Optimization of parameters for a fringe projection measurement system by use of an improved differential evolution method

Zebo Wu; Wei Tao; Na Lv; Hui Zhao

doi:10.1364/OE.507602

1. Introduction

Fringe projection profilometry is an optical measurement technology based on structured light, which can be used to reconstruct the surface of the measured object. Due to its advantages of low cost, non-contact and high precision, it is widely used in industrial detection, biometric identification, digitization of cultural relics and other fields [1–5]. Improving measurement accuracy is a critical task. The fringe projection measurement system is categorized under binocular vision according to the imaging principle. From the perspective of measurement principles, it is associated with optical triangulation. Consequently, the geometric optical structural parameters significantly influence the measurement accuracy. To address this challenge, researchers have recently focus on investigating methods to enhance measurement accuracy by considering optical structural parameters.

Researchers have conducted geometric analysis of fringe projection system and revealed the influence on measurement accuracy [6–11]. Zhang [6] employed peripolar geometry for a geometric analysis of the system. They studied the relationship between phase-depth sensitivity and fringe direction, identifying the optimal fringe direction under high-sensitivity conditions, thus improving accuracy. Zhang [7] established an imaging model based on the principle of perspective projection and proposed a method to enhance measurement accuracy by optimizing the projection angle of the projector. After analyzing the imaging principle of the measurement system, Zhou [8] proposed that the measurement accuracy can be improved when the phase change direction of the fringe pattern is parallel to the projector-camera baseline direction. Rao [9] analyzed the relationship between the frequency variation and pixel-matching errors in sinusoidal fringe patterns. Then they proposed a method to select the optimal frequency for fringe pattern. However, the geometric principle of the fringe projection system involves multiple parameters. The quantitative research on theoretically obtaining optimal values for multiple optical structural parameters of fringe projection system simultaneously is insufficient. Patterns are projected onto the measured object using a projector, with height information being encoded in the phase of the sinusoidal light intensity. Researchers have investigated phase to height conversion method to improve measurement accuracy [12–16]. Rigoberto [15] formalized the imaging principle of fringe projection system, considering overall optical structure parameters and then proposed a generalized method for phase-to-coordinate conversion. Pei [16] enhanced the fitting ability of the phase-to-coordinate model through the application of Gaussian process regression and improves the parameters acquirement accuracy. Researchers also investigated various calibration methods to acquire the accurate optical structure parameters, aiming to improve measurement accuracy [17–21]. Yu [17] modeled the projector projecting as the inverse process of camera imaging. By fixing the position of the camera and calibration board, the perspective transformation error can be effectively eliminated, acquiring accurate optical structure parameters. Sun [18] aimed at the correspondence error problem of planar target and imaging points, proposed that using the constraints of vanishing points and lines and estimation by the objective function to obtain the reliable and accurate optical structure parameters. However, it is worth noting that the initial configuration process of the system currently lacks theoretical guidelines and heavily relies on the operators’ experience, which can introduce human-induced errors that are difficult to avoid.

This paper proposes an optimization theoretical model involving comprehensive structural parameters of the fringe projection measurement system and attains automatic parameters optimization with the help of improved differential evolution and the Levy flight optimization algorithm. A mathematical model of the measurement system is established using optical triangulation. The optimization merit function is formulated by combining three positions across the measuring range and using the phase sensitivity criterion as a foundation. The imaging effectiveness criteria and sensor geometric dimensions are employed as constraints. Secondly, A combined optimization algorithm is used to find the optimal values of five structural parameters, aimed to jump out of the local extremum and obtain optimization results quickly. Finally, we constructed an experimental platform to validate the improvement in measurement accuracy achieved through the proposed optimization process. The results demonstrated significant improvement in measurement accuracy.

2. Optimization principle

2.1 Imaging principle of fringe projection system

The fringe projection-based 3D measurement utilizes the imaging principle of optical triangulation [22]. Multiple light beams are projected onto the object’s surface using a planar light source, while a camera with a lens focuses the reflected light beams onto a two-dimensional imaging plane. In a classical single projector-camera optical system (Fig. 1(a)), the red lines indicate that one of the rays from the planar light source irradiates the measured object, causing the displacement on the imaging device. The relationship between the object’s height h and the imaging displacement can be expressed as [23]:

(1)$$h = \frac{{\frac{l}{{\cos \alpha }}x\cos \varphi }}{{s\sin (\alpha + \varphi ) - x\cos (\alpha + \varphi )}}$$

where l is the perpendicular distance between the lens and the reference plane, s is the perpendicular distance between the lens and the CCD, φ is the angle between the beam projected by the projector and the vertical reference line, and α is the angle between the beam received by the camera and the vertical reference line. It is worth noting that the angle φ and α vary for different spatial locations.

Fig. 1. (a) optical triangulation of a single projector-camera system. (b) three selected positions for optimization merit function construction. (c) optical structure of fringe projection system and parameters definition.

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Define k as the perpendicular distance between the projector's center and the reference plane, and c as the horizontal distance between the projector's center and the camera's center. Then the angle φ can be expressed as:

(2)$$\tan \varphi = \frac{{c - l\tan \alpha }}{k}$$

The phase difference ΔΦ between the reference plane and the measuring surface can be determined using the most popular phase-shifting algorithms [2]. The relationship between phase difference and image displacement x is represented as [24]:

(3)$$x = \frac{{\Delta \phi \cdot \textrm{s} \cdot \lambda }}{{2\pi l}}$$

where λ denotes the spatial pitch of the projected sinusoidal patterns. Substituting Eq. (3) and Eq. (2) into Eq. (1), the relationship between the phase difference ΔΦ and the object’s height h can be obtained as:

(4)$$\Delta \phi = \frac{{2\pi hl\cos \alpha (k\sin \alpha + c\cos \alpha - l\sin \alpha )}}{{lk\lambda + \lambda h\cos \alpha (k\cos \alpha - c\sin \alpha + l\sin \alpha \cdot \tan \alpha )}}$$

2.2 Optimization merit function construction

Phase sensitivity is an important indicator to validate the accuracy of fringe projection system. The phase sensitivity function represents the degree of pattern deformation caused by changes in object depth [25]. It is defined as the ratio of the phase difference and the height of the measured object. The phase sensitivity δ_φ in the fringe projection system is expressed cording to Eq. (4):

(5)$${\delta _\phi } = \frac{{d\Delta \phi }}{{dh}} = \frac{{2\pi {l^2}k\cos \alpha (k\sin \alpha + c\cos \alpha - l\sin \alpha )}}{{\lambda {{(lk + h(k{{\cos }^2}\alpha - c\sin \alpha \cos \alpha + l{{\sin }^2}\alpha ))}^2}}}$$

Simultaneously, it is noteworthy that the height of the measured object significantly influences the phase sensitivity function. Define the maximum height of the object to the reference plane as h₀, considering the midpoint of this elevation range as the zero position. Therefore, the height of the object varies within the range [-h₀/2, h₀/2]. Notably, when the height of the object equals h₀/2, the phase sensitivity function reaches its minimum value. Conversely, when the height of the object equals -h₀/2, the phase sensitivity function attains its maximum value. This can be expressed as follows:

(6)$$\left\{ {\begin{array}{cc} {\min ({\delta_\phi }) = \frac{{2\pi {l^2}k\cos \alpha (k\sin \alpha + c\cos \alpha - l\sin \alpha )}}{{\lambda {{(lk + {h_0}(k{{\cos }^2}\alpha - c\sin \alpha \cos \alpha + l{{\sin }^2}\alpha )/2)}^2}}}}&{h ={+} \frac{{{h_0}}}{2}}\\ {\max ({\delta_\phi }) = \frac{{2\pi {l^2}k\cos \alpha (k\sin \alpha + c\cos \alpha - l\sin \alpha )}}{{\lambda {{(lk - {h_0}(k{{\cos }^2}\alpha - c\sin \alpha \cos \alpha + l{{\sin }^2}\alpha )/2)}^2}}}}&{h ={-} \frac{{{h_0}}}{2}} \end{array}} \right.$$

A higher phase sensitivity value indicates a reduced influence of the height measurement results on phase errors. That means the measurement accuracy is maximized when the phase sensitivity function reaches maximum. Consequently, the optimization goal is to attain the highest possible value for the minimum phase sensitivity function. Therefore, we define the function f as the optimization criterion:

(7)$$f = \max [\min ({\delta _\phi })] = \max \left\{ {\frac{{2\pi {l^2}k\cos \alpha (k\sin \alpha + c\cos \alpha - l\sin \alpha )}}{{\lambda {{(lk + 0.5{h_0}(k{{\cos }^2}\alpha - c\sin \alpha \cos \alpha + l{{\sin }^2}\alpha ))}^2}}}} \right\}$$

As the projector projects a surface light source, the phase sensitivity function value varies for different spatial locations. Therefore, three representative locations of the measured position are selected for merit function calculation. As shown in Fig. 1(b), P1 means the imaging point for this location is at the central position of the CCD, indicating that the reflected light ray for this location coincides with the optical axis of the camera. P2 means that the imaging point for this location is at the leftmost boundary of the CCD, and P3 means that the imaging point for this location is at the rightmost boundary of the CCD. α_m denotes the angle between the incident beam and the vertical reference line at position P1. It is noticeable that α_m also represents the tilt angle of the camera's installation. The angle related to position P3 is expressed as α_r=α_m+θ. θ represents the tangent angle between half of the CCD's constant width and the constant length of the camera. The angle related to position P2 is expressed as α_l=α_m-θ. Each optimization criterion is represented in sequence as follows:

(8)$$\begin{array}{c} {{f_1} = \max [\min ({\delta _\phi })] = \max \left\{ {\frac{{2\pi {l^2}k\cos {\alpha_m}(k\sin {\alpha_m} + c\cos {\alpha_m} - l\sin {\alpha_m})}}{{\lambda {{(lk + 0.5{h_0}(k{{\cos }^2}{\alpha_m} - c\sin {\alpha_m}\cos {\alpha_m} + l{{\sin }^2}{\alpha_m}))}^2}}}} \right\}}\\ {{f_2} = \max \left\{ {\frac{{2\pi {l^2}k\cos ({\alpha_\textrm{m}} + \theta )(k\sin ({\alpha_m} + \theta ) + c\cos ({\alpha_m} + \theta ) - l\sin ({\alpha_m} + \theta ))}}{{\lambda {{(lk + {h_0}(k{{\cos }^2}({\alpha_m} + \theta ) - c\sin ({\alpha_m} + \theta )\cos ({\alpha_m} + \theta ) + l{{\sin }^2}({\alpha_m} + \theta ))/2)}^2}}}} \right\}}\\ {{f_3} = \max \left\{ {\frac{{2\pi {l^2}k\cos ({\alpha_m} - \theta )(k\sin ({\alpha_m} - \theta ) + c\cos ({\alpha_m} - \theta ) - l\sin ({\alpha_m} - \theta ))}}{{\lambda {{(lk + {h_0}(k{{\cos }^2}({\alpha_m} - \theta ) - c\sin ({\alpha_m} - \theta )\cos ({\alpha_m} - \theta ) + l{{\sin }^2}({\alpha_m} - \theta ))/2)}^2}}}} \right\}} \end{array}$$

Therefore, we establish the optimization merit function as shown in Eq. (9):

(9)$${f_{opt}} = {a_1}{f_1} + {a_2}{f_2} + {a_3}{f_3}$$

where a_i (i = 1,2,3) denotes the normalized weight coefficients. The relationship between the coefficients is: a₁+ a₂+ a₃= 1. The values of the three weighting coefficients can be reasonably assigned based on specific requirements. Here we set a1 = a2 = a3 = 1/3. The size of the coefficient determines the proportion of the respective optimization objective f_i (i = 1,2,3) in the entire optimization merit function. The optimization merit function systematically explores and evaluates the optimal phase sensitivity across the entire field of view and measurement range, ultimately impacting the measurement accuracy of the fringe projection system.

It is evident that the optimization merit function in Eq. (9) is related to several optical structure parameters. The image distance s is confirmed once the lens is selected. Therefore, the parameters to be optimized (Fig. 1(c)) include: the horizontal distance of the two optical centers c, the installation height of the projector k, the installation height of the camera l, the spatial pitch of the pattern λ, and the camera tilt angle α_m. It is essential to collectively consider comprehensive constraints on these structural parameters in order to achieve the maximum optimization merit function.

2.3 Boundary condition constraints of system

The boundary constraints of the measurement system are approached from two perspectives. Firstly, the capability of the CCD to distinctly resolve the pattern is considered to ensure accurate phase recovery from the captured patterns. Secondly, as a compact imaging sensor, the geometric dimensions of the mechanical structures are also subjected to limitations.

(1) Imaging effectiveness constraint

The projector projects patterns onto the object and the deformed images are captured by the camera. The spatial pitch of the pattern represents the width of two adjacent variations of the pattern projected onto the object. Consequently, the CCD must be capable to resolve this width to ensure a clear distinction of the received pattern. Operating as a digital sampling device, the CCD's sampling frequency adheres to the Nyquist sampling theorem [26]. The Nyquist sampling theorem dictates that the original signal can be completely reconstructed from the samples by sampling at a frequency higher than twice the highest frequency of the signal. Therefore, the width of one pattern pitch imaged on the CCD must consist of a minimum of two pixels. The camera is a typical small-aperture imaging optical system (Fig. 2(a)) whose imaging principle obeys the Gaussian formula. The imaging height of one pattern pitch can be calculated using the Gaussian magnification formula. The minimal value should be bigger than 2ε. ε denotes the width of one pixel of the CCD. Therefore, the imaging effectiveness criteria constraint can be expressed as:

(10)$$R = \frac{{\lambda \cdot s \cdot k \cdot \cos ({\alpha _m} - \theta )}}{{l \cdot \sqrt {{{(c - l\tan ({\alpha _m} - \theta ))}^2} + {k^2}} }} > 2\varepsilon$$

(2) Geometry dimensions constraint

Fig. 2. (a) magnification by optical system in imaging effectiveness constraints. (b) geometric constraints calculation of the imaging sensor.

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While extending the structural parameters such as c could enhance the phase sensitivity of the sensor system, the imaging sensor's size occasionally faces constraints related to mechanical installation space. Consequently, it is essential to restrict the geometric dimensions during the measurement system design. Furthermore, the optical structural parameters play a critical role in determining the dimensions of imaging sensor. When the height of the measured object changes from h₀/2 to - h₀/2, the position of the corresponding imaging point moves from point B to point C. The maximum geometric dimensions increment of |AB| and |BC| are calculated as follows:

(11)$$\begin{array}{l} {m_1} = {x_{ - h/2}} \cdot \cos ({\alpha _m} + \theta ) = \frac{{hs{{\cos }^2}({\alpha _m} + \theta )(k\sin ({\alpha _m} + \theta ) + c\cos ({\alpha _m} + \theta ) - l\sin ({\alpha _m} + \theta ))}}{{2lk - h(k{{\cos }^2}({\alpha _m} + \theta ) - c\sin ({\alpha _m} + \theta )\cos ({\alpha _m} + \theta ) + l{{\sin }^2}({\alpha _m} + \theta ))}}\\ {m_2} = {x_{ - h/2}} \cdot \sin ({\alpha _m} - \theta ) = \frac{{hs\sin ({\alpha _m} - \theta )\cos ({\alpha _m} - \theta )(k\sin ({\alpha _m} - \theta ) + c\cos ({\alpha _m} - \theta ) - l\sin ({\alpha _m} - \theta ))}}{{2lk + h(k{{\cos }^2}({\alpha _m} - \theta ) - c\sin ({\alpha _m} - \theta )\cos ({\alpha _m} - \theta ) + l{{\sin }^2}({\alpha _m} - \theta ))}} \end{array}$$

Therefore, the transverse distance W and the axial distance H of the imaing sensor can be constrainted as follows, where d₂ is the installation height of the imaging sensor:

(12)$$\begin{array}{c} {W > c + {m_1}}\\ {H > l - {d_2} + s \cdot \cos {\alpha _m} + {m_2}} \end{array}$$

To sum up, we have established the theoretical model of the optimization merit function in Eq. (9) to identify the multiple optimal structural parameters for improving measurement accuracy. The boundary constraints in Eq. (10) and Eq. (12) have been constructed from two aspects: imaging effectiveness constraint and geometric dimensions constraint. The defined optimization parameters involve five factors: the horizontal distance of the two optical centers c, the installation height of the projector k, the installation height of the camera l, the spatial frequency of the pattern λ, the camera tilt angle α_m. In the boundary condition space, we employed a combination of the improved differential evolution and Levy flight algorithm(L-IDE) to discover the optimal parameters solutions.

3. Parameters optimization process

3.1 Joint of improved differential evolution with levy flight algorithm (L-IDE)

(1) Improved differential evolution algorithm

The differential evolution (DE) algorithm is a globally optimized algorithm with low complexity and good robustness [27–29]. The process of finding the optimal solution is compared to the evolutionary process of natural selection in populations. The particles are randomly distributed in the feasible solution space of the problem and then selected with a certain formula. Each iteration of the algorithm consists of three progressive steps: the crossover and mutation operations start the blind search process of the algorithm; the selection operation guides the direction of evolution. The fitness function is employed to evaluate the optimal particles. The algorithm proceeds to the next iteration when all particles in the group have undergone changes. The whole particles converge toward optimal values, similar to the evolutionary process observed in natural populations.

Assuming the D-dimensional search space R^D, the total number of particles is n, the i-th particle after initialization is defined as the vector X_i= (x_i1, x_i2,…x_iD), the current individual optimal position searched by the i-th particle is P_i = (p_i₁,p_i₂,. . ., p_i_D), the current population optimal position P_g = (p_g₁,p_g₂,. . ., p_g_D) is the optimal value. The i-th particle after mutation operation is V_i= (v_i1, v_i2,…v_iD); after crossover operation is U_i= (u_i1, u_i2,…u_iD).

The initialization operation of DE algorithm is expressed according to Eq. (13).

(13)$${x_i}(0) = x_i^{\min } + {r_i} \times (x_i^{\max } - x_i^{\min })$$

where x^min and x^max represent the maximum and minimum values of the defined parameter, r is the random value. The typical DE algorithm uses a Gaussian process to determine r. However, different random mappings directly affect the subsequent search process. We substitute the Sobol sequence (Fig. 3(b)) for Gaussian sequence (Fig. 3(a)) to generate more symmetrical distