A novel approach to obtain optimal exposure for 3D shape reconstruction of high dynamic range objects

2.1.1. Phase shift method.

The phase shift method is utilized as the 3D reconstruction method in this work. In general, image acquisition can be divided into three steps [23]: fringe projection, fringe reflection, and fringe acquisition. Mathematically, a typical sinusoidal fringe pattern can be described as:

$$\begin{align} I(x, y) = A(x, y)+B(x, y) \cos \left[\phi(x, y)+\delta_{k}\right], \end{align} \tag{ 1 }$$

where A(x, y) is the average value, B(x, y) is the amplitude (or projector modulation), (x, y) represents the pixel coordinates of the projector, φ(x, y) is the phase to be determined, and δ_k is the phase shift. The phase of a three-step phase-shifting algorithm with an equal phase shift of 2π/3, i.e. $\delta_{k} = 2\pi(k-1)/3$, can be computed as:

$$\begin{align} \phi(x, y) = \tan ^{-1}\left\{\frac{\sqrt{3}\left[I_{1}(x, y)-I_{3}(x, y)\right]}{2 I_{2}(x, y)-I_{1}(x, y)-I_{3}(x, y)}\right\}. \end{align} \tag{ 2 }$$

The value of φ(x, y) ranges from −π to π with 2π discontinuities, which is usually called the wrapped phase. The discontinuities can be handled for a continuous phase map by adopting a phase unwrapping algorithm [6]. Once the continuous phase map is retrieved, the 3D shape of the target can be reconstructed from this phase after calibration. The same set of three equations can also generate the surface texture illuminated by the average projected and ambient light:

$$\begin{align} T(x,y) = A(x, y)+B(x, y). \end{align} \tag{ 3 }$$

2.1.2. Exposure fusion.

To find the optimal exposure sequence for the HDR objects, a novel auto-exposure selection approach is proposed. Figure 1 shows the architecture of the proposed method. Our main idea for searching the optimal exposure sequence is gradually reducing the possible range by analyzing the grayscale histogram of the target. Therefore, the grayscale histogram of the HDR target is computed to calculate the probability density function. Then, the sequence with the demanded exposure style can be obtained by the area-weighted method, which will be described later. The time efficiency of exposure times is considered by recording the running time of 3D reconstruction and the number of generated 3D cloud points as indicators. By evaluating these two indicators, the sequence with the highest score is selected as the optimal exposure sequence.

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For a HDR object, the relationship among the pixel intensity on the image, ambient light, and projection light is as illustrated in figure 2 and can be described as:

$$\begin{align} I = \rho t\left[\alpha\left(I^{p}+\beta_{1}\right)+\beta_{2}\right], \end{align} \tag{ 4 }$$

where I^p represents the projected light intensity on the surface of the target, β₁ represents the ambient light with an intensity of β₁ reflected by the object, and α represents the surface reflectivity. β₂ represents the ambient light coming directly to the camera sensor with an intensity of β₂. ρ represents the camera gain, and when the camera gain is set to 0 dB, ρ = 1. t represents the exposure time. Letting $x_{1} = \rho tI^{p}$, x₂ = ρt, b₁ = α, and $b_{2} = \alpha \beta_{1}+\beta_{2}$. The equation (4) can be simplified as:

$$\begin{align} I = b_{1}x_{1}+b_{2}x_{2}. \end{align} \tag{ 5 }$$

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Researchers in [26] have introduced an approach to determine the parameters in equation (5). To solve this problem, three important procedures are needed to be conducted. The first step is to set the camera gain as 0 dB, i.e. ρ = 1. The second step is to set a certain camera exposure time t_c . The last step is to project five uniform gray level patterns $I^{p}_{i},i = 1,2,\ldots,5$ onto the target surface by the projector, and to capture the images $I^{c}_{i},i = 1,2,\ldots,5,$ by the camera. Based on the captured images $I^{c}_{i},i = 1,2,\ldots,5,$ for each pixel, the parameters b₁ and b₂ in equation (5) can be estimated with the least square method. By this method, the relationships among exposure time t, the intensity of projection I^p and the intensity of the image pixel can be uniquely determined. The relationship between the intensity I and exposure time t is mainly analyzed in this work. According to equation (4), the pixel intensity is linearly related to exposure time when the ambient environment is constant. However, it is difficult to find the optimal exposure sequence because there are too many combinations of exposure times. To reduce the search range, the upper and lower bounds of exposure times, as well as the length of the exposure time sequence, are limited in this work. The sequence composed of a group of exposure times from the shortest to the longest exposure time is called the exposure sequence in this work for convenience. For all exposure sequences, the length n, shortest exposure $t_{\min}$, and longest exposure $t_{\max}$ are the basic elements. $t_{\min}$ is defined as the exposure time for which details can be observed clearly in a specular area, and nothing can be seen with reduced exposure time. $t_{\max}$ is defined as the exposure time in which the shape of the target is clear in the captured image and there is no under-exposure area on the surface of the target. The length of the preliminary exposure sequence n is usually between 5 and 13. This value will not affect the length of the final optimal exposure sequence. With these constraints, the problem can be transferred to finding an optimal exposure sequence with known $t_{\min}$, $t_{\max}$ and n.

Three types of objects with different surface grayscale distributions are listed in figure 3. For the first object, the black metal nut as shown in figure 3(a), its grayscale histogram exhibits separate concentrations in the range of 20–80 and the range of 150–180. The second object, the soft package lithium battery as shown in figure 3(c), has a brighter surface than the nut, and its grayscale histogram is mainly concentrated in the range of 170–255. The third object, the stainless stamping as shown in figure 3(e), has the brightest surface, and its grayscale histogram is mainly concentrated in the range of 250–255. We find that when the exposure time tends to be distributed under some specific distribution, these exposure sequences perform well. To describe these unique exposure sequences, the combination of different exposure times with unique distribution is called 'exposure style'. Exposure style is proposed to help us find the optimal exposure sequence as a guide. Once the shortest and the longest exposure time are fixed, the exposure sequence with different exposure style shows its exposure style. It is not difficult to ascertain that the optimal exposure sequence of the three objects must have a complementary exposure style with their surface grayscale distributions in order to help display more surface detail.

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To make it more normative, a uniformly distributed exposure sequence is defined as a standard exposure sequence that has an even exposure style. Here we introduce the exposure value concept to explain this.

Under the same ISO value and iris aperture, the luminous flux will be doubled when the exposure time increases two-fold. It means that the luminous flux positively correlates with exposure time:

$$\begin{align} EV = \log_{2}({F}^{2}/T), \end{align} \tag{ 6 }$$

where F refers to the iris aperture, T refers to exposure time, and EV refers to exposure value. When the value of T increases by $\sqrt{2}$ times, EV decreases linearly. Then, we obtain an exposure sequence as follows:

$$\begin{align} T_{i} = t_{\min } 2^{\frac{i-1}{n-1} \log _{2} \frac{t_{\max }}{t_{\min }}}, i = 1,2, \ldots \ldots, n, \end{align} \tag{ 7 }$$

where i refers to the number of exposure times in the sequence. T_i refers to the ith time in the sequence. Larger order numbers correspond to longer exposure times. We define that the exposure sequence that makes EVs vary linearly is the standard exposure sequence. The exposure times in this sequence can be regarded as evenly distributed between $t_{\min}$ and $t_{\max}$.

According to this definition, the exposure sequence computed by equation (7) is uniformly distributed. The exposure sequence has an even exposure style. This sequence is a geometric series. The reason for this definition is that the dynamic range of the HDR object's luminance is large, and it is necessary to increase the exposure sequence proportionally in order to capture the details of the HDR object's surface better.

According to equation (4), we know that I is linearly related to the exposure time t. Therefore, the subscript of the selected time represents the light intensity level corresponding to that time.

Supposing that the environment is constant, the footnotes at each time in the standard exposure time series can represent the corresponding intensity level.

With the standard exposure sequence generated by equation (7), the exposure sequences with different exposure styles can be obtained by selecting exposure times from this standard exposure sequence. For example, supposing we have generated an exposure sequence with length of 8 and that the sequence is $t_{1}, t_{2}, t_{3}, t_{4} , t_{5}, t_{6}, t_{7}$, t₈; the exposure sequence with dark style can be represented by selecting $t_{1}, t_{2}, t_{3}, t_{4}, t_{6}$, t₈; the exposure sequence with even style can be represented by selecting $t_{1}, t_{2}, t_{4}, t_{5}, t_{7}$, t₈; the bright style can be represented by selecting $t_{1}, t_{3}, t_{5}, t_{6}, t_{7}$, t₈.

The experiment results support our speculation that the optimal exposure sequence for a target shows a complementary character link with the surface grayscale distribution. To illustrate the validity of this claim, a set of comparison experiments are designed with a soft packing lithium battery as an example. The related experimentation is presented in section 3.

By analyzing the exposure style which the target prefers, the searching field of the optimal exposure sequence is narrowed. The exposure style shows a distribution of exposure times for the optimal exposure sequence. Therefore, finding the exposure sequence with a demanded exposure style is important for automatically seeking the optimal exposure sequence.

To find the exposure sequence with the demanded exposure style of a given target automatically, an area-weighted method is proposed in this work.

In the area-weighted method, the grayscale histogram of the HDR target is divided into n equally divided regions, and the ratio $area_{k}(k = 1,2,\dots, n)$ between the area of each region and the area of all regions is recorded. For each node $i\,(i = 1,2,\dots, n)$, the final ratio r_i can be calculated by $r_{i} = \sum_{k = 1}^{i} area_{k}$. These ratios can also be obtained from the probability accumulation function by integral as follows:

$$\begin{align} P\left(k_{j}\right) = \int_{0}^{k_{j}} f\,(x) d x, \end{align} \tag{ 8 }$$

where f(x) is the probability density function of the image (i.e. the percentage of points with a given grayscale value). With these ratios, the exposure sequence with demanded exposure style can be computed as:

$$\begin{align} t_{j} = t_{\min }+P(k_{j})\left(t_{\max }-t_{\min }\right), \end{align} \tag{ 9 }$$

where k_j ∈[0, 255],  j = 1, 2, ..., n for an 8-bit camera image.

Figure 4 shows that the exposure sequence selected by the area-weighted method can generate an exposure sequence with the expected exposure style. As illustrated in figure 4, the area-weighted method finds the preference of the HDR target well and offers a reference for exposure sequence selection.

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The proposed auto-exposure strategy can be conducted in four steps as following:

Step 1: The camera gain is set as 0 dB, i.e. ρ = 1, then the aperture of the camera lens is adjusted to a proper position where there are as few saturated areas as possible in the captured image while projecting a uniform pattern with the gray value of 250. The shortest exposure time $t_{\min}$, the longest exposure time $t_{\max}$ and the length of the preliminary exposure sequence $n(n\gt3)$ can be obtained from the definitions in the previous section.

Step 2: After setting $t_{\min}$, $t_{\max}$, and n, the uniform exposure sequence can be generated by equation (7). Then, finding a proper exposure time in the uniform exposure sequence ensures that there are no saturated areas in the captured image. The probability density function f(x) is then calculated by normalizing the grayscale histogram of the captured image. The preliminary exposure sequence is generated by equation (9).

For example, supposing that the target is the black metal nut as mentioned in figure 3, we set $t_{\min} = 2$ ms, $t_{\max} = 30$ ms and n = 6. As illustrated in figure 5, the histogram is divided into six parts separately indexed as k = 0, 51, 102, 153, 204, and 255. According to equation (8), we obtain a sequence with percentage P(k_j ) calculated as 0.0%, 32.0%, 55.0%, 58.0%, 97.0%, 99%, and 100%. As calculated by equation (9), the preliminary exposure sequence is 2, 10.96, 17.4, 18.24, 29.16, and 29.72 ms.

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Step 3: In order to find the optimal exposure sequence with the desired exposure style. The preliminary exposure sequence is recorded as M, where $M = \left\{t_{j} \mid j = 1,2, \dots, n\right\}$. The lengths of the alternate exposure sequences obtained from uniform sampling were taken as n = 3, 4, ... , m, respectively. They are recorded as $M_{j},\ j = 1,2, \dots, n$.

Step 4: The running time of 3D measurement time_j and the number of the 3D cloud points p_j are recorded when the alternate exposure sequence M_j is used to reconstruct the 3D shape of the target. At this step, we find the final optimal exposure sequence from the alternate exposure sequence by evaluating time_j and p_j , as described below:

We set $Q = \left\{\frac{1}{t i m e_{j}},\ j = 1,2,\ldots, m\right\}$, $R = \left\{p_{j}-p_{0},\ j = 1,2,\right.$ $\left.\dots, m\right\}$, $p_0 = \gamma \min \left\{p_j\right\} ,\ j = 1,2,\ldots, m$, and γ  $=$ 0.8. The time index $f_{t_j}$ can be described as $f_{t_j} = f_{norm}(\frac{1}{t i m e_{j}}, Q)$. The point index can be described as $f_{p_j} = f_{norm}(p_{j}-p_{0}, R)$. f_norm represents the normalization function:

$$\begin{align} f_{norm}(x_i, X) = \frac{x_i}{\sum X }. \end{align} \tag{ 10 }$$

The score of the alternate exposure sequence M_j can be calculated as follows:

$$\begin{align} v_j = \lambda_{1}\,\,f_{t_j}+\lambda_{2}\,\,f_{p_j}. \end{align} \tag{ 11 }$$

Generally, the running time index and cloud point index are treated with the same importance. Therefore, we set λ₁, λ₂ to 0.5.

By ranking the scores, the optimal exposure sequence with the highest score can be found.

As mentioned in section 2, the message we seek to convey is that the optimal exposure sequence for a target must have a complementary exposure style with the surface grayscale distribution. An object with a dark appearance (overall darkness in the grayscale histogram distribution) will not choose an exposure sequence with the same dark style but will choose an exposure sequence with a light style that is complementary to it; similarly, an object with a uniform grayscale distribution will prefer a style with an even exposure time distribution, and an object with a bright grayscale distribution will prefer a style with a dark exposure time distribution.

The soft packing lithium battery and cylindrical lithium battery are selected as the targets. Four kinds of exposure styles are designed for comparison: dark, slightly dark, slightly light, and light, as shown in table 1. The result for the cylindrical lithium battery is shown in table 3. In this experiment, the $t_{\min}$, $t_{\max}$ and n are set as 2 ms, 128 ms and 13, respectively.

Table 1. Four exposure sequences with different exposure styles.

Exposure style	Exposure sequence
Dark	$t_{1},t_{2},t_{3},t_{4},t_{5},t_{7},t_{9},t_{10}$
Slightly dark	$t_{1},t_{2},t_{3},t_{4},t_{5},t_{7},t_{10},t_{13}$
Slightly light	$t_{1},t_{4},t_{7},t_{9},t_{10},t_{11},t_{12},t_{13}$
Light	$t_{4},t_{5},t_{7},t_{9},t_{10},t_{11},t_{12},t_{13}$

Results in tables 2 and 3 show that for the slightly dark style objects (soft packing battery and cylindrical lithium battery), the slightly light style exposure sequence is more suitable than the others.

From the experimental results, it can be seen that the soft pack battery and the cylindrical lithium battery, both belonging to the slightly dark style, are more suitable for the selection of exposure time in the slightly bright style. The experiment shows that the different exposure styles of the exposure sequence greatly influence the 3D reconstruction results of the same measurement object and that the suitable exposure style can enable a better measurement effect of the selected exposure sequences.

However, finding an exposure sequence with a demanded exposure style requires significant time if we adopt the method mentioned above. Therefore, the area weighted method is used instead of manual selection. To show the value of this method, we design an experiment for comparison.

The area-weighted method is used to select the exposure sequence automatically. The results in tables 4 and 5 show that the exposure sequence generated by area-weighted method can be close to the demanded exposure sequence and have the same exposure style as the expected exposure style and that the proposed area-weighted method can serve as a guide to find the target's exposure preference, i.e. the exposure sequence with the demanded exposure style.

Table 4. Comparison result for the exposure sequences generated by manual and area-weighted methods for the soft packing lithium battery.

Generated method	Exposure sequence
	$t_{1},t_{4},t_{7},t_{9},t_{10},t_{11},t_{12},t_{13}$
Manually designed	2.0, 3.0, 23.0, 32.0, 45.0, 64.0, 90.0, 128.0
Area-weighted method	2.0, 5.2, 17.6, 26.9, 37.8, 47.5, 52.2, 128.0

Table 5. Comparison result for the exposure sequences generated by manual and area-weighted methods for the cylindrical lithium battery.

Generated method	Exposure sequence
	$t_{1},t_{4},t_{7},t_{9},t_{10},t_{11},t_{12},t_{13}$
Manually designed	2.0, 3.0, 23.0, 32.0, 45.0, 64.0, 90.0, 128.0
Area-weighted method	2.0, 9.1, 17.3, 25.0, 37.8, 55.5, 77.6, 128.0

We verify our auto-exposure method by testing on the soft packing lithium battery, which is a classical HDR target. To reflect the advantages of our method, two representative methods: the auto-HDR [26] method and Zhang [27] method are selected for comparison.

The setup procedure of our method includes setting the camera gain as 0 dB, i.e. ρ = 1, then adjusting the aperture of the camera lens to a proper position where there are as few saturated areas in the captured image as possible. $t_{\min}$ and $t_{\max}$ can be obtained by the criterion introduced in section 2. For the soft packing lithium battery, $t_{\min}$ and $t_{\max}$ are obtained as 2 ms and 128 ms, respectively. The length of the preliminary exposure sequence n is set as 13. With the help of the area-weighted method, the ratio of each exposure time in the demanded sequence is calculated as shown in figure 6, the preliminary exposure sequence M can be obtained by equation (9), and the sequence is shown in table 6.

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Then, the alternate exposure sequence M_j , j = 1, 2, ..., 13 with different length is selected by uniform sampling of the preliminary exposure sequence M. The results are shown in table 7. For each alternate sequence M_j , the running time index $f_{t_j}$ of 3D reconstruction and the points index $f_{p_j}$ are recorded in table 7. The related parameters (Q, R, p₀, $f_{t_j})$ and the point index $\ f_{p_j}$ are calculated in table 7. Scores of the alternate exposure sequence are calculated by equation (11), and the result is shown in table 8.

Table 7. Alternate exposure sequence for the soft packing lithium battery.

n	Sequences	Exposure sequences
3	s1	$t_{1},t_{7},t_{13}$
4	s2	$t_{1},t_{5},t_{7},t_{13}$
5	s3	$t_{1},t_{4},t_{7},t_{10},t_{13}$
6	s4	$t_{1},t_{3},t_{6},t_{8},t_{10},t_{13}$
7	s5	$t_{1},t_{3},t_{5},t_{7},t_{9},t_{11},t_{13}$
8	s6	$t_{1},t_{3},t_{5},t_{7},t_{9},t_{11},t_{12},t_{13}$
10	s7	$t_{1},t_{2},t_{4},t_{6},t_{7},t_{8},t_{9},t_{10},t_{11},t_{12},t_{13}$
12	s8	$t_{1},t_{2},t_{3},t_{5},t_{6},t_{7},t_{8},t_{9},t_{10},t_{11},t_{12},t_{13}$

Table 8. Recorded results and scores of the soft packing lithium battery.

n	Sequences	Time (s)	Points	$f_{t_j}$	$f_{p_j}$	Scores
3	s1	2007.4	1461 099	0.195	0.009	0.102
4	s2	2324.3	1750 953	0.168	0.053	0.111
5	s3	2740.3	2306 022	0.143	0.137	0.140
6	s4	3029.8	2286 404	0.129	0.134	0.132
7	s5	3633.7	2511 699	0.108	0.168	0.138
8	s6	3952.0	2541 466	0.099	0.173	0.136
10	s7	4525.6	2483 807	0.087	0.164	0.125
12	s8	5511.1	2458 261	0.071	0.160	0.116

Note: The bold number means the highest score.

The results in table 8 show that exposure sequence s₃ in table 7 achieved the highest score of 0.140 in the comprehensive evaluation, which means that s₃ is the optimal exposure sequence after considering both running time and cloud points. The final exposure sequence for the soft pack lithium battery calculated by our method is 2.0, 16.4, 31.4, 61.2, and 128.0 ms as shown in table 9. The final exposure sequence of the auto-HDR method is 3.5, 6.4, 11.5, 20.7, 37.2, and 67.0 ms. The final exposure sequence of Zhang's method is 2.0, 2.3, 3.2, 6.0, 21.3, and 265.0 ms. The 3D reconstruction results of the three methods are illustrated in figure 7.

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The experiment results in table 10 show that our method obtains the highest score of 0.365 when comparing with other methods. To evaluate the precision of the reconstructed results, the objects sprayed with white powder are reconstructed as the ground truth. By comparing the experiment results with the standard model, the average absolute error of different methods are calculated. The average absolute error can be calculated by equation (12). The comparison of the three methods is shown in tables 10, 15 and 20.

$$\begin{align} acc = \frac{1}{n}\sum\limits_{i = 1}^n {\left\| {{X_i} - {Y_i}} \right\|,} \end{align} \tag{ 12 }$$

where the X represents the 3D coordinates of the measured cloud points and Y represents the 3D coordinates of the standard model. acc is the average absolute error. n is the valid number of cloud points.

The 3D reconstruction result illustrated in figure 7 shows that the 3D shape of the HDR target can be effectively recovered by the proposed auto-exposure method. As illustrated in table 7, the number of 3D points grows slowly when the length of the exposure sequence is larger than 5. At the meanwhile, the processing time keeps on increasing. It means that the advantage of this sequence in generating the 3D cloud points is being replaced by the disadvantage in spending time. By comprehensive evaluation, the sequence with good performance in both time efficiency and 3D reconstruction is selected.

To testify the practicability of our method, another two HDR objects with different grayscale distributions are chosen for the experiments. The first one is a black metal bracket as shown in figure 8. The second one is shown in figure 9: a stainless stamping plate that has a strong reflection property. Similar to the experiment mentioned above, by analyzing the grayscale distribution of these two targets, the preliminary exposure sequence can be calculated with the area-weighted method. Notice that the camera sensitivity is set to 0 dB, the aperture of the camera lens is fixed at the same value as in the first experiment, and the initial parameters are set as $t_{\min} = 3$ ms, $t_{\max} = 200$ ms, n = 8 and $t_{\min} = 2$ ms, $t_{\max} = 50$ ms, n = 8, respectively. The area weights figures are shown in figures 10 and 11. All of the exposure times and alternate exposure sequences can then be calculated by our auto-exposure strategy, as listed in tables 11, 12, 16 and 17. The recorded results and scores are listed in tables 13 and 18. The final exposure sequences are listed in tables 14 and 19.

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Table 12. Alternate exposure sequence for the black metal bracket.

n	Sequences	Exposure sequences
3	s1	$t_{1},t_{4},t_{8}$
4	s2	$t_{1},t_{3},t_{5},t_{8}$
5	s3	$t_{1},t_{2},t_{4},t_{6},t_{8}$
6	s4	$t_{1},t_{2},t_{4},t_{5},t_{7},t_{8}$
7	s5	$t_{1},t_{2},t_{3},t_{4},t_{5},t_{7},t_{8}$
8	s6	$t_{1},t_{2},t_{3},t_{4},t_{5},t_{6},t_{7},t_{8}$

Table 13. Recorded results and scores for the black metal bracket.

n	Sequences	Time (s)	Points	$f_{t_j}$	$f_{p_j}$	Scores
3	s1	2075.796	1308 549	0.245	0.092	0.168
4	s2	2471.472	1365 256	0.205	0.112	0.159
5	s3	2867.369	1626 068	0.177	0.204	0.190
6	s4	3405.297	1618 628	0.149	0.201	0.175
7	s5	4236.619	1606 001	0.120	0.197	0.158
8	s6	4871.71	1601 511	0.104	0.195	0.150

Note: The bold number means the highest score.

Table 17. Alternate exposure sequence for the stainless stamping plate.

n	Sequences	Exposure sequences
3	s1	$t_{1},t_{5},t_{8}$
4	s2	$t_{1},t_{4},t_{6},t_{8}$
5	s3	$t_{1},t_{4},t_{5},t_{6},t_{8}$
6	s4	$t_{1},t_{3},t_{5},t_{6},t_{7},t_{8}$
7	s5	$t_{1},t_{3},t_{4},t_{5},t_{6},t_{7},t_{8}$

Table 18. Recorded results and scores for the stainless stamping plate.

n	Sequences	Time (s)	Points	$f_{t_j}$	$f_{p_j}$	Scores
3	s1	2394.237	2254 508	0.270	0.187	0.232
4	s2	2994.207	2270 467	0.214	0.193	0.207
5	s3	3388.97	2324 720	0.176	0.216	0.206
6	s4	3919.202	2296 099	0.159	0.204	0.187
7	s5	4910.569	2289 613	0.130	0.201	0.168

Note: The bold number means the highest score.

With the computed exposure time, the related fringe images for these two targets can be captured automatically. The fused images for these two objects by three methods are shown in figures 8(a)–(c) and 9(a)–(c), respectively. The experiment results in tables 10, 15 and 20 show that our proposed method acquired a higher score than the other two methods. Especially in table 15, the exposure sequence selected by our method takes the least time and generates the most 3D cloud points. In tables 10 and 20, the number of the 3D clouds points generated by the exposure sequence selected by our method decreases about 3% and 16% to the most one. However, the running time of our selected exposure sequence decreases about 25% and 34% when comparing with the longest one. It shows the advantage of our method that the time efficiency and the quality of the 3D reconstruction are considered at the same time. From these images, it can be found that the reconstructed 3D shapes for these two HDR targets are accurate and complete. These experimental results demonstrate that our proposed auto-exposure technique has potential practicability in industrial applications.