1. Introduction

Three-dimensional (3D) optical measurement technologies are used in many different fields such as 3D sensing, mechanical engineering, machine vision, intelligence robot control, industry monitoring, biomedicine, dressmaking, and so on. The main advantages of these technologies include: being noncontact and having dense measurement, high speed, high automation degree, and high accuracy [1].

There are two categories in noncontact 3D optical measurement methods: passive and active [2,3]. Passive methods use natural light (i.e., a stereoscopic scan), whereas active methods use structured light (i.e., laser scan or fringe analysis methods). Fringe analysis techniques are widely used in active optical measurement where a grating is projected on the surface of the object and viewed through digital cameras, and then the fringe image resulting from the grating is processed to obtain the height distribution.

The core idea of the fringe analysis technique is that if a pattern of lines, usually parallel to each other, is projected onto the surface of an object and viewed from an offset angle, the observed fringe is distorted by the surface shape of the object. The distortion of the fringe pattern contains information about the height of the object. The fringe images are analyzed to obtain height information through phase-difference computation [2–5]. By employing a single fringe analysis, the fractional phase value can be calculated by either phase-shift [4–6] or Fourier-transform techniques [7]. Fourier transforms are widely used to extract phase information and have the advantage of using only one fringe pattern image to extract the phase distribution [7]. The image is processed in the frequency domain using fast Fourier transforms (FFTs). Mathematically, this method is closely related to the Hilbert transform used in [8], which adopted the phase extraction method of a modified Hilbert transform with Laplacian pyramid algorithms to improve phase extraction accuracy. The algorithm in [9] uses a square wave to demodulate phase and moving averages and comb-shaped filters to extract phase information from low frequency. These methods mainly focus on how to extract phase value (the fractional phase) accurately and quickly from one fringe pattern image. Since the obtained fractional phase is computed in a range between -π and π, the intrinsic periodic nature of the fringe pattern is lost. Thus one of the most critical steps of the fringe analysis method is to find out the integral fringe order information, i.e., the process of eliminating the 2π discontinuities (phase unwrapping) so as to obtain an absolute phase measurement [6,8,10,11]. The key to reliable phase unwrapping is the ability to accurately detect the 2π jumps. However, for complex geometric surfaces, noisy images, and sharp changing surfaces the phase-unwrapping procedure is usually difficult.

In order to relax this fundamental limitation, multi-fringe analyses are used to get the fringe order. In these techniques, additional encoded fringe patterns are added to perform a spatio-temporal encoding of the measurement range. In the method in [13], excluding the three phase-shift fringes, one additional fringe pattern is used to define the centerline of the fringe grating. In [14], a strip marker is added in the phase-shifting algorithm, which gives an indication for the fringe order. These methods can distinguish the fringe order clearly in the center area of the fringe pattern. However, it may be less effective in other areas in some cases. To deal with this problem, more additional code fringes are used. The Gray-code method uses a series of fringes according to the binary mode to define the fringe order [15,16]. The Gray-code projection, especially in combination with the phase-shift method, has been proposed by many researchers [1,15]. Temporal phase unwrapping is a well-established technique to retrieve the absolute phase [17–21]. The method makes use of a sequence of at least two fringe maps with varied fringe pitches. Several analysis methods have been developed to determine the absolute phase and to simplify the temporal unwrapping method. Kinell and Sjödahl [19] introduced the reduced temporal-phase unwrapping method. This method uses an arbitrary number of at least two fringe maps for the phase calculation. Each of the fringe maps is obtained from phase-shift patterns. Zhao et al. [20] used two sets of phase-shift patterns with different frequencies in the unwrapping process. One higher-frequency phase-shift pattern is used to obtain the primary phase value. The other lower-frequency phase-shift pattern determines the fringe orders. The two frequency patterns are projected by two different gratings. To achieve automatic phase unwrapping, Li et al. [22] made two sets of fringes with different frequencies on a single grating. The high frequency is N (N = 3,4,5...) times greater than the low frequency. In order to optimize the two-frequency phase measuring, Li et al. [23] presented a rigorous stochastic analysis of phase-measuring-profilometry temporal noise and give a solution to find the higher frequency, where intensity noise variance is at its minimum. Kakunai et al. [24] used liquid-crystal gratings to make two-frequency phase-shift patterns. The liquid-crystal grating can give an arbitrary projection pitch, an accurate phase shift, and a constant surface brightness compared with conventional gratings. Furthermore, multichannel technologies can also be introduced to reduce the number of acquired fringes. In [24], a two-colored projection system is used to produce such gratings simultaneously. In [25], reduced temporal-phase unwrapping is combined with the multichannel method. This method uses a color video projector with three LCDs. For each pixel, it is possible to calculate three wrapped phase maps, which can be combined to obtain the absolute phase value. As Kinell et al. [19] pointed out, a better use of the three channels would have been to code the finger frequency rather than the phase shift in color.

