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Absolute Phase Retrieval Using One Coded Pattern and Geometric Constraints of Fringe Projection System

Absolute Phase Retrieval Using One Coded Pattern and Geometric Constraints of Fringe Projection... applied sciences Article Absolute Phase Retrieval Using One Coded Pattern and Geometric Constraints of Fringe Projection System 1 1 1 1 2 1 , Xu Yang , Chunnian Zeng , Jie Luo , Yu Lei , Bo Tao and Xiangcheng Chen * School of Automation, Wuhan University of Technology, Wuhan 430070, China; yx_auto@whut.edu.cn (X.Y.); zengchn@whut.edu.cn (C.Z.); luo_jie@whut.edu.cn (J.L.); leiyu9087@whut.edu.cn (Y.L.) Key Laboratory of Metallurgical Equipment and Control Technology, Ministry of Education, Wuhan University of Science and Technology, Wuhan 430081, China; taoboq@wust.edu.cn * Correspondence: chenxgcg@ustc.edu; Tel.: +86-139-66733394 Received: 30 October 2018; Accepted: 14 December 2018; Published: 18 December 2018 Abstract: Fringe projection technologies have been widely used for three-dimensional (3D) shape measurement. One of the critical issues is absolute phase recovery, especially for measuring multiple isolated objects. This paper proposes a method for absolute phase retrieval using only one coded pattern. A total of four patterns including one coded pattern and three phase-shift patterns are projected, captured, and processed. The wrapped phase, as well as average intensity and intensity modulation, are calculated from three phase-shift patterns. A code word encrypted into the coded pattern can be calculated using the average intensity and intensity modulation. Based on geometric constraints of fringe projection system, the minimum fringe order map can be created, upon which the fringe order can be calculated from the code word. Compared with the conventional method, the measurement depth range is significantly improved. Finally, the wrapped phase can be unwrapped for absolute phase map. Since only four patterns are required, the proposed method is suitable for real-time measurement. Simulations and experiments have been conducted, and their results have verified the proposed method. Keywords: absolute phase retrieval; phase-shift; fringe order; geometric constraints 1. Introduction Optical 3D measurement plays a pivotal role in all aspects of our lives, such as industrial production, biological medicine, and consumer entertainment [1–5]. Many optical technologies including structured light, stereo vision, and digital fringe projection (DFP) have been exploited to achieve high-density and full-field 3D measurement [6]. Among those technologies, DFP has become the most popular one because of its speed, accuracy, and flexibility [7]. Fourier transform and phase-shift are two main methods applied in the DFP system [8]. The former method only uses one pattern for computing phase map, but the measured surfaces must be rather simple to avoid a spectral overlapping problem. On the other hand, the phase-shift method exploits at least three patterns to compute the phase map pixel-by-pixel, which can achieve higher accuracy and stronger robustness, especially for complex surfaces. However, those two methods can only work out wrapped phases which need to be unwrapped for absolute phase maps. Ideally, when referring to the neighboring pixels, the wrapped phase can be unwrapped by adding integral multiple of 2p at each pixel. In reality, however, local shadows, random noises, and isolated objects are very usual occurrences that make the unwrapping phase difficult [9]. Thus, many absolute phase retrieval algorithms have been proposed, which can be divided into two major classes: spatial algorithms and temporal algorithms [7]. The spatial algorithms are Appl. Sci. 2018, 8, 2673; doi:10.3390/app8122673 www.mdpi.com/journal/applsci Appl. Sci. 2018, 8, 2673 2 of 11 generally used for smooth surfaces, while the temporal algorithms are more suitable for complex surfaces and attract more attention [10]. Research conducted in this field brings forth several typical examples. Chen et al. [11,12] first proposed two-wavelength phase-shift interferometry, and then developed multi-wavelength phase-shift interferometry to enhance the measurement capability. Sansoni et al. [13] combined phase-shift and gray-code into the 3D vision system, which greatly improved the measurement performance. Wang et al. [14] put forward an effective and robust phase-coding method. Zheng et al. [15] improved the phase-coding method for a large number of code words. Chen et al. [16,17] successively developed a quantized phase-coding method and a modified gray-level coding method, which achieved good results when measuring isolated objects. Nevertheless, all the aforementioned methods require three or more extra patterns, which will limit the speed of measurement. To reduce the number of patterns, some researchers have utilized color patterns for 3D measurement [18–20]. However, these methods have always failed for colorful objects. Other researchers have employed more cameras to capture the patterns from different perspectives, such that the multi-view geometric constraints can be used for absolute phase calculation [21–23]. However, the measurement field reduces because of the multiple perspectives, and the cost and complexity of the system increase due to additional cameras [24]. To realize high-speed measurement, An et al. [25] recently proposed a pixel-wise phase unwrapping method with no additional pattern. Based on the geometric constraints of fringe projection system, an artificial phase map F at the closest depth plane z is generated, and then the phase min min unwrapping can be executed by referring to F . Subsequently, a number of algorithms were min developed for phase unwrapping based on An’s method [26–29]. However, the maximum depth range this method can handle is within 2p in phase domain. When the object points far away from depth plane z brings more than 2p changes, this method is no longer applicable. min Inspired by An’s method, this paper presents an absolute phase retrieval method using only one additional coded pattern to improve the measurement depth range. Firstly, the wrapped phase is calculated from three phase-shift patterns, and the code word is extracted from the coded pattern. Secondly, an artificial fringe order map k of depth plane z is generated, and then the code word min min is mapped to the fringe order by referring to the fringe order map k . Finally, the wrapped phase is min unwrapped for the absolute phase map. Simulations and experiments have been conducted to verify the proposed method. 2. Principle 2.1. Fringe Projection System The setup of a typical fringe projection system is shown in Figure 1. This system mainly includes a projector, a camera, and measured objects. The patterns are projected by the projector onto the measured objects from one direction, modulated by the objects’ surfaces, and then captured by the camera from another direction. In Figure 1, Points O and O respectively denote the optical centers of c p the camera and the projector. The optical axes of the projector and the camera intersect at point O on the reference plane. Note that line O O is parallel to the reference plane, so points O and O have c p c p the same distance L from the reference surface. Based on the triangulation principle, the height of the measured objects can be computed as [30]: L Df h = (1) 2p f d + Df where Df denotes the phase difference between the point P on the object and the point B on the reference plane, f denotes the frequency of the fringe on the reference plane. For a specific system, parameters L, d and f are fixed, which can be obtained by calibration [31]. 0 0 Appl. Sci. 2018, 8, 2673 3 of 11 Appl. Sci. 2018, 8, x FOR PEER REVIEW 3 of 11 Figure 1. Fringe projection system. Figure 1. Fringe projection system. 2.2. Phase-Shift and Coded Patterns 2.2. Phase-Shift and Coded Patterns Phase-shift methods have been widely used for optical measurement because of their Phase-shift methods have been widely used for optical measurement because of their measurement accuracy, spatial resolution, and data density [8]. The three-step phase-shift method measurement accuracy, spatial resolution, and data density [8]. The three-step phase-shift method requires the fewest number of patterns among various phase-shift methods, thus it is desirable for requires the fewest number of patterns among various phase-shift methods, thus it is desirable for high-speed applications. Three-step phase-shift patterns can be described as: high-speed applications. Three-step phase-shift patterns can be described as: I (x, y)= A(x, y)+ B(x, y) cos (x, y)− 2 / 3   8 1 I (x, y) = A(x, y) + B(x, y) cos[f(x, y) 2p/3] < 1 I (x, y)=+ A(x, y) B(x, y) cos(x, y) (2) I (x, y) = A(x, y) + B(x, y) cos[f(x, y)] (2) > I (x, y)= A(x, y)+ B(x, y) cos (x, y)+ 2 / 3   :  I (x, y) = A(x, y) + B(x, y) cos[f(x, y) + 2p/3] where A(x, y) denotes the average intensity, B(x, y) denotes the intensity modulation, and where A(x, y) denotes the average intensity, B(x, y) denotes the intensity modulation, and f(x, y)  (xy , ) denotes the phase to be solved for. Figure 2a–c shows three phase-shift patterns generated denotes the phase to be solved for. Figure 2a–c shows three phase-shift patterns generated using using the above equations, and same rows of the three patterns are shown in Figure 3a. Solving the the above equations, and same rows of the three patterns are shown in Figure 3a. Solving the above above equations, the three variables can be calculated as: equations, the three variables can be calculated as: A(x, y)= (I + I + I ) / 3 1 2 3 2 2 1/ 2 B(x, y)= [(I − I ) / 3+ (2I − I − I ) / 9] (3) > A(x , y) = (I + I + I )/3 1 1 2 3 3 2 1 3 1/2 −1 2 2  (x, y)= tan [ 3(I − I ) / (2I − I − I )] (3) B(x, y) = [(I I ) /3 + (2I I I ) /9]  1 3 2 1 3 1 3 2 1 3 > p f(x, y) = tan [ 3(I I )/(2I I I )] 1 3 2 1 3 Because of the arctangent operation, the wrapped phase  (xy , ) is limited in range of [−π, π]. Thus, phase unwrapping should be carried out to recover the absolute phase. If the fringe order Because of the arctangent operation, the wrapped phase f(x, y) is limited in range of [p, p]. k (x, y) is determined, the absolute phase (xy , ) can be calculated as: Thus, phase unwrapping should be carried out to recover the absolute phase. If the fringe order k(x, y) (x, y)= (x, y)+ 2 k(x, y) is determined, the absolute phase F(x, y) can be calculated as: (4) To determine the fringe order, we designed one coded pattern. Figure 2d shows the coded F(x, y) = f(x, y) + 2p k(x, y) (4) pattern, and one row of this pattern is shown in Figure 3b. The coded pattern can be described as: To determine the fringe order, we designed one coded pattern. Figure 2d shows the coded pattern, I (x, y)= A(x, y)+ B(x, y) *M (x, y) = A(x, y)+ B(x, y) * 2 * mod x / P , N / N −1 ( ) (5) M   and one row of this pattern is shown in Figure 3b. The