms error is.86,.61, and.39 rad, respectivey).

5 8714 Vo. 56, No. 31 / November / Appied Optics Research Artice Fig. 7. Φ ( i, Φ ) j c ( i, Φ ) j j j m Φ j j j + 1 j + 2 j + 3 j + 4 j + 5 Iustration of the proposed boundary correction method. Figure 7 iustrates how the boundary correction agorithm works on the first r pixes. The red soid circe i j ; Φ j is the pixe to be corrected, and the bue circes are those inner neighbors. Because of the surface smooth assumption, the correct phase Φ c j is supposed to be cose to the phase of its neighbors. The corrected phase was determined by Φ m Φ c j j Φ j 2π Round Φ j 2π where Φ m j is the median phase vaue of m inner neighbors i j 1 ;i j 2 ; ;i j m. We used the median function to avoid a arge error from spike noise on an individua pixe. After correction, we updated corrected pixe phase i j ; Φ c j (red hoow circe shown in Fig. 7) and used the updated vaue for the foowing pixe correction. For the first r pixes on each row, we started the correction process from pixe i r ; Φ r and repeated the same correction process for the rest of pixes i j ; Φ j in the order of j r 1, r 2; ; 1. For the ast r pixes we impemented the simiar boundary correction agorithm. Note that now m inner neighbors for pixe i j ; Φ j ie on the interva i j m ;i j 1, and the order to process data is from right (i.e., outside) to eft (i.e., inner). After boundary correction on each row, we aso appied the same boundary correction process for each coumn to further reduce the unwrapping artifacts. It is important to note that the LUT was generated using the idea binary patterns from simuation data. Such an LUT was used to compensate phase error for the phase obtained from camera-captured fringe patterns. The size of the Gaussian fiter used to generate LUT coud be different from that appied to experimenta fringe patterns. For exampe, we suggest using a Gaussian fiter with a size of 1/6 fringe period, and a standard deviation of 1/18 fringe period to generate LUT. To maximize the effectiveness of the LUT-compensation agorithm, the fiter size shoud be adapted based on a given hardware system setup. The reason is that to generate the same eve of burring, the Gaussian fiter size appied to the projected image shoud be different from that appied to the capture image if the projected image and capture image has a different pixe size. Practicay, such a scaing factor can be determined through caibration. For exampe, one can capture a uniform fat board and determine the fringe period on the camera space. The Gaussian fiter size shoud be scaed by a T c T p, where T c is the camera captured fringe period and T p is the projected fringe period. 3. SIMULATION We first performed some simuations to verify the performance of our proposed method. In these simuations, three phaseshifted square binary patterns with ow frequency and three i ; phase-shifted square binary patterns with high frequency are generated. The ow-frequency binary pattern period is T 52 pixes, and the high-frequency period is T h 18 pixes. The fringe period ratio, T T h , is very arge and often fais the conventiona two-frequency phase unwrapping agorithm. In addition, Gaussian noise with zero mean and.1 standard deviation was added to the fringe pattern to evauate the robustness of the proposed agorithm. Figures 8 and 9 show simuation resuts. Figures 8 and 9, respectivey, show one of the ow-frequency patterns with added noise and one cross-section of the pattern; and Figs. 8 and 9, respectivey, show one of the high-frequency patterns and its cross-section. It shoud be noted that a these fringe patterns were appied to a 9 9 pixe Gaussian fiter with a standard deviation of 3 pixes before adding Gaussian noise. Athough the high-frequency pattern appears sinusoida, the binary structures of the ow-frequency pattern is very obvious, as expected. We appied a arge Gaussian fiter (9 9 pixes with a standard deviation of 3 pixes) to the ow-frequency patterns. Figures 8(c) and 9(c), respectivey, show the fitered pattern and its cross-section. Even after fitering, the structured pattern is sti not sinusoida, and thus the resutant phase, as shown in Figs. 8(e) and 9(e) sti cannot be directy used for tempora phase unwrapping without probems. We then appied the LUT compensation agorithm to reduce the ow-frequency phase error and the geometric constraint-based phase unwrapping agorithm to unwrap the ow-frequency phase. The owfrequency phase was then used to unwrap the high-frequency phase shown in Fig. 8(d) and one cross section of the phase shown in Fig. 9(d). Figures 8(f) and 9(f) show the resuts. Ceary, even with such a arge fringe period ratio and such a arge noise, the proposed agorithm can sti work propery. As a comparison, Fig. 1 shows the resuts if the arge Gaussian fiter is appied to the ow-frequency patterns, and the LUT error compensation is not impemented. Obviousy, phase unwrapping competey fais if the fiter is not appied, and some phase unwrapping artifacts occur [e.g., those bumps shown in Fig. 1(d)] if the LUT is not used. Fig. 8. Simuation resuts. One of three phase-shifted owfrequency binary patterns (T 54 pixes) with added Gaussian noise (zero mean,.1 standard deviation); one of three phase-shifted highfrequency binary patterns (T h 18 pixes) with added Gaussian noise (zero mean,.1 standard deviation); (c) ow-frequency pattern shown in is fitered by a arge Gaussian fiter; (d) wrapped phase from highfrequency patterns; (e) wrapped phase from fitered ow-frequency patterns; (f) unwrapped high-frequency phase.

6 Research Artice Vo. 56, No. 31 / November / Appied Optics 8715 intensity x intensity x intensity x (c) h (d) Fig. 11. Photograph of the experimenta system setup (e) h (f) Fig. 9. Cross-sections of the resuts shown in Fig. 8. For better visuaization, ony 6 points are potted on a these figures. (f) shows corresponding cross section of Figs. 8 8(f), respectivey. 1 h (c) These simuation resuts confirmed that the proposed method can successfuy unwrap phase with arge noise and with a arge fringe period ratio. 4. EXPERIMENT To test the performance of our proposed method for practica appications, we deveoped a 3D shape measurement system, shown in Fig. 11, which incudes a compementary metaoxide-semiconductor (CMOS) camera (Mode: Pointgrey GS3-U3-23S6M-C) and a DLP projector (Mode: Texas Instruments LightCrafter 45). The camera was attached with a 25 mm foca ength ens (Mode: Fujinon CF25HA-1) and the resoution was pixes. The projector s h (d) Fig. 1. Unwrapping resuts with impementing the proposed method. Unwrapped high-frequency phase without appying a arge Gaussian fiter to the ow-frequency patterns; unwrapped phase without impementing the LUT-based error compensation agorithm; (c) one cross-section of ; (d) one cross-section of. resoution was pixes. The entire system was caibrated by the structured ight system caibration method deveoped by Li et a. [18]. For a experiments, we used a fringe period of T h 18 pixes for three phase-shifted high-frequency binary patterns, and a fringe period of T 42 pixes for three phase-shifted ow-frequency binary patterns. We first empoyed the proposed absoute phase recovery framework to measure a white board. Figure 12 shows the measurement resuts. Figure 12 shows one of the three phaseshifted high-frequency fringe patterns. Figure 12 shows one of the three phase-shifted ow-frequency binary patterns. After appying a arge Gaussian fiter (71 71pixes with a standard deviation of pixes) to the ow-frequency binary patterns, the fitered image is shown in Fig. 12(c). By appying the three-step phase-shifting agorithm to the high-frequency fringe patterns, the wrapped phase map can be obtained, as shown in Fig. 12(d). We aso cacuated the ow-frequency wrapped phase based on fitered phase-shifted fringe images and appied an LUT-based error compensation approach discussed in Subsection 2.C to obtain the compensated phase map, which is shown in Fig. 12(e). We then empoyed the enhanced two-frequency phase unwrapping agorithm expained in Section 2.B to obtain the absoute unwrapped phase that was further used to reconstruct a 3D shape using the caibrated system parameters. Figure 12(f) shows the 3D resuts. As a comparison, Fig. 12(g) shows the resut when the LUT error compensation was not appied. To better visuaize the difference between the resuts with and without adopting the LUT-based error compensation, the same cross-section of these two 3D shapes is potted in Fig. 13. Note that we removed the gross sope of the pane depth curves for a pots in Fig. 13. For such a fat board measurement with sma depth variance, the reconstructed 3D resut without LUT error compensation is simiar to the one with LUT error compensation. Athough the fat board is successfuy reconstructed overa, one can observe that there is some incorrecty reconstructed geometry near the boundaries, and this probem was the unwrapping artifacts discussed in Subsection 2.C. We then empoyed the boundary error correction agorithm. In a our experiments, we chose r 81pixes from each image boundary as unreiabe pixes, and m 5 neighboring pixe to

7 8716 Vo. 56, No. 31 / November / Appied Optics Research Artice Fig. 12. Measurement resuts of a simpe white board. One of three high-frequency phase-shifted fringe patterns; one of three ow-frequency phase-shifted fringe patterns; (c) the fringe image after appying Gaussian fiter to ; (d) high-frequency wrapped phase; (e) ow-frequency wrapped phase with LUT error compensation; (f) 3D resut before boundary correction; (g) 3D resut without LUT error compensation; (h) fina 3D resut of proposed method; (i) 3D resut of conventiona method. determine median phase vaue that was used as reference for correction. Figure 12(h) shows the fina 3D resut after empoying the aforementioned boundary correction agorithm. Ceary, the reconstructed resut has smooth geometry for the entire 3D surface, demonstrating that proposed boundary correction can effectivey correct the boundary probems. To further vaidate the performance of our agorithm, we aso appied a conventiona tempora phase unwrapping agorithm [8] to measure the same pane. Figure 12(i) shows the 3D measurement resuts using the tempora phase unwrapping agorithm. Figure 13 shows the cross-sections for both the 3D resut reconstructed from our agorithm and that from the tempora phase unwrapping agorithm. Ceary, they perfecty overap with each other, demonstrating the success of our proposed method for measuring an absoute 3D smooth surface ike a pane. We aso measured an ow statue with more compex surface geometry to further verify the performance of our proposed agorithm. Figure 14 shows one of the high-frequency phase-shifted fringe images, Fig. 14 shows one of the ow-frequency fringe images, and Fig. 14(c) shows the resutant ow-frequency image after appying a arge Gaussian fiter. Figure 14(d) is the fina high-frequency unwrapped phase obtained from the proposed agorithm. The reconstructed 3D geometry without LUT compensation is iustrated in Fig. 14(e), having obvious surface discontinuities caused by incorrect phase unwrapping. Figure 14(f) shows the resut after empoying the LUT error compensation step. It ceary shows that the 3D centra area is significanty improved. However, some areas near the boundary regions are not correcty reconstructed. We then empoyed the aforementioned boundary correction agorithm, and Fig. 14(g) shows the resut. This figure shows that most boundary probems are propery addressed. Simiary, we empoyed the tempora phase unwrapping agorithm to measure the same statue, and the resut is shown in Fig. 14(h). Figure 15 shows the cose-up views for these 3D reconstructions for better visua comparisons. Figures 15 and 15, respectivey, show the zoom-in view of the 3D resut before and after LUT compensation. Again, the LUT-based error compensation agorithm can improve measurement quaity by reducing unwrapping artifacts. Figure 15(c) shows the zoom-in view of fina measurement resut whose boundary artifacts are significanty aeviated compared to the resut before empoying the boundary correction agorithm. Figure 15(d) shows the same region of the zoom-in view for the resut obtained from the conventiona tempora phase unwrapping agorithm. Compared to the ground truth shown in Fig. 15(d), the region near the right ear of our fina resut (highighted within the red-dashed window) shown in Fig. 15(c) has an overa shift in depth. In this area, the neighbors we used for boundary correction have arge depth difference from the pixe 25 Before LUT After LUT Proposed method Conventiona method Fig. 13. Cross-section of 3D resuts shown in Fig. 12. One cross-section of Figs. 12(g) and 12(f); one cross-section of Figs. 12(h) and 12(i). Fig. 14. Experiment resuts of an ow statue with compex geometry. One of three high-frequency phase-shifted fringe patterns; one of three ow-frequency phase-shifted patterns; (c) the fringe image after appying Gaussian fiter to ; (d) fina high-frequency absoute unwrapped phase; (e) 3D resut without LUT error compensation; (f) 3D resut before boundary correction; (g) fina 3D resut of proposed method; (h) 3D resut of conventiona method.

8 Research Artice Vo. 56, No. 31 / November / Appied Optics 8717 Fig. 15. Cose-up views of the 3D reconstructions of the ow statue. Zoom-in view of Fig. 14(e); zoom-in view of Fig. 14(f); (c) zoom-in view of Fig. 14(g); (d) zoom-in view of Fig. 14(h). (c) and (d) highight one region that our proposed method fais to correcty reconstruct the 3D absoute geometry. to its neighboring pixes that does not satisfy our assumption that the oca geometry near the corrected boundary must be smooth. Figure 16 shows the same cross-sections for a those 3D reconstructions shown in Fig. 15 to further visuaize the differences. Figure 16 shows that these two curves overap we except for about 1 pixes near the most eft end where the boundary correction agorithm faied. This, once again, was caused by the vioation of surface smoothness assumption. This compex statue experiment demonstrated that, for a compex object measurement, the LUT-based error compensation can effectivey reduce unwrapping artifacts within the centra area, and the boundary correction agorithm can further reduce the unwrapping artifacts near the boundary region. However, there might sti remain unwrapping artifacts near the boundary if the surface smoothness assumption is not satisfied near the boundary regions; and this is the imitation of our proposed method. Finay, we tested our agorithm on mutipe isoated objects: two separated bas were measured at the same time. One of the captured high-frequency fringe patterns is shown in Fig. 17. For mutipe-object measurement, we first removed the background and then found the argest two connected components to separate these two bas. The separated resuts based on the texture images are shown in Figs. 17 and 17(c). We appied our agorithm to cacuate the absoute phase for the separated Before LUT After LUT Proposed method Conventiona method Fig. 16. Cross-section of 3D resuts shown in Fig. 14. One cross-section of Figs. 15 and 15; one cross-section of Figs. 15(c) and 15(d). Fig. 17. Experimenta resuts on simutaneousy measuring two separated bas. One of three high-frequency phase-shifted fringe patterns; separation of the first ba; (c) separation of the second ba; (d) high-frequency absoute unwrapped phase of the first ba; (e) high-frequency absoute unwrapped phase of the second ba; (f) fina high-frequency absoute unwrapped phase; (g) fina 3D resut of proposed method; (h) 3D resut of conventiona method Proposed method Conventiona method Fig. 18. Cross-sections of 3D measurement resuts of mutipe objects shown in Figs. 17(g) and 17(h). image of each object whose unwrapped phase is shown in Figs. 17(d) and 17(e), respectivey. The unwrapped phase maps were combined into one fina absoute phase map, as shown in Fig. 17(f). Figure 17(g) shows the reconstructed geometry using our proposed method. As a comparison, Fig. 17(h) shows the 3D measurement resut using the conventiona tempora phase unwrapping agorithm. Once again, we potted the cross-sections of the measurement resut using our proposed method and that using the conventiona tempora phase-unwrapping method. Figure 18 shows two overapped cross-sections. They are perfecty overapped, as expected, since the sphere surfaces are competey smooth. In summary, a these experimenta resuts ceary demonstrated that our proposed method can satisfactoriy recover absoute 3D geometry if the surface geometry is smooth near the boundary regardess whether it is a simpe object or mutipe objects. However, the proposed method coud sti resut in some artifacts near boundary if surface geometry is not smooth on the boundary.

9 8718 Vo. 56, No. 31 / November / Appied Optics Research Artice 5. SUMMARY This paper has presented a nove method for absoute 3D shape measurement that uses ony six square binary patterns for an enhanced two-frequency phase unwrapping agorithm. The projector is ony required to be defocused so that highfrequency patterns can generate a high-quaity phase. A Gaussian fiter is appied to ow-frequency fringe images for artificiay reducing phase error caused by the binary structures of the ow-frequency patterns. Then an LUT-based error compensation was deveoped to reduce harmonic error impact, and a boundary correction agorithm was aso proposed to further aeviate boundary unwrapping artifacts introduced by Gaussian fitering. Experimenta resuts demonstrated the success of our proposed method to measure the absoute shape of compex geometry objects, as we as the mutipe isoated objects, despite that there might sti be unwrapping artifacts if the surface geometry near boundary is not smooth. Funding ). Nationa Science Foundation (NSF) (CMMI- Acknowedgment. The authors woud ike to thank other graduate and undergraduate students, especiay Beatrice Lim, in our aboratory for their vauabe discussions and for the assistance on experiments. This work was not possibe without their support and hep. REFERENCES 1. S. Zhang, Recent progresses on rea-time 3-d shape measurement using digita fringe projection techniques, Opt. Lasers Eng. 48, (21). 2. D. Maacara, ed., Optica Shop Testing, 3rd ed. (Wiey, 27) 3. D. C. Ghigia and M. D. Pritt, eds., Two-Dimensiona Phase Unwrapping: Theory, Agorithms, and Software (Wiey, 1998). 4. X. Su and W. Chen, Reiabiity-guided phase unwrapping agorithm: a review, Opt. Lasers Eng. 42, (24). 5. Y.-Y. Cheng and J. C. Wyant, Two-waveength phase shifting interferometry, App. Opt. 23, (1984). 6. Y.-Y. Cheng and J. C. Wyant, Mutipe-waveength phase shifting interferometry, App. Opt. 24, (1985). 7. D. P. Towers, J. D. C. Jones, and C. E. Towers, Optimum frequency seection in muti-frequency interferometry, Opt. Lett. 28, (23). 8. G. Sansoni, M. Carocci, and R. Rodea, Three-dimensiona vision based on a combination of gray-code and phase-shift ight projection: anaysis and compensation of the systematic errors, App. Opt. 38, (1999). 9. S. Zhang, Fexibe 3D shape measurement using projector defocusing: extended measurement range, Opt. Lett. 35, (21). 1. K. Creath, Step height measurement using two-waveength phaseshifting interferometry, App. Opt. 26, (1987). 11. J.-S. Hyun and S. Zhang, Enhanced two-frequency phase-shifting method, App. Opt. 55, (216). 12. Y. An, J.-S. Hyun, and S. Zhang, Pixe-wise absoute phase unwrapping using geometric constraints of structured ight system, Opt. Express 24, (216). 13. M. Fujigaki, Y. Oura, D. Asai, and Y. Murata, High-speed height measurement by a ight-source-stepping method using a inear ed array, Opt. Express 21, (213). 14. M. Schaffer, M. Große, B. Harendt, and R. Kowarschik, Coherent two-beam interference fringe projection for high-speed threedimensiona shape measurements, App. Opt. 52, (213). 15. S. Heist, P. Lutzke, I. Schmidt, P. Dietrich, P. Kühmstedt, A. Tünnermann, and G. Notni, High-speed three-dimensiona shape measurement using GOBO projection, Opt. Lasers Eng. 87, 9 96 (216). 16. S. Lei and S. Zhang, Fexibe 3-D shape measurement using projector defocusing, Opt. Lett. 34, (29). 17. B. Li, Y. Wang, J. Dai, W. Lohry, and S. Zhang, Some recent advances on superfast 3D shape measurement with digita binary defocusing techniques, Opt. Lasers Eng. 54, (214). 18. B. Li, N. Karpinsky, and S. Zhang, Nove caibration method for structured ight system with an out-of-focus projector, App. Opt. 53, (214).