Generally speaking, in temporal-phase unwrapping methods, a sequence of phase-shift patterns is projected onto a surface. In a two-frequency phase measurement, two sets of phase-shift patterns are used. In the Gray-code method, a series of fringes according to the binary mode is projected. Actually, these methods belong to frequency-modulated methods. In general, the more that additional fringe patterns are used the better the measurement results that can be obtained. However, using more fringe patterns will reduce the data acquisition speed.

In this paper we propose a solution to this problem by employing an amplitude-modulation fringe pattern instead of a series of code fringe patterns. This novel fringe pattern is designed to identify the fringe order by the amplitude of the fringes. Compared with the spatio-temporal methods mentioned above, the presented method has the following two novelties:

A detailed algorithm is given in the paper. The system error and application are discussed. Experiments show that the proposed method is effective in many applications.

2. Amplitude-modulation fringe pattern

The core purpose of the proposed method is to distinguish the fringe order by an amplitude-modulation fringe pattern.

2.1. Amplitude-modulation fringe pattern

Generally, the sinusoidal fringe grating has the following form:

Considering the periodicity of the trigonometric function, θ(x, y) can be express as

The new amplitude-modulation fringe pattern has the following form:

The pattern of the amplitude-modulation fringe is shown in Fig. 1 , where the solid line denotes the amplitude-modulation fringe pattern and the dash-dotted line denotes the normal sinusoidal pattern. The horizontal axis of Fig. 1 denotes phase θ, the vertical axis denotes gray intensity I, the range of θ is [0,20π], and the pattern coefficients a and b are set as 0 and 1, respectively. In this example, the first period is unchanged, which means that $\lambda [0]=1$. In this paper, we will take this example to demonstrate our method.

 

Fig. 1 Amplitude-modulation fringe pattern.

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It can be seen that the fringe function in Eq. (1) has an intrinsic periodic nature but in Eq. (3) the amplitude of the sinusoidal pattern is related to λ[k] (as shown in Fig. 1); thus, the periodic order is defined by λ[k]. Accordingly, we can use the amplitude-modulation fringe to get the fringe order and then obtain the final absolute phase map, combing with the phase-shift fringes.

2.2. Phase shift and fringe order detection

Absolute phase calculation is based on the phase-shift fringe and an amplitude-modulation fringe pattern. The former is used to get the primary phase value, and the later is used to obtain phase-order information.

Shifting the sinusoidal pattern in Eq. (1) by π/2, we can get the phase-shift fringe images as follows:

With Eqs. (6)–(9) we have

The original phase can be deduced from Eq. (10) as

Equation (11) is the four-step phase-shifting formula, and φ(x, y) is in a range between 0 and 2π. The intrinsic periodic nature of the fringe function means that fringe order information is lost in Eq. (11).

With Eqs. (6–(9), we can obtain data modulation as

Data modulation has a value between 0 and 1 and can be used to determine the quality of the phase data at each pixel, with 1 being the best.

In fringe analysis technology, we need to get the absolute phase θ(x, y) on the basis of the original phase φ(x, y). The key issue is how to get fringe order k(x, y); here, we use the amplitude-modulation fringe pattern.

With Eqs. (6–(9), we have

With Eq. (6) and Eq. (3), we have

With Eqs. (13) and (14), we can get the parameter λ(x, y), which can be seen in Fig. 2 , where the horizontal axis denotes phase θ and the vertical axis denotes λ.

 

Fig. 2 Parameter λ.

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Then with Eq. (4), we can get the fringe order k(x, y),

With Eq. (15) we can get the fringe order k, which demonstrates the fringe order in the amplitude-modulation area of the fringe image. Then we will find the fringe order of the whole image in Subsection 2.3.