coded pattern can be described as: where P represents the fringe period, the truncated integer k = x / P represents the fringe order,   and I th (xe , y rema ) =inder A(x, yC)=+mB o( dx(,ky , N)) M repres (x, yent ) = s th Ae( x code , y) + word B(x ; ,not y) e th [2at  it mod is a (per dx/ ioP dice, N function )/N with 1] (5) a period of N. Once these four patterns are captured, the coded coefficient M (x, y) ranging from −1 where P represents the fringe period, the truncated integer k = x/P represents the fringe order, d e to 1 can be calculated as: and the remainder C = mod(k, N) represents the code word; note that it is a periodic function with a −1 M (x, y)=− cos [(I A) / B] (6) period of N. Once these four patterns are captured, the coded coefficient M(x, y) ranging from 1 to 1 can be calculated as: Then the code word C(x, y) can be computed as: M(x, y) = cos [(I A)/B] (6) Then the code word C(x, y) can be computed as: Appl. Sci. 2018, 8, x FOR PEER REVIEW 4 of 11 Ap Appl. pl. Sci. Sci.2018 2018, ,88,, x FOR 2673 PEER REVIEW 4 4of of11 11 C(x, y)=+ round (M 1) *N / 2 C(x, y)=+ round (M 1) *N / 2 (7)   (7) C(x, y) = round (M + 1) N/2 (7) [ ] Figure 2. Projected patterns. (a–c) phase-shift patterns; (d) coded pattern. Figure 2. Projected patterns. (a–c) phase-shift patterns; (d) coded pattern. Figure 2. Projected patterns. (a–c) phase-shift patterns; (d) coded pattern. Figure 3. Same rows as in Figure 2. (a) Phase-shift patterns; (b) coded pattern. Figure Figure 33 . .Same Same rows rows aa s s in in Figu Figu re 2 re 2 . ( . ( aa ) ) P P hase hase -- shif shif t t patterns patterns ; ;( ( b b ) ) cc od od ed ed patte patte rn rn . . 2.3. Geometric Constraints for Phase Unwrapping 2.3. Geometric Constraints for Phase Unwrapping 2.3. Geometric Constraints for Phase Unwrapping An et al. [25] have recently proposed a pixel-wise phase unwrapping method based on geometric An et al. [25] have recently proposed a pixel-wise phase unwrapping method based on An et al. [25] have recently proposed a pixel-wise phase unwrapping method based on constraints of the fringe projection system. The main idea is to create the minimum phase map F min geometric constraints of the fringe projection system. The main idea is to create the minimum phase geometric constraints of the fringe projection system. The main idea is to create the minimum phase at the closest depth plane z of the measured volume. Then phase unwrapping can be performed min map Фmin at the closest depth plane zmin of the measured volume. Then phase unwrapping can be map Фmin at the closest depth plane zmin of the measured volume. Then phase unwrapping can be with reference to minimum phase map F . The details of this method have been described in [25]. min performed with reference to minimum phase map Фmin. The details of this method have been performed with reference to minimum phase map Фmin. The details of this method have been The following briefly introduces the main idea of this method. described in [25]. The following briefly introduces the main idea of this method. described in [25]. The following briefly introduces the main idea of this method. Figure 4 illustrates the phase unwrapping method using the minimum phase map F . If the min Figure 4 illustrates the phase unwrapping method using the minimum phase map Фmin. If the Figure 4 illustrates the phase unwrapping method using the minimum phase map Фmin. If the wrapped phase f is less than F , we need to add k times of 2p to the wrapped phase f to obtain the min wrapped phase ϕ is less than Фmin, we need to add k times of 2π to the wrapped phase ϕ to obtain the wrapped phase ϕ is less than Фmin, we need to add k times of 2π to the wrapped phase ϕ to obtain the absolute phase F. The fringe order k can be computed as: absolute phase Ф. The fringe order k can be computed as: absolute phase Ф. The fringe order k can be computed as: F f − min   −  min min k(x, y) = ceil (8) k (x, y)= ceil k (x, y)= ceil (8)  (8)  2p 2 2   where function ceil() returns the closest upper integer value. It should be noted that the above equation where function ceil() returns the closest upper integer value. It should be noted that the above where function ceil() returns the closest upper integer value. It should be noted that the above must satisfy the following condition: equation must satisfy the following condition: equation must satisfy the following condition: 02 −   02 −   (9) min (9) 0  F Fmin < 2p (9) min Its physics signification is that the measured objects should be close to the depth plane zmin and Its physics signification is that the measured objects should be close to the depth plane zmin and Its physics signification is that the measured objects should be close to the depth plane z and min within 2π in phase domain. In other words, the maximum depth range should be less than 2π within 2π in phase domain. In other words, the maximum depth range should be less than 2π within 2p in phase domain. In other words, the maximum depth range should be less than 2p changes changes which will limit the applications of this method. For example, at point A,−  2 , and changes which will limit the applications of this method. For example, at point A,−  2 , and mminin Appl. Sci. 2018, 8, x FOR PEER REVIEW 5 of 11 Appl. Sci. 2018, 8, 2673 5 of 11 wrapped phase  is correctly unwrapped for the absolute phase = ; at point B, −  2 , min Appl. Sci. 2018, 8, x FOR PEER REVIEW 5 of 11 which will limit the applications of this method. For example, at point A,F F < 2p, and wrapped  min but wrapped phase  is wrongly unwrapped for the absolute phase  . phase f is correctly unwrapped for the absolute phase F = F; at point B, F F > 2p, but wrapped wrapped phase  is correctly unwrapped for the absolute phase = ; at point B, −  2 , min min phase f is wrongly unwrapped for the absolute phase F 6= F. but wrapped phase  is wrongly unwrapped for the absolute phase  . Figure 4. Phase unwrapping using the minimum phase map Фmin. Figure 4. Phase unwrapping using the minimum phase map Фmin. Figure 4. Phase unwrapping using the minimum phase map F . min 2.4. Phase Unwrapping with One Coded Pattern 2.4. Phase 2.4. Phas Unwrapping e Unwrappinwith g witOne h One Co Coded ded Pattern Pattern To improve the measurement depth range, we utilized an additional coded pattern to provide To improve the measurement depth range, we utilized an additional coded pattern to provide To improve the measurement depth range, we utilized an additional coded pattern to provide more information for fringe order determination. Assume that the camera captures an object placed more information for fringe order determination. Assume that the camera captures an object placed more information for fringe order determination. Assume that the camera captures an object placed at at the depth plane zmin, there exists a one-to-one mapping between the camera sensor and the at the depth plane zmin, there exists a one-to-one mapping between the camera sensor and the the depth plane z , there exists a one-to-one mapping between the camera sensor and the projector min projector sensor, and the minimum fringe order kmin can be uniquely defined on the projector sensor. projector sensor, and the minimum fringe order kmin can be uniquely defined on the projector sensor. sensor, and the minimum fringe order k can be uniquely defined on the projector sensor. Figure 5 min Figure 5 illustrates the main idea to determine the fringe order k, in which line kmin plots the Figure 5 illustrates the main idea to determine the fringe order k, in which line kmin plots the illustrates the main idea to determine the fringe order k, in which line k plots the minimum fringe min minimum fringe order, the line C plots the code word at depth plane z, and line k plots the minimum fringe order, the line C plots the code word at depth plane z, and line k plots the order, the line C plots the code word at depth plane z, and line k plots the corresponding fringe order. corresponding fringe order. The relationship between the three variables can be described as: corresponding fringe order. The relationship between the three variables can be described as: The relationship between the three variables can be described as:  kC − min k=+ C N *ceil (10)  kC −  min  k=+ C N *ceil k C (10)  min k = C + N ceil (10)  For example, at point D, kC− 0 , thus kC = ; at point E, 0 k −C N , thus k=+ C N ; min min For example, at point D, kC− 0 , thus kC = ; at point E, 0 k −C N , thus k=+ C N ; at point F, N  k −C  2* N , thus k=+ C 2* N . Similarly, the above equation must satisfy the min min min For example, at point D, k C < 0, thus k = C; at point E, 0 < k C < N, thus k = C + N; min min at point F, N  k −C  2* N , thus k=+ C 2* N . Similarly, the above equation must satisfy the following condition: min at point F, N < k C < 2 N, thus k = C + 2 N. Similarly, the above equation must satisfy the min following condition: following condition: 0 k− k  N (11) min 0  k k < N (11) min 0 k− k  N (11) In other words, the measured objects should be close to the depth plane zmin within 2πN in min phase domain. Through the above analysis, the proposed method raises the measurement depth In other words, the measured objects should be close to the depth plane z within 2pN in phase min In other words, the measured objects should be close to the depth plane zmin within 2πN in range by N times compared with the traditional method. domain. Through the above analysis, the proposed method raises the measurement depth range by N phase domain. Through the above analysis, the proposed method raises the measurement depth times compared with the traditional method. range by N times compared with the traditional method. Figure 5. Fringe order determination using the minimum fringe order kmin. Figure 5. Fringe order determination using the minimum fringe order kmin. Figure 5. Fringe order determination using the minimum fringe order k . min Appl. Sci. 2018, 8, 2673 6 of 11 Appl. Sci. 2018, 8, x FOR PEER REVIEW 6 of 11 Appl. Sci. 2018, 8, x FOR PEER REVIEW 6 of 11 3. Simulation 3. Simulation 3. Simulation To test the performance of the proposed method, some simulations were carried out. Figure 6 To test the performance of the proposed method, some simulations were carried out. Figure 6 To test the performance of the proposed method, some simulations were carried out. Figure 6 shows the simulation of the closet depth plane z . Specifically, Figure 6a–c shows three phase-shift min shows the simulation of the closet depth plane zmin. Specifically, Figure 6a–c shows three phase-shift shows the simulation of the closet depth plane zmin. Specifically, Figure 6a–c shows three phase-shift patterns with eight periods; Figure 6d shows the corresponding wrapped phase ranging fromp to p; patterns with eight periods; Figure 6d shows the corresponding wrapped phase ranging from −π to patterns with eight periods; Figure 6d shows the corresponding wrapped phase ranging from −π to Figure 6e shows the fringe order map regarded as k ; and Figure 6f shows the absolute phase map min π; Figure 6e shows the fringe order map regarded as kmin; and Figure 6f shows the absolute phase π; Figure 6e shows the fringe order map regarded as kmin; and Figure 6f shows the absolute phase regarded as F . min map regarded as Фmin. map regarded as Фmin. Figure 6. Simulation of depth plane zmin. (a–c) Phase-shift patterns; (d) wrapped phase map; (e) Figure 6. Simulation of depth plane z . (a–c) Phase-shift patterns; (d) wrapped phase map; min Figure 6. Simulation of depth plane zmin. (a–c) Phase-shift patterns; (d) wrapped phase map; (e) minimum fringe order map kmin; (f) minimum phase map Фmin. (e) minimum fringe order map k ; (f) minimum phase map F . min min minimum fringe order map kmin; (f) minimum phase map Фmin. Then, a hemisphere was selected as the measure object and simulated, as shown in Figure 7. Then, a hemisphere was selected as the measure object and simulated, as shown in Figure 7. Then, a hemisphere was selected as the measure object and simulated, as shown in Figure 7. Specifically, Figure 7a–c shows the three phase-shift patterns; Figure 7d shows the coded pattern Specifically, Figure 7a–c shows the three phase-shift patterns; Figure 7d shows the coded pattern with Specifically, Figure 7a–c shows the three phase-shift patterns; Figure 7d shows the coded pattern with N = 4; Figure 7e shows the fringe order determined by the proposed method; Figure 7f shows N = 4; Figure 7e shows the fringe order determined by the proposed method; Figure 7f shows the with the N fri = nge 4; Fo igure rder map 7e show determined s the fring by e An’ order s met determ hod for ined com by par the iso pro n; po Fig sed ure met 7g shows hod; Figure the ab 7f solu shows te fringe order map determined by An’s method for comparison; Figure 7g shows the absolute phase phase map recovered by the proposed method; Figure 7h shows the absolute phase map recovered the fringe order map determined by An’s method for comparison; Figure 7g shows the absolute map recovered by the proposed method; Figure 7h shows the absolute phase map recovered by An’s by An’s method. Obviously, the fringe order and the absolute phase map are correctly determined phase map recovered by the proposed method; Figure 7h shows the absolute phase map recovered method. Obviously, the fringe order and the absolute phase map are correctly determined by the by the proposed method. However, An’s method fails in contrast. The 3D reconstruction results of by An’s method. Obviously, the fringe order and the absolute phase map are correctly determined proposed method. However, An’s method fails in contrast. The 3D reconstruction results of the the hemisphere using the two methods are shown in Figure 8. As we can see, the proposed method by the proposed method. However, An’s method fails in contrast. The 3D reconstruction results of can accurately recover the whole surface of the hemisphere, but An’s method fails to measure the hemisphere using the two methods are shown in Figure 8. As we can see, the proposed method can the hemisphere using the two methods are shown in Figure 8. As we can see, the proposed method overall hemisphere surface. The maximum depth range of the proposed method can deal with is accurately recover the whole surface of the hemisphere, but An’s method fails to measure the overall can accurately recover the whole surface of the hemisphere, but An’s method fails to measure the 2πN, which is four times that of An’s method. hemisphere surface. The maximum depth range of the proposed method can deal with is 2pN, which overall hemisphere surface. The maximum depth range of the proposed method can deal with is is four times that of An’s method. 2πN, which is four times that of An’s method. Figure 7. Simulation of a hemisphere. (a–c) Phase-shift patterns; (d) coded pattern; (e) fringe order map using the proposed method; (f) fringe order map using An’s method; (g) absolute phase map using the proposed method; (h) absolute phase map using An’s method. Figure 7. Simulation of a hemisphere. (a–c) Phase-shift patterns; (d) coded pattern; (e) fringe order Figure 7. Simulation of a hemisphere. (a–c) Phase-shift patterns; (d) coded pattern; (e) fringe order map using the proposed method; (f) fringe order map using An’s method; (g) absolute phase map map using the proposed method; (f) fringe order map using An’s method; (g) absolute phase map using the proposed method; (h) absolute phase map using An’s method. using the proposed method; (h) absolute phase map using An’s method. Appl. Sci. 2018, 8, 2673 7 of 11 Appl. Sci. 2018, 8, x FOR PEER REVIEW 7 of 11 Appl. Sci. 2018, 8, x FOR PEER REVIEW 7 of 11 Figure Figure 8. 8. A A 3D 3D recons reconstr tru uction ction of of the the hem hemispher isphere e.. ( (a a) ) T The he pro proposed posed m method; ethod; ( (b b) ) An’s An’s method. method. Figure 8. A 3D reconstruction of the hemisphere. (a) The proposed method; (b) An’s method. 4. Experimental Setup 4. Experimental Setup 4. Experimental Setup To test the proposed method in real condition, a fringe projection system was set up. The system To test the proposed method in real condition, a fringe projection system was set up. The To test the proposed method in real condition, a fringe projection system was set up. The consisted of a projector (Light Crafter 4500) with resolution of 912 1140 pixels, and a camera (IOI Flare system consisted of a projector (Light Crafter 4500) with resolution of 912 × 1140 pixels, and a camera system consisted of a projector (Light Crafter 4500) with resolution of 912 × 1140 pixels, and a camera 2M360-CL) with resolution of 1280  1024 pixels. A flat board was placed at the closest depth plane (IOI Flare 2M360-CL) with resolution of 1280 × 1024 pixels. A flat board was placed at the closest (IOI Flare 2M360-CL) with resolution of 1280 × 1024 pixels. A flat board was placed at the closest of the measured volume, and used as the reference plane. Two isolated objects were selected as the depth plane of the measured volume, and used as the reference plane. Two isolated objects were depth plane of the measured volume, and used as the reference plane. Two isolated objects were measured objects. Total four patterns, including three phase-shift patterns and one coded pattern, selected as the measured objects. Total four patterns, including three phase-shift patterns and one selected as the measured objects. Total four patterns, including three phase-shift patterns and one were projected onto the reference plane and the measured objects by the projector, and sequentially coded pattern, were projected onto the reference plane and the measured objects by the projector, coded pattern, were projected onto the reference plane and the measured objects by the projector, captured by the camera. and sequentially captured by the camera. and sequentially captured by the camera. Figure 9a–c shows three phase-shift patterns projected onto the reference plane, respectively. Figure 9a–c shows three phase-shift patterns projected onto the reference plane, respectively. Figure 9a–c shows three phase-shift patterns projected onto the reference plane, respectively. Figure 9d shows the corresponding wrapped phase. Figure 9e shows the fringe order, also regarded Figure 9d shows the corresponding wrapped phase. Figure 9e shows the fringe order, also regarded Figure 9d shows the corresponding wrapped phase. Figure 9e shows the fringe order, also regarded as the minimum fringe order map k . Figure 9f shows the absolute phase map also regarded as the min as the minimum fringe order map kmin. Figure 9f shows the absolute phase map also regarded as the as the minimum fringe order map kmin. Figure 9f shows the absolute phase map also regarded as the minimum phase map F . Similarly, Figure 10a–c shows the images of three phase-shift patterns min minimum phase map Фmin. Similarly, Figure 10a–c shows the images of three phase-shift patterns minimum phase map Фmin. Similarly, Figure 10a–c shows the images of three phase-shift patterns projected onto the measured objects, respectively. Figure 10d shows the corresponding wrapped phase projected onto the measured objects, respectively. Figure 10d shows the corresponding wrapped projected onto the measured objects, respectively. Figure 10d shows the corresponding wrapped map calculated from the three phase-shift patterns. Meanwhile, the average intensity and intensity phase map calculated from the three phase-shift patterns. Meanwhile, the average intensity and phase map calculated from the three phase-shift patterns. Meanwhile, the average intensity and modulation were calculated. Figure 10e shows the coded pattern with N = 4, and Figure 10f shows the intensity modulation were calculated. Figure 10e shows the coded pattern with N = 4, and Figure 10f intensity modulation were calculated. Figure 10e shows the coded pattern with N = 4, and Figure 10f corresponding code word map. shows the corresponding code word map. shows the corresponding code word map. Figure Figure 9 9. . Ima Images ges of ofthe therefe refer rence enceplane plane. . (a–( ca )– Phas c) Phase-shift e-shift patterns patterns; ; (d) (w d) rapp wrapped ed phase phase map; map; (e) Figure 9. Images of the reference plane. (a–c) Phase-shift patterns; (d) wrapped phase map; (e) ( m e) inimu minimum m fringe fringe order order map map kmink ; (f) ;m (finimu ) minimum m phase phase map map ФminF . . minimum fringe order map kmin; min (f) minimum phase map Фmin. min In order to compare the proposed method and An’s method, Equations (12) and (14) were both used for computing fringe order. Figure 11a,b shows the fringe order maps recovered by the two methods. As we can see, the proposed method recovered the fringe order map F correctly; however, An’s method led to the wrong fringe order map F at some areas. There are obvious differences between the two fringe order maps within the two circular areas plotted in Figure 11. The pixels of the Appl. Sci. 2018, 8, x FOR PEER REVIEW 8 of 11 Appl. Sci. 2018, 8, x FOR PEER REVIEW 8 of 11 Appl. Sci. 2018, 8, 2673 8 of 11 same stripe had the same fringe order k in Figure 11a. However, the pixels of the same stripe had a different fringe order k’ in Figure 11b. Appl. Sci. 2018, 8, x FOR PEER REVIEW 8 of 11 Figure 10. Images of the measured objects. (a–c) Phase-shift patterns; (d) wrapped phase map; (e) Figure 10. Images of the measured objects. (a–c) Phase-shift patterns; (d) wrapped phase map; (e) coded pattern; (f) code-word map. coded pattern; (f) code-word map. In order to compare the proposed method and An’s method, Equations (12) and (14) were both In order to compare the proposed method and An’s method, Equations (12) and (14) were both used for computing fringe order. Figure 11a,b shows the fringe order maps recovered by the two used for computing fringe order. Figure 11a,b shows the fringe order maps recovered by the two methods. As we can see, the proposed method recovered the fringe order map Ф correctly; however, methods. As we can see, the proposed method recovered the fringe order map Ф correctly; however, An’s method led to the wrong fringe order map Ф′ at some areas. There are obvious differences An’s method led to the wrong fringe order map Ф′ at some areas. There are obvious differences between the two fringe order maps within the two circular areas plotted in Figure 11. The pixels of between the two fringe order maps within the two circular areas plotted in Figure 11. The pixels of Figure Figure 10. 