2.3. Absolute phase map

From the distribution of φ(x, y), it is easy to see that there is a phase jump from 2π to 0 where the fringe order k(x, y) increases to 1. Then based on the result of λ(k), we traverse the image by rows and get the fringe order of the other points as follows.

Assuming the point (x₀, y₀) is known for its fringe order, k(x₀, y₀) = Q, let Δφ(x₀ − 1, y₀) = φ(x₀ − 1, y₀) − φ(x₀, y₀), then

With Eq. (16) we get the fringe order of (x₀ − 1, y₀). If π/2 ≤ Δφ(x₀ − 1, y₀) ≤ 3π/2, it is considered that Δφ is too large to deduce k(x₀ − 1, y₀). Point-by-point, we can calculate the left of (x₀, y₀) until π/2 ≤ Δφ ≤ 3π/2.

For the right of (x₀, y₀), let Δφ(x₀ + 1, y₀) = φ(x₀, y₀) − φ(x₀ + 1, y₀), then

With Eq. (17) we can calculate the right of (x₀, y₀) point-by-point until Δφ≤π/2 or Δφ≥3π/2.

Then in the same way we traverse the image row-by-row and then column-by-column to get the fringe order of the other points. Thus we can get the fringe order k(x, y) in the whole image. With the obtained k(x, y) and φ(x, y), we can get the absolute phase map θ(x, y).

In Subsection 2.4 we will discuss the error of the proposed algorithm.

2.4. Error analysis of amplitude-modulation fringe pattern

Since the fringe order k is obtained by λ, the error in the calculation of λ is closely related to the precision of the final absolute phase value (here we omit (x, y) in the expression for convenience).

In Eq. (3), since Î(x, y) and k(x, y) are both related to θ(x, y), we can consider the expression as a function of θ(x, y):

Then we can get the total differentiation of λ,

In Eq. (20), most of the coefficients remain in a limited range such as 1/b, λ, and cosθ. But the coefficient $\genfrac{}{}{0.1ex}{}{1}{\text{sin}\left(\theta \right)}$ would lead to confusion at sin(θ) = 0; furthermore, δ(kπ) would cause a large error around the neighborhood of θ = kπ and k = 1,2,…n, which is the jump position of λ. In order to demonstrate the influence of noise, we add a random noise (with zero mean and variance 0.0058) to the amplitude-modulation fringe pattern, and the calculation result of λ can be seen in Fig. 3 . In order to decrease the noise, we present the following method.

 

Fig. 3 Calculation result of λ.

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First, with Eq. (10), we have

Thus we can define λ(x, y) = 1 when I₁(x, y) = I₃(x, y). Combining this with the computation of Eq. (14), we can take the following measure: if the denominator of Eq. (14) is zero, then define λ(x, y) = 1.

Second, considering the noise in the neighborhoods of θ = kπ and k = 1,2,…n, we can apply a reshape method to λ(x, y).

With Eqs. (6)–(10) we have

Equation (22) means that we can judge if θ(x, y) belongs to (2kπ, 2kπ + π) or (2kπ + π,2kπ + 2π). Thus let λ(x, y) = 1 if I₁(x, y) ≤ I₃(x, y), which means point (x, y) is not in the amplitude-modulation area. Then for the area of I₁(x, y) > I₃(x, y) we have

Equation (23) means that we can judge if θ(x, y) is in the left or right hand of 2kπ + 0.5π. Thus we can apply the following reshape method to λ(x, y):

To show the relationship more precisely, a part of the data in Fig. 3 is shown in Fig. 4 , where the horizontal axis denotes phase θ$(5\pi ,10\pi )$. At the top of Fig. 4 a part of λ(x, y) is shown, the corresponding I₁ and I₃ are shown at the middle of Fig. 4, and I₂ and I₄ are shown at the bottom of Fig. 4. The relationship with Eq. (22) can be seen in the middle of Fig. 4, and the relationship with Eq. (23) and the reshaped algorithm of Eq. (24) can be seen in the bottom of Fig. 4.

 

Fig. 4 Relationships in the reshape method.

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This reshape method can avoid the impulse influence in the neighborhoods of $\theta =k\pi$. The result is shown in Fig. 5 . Furthermore, the reshape method can be applied iteratively for the second time to get a better result.

 

Fig. 5 Result of λ (reshaped).

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3. Experiment

In order to test the proposed method, an experimental system has been fabricated and assembled as shown in Fig. 6 , which consists of a black-and-white CCD camera (UNIQ UP1800), a digital light processing (DLP) projector (Optoma EP737), an image processor board (Matrox Meteor II), and a personal computer. The tested object is placed in front of the projector at a distance of 90 cm. The measurement range is 50*45 cm. In the experiment, the fringe pattern is designed in the computer and then projected to the tested object by the projector. The spatial period of signal I is 4.2cm. According to the size of the object and the measurement range, we set n = 9, which is suitable to distinguish the fringe order in the experiments. The fringe images are snapped by a CCD camera with a resolution of 1380* 1300.

 

Fig. 6 System layout.

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First, we apply the proposed method on an object that consists of three boxes with different heights. The sizes of the boxes are 375*160*178(mm), 280*67*158(mm), and 272*105*139(mm). The obtained fringe images are shown in Fig. 7 . Figure 7(a) is the phase-shift fringe image. It can be seen from the image that the surface shape distorts the obtained fringe. Thus the fringe order is confused by the height of the boxes. Figure 7(b) is the amplitude-modulation fringe image. We will find the correct fringe order using our method.

 

Fig. 7 Fringe images of object 1.

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The fringe images are analyzed by the proposed method. Figures 8 –10 show the image processing on row 570. Figure 8 shows the pixel gray of the amplitude-modulation fringe $\widehat{I}$, which is indicated by a cyan line, and phase-shift image $I_1$ is indicated by a black line. From the pixel gray values, we can use the formulations of Section 2.2 to calculate $\lambda (k)$, as Fig. 9 shows. It can be seen that there are some noises in the results for Fig. 9. Then by using the reshape algorithm of Section 2.4 to the fringe pattern, we can obtain the fringe order, as Fig. 10 shows.

 

Fig. 8 Pixel gray of $\widehat{I}$ and $I_1$.

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Fig. 10 Calculated $\lambda (k)$ (reshaped).

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Fig. 9 Calculated $\lambda (k)$.

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If we apply the algorithm to the whole amplitude-modulation fringe pattern, we can get the fringe order of the image, as Fig. 11(a) shows. It can be seen that the fringe order is carried out correctly. Then the absolute phase map is obtained, as Fig. 11(b) shows. With the absolute phase map and the system geometry, the 3D coordinates of the object points can be calculated. The final 3D shape reconstruction result, the point cloud, can be seen in Figs. 11(c)–11(e).

 

Fig. 11 3D shape reconstruction result of object 1.

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Figure 12 shows another example. The experiment is done on a model head. The size of the object is 345*364*300(mm). Figure 12(a) is the image of the fringe pattern, Fig. 12(b) is the fringe order of the image, and the final 3D shape reconstruction result is shown in Figs. 12(c)–12(e).

 

Fig. 12 Experiment result of object 2.

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Then a ladder-shaped work piece is tested using the proposed method. Figure 13 shows the fringe image of the work piece. The work piece is made of steel and is machined by a computer numerical control (CNC) milling machine. The size of the work piece is 200*78*104(mm), and it consists of five parallel plates, the heights of which are given in Table 1 . We measure the work piece using the proposed method and obtain the 3D shape reconstruction result that is the 3D point data of the work piece, as Fig. 14 shows. If we fit the point data of each plate with a plane and calculate the distances between the planes, we can get the measurement heights of the plates, which are shown in Table 1. Then we can calculate the height errors and standard deviations of each plate, which are shown in Table 1. Comparing the measurement height with the true height of the plates, we can obtain the error map of the piece, which is shown in Fig. 15 . It can be seen that the maximum height of the object is 105 mm, while the height error is less than 0.2 mm. The result shows that the proposed method is effective and robust.

 

Fig. 13 Fringe image of object 3.

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Table 1. Experiment Results of Object 3 (Unit: mm)

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Fig. 14 3D shape reconstruction result of object 3.

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Fig. 15 Error map of object 3.

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4. Conclusion

In this paper, an algorithm based on an amplitude-modulation fringe pattern for phase analysis has been thoroughly discussed. Phase-shift fringe is used to get the primary phase value, and amplitude-modulation fringe is used to obtain phase-order information. In this solution, the fringe order is encoded by alternating the amplitude of the sinusoidal fringe pattern. Thus using one additional fringe pattern, the absolute phase can be obtained quickly and correctly. Experiments show that the proposed method gets a nice result in many applications.