10. Images Images of of the the measur measured ed objects. object (s a.– c (a )– Phase-shift c) Phase-shif patterns; t patterns; (d) wrapped (d) wrapp phase ed ph map; ase (m e)ap; coded (e) the same stripe had the same fringe order k in Figure 11a. However, the pixels of the same stripe had the same stripe had the same fringe order k in Figure 11a. However, the pixels of the same stripe had pattern; coded pattern; (f) code-wor (f) cod demap. -word map. a different fringe order k’ in Figure 11b. a different fringe order k’ in Figure 11b. For better illustration, Figure 12a,b shows the 600th rows of the two fringe order maps and In For order better to ill coustra mpar ti eon th , e Fig prop ure osed 12a,met b sho hod wsand the An’ 600th s met rows hod, of Eq th ue ation two s fri (12 n)ge ano dr ( der 14) w me aps re ban oth d For better illustration, Figure 12a,b shows the 600th rows of the two fringe order maps and absolute phase maps. Clearly, F F < 8p and F F < 2p. This indicates that the maximum used absolut for e com phase put m ing apsfri . C nle ge ar order ly, . − Fi m giur n e  11 8a ,b and show s− thm e ifr n inge 2 . oTh rder is indi maps cate rescov that ered the by m ax the imum two absolute phase maps. Clearly, −  8 and  −  2 . This indicates that the maximum min min min min depth range of the proposed method is up to 8p, and that of An’s method is only 2p. Therefore, the methods. As we can see, the proposed method recovered the fringe order map Ф correctly; however, depth range of the proposed method is up to 8π, and that of An’s method is only 2π. Therefore, the depth range of the proposed method is up to 8π, and that of An’s method is only 2π. Therefore, the proposed method can obtain much larger depth range than An’s method. Finally, we reconstructed the An’s method led to the wrong fringe order map Ф′ at some areas. There are obvious differences proposed method can obtain much larger depth range than An’s method. Finally, we reconstructed proposed method can obtain much larger depth range than An’s method. Finally, we reconstructed 3D shapes of the two isolated objects, as shown in Figure 13. between the two fringe order maps within the two circular areas plotted in Figure 11. The pixels of the 3D shapes of the two isolated objects, as shown in Figure 13. the 3D shapes of the two isolated objects, as shown in Figure 13. the same stripe had the same fringe order k in Figure 11a. However, the pixels of the same stripe had a different fringe order k’ in Figure 11b. For better illustration, Figure 12a,b shows the 600th rows of the two fringe order maps and absolute phase maps. Clearly, −  8 and −  2 . This indicates that the maximum min min depth range of the proposed method is up to 8π, and that of An’s method is only 2π. Therefore, the proposed method can obtain much larger depth range than An’s method. Finally, we reconstructed the 3D shapes of the two isolated objects, as shown in Figure 13. Figure 11. Fringe order maps. (a) The proposed method; (b) An’s method. Figure 11. Fringe order maps. (a) The proposed method; (b) An’s method. Figure 11. Fringe order maps. (a) The proposed method; (b) An’s method. Figure 11. Fringe order maps. (a) The proposed method; (b) An’s method. Figure 12. The 600th rows. (a) Fringe order maps; (b) absolute phase maps. Figure Figure 12 12. . The The 600th rows 600th rows. . ((a a) ) Fr Fringe inge ord order er m maps; aps; ( (b b) ) a absolute bsolute ph phase ase maps. maps. Figure 12. The 600th rows. (a) Fringe order maps; (b) absolute phase maps. Appl. Sci. 2018, 8, 2673 9 of 11 Appl. Sci. 2018, 8, x FOR PEER REVIEW 9 of 11 Appl. Sci. 2018, 8, x FOR PEER REVIEW 9 of 11 Appl. Sci. 2018, 8, x FOR PEER REVIEW 9 of 11 Figure Figure 1 13. 3. Mea Measur surem ement ent res result ult of of two two iisolated solated obje objects. cts. Figure 13. Measurement result of two isolated objects. Figure 13. Measurement result of two isolated objects. In order to further verify our method, two separate planes were also measured using the proposed In order to further verify our method, two separate planes were also measured using the In order to further verify our method, two separate planes were also measured using the In order to further verify our method, two separate planes were also measured using the method. Figure 14a–c shows three phase-shift patterns projected onto the two planes, respectively. proposed method. Figure 14a–c shows three phase-shift patterns projected onto the two planes, proposed method. Figure 14a–c shows three phase-shift patterns projected onto the two planes, proposed method. Figure 14a–c shows three phase-shift patterns projected onto the two planes, Figure 14d shows the corresponding wrapped phase map. Figure 14e shows the coded pattern, respectively. Figure 14d shows the corresponding wrapped phase map. Figure 14e shows the coded respectively. Figure 14d shows the corresponding wrapped phase map. Figure 14e shows the coded respectively. Figure 14d shows the corresponding wrapped phase map. Figure 14e shows the coded and Figure 14f shows the corresponding code-word map. Then the fringe order was calculated, pattern, and Figure 14f shows the corresponding code-word map. Then the fringe order was pattern, and Figure 14f shows the corresponding code-word map. Then the fringe order was pattern, and Figure 14f shows the corresponding code-word map. Then the fringe order was as shown in Figure 15a. Using Equation (4), the absolute phase map was recovered, as shown in calculated, as shown in Figure 15a. Using Equation (4), the absolute phase map was recovered, as calculated, as shown in Figure 15a. Using Equation (4), the absolute phase map was recovered, as calculated, as shown in Figure 15a. Using Equation (4), the absolute phase map was recovered, as Figure 15b. Finally, the 3D shapes of two planes were reconstructed, as shown in Figure 16. There are shown in Figure 15b. Finally, the 3D shapes of two planes were reconstructed, as shown in Figure 16. shown in Figure 15b. Finally, the 3D shapes of two planes were reconstructed, as shown in Figure 16. shown in Figure 15b. Finally, the 3D shapes of two planes were reconstructed, as shown in Figure 16. no obvious mistakes in the measurement results. The experimental results illustrate the performance There are no obvious mistakes in the measurement results. The experimental results illustrate the There are no obvious mistakes in the measurement results. The experimental results illustrate the There are no obvious mistakes in the measurement results. The experimental results illustrate the of the proposed method. performance of the proposed method. performance of the proposed method. performance of the proposed method. Figure 14. Images of two planes. (a–c) Phase-shift patterns; (d) wrapped phase map; (e) coded Figure 14. Images of two planes. (a–c) Phase-shift patterns; (d) wrapped phase map; (e) coded Figure Figure 14. 14. Images Images ofof two two pla pl nes. ane (s. a–c (a ) – Phase-shift c) Phase-shif patterns; t patter (d ns; ) wrapped (d) wrapp phase ed ph map; ase( em ) coded ap; (e) pattern; coded pattern; (f) code-word map. ( pattern; f) code-wor (f) cd od map. e-word map. pattern; (f) code-word map. Figure 15. (a) Fringe order map; (b) absolute phase map. Figure 15. (a) Fringe order map; (b) absolute phase map. Figure Figure 15. 15. ( (a a) ) F Fringe ringe order order map; map; ( (b b) ) absolute absolute p phase hase map. map. Appl. Sci. 2018, 8, 2673 10 of 11 Appl. Sci. 2018, 8, x FOR PEER REVIEW 10 of 11 Figure Figure 16. 16. Measur Measurem ement ent res result ult of of two two planes. planes. 5. Conclusions 5. Conclusions This paper has presented an absolute phase retrieval method using only one coded pattern. A total This paper has presented an absolute phase retrieval method using only one coded pattern. A of four patterns are used for 3D shape measurement, which is suitable for high-speed applications. total of four patterns are used for 3D shape measurement, which is suitable for high-speed The code words are encoded into the coded pattern, which can be correctly recovered using the average applications. The code words are encoded into the coded pattern, which can be correctly recovered intensity and intensity modulation of phase-shift patterns. Based on the geometric constraints of fringe using the average intensity and intensity modulation of phase-shift patterns. Based on the geometric projection system, the minimum fringe order map is generated, then the code word can be easily constraints of fringe projection system, the minimum fringe order map is generated, then the code converted into fringe order. Compared with the conventional method, the proposed method can word can be easily converted into fringe order. Compared with the conventional method, the significantly enhance the measurement depth range. proposed method can significantly enhance the measurement depth range. Author Contributions: X.C. and B.T. conceived and designed the experiments; X.Y. and J.L. performed the Author Contributions: X.C. and B.T. conceived and designed the experiments; X.Y. and J.L. performed the experiments; X.C. and C.Z. analyzed the data; X.Y. and Y.L. wrote the paper. experiments; X.C. and C.Z. analyzed the data; X.Y. and Y.L. wrote the paper. Funding: This research was funded by National Natural Science Foundation of China (51605130), Hubei Provincial Natural Funding: Science This rese Foundation arch was of fun China ded (2018CFB656), by National Na Fundamental tural Science Resear Fouch ndati Funds on of for Ch the ina Central (51605130), Universities Hubei (WUT: 2017IVA059), Open Fund of the Key Laboratory for Metallurgical Equipment and Control of Ministry of Provincial Natural Science Foundation of China (2018CFB656), Fundamental Research Funds for the Central Education in Wuhan University of Science and Technology (2018B03). Universities (WUT: 2017IVA059), Open Fund of the Key Laboratory for Metallurgical Equipment and Control Conflicts of Interest: The authors declare no conflicts of interest. of Ministry of Education in Wuhan University of Science and Technology (2018B03). Conflicts of Interest: The authors declare no conflicts of interest. References References 1. Geng, J. 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Absolute Phase Retrieval Using One Coded Pattern and Geometric Constraints of Fringe Projection System

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applied sciences Article Absolute Phase Retrieval Using One Coded Pattern and Geometric Constraints of Fringe Projection System 1 1 1 1 2 1 , Xu Yang , Chunnian Zeng , Jie Luo , Yu Lei , Bo Tao and Xiangcheng Chen * School of Automation, Wuhan University of Technology, Wuhan 430070, China; yx_auto@whut.edu.cn (X.Y.); zengchn@whut.edu.cn (C.Z.); luo_jie@whut.edu.cn (J.L.); leiyu9087@whut.edu.cn (Y.L.) Key Laboratory of Metallurgical Equipment and Control Technology, Ministry of Education, Wuhan University of Science and Technology, Wuhan 430081, China; taoboq@wust.edu.cn * Correspondence: chenxgcg@ustc.edu; Tel.: +86-139-66733394 Received: 30 October 2018; Accepted: 14 December 2018; Published: 18 December 2018 Abstract: Fringe projection technologies have been widely used for three-dimensional (3D) shape measurement. One of the critical issues is absolute phase recovery, especially for measuring multiple isolated objects. This paper proposes a method for absolute phase retrieval using only one coded pattern. A total of four patterns including one coded pattern and three phase-shift patterns are projected, captured, and processed. The wrapped phase, as well as average intensity and intensity modulation, are calculated from three phase-shift patterns. A code word encrypted into the coded pattern can be calculated using the average intensity and intensity modulation. Based on geometric constraints of fringe projection system, the minimum fringe order map can be created, upon which the fringe order can be calculated from the code word. Compared with the conventional method, the measurement depth range is significantly improved. Finally, the wrapped phase can be unwrapped for absolute phase map. Since only four patterns are required, the proposed method is suitable for real-time measurement. Simulations and experiments have been conducted, and their results have verified the proposed method. Keywords: absolute phase retrieval; phase-shift; fringe order; geometric constraints 1. Introduction Optical 3D measurement plays a pivotal role in all aspects of our lives, such as industrial production, biological medicine, and consumer entertainment [1–5]. Many optical technologies including structured light, stereo vision, and digital fringe projection (DFP) have been exploited to achieve high-density and full-field 3D measurement [6]. Among those technologies, DFP has become the most popular one because of its speed, accuracy, and flexibility [7]. Fourier transform and phase-shift are two main methods applied in the DFP system [8]. The former method only uses one pattern for computing phase map, but the measured surfaces must be rather simple to avoid a spectral overlapping problem. On the other hand, the phase-shift method exploits at least three patterns to compute the phase map pixel-by-pixel, which can achieve higher accuracy and stronger robustness, especially for complex surfaces. However, those two methods can only work out wrapped phases which need to be unwrapped for absolute phase maps. Ideally, when referring to the neighboring pixels, the wrapped phase can be unwrapped by adding integral multiple of 2p at each pixel. In reality, however, local shadows, random noises, and isolated objects are very usual occurrences that make the unwrapping phase difficult [9]. Thus, many absolute phase retrieval algorithms have been proposed, which can be divided into two major classes: spatial algorithms and temporal algorithms [7]. The spatial algorithms are Appl. Sci. 2018, 8, 2673; doi:10.3390/app8122673 www.mdpi.com/journal/applsci Appl. Sci. 2018, 8, 2673 2 of 11 generally used for smooth surfaces, while the temporal algorithms are more suitable for complex surfaces and attract more attention [10]. Research conducted in this field brings forth several typical examples. Chen et al. [11,12] first proposed two-wavelength phase-shift interferometry, and then developed multi-wavelength phase-shift interferometry to enhance the measurement capability. Sansoni et al. [13] combined phase-shift and gray-code into the 3D vision system, which greatly improved the measurement performance. Wang et al. [14] put forward an effective and robust phase-coding method. Zheng et al. [15] improved the phase-coding method for a large number of code words. Chen et al. [16,17] successively developed a quantized phase-coding method and a modified gray-level coding method, which achieved good results when measuring isolated objects. Nevertheless, all the aforementioned methods require three or more extra patterns, which will limit the speed of measurement. To reduce the number of patterns, some researchers have utilized color patterns for 3D measurement [18–20]. However, these methods have always failed for colorful objects. Other researchers have employed more cameras to capture the patterns from different perspectives, such that the multi-view geometric constraints can be used for absolute phase calculation [21–23]. However, the measurement field reduces because of the multiple perspectives, and the cost and complexity of the system increase due to additional cameras [24]. To realize high-speed measurement, An et al. [25] recently proposed a pixel-wise phase unwrapping method with no additional pattern. Based on the geometric constraints of fringe projection system, an artificial phase map F at the closest depth plane z is generated, and then the phase min min unwrapping can be executed by referring to F . Subsequently, a number of algorithms were min developed for phase unwrapping based on An’s method [26–29]. However, the maximum depth range this method can handle is within 2p in phase domain. When the object points far away from depth plane z brings more than 2p changes, this method is no longer applicable. min Inspired by An’s method, this paper presents an absolute phase retrieval method using only one additional coded pattern to improve the measurement depth range. Firstly, the wrapped phase is calculated from three phase-shift patterns, and the code word is extracted from the coded pattern. Secondly, an artificial fringe order map k of depth plane z is generated, and then the code word min min is mapped to the fringe order by referring to the fringe order map k . Finally, the wrapped phase is min unwrapped for the absolute phase map. Simulations and experiments have been conducted to verify the proposed method. 2. Principle 2.1. Fringe Projection System The setup of a typical fringe projection system is shown in Figure 1. This system mainly includes a projector, a camera, and measured objects. The patterns are projected by the projector onto the measured objects from one direction, modulated by the objects’ surfaces, and then captured by the camera from another direction. In Figure 1, Points O and O respectively denote the optical centers of c p the camera and the projector. The optical axes of the projector and the camera intersect at point O on the reference plane. Note that line O O is parallel to the reference plane, so points O and O have c p c p the same distance L from the reference surface. Based on the triangulation principle, the height of the measured objects can be computed as [30]: L Df h = (1) 2p f d + Df where Df denotes the phase difference between the point P on the object and the point B on the reference plane, f denotes the frequency of the fringe on the reference plane. For a specific system, parameters L, d and f are fixed, which can be obtained by calibration [31]. 0 0 Appl. Sci. 2018, 8, 2673 3 of 11 Appl. Sci. 2018, 8, x FOR PEER REVIEW 3 of 11 Figure 1. Fringe projection system. Figure 1. Fringe projection system. 2.2. Phase-Shift and Coded Patterns 2.2. Phase-Shift and Coded Patterns Phase-shift methods have been widely used for optical measurement because of their Phase-shift methods have been widely used for optical measurement because of their measurement accuracy, spatial resolution, and data density [8]. The three-step phase-shift method measurement accuracy, spatial resolution, and data density [8]. The three-step phase-shift method requires the fewest number of patterns among various phase-shift methods, thus it is desirable for requires the fewest number of patterns among various phase-shift methods, thus it is desirable for high-speed applications. Three-step phase-shift patterns can be described as: high-speed applications. Three-step phase-shift patterns can be described as: I (x, y)= A(x, y)+ B(x, y) cos (x, y)− 2 / 3   8 1 I (x, y) = A(x, y) + B(x, y) cos[f(x, y) 2p/3] < 1 I (x, y)=+ A(x, y) B(x, y) cos(x, y) (2) I (x, y) = A(x, y) + B(x, y) cos[f(x, y)] (2) > I (x, y)= A(x, y)+ B(x, y) cos (x, y)+ 2 / 3   :  I (x, y) = A(x, y) + B(x, y) cos[f(x, y) + 2p/3] where A(x, y) denotes the average intensity, B(x, y) denotes the intensity modulation, and where A(x, y) denotes the average intensity, B(x, y) denotes the intensity modulation, and f(x, y)  (xy , ) denotes the phase to be solved for. Figure 2a–c shows three phase-shift patterns generated denotes the phase to be solved for. Figure 2a–c shows three phase-shift patterns generated using using the above equations, and same rows of the three patterns are shown in Figure 3a. Solving the the above equations, and same rows of the three patterns are shown in Figure 3a. Solving the above above equations, the three variables can be calculated as: equations, the three variables can be calculated as: A(x, y)= (I + I + I ) / 3 1 2 3 2 2 1/ 2 B(x, y)= [(I − I ) / 3+ (2I − I − I ) / 9] (3) > A(x , y) = (I + I + I )/3 1 1 2 3 3 2 1 3 1/2 −1 2 2  (x, y)= tan [ 3(I − I ) / (2I − I − I )] (3) B(x, y) = [(I I ) /3 + (2I I I ) /9]  1 3 2 1 3 1 3 2 1 3 > p f(x, y) = tan [ 3(I I )/(2I I I )] 1 3 2 1 3 Because of the arctangent operation, the wrapped phase  (xy , ) is limited in range of [−π, π]. Thus, phase unwrapping should be carried out to recover the absolute phase. If the fringe order Because of the arctangent operation, the wrapped phase f(x, y) is limited in range of [p, p]. k (x, y) is determined, the absolute phase (xy , ) can be calculated as: Thus, phase unwrapping should be carried out to recover the absolute phase. If the fringe order k(x, y) (x, y)= (x, y)+ 2 k(x, y) is determined, the absolute phase F(x, y) can be calculated as: (4) To determine the fringe order, we designed one coded pattern. Figure 2d shows the coded F(x, y) = f(x, y) + 2p k(x, y) (4) pattern, and one row of this pattern is shown in Figure 3b. The coded pattern can be described as: To determine the fringe order, we designed one coded pattern. Figure 2d shows the coded pattern, I (x, y)= A(x, y)+ B(x, y) *M (x, y) = A(x, y)+ B(x, y) * 2 * mod x / P , N / N −1 ( ) (5) M   and one row of this pattern is shown in Figure 3b. The coded pattern can be described as: where P represents the fringe period, the truncated integer k = x / P represents the fringe order,   and I th (xe , y rema ) =inder A(x, yC)=+mB o( dx(,ky , N)) M repres (x, yent ) = s th Ae( x code , y) + word B(x ; ,not y) e th [2at  it mod is a (per dx/ ioP dice, N function )/N with 1] (5) a period of N. Once these four patterns are captured, the coded coefficient M (x, y) ranging from −1 where P represents the fringe period, the truncated integer k = x/P represents the fringe order, d e to 1 can be calculated as: and the remainder C = mod(k, N) represents the code word; note that it is a periodic function with a −1 M (x, y)=− cos [(I A) / B] (6) period of N. Once these four patterns are captured, the coded coefficient M(x, y) ranging from 1 to 1 can be calculated as: Then the code word C(x, y) can be computed as: M(x, y) = cos [(I A)/B] (6) Then the code word C(x, y) can be computed as: Appl. Sci. 2018, 8, x FOR PEER REVIEW 4 of 11 Ap Appl. pl. Sci. Sci.2018 2018, ,88,, x FOR 2673 PEER REVIEW 4 4of of11 11 C(x, y)=+ round (M 1) *N / 2 C(x, y)=+ round (M 1) *N / 2 (7)   (7) C(x, y) = round (M + 1) N/2 (7) [ ] Figure 2. Projected patterns. (a–c) phase-shift patterns; (d) coded pattern. Figure 2. Projected patterns. (a–c) phase-shift patterns; (d) coded pattern. Figure 2. Projected patterns. (a–c) phase-shift patterns; (d) coded pattern. Figure 3. Same rows as in Figure 2. (a) Phase-shift patterns; (b) coded pattern. Figure Figure 33 . .Same Same rows rows aa s s in in Figu Figu re 2 re 2 . ( . ( aa ) ) P P hase hase -- shif shif t t patterns patterns ; ;( ( b b ) ) cc od od ed ed patte patte rn rn . . 2.3. Geometric Constraints for Phase Unwrapping 2.3. Geometric Constraints for Phase Unwrapping 2.3. Geometric Constraints for Phase Unwrapping An et al. [25] have recently proposed a pixel-wise phase unwrapping method based on geometric An et al. [25] have recently proposed a pixel-wise phase unwrapping method based on An et al. [25] have recently proposed a pixel-wise phase unwrapping method based on constraints of the fringe projection system. The main idea is to create the minimum phase map F min geometric constraints of the fringe projection system. The main idea is to create the minimum phase geometric constraints of the fringe projection system. The main idea is to create the minimum phase at the closest depth plane z of the measured volume. Then phase unwrapping can be performed min map Фmin at the closest depth plane zmin of the measured volume. Then phase unwrapping can be map Фmin at the closest depth plane zmin of the measured volume. Then phase unwrapping can be with reference to minimum phase map F . The details of this method have been described in [25]. min performed with reference to minimum phase map Фmin. The details of this method have been performed with reference to minimum phase map Фmin. The details of this method have been The following briefly introduces the main idea of this method. described in [25]. The following briefly introduces the main idea of this method. described in [25]. The following briefly introduces the main idea of this method. Figure 4 illustrates the phase unwrapping method using the minimum phase map F . If the min Figure 4 illustrates the phase unwrapping method using the minimum phase map Фmin. If the Figure 4 illustrates the phase unwrapping method using the minimum phase map Фmin. If the wrapped phase f is less than F , we need to add k times of 2p to the wrapped phase f to obtain the min wrapped phase ϕ is less than Фmin, we need to add k times of 2π to the wrapped phase ϕ to obtain the wrapped phase ϕ is less than Фmin, we need to add k times of 2π to the wrapped phase ϕ to obtain the absolute phase F. The fringe order k can be computed as: absolute phase Ф. The fringe order k can be computed as: absolute phase Ф. The fringe order k can be computed as: F f − min   −  min min k(x, y) = ceil (8) k (x, y)= ceil k (x, y)= ceil (8)  (8)  2p 2 2   where function ceil() returns the closest upper integer value. It should be noted that the above equation where function ceil() returns the closest upper integer value. It should be noted that the above where function ceil() returns the closest upper integer value. It should be noted that the above must satisfy the following condition: equation must satisfy the following condition: equation must satisfy the following condition: 02 −   02 −   (9) min (9) 0  F Fmin < 2p (9) min Its physics signification is that the measured objects should be close to the depth plane zmin and Its physics signification is that the measured objects should be close to the depth plane zmin and Its physics signification is that the measured objects should be close to the depth plane z and min within 2π in phase domain. In other words, the maximum depth range should be less than 2π within 2π in phase domain. In other words, the maximum depth range should be less than 2π within 2p in phase domain. In other words, the maximum depth range should be less than 2p changes changes which will limit the applications of this method. For example, at point A,−  2 , and changes which will limit the applications of this method. For example, at point A,−  2 , and mminin Appl. Sci. 2018, 8, x FOR PEER REVIEW 5 of 11 Appl. Sci. 2018, 8, 2673 5 of 11 wrapped phase  is correctly unwrapped for the absolute phase = ; at point B, −  2 , min Appl. Sci. 2018, 8, x FOR PEER REVIEW 5 of 11 which will limit the applications of this method. For example, at point A,F F < 2p, and wrapped  min but wrapped phase  is wrongly unwrapped for the absolute phase  . phase f is correctly unwrapped for the absolute phase F = F; at point B, F F > 2p, but wrapped wrapped phase  is correctly unwrapped for the absolute phase = ; at point B, −  2 , min min phase f is wrongly unwrapped for the absolute phase F 6= F. but wrapped phase  is wrongly unwrapped for the absolute phase  . Figure 4. Phase unwrapping using the minimum phase map Фmin. Figure 4. Phase unwrapping using the minimum phase map Фmin. Figure 4. Phase unwrapping using the minimum phase map F . min 2.4. Phase Unwrapping with One Coded Pattern 2.4. Phase 2.4. Phas Unwrapping e Unwrappinwith g witOne h One Co Coded ded Pattern Pattern To improve the measurement depth range, we utilized an additional coded pattern to provide To improve the measurement depth range, we utilized an additional coded pattern to provide To improve the measurement depth range, we utilized an additional coded pattern to provide more information for fringe order determination. Assume that the camera captures an object placed more information for fringe order determination. Assume that the camera captures an object placed more information for fringe order determination. Assume that the camera captures an object placed at at the depth plane zmin, there exists a one-to-one mapping between the camera sensor and the at the depth plane zmin, there exists a one-to-one mapping between the camera sensor and the the depth plane z , there exists a one-to-one mapping between the camera sensor and the projector min projector sensor, and the minimum fringe order kmin can be uniquely defined on the projector sensor. projector sensor, and the minimum fringe order kmin can be uniquely defined on the projector sensor. sensor, and the minimum fringe order k can be uniquely defined on the projector sensor. Figure 5 min Figure 5 illustrates the main idea to determine the fringe order k, in which line kmin plots the Figure 5 illustrates the main idea to determine the fringe order k, in which line kmin plots the illustrates the main idea to determine the fringe order k, in which line k plots the minimum fringe min minimum fringe order, the line C plots the code word at depth plane z, and line k plots the minimum fringe order, the line C plots the code word at depth plane z, and line k plots the order, the line C plots the code word at depth plane z, and line k plots the corresponding fringe order. corresponding fringe order. The relationship between the three variables can be described as: corresponding fringe order. The relationship between the three variables can be described as: The relationship between the three variables can be described as:  kC − min k=+ C N *ceil (10)  kC −  min  k=+ C N *ceil k C (10)  min k = C + N ceil (10)  For example, at point D, kC− 0 , thus kC = ; at point E, 0 k −C N , thus k=+ C N ; min min For example, at point D, kC− 0 , thus kC = ; at point E, 0 k −C N , thus k=+ C N ; at point F, N  k −C  2* N , thus k=+ C 2* N . Similarly, the above equation must satisfy the min min min For example, at point D, k C < 0, thus k = C; at point E, 0 < k C < N, thus k = C + N; min min at point F, N  k −C  2* N , thus k=+ C 2* N . Similarly, the above equation must satisfy the following condition: min at point F, N < k C < 2 N, thus k = C + 2 N. Similarly, the above equation must satisfy the min following condition: following condition: 0 k− k  N (11) min 0  k k < N (11) min 0 k− k  N (11) In other words, the measured objects should be close to the depth plane zmin within 2πN in min phase domain. Through the above analysis, the proposed method raises the measurement depth In other words, the measured objects should be close to the depth plane z within 2pN in phase min In other words, the measured objects should be close to the depth plane zmin within 2πN in range by N times compared with the traditional method. domain. Through the above analysis, the proposed method raises the measurement depth range by N phase domain. Through the above analysis, the proposed method raises the measurement depth times compared with the traditional method. range by N times compared with the traditional method. Figure 5. Fringe order determination using the minimum fringe order kmin. Figure 5. Fringe order determination using the minimum fringe order kmin. Figure 5. Fringe order determination using the minimum fringe order k . min Appl. Sci. 2018, 8, 2673 6 of 11 Appl. Sci. 2018, 8, x FOR PEER REVIEW 6 of 11 Appl. Sci. 2018, 8, x FOR PEER REVIEW 6 of 11 3. Simulation 3. Simulation 3. Simulation To test the performance of the proposed method, some simulations were carried out. Figure 6 To test the performance of the proposed method, some simulations were carried out. Figure 6 To test the performance of the proposed method, some simulations were carried out. Figure 6 shows the simulation of the closet depth plane z . Specifically, Figure 6a–c shows three phase-shift min shows the simulation of the closet depth plane zmin. Specifically, Figure 6a–c shows three phase-shift shows the simulation of the closet depth plane zmin. Specifically, Figure 6a–c shows three phase-shift patterns with eight periods; Figure 6d shows the corresponding wrapped phase ranging fromp to p; patterns with eight periods; Figure 6d shows the corresponding wrapped phase ranging from −π to patterns with eight periods; Figure 6d shows the corresponding wrapped phase ranging from −π to Figure 6e shows the fringe order map regarded as k ; and Figure 6f shows the absolute phase map min π; Figure 6e shows the fringe order map regarded as kmin; and Figure 6f shows the absolute phase π; Figure 6e shows the fringe order map regarded as kmin; and Figure 6f shows the absolute phase regarded as F . min map regarded as Фmin. map regarded as Фmin. Figure 6. Simulation of depth plane zmin. (a–c) Phase-shift patterns; (d) wrapped phase map; (e) Figure 6. Simulation of depth plane z . (a–c) Phase-shift patterns; (d) wrapped phase map; min Figure 6. Simulation of depth plane zmin. (a–c) Phase-shift patterns; (d) wrapped phase map; (e) minimum fringe order map kmin; (f) minimum phase map Фmin. (e) minimum fringe order map k ; (f) minimum phase map F . min min minimum fringe order map kmin; (f) minimum phase map Фmin. Then, a hemisphere was selected as the measure object and simulated, as shown in Figure 7. Then, a hemisphere was selected as the measure object and simulated, as shown in Figure 7. Then, a hemisphere was selected as the measure object and simulated, as shown in Figure 7. Specifically, Figure 7a–c shows the three phase-shift patterns; Figure 7d shows the coded pattern Specifically, Figure 7a–c shows the three phase-shift patterns; Figure 7d shows the coded pattern with Specifically, Figure 7a–c shows the three phase-shift patterns; Figure 7d shows the coded pattern with N = 4; Figure 7e shows the fringe order determined by the proposed method; Figure 7f shows N = 4; Figure 7e shows the fringe order determined by the proposed method; Figure 7f shows the with the N fri = nge 4; Fo igure rder map 7e show determined s the fring by e An’ order s met determ hod for ined com by par the iso pro n; po Fig sed ure met 7g shows hod; Figure the ab 7f solu shows te fringe order map determined by An’s method for comparison; Figure 7g shows the absolute phase phase map recovered by the proposed method; Figure 7h shows the absolute phase map recovered the fringe order map determined by An’s method for comparison; Figure 7g shows the absolute map recovered by the proposed method; Figure 7h shows the absolute phase map recovered by An’s by An’s method. Obviously, the fringe order and the absolute phase map are correctly determined phase map recovered by the proposed method; Figure 7h shows the absolute phase map recovered method. Obviously, the fringe order and the absolute phase map are correctly determined by the by the proposed method. However, An’s method fails in contrast. The 3D reconstruction results of by An’s method. Obviously, the fringe order and the absolute phase map are correctly determined proposed method. However, An’s method fails in contrast. The 3D reconstruction results of the the hemisphere using the two methods are shown in Figure 8. As we can see, the proposed method by the proposed method. However, An’s method fails in contrast. The 3D reconstruction results of can accurately recover the whole surface of the hemisphere, but An’s method fails to measure the hemisphere using the two methods are shown in Figure 8. As we can see, the proposed method can the hemisphere using the two methods are shown in Figure 8. As we can see, the proposed method overall hemisphere surface. The maximum depth range of the proposed method can deal with is accurately recover the whole surface of the hemisphere, but An’s method fails to measure the overall can accurately recover the whole surface of the hemisphere, but An’s method fails to measure the 2πN, which is four times that of An’s method. hemisphere surface. The maximum depth range of the proposed method can deal with is 2pN, which overall hemisphere surface. The maximum depth range of the proposed method can deal with is is four times that of An’s method. 2πN, which is four times that of An’s method. Figure 7. Simulation of a hemisphere. (a–c) Phase-shift patterns; (d) coded pattern; (e) fringe order map using the proposed method; (f) fringe order map using An’s method; (g) absolute phase map using the proposed method; (h) absolute phase map using An’s method. Figure 7. Simulation of a hemisphere. (a–c) Phase-shift patterns; (d) coded pattern; (e) fringe order Figure 7. Simulation of a hemisphere. (a–c) Phase-shift patterns; (d) coded pattern; (e) fringe order map using the proposed method; (f) fringe order map using An’s method; (g) absolute phase map map using the proposed method; (f) fringe order map using An’s method; (g) absolute phase map using the proposed method; (h) absolute phase map using An’s method. using the proposed method; (h) absolute phase map using An’s method. Appl. Sci. 2018, 8, 2673 7 of 11 Appl. Sci. 2018, 8, x FOR PEER REVIEW 7 of 11 Appl. Sci. 2018, 8, x FOR PEER REVIEW 7 of 11 Figure Figure 8. 8. A A 3D 3D recons reconstr tru uction ction of of the the hem hemispher isphere e.. ( (a a) ) T The he pro proposed posed m method; ethod; ( (b b) ) An’s An’s method. method. Figure 8. A 3D reconstruction of the hemisphere. (a) The proposed method; (b) An’s method. 4. Experimental Setup 4. Experimental Setup 4. Experimental Setup To test the proposed method in real condition, a fringe projection system was set up. The system To test the proposed method in real condition, a fringe projection system was set up. The To test the proposed method in real condition, a fringe projection system was set up. The consisted of a projector (Light Crafter 4500) with resolution of 912 1140 pixels, and a camera (IOI Flare system consisted of a projector (Light Crafter 4500) with resolution of 912 × 1140 pixels, and a camera system consisted of a projector (Light Crafter 4500) with resolution of 912 × 1140 pixels, and a camera 2M360-CL) with resolution of 1280  1024 pixels. A flat board was placed at the closest depth plane (IOI Flare 2M360-CL) with resolution of 1280 × 1024 pixels. A flat board was placed at the closest (IOI Flare 2M360-CL) with resolution of 1280 × 1024 pixels. A flat board was placed at the closest of the measured volume, and used as the reference plane. Two isolated objects were selected as the depth plane of the measured volume, and used as the reference plane. Two isolated objects were depth plane of the measured volume, and used as the reference plane. Two isolated objects were measured objects. Total four patterns, including three phase-shift patterns and one coded pattern, selected as the measured objects. Total four patterns, including three phase-shift patterns and one selected as the measured objects. Total four patterns, including three phase-shift patterns and one were projected onto the reference plane and the measured objects by the projector, and sequentially coded pattern, were projected onto the reference plane and the measured objects by the projector, coded pattern, were projected onto the reference plane and the measured objects by the projector, captured by the camera. and sequentially captured by the camera. and sequentially captured by the camera. Figure 9a–c shows three phase-shift patterns projected onto the reference plane, respectively. Figure 9a–c shows three phase-shift patterns projected onto the reference plane, respectively. Figure 9a–c shows three phase-shift patterns projected onto the reference plane, respectively. Figure 9d shows the corresponding wrapped phase. Figure 9e shows the fringe order, also regarded Figure 9d shows the corresponding wrapped phase. Figure 9e shows the fringe order, also regarded Figure 9d shows the corresponding wrapped phase. Figure 9e shows the fringe order, also regarded as the minimum fringe order map k . Figure 9f shows the absolute phase map also regarded as the min as the minimum fringe order map kmin. Figure 9f shows the absolute phase map also regarded as the as the minimum fringe order map kmin. Figure 9f shows the absolute phase map also regarded as the minimum phase map F . Similarly, Figure 10a–c shows the images of three phase-shift patterns min minimum phase map Фmin. Similarly, Figure 10a–c shows the images of three phase-shift patterns minimum phase map Фmin. Similarly, Figure 10a–c shows the images of three phase-shift patterns projected onto the measured objects, respectively. Figure 10d shows the corresponding wrapped phase projected onto the measured objects, respectively. Figure 10d shows the corresponding wrapped projected onto the measured objects, respectively. Figure 10d shows the corresponding wrapped map calculated from the three phase-shift patterns. Meanwhile, the average intensity and intensity phase map calculated from the three phase-shift patterns. Meanwhile, the average intensity and phase map calculated from the three phase-shift patterns. Meanwhile, the average intensity and modulation were calculated. Figure 10e shows the coded pattern with N = 4, and Figure 10f shows the intensity modulation were calculated. Figure 10e shows the coded pattern with N = 4, and Figure 10f intensity modulation were calculated. Figure 10e shows the coded pattern with N = 4, and Figure 10f corresponding code word map. shows the corresponding code word map. shows the corresponding code word map. Figure Figure 9 9. . Ima Images ges of ofthe therefe refer rence enceplane plane. . (a–( ca )– Phas c) Phase-shift e-shift patterns patterns; ; (d) (w d) rapp wrapped ed phase phase map; map; (e) Figure 9. Images of the reference plane. (a–c) Phase-shift patterns; (d) wrapped phase map; (e) ( m e) inimu minimum m fringe fringe order order map map kmink ; (f) ;m (finimu ) minimum m phase phase map map ФminF . . minimum fringe order map kmin; min (f) minimum phase map Фmin. min In order to compare the proposed method and An’s method, Equations (12) and (14) were both used for computing fringe order. Figure 11a,b shows the fringe order maps recovered by the two methods. As we can see, the proposed method recovered the fringe order map F correctly; however, An’s method led to the wrong fringe order map F at some areas. There are obvious differences between the two fringe order maps within the two circular areas plotted in Figure 11. The pixels of the Appl. Sci. 2018, 8, x FOR PEER REVIEW 8 of 11 Appl. Sci. 2018, 8, x FOR PEER REVIEW 8 of 11 Appl. Sci. 2018, 8, 2673 8 of 11 same stripe had the same fringe order k in Figure 11a. However, the pixels of the same stripe had a different fringe order k’ in Figure 11b. Appl. Sci. 2018, 8, x FOR PEER REVIEW 8 of 11 Figure 10. Images of the measured objects. (a–c) Phase-shift patterns; (d) wrapped phase map; (e) Figure 10. Images of the measured objects. (a–c) Phase-shift patterns; (d) wrapped phase map; (e) coded pattern; (f) code-word map. coded pattern; (f) code-word map. In order to compare the proposed method and An’s method, Equations (12) and (14) were both In order to compare the proposed method and An’s method, Equations (12) and (14) were both used for computing fringe order. Figure 11a,b shows the fringe order maps recovered by the two used for computing fringe order. Figure 11a,b shows the fringe order maps recovered by the two methods. As we can see, the proposed method recovered the fringe order map Ф correctly; however, methods. As we can see, the proposed method recovered the fringe order map Ф correctly; however, An’s method led to the wrong fringe order map Ф′ at some areas. There are obvious differences An’s method led to the wrong fringe order map Ф′ at some areas. There are obvious differences between the two fringe order maps within the two circular areas plotted in Figure 11. The pixels of between the two fringe order maps within the two circular areas plotted in Figure 11. The pixels of Figure Figure 10. 10. Images Images of of the the measur measured ed objects. object (s a.– c (a )– Phase-shift c) Phase-shif patterns; t patterns; (d) wrapped (d) wrapp phase ed ph map; ase (m e)ap; coded (e) the same stripe had the same fringe order k in Figure 11a. However, the pixels of the same stripe had the same stripe had the same fringe order k in Figure 11a. However, the pixels of the same stripe had pattern; coded pattern; (f) code-wor (f) cod demap. -word map. a different fringe order k’ in Figure 11b. a different fringe order k’ in Figure 11b. For better illustration, Figure 12a,b shows the 600th rows of the two fringe order maps and In For order better to ill coustra mpar ti eon th , e Fig prop ure osed 12a,met b sho hod wsand the An’ 600th s met rows hod, of Eq th ue ation two s fri (12 n)ge ano dr ( der 14) w me aps re ban oth d For better illustration, Figure 12a,b shows the 600th rows of the two fringe order maps and absolute phase maps. Clearly, F F < 8p and F F < 2p. This indicates that the maximum used absolut for e com phase put m ing apsfri . C nle ge ar order ly, . − Fi m giur n e  11 8a ,b and show s− thm e ifr n inge 2 . oTh rder is indi maps cate rescov that ered the by m ax the imum two absolute phase maps. Clearly, −  8 and  −  2 . This indicates that the maximum min min min min depth range of the proposed method is up to 8p, and that of An’s method is only 2p. Therefore, the methods. As we can see, the proposed method recovered the fringe order map Ф correctly; however, depth range of the proposed method is up to 8π, and that of An’s method is only 2π. Therefore, the depth range of the proposed method is up to 8π, and that of An’s method is only 2π. Therefore, the proposed method can obtain much larger depth range than An’s method. Finally, we reconstructed the An’s method led to the wrong fringe order map Ф′ at some areas. There are obvious differences proposed method can obtain much larger depth range than An’s method. Finally, we reconstructed proposed method can obtain much larger depth range than An’s method. Finally, we reconstructed 3D shapes of the two isolated objects, as shown in Figure 13. between the two fringe order maps within the two circular areas plotted in Figure 11. The pixels of the 3D shapes of the two isolated objects, as shown in Figure 13. the 3D shapes of the two isolated objects, as shown in Figure 13. the same stripe had the same fringe order k in Figure 11a. However, the pixels of the same stripe had a different fringe order k’ in Figure 11b. For better illustration, Figure 12a,b shows the 600th rows of the two fringe order maps and absolute phase maps. Clearly, −  8 and −  2 . This indicates that the maximum min min depth range of the proposed method is up to 8π, and that of An’s method is only 2π. Therefore, the proposed method can obtain much larger depth range than An’s method. Finally, we reconstructed the 3D shapes of the two isolated objects, as shown in Figure 13. Figure 11. Fringe order maps. (a) The proposed method; (b) An’s method. Figure 11. Fringe order maps. (a) The proposed method; (b) An’s method. Figure 11. Fringe order maps. (a) The proposed method; (b) An’s method. Figure 11. Fringe order maps. (a) The proposed method; (b) An’s method. Figure 12. The 600th rows. (a) Fringe order maps; (b) absolute phase maps. Figure Figure 12 12. . The The 600th rows 600th rows. . ((a a) ) Fr Fringe inge ord order er m maps; aps; ( (b b) ) a absolute bsolute ph phase ase maps. maps. Figure 12. The 600th rows. (a) Fringe order maps; (b) absolute phase maps. Appl. Sci. 2018, 8, 2673 9 of 11 Appl. Sci. 2018, 8, x FOR PEER REVIEW 9 of 11 Appl. Sci. 2018, 8, x FOR PEER REVIEW 9 of 11 Appl. Sci. 2018, 8, x FOR PEER REVIEW 9 of 11 Figure Figure 1 13. 3. Mea Measur surem ement ent res result ult of of two two iisolated solated obje objects. cts. Figure 13. Measurement result of two isolated objects. Figure 13. Measurement result of two isolated objects. In order to further verify our method, two separate planes were also measured using the proposed In order to further verify our method, two separate planes were also measured using the In order to further verify our method, two separate planes were also measured using the In order to further verify our method, two separate planes were also measured using the method. Figure 14a–c shows three phase-shift patterns projected onto the two planes, respectively. proposed method. Figure 14a–c shows three phase-shift patterns projected onto the two planes, proposed method. Figure 14a–c shows three phase-shift patterns projected onto the two planes, proposed method. Figure 14a–c shows three phase-shift patterns projected onto the two planes, Figure 14d shows the corresponding wrapped phase map. Figure 14e shows the coded pattern, respectively. Figure 14d shows the corresponding wrapped phase map. Figure 14e shows the coded respectively. Figure 14d shows the corresponding wrapped phase map. Figure 14e shows the coded respectively. Figure 14d shows the corresponding wrapped phase map. Figure 14e shows the coded and Figure 14f shows the corresponding code-word map. Then the fringe order was calculated, pattern, and Figure 14f shows the corresponding code-word map. Then the fringe order was pattern, and Figure 14f shows the corresponding code-word map. Then the fringe order was pattern, and Figure 14f shows the corresponding code-word map. Then the fringe order was as shown in Figure 15a. Using Equation (4), the absolute phase map was recovered, as shown in calculated, as shown in Figure 15a. Using Equation (4), the absolute phase map was recovered, as calculated, as shown in Figure 15a. Using Equation (4), the absolute phase map was recovered, as calculated, as shown in Figure 15a. Using Equation (4), the absolute phase map was recovered, as Figure 15b. Finally, the 3D shapes of two planes were reconstructed, as shown in Figure 16. There are shown in Figure 15b. Finally, the 3D shapes of two planes were reconstructed, as shown in Figure 16. shown in Figure 15b. Finally, the 3D shapes of two planes were reconstructed, as shown in Figure 16. shown in Figure 15b. Finally, the 3D shapes of two planes were reconstructed, as shown in Figure 16. no obvious mistakes in the measurement results. The experimental results illustrate the performance There are no obvious mistakes in the measurement results. The experimental results illustrate the There are no obvious mistakes in the measurement results. The experimental results illustrate the There are no obvious mistakes in the measurement results. The experimental results illustrate the of the proposed method. performance of the proposed method. performance of the proposed method. performance of the proposed method. Figure 14. Images of two planes. (a–c) Phase-shift patterns; (d) wrapped phase map; (e) coded Figure 14. Images of two planes. (a–c) Phase-shift patterns; (d) wrapped phase map; (e) coded Figure Figure 14. 14. Images Images ofof two two pla pl nes. ane (s. a–c (a ) – Phase-shift c) Phase-shif patterns; t patter (d ns; ) wrapped (d) wrapp phase ed ph map; ase( em ) coded ap; (e) pattern; coded pattern; (f) code-word map. ( pattern; f) code-wor (f) cd od map. e-word map. pattern; (f) code-word map. Figure 15. (a) Fringe order map; (b) absolute phase map. Figure 15. (a) Fringe order map; (b) absolute phase map. Figure Figure 15. 15. ( (a a) ) F Fringe ringe order order map; map; ( (b b) ) absolute absolute p phase hase map. map. Appl. Sci. 2018, 8, 2673 10 of 11 Appl. Sci. 2018, 8, x FOR PEER REVIEW 10 of 11 Figure Figure 16. 16. Measur Measurem ement ent res result ult of of two two planes. planes. 5. Conclusions 5. Conclusions This paper has presented an absolute phase retrieval method using only one coded pattern. A total This paper has presented an absolute phase retrieval method using only one coded pattern. A of four patterns are used for 3D shape measurement, which is suitable for high-speed applications. total of four patterns are used for 3D shape measurement, which is suitable for high-speed The code words are encoded into the coded pattern, which can be correctly recovered using the average applications. The code words are encoded into the coded pattern, which can be correctly recovered intensity and intensity modulation of phase-shift patterns. Based on the geometric constraints of fringe using the average intensity and intensity modulation of phase-shift patterns. Based on the geometric projection system, the minimum fringe order map is generated, then the code word can be easily constraints of fringe projection system, the minimum fringe order map is generated, then the code converted into fringe order. Compared with the conventional method, the proposed method can word can be easily converted into fringe order. Compared with the conventional method, the significantly enhance the measurement depth range. proposed method can significantly enhance the measurement depth range. Author Contributions: X.C. and B.T. conceived and designed the experiments; X.Y. and J.L. performed the Author Contributions: X.C. and B.T. conceived and designed the experiments; X.Y. and J.L. performed the experiments; X.C. and C.Z. analyzed the data; X.Y. and Y.L. wrote the paper. experiments; X.C. and C.Z. analyzed the data; X.Y. and Y.L. wrote the paper. Funding: This research was funded by National Natural Science Foundation of China (51605130), Hubei Provincial Natural Funding: Science This rese Foundation arch was of fun China ded (2018CFB656), by National Na Fundamental tural Science Resear Fouch ndati Funds on of for Ch the ina Central (51605130), Universities Hubei (WUT: 2017IVA059), Open Fund of the Key Laboratory for Metallurgical Equipment and Control of Ministry of Provincial Natural Science Foundation of China (2018CFB656), Fundamental Research Funds for the Central Education in Wuhan University of Science and Technology (2018B03). Universities (WUT: 2017IVA059), Open Fund of the Key Laboratory for Metallurgical Equipment and Control Conflicts of Interest: The authors declare no conflicts of interest. of Ministry of Education in Wuhan University of Science and Technology (2018B03). Conflicts of Interest: The authors declare no conflicts of interest. References References 1. Geng, J. 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Applied Sciences – Multidisciplinary Digital Publishing Institute

Published: Dec 18, 2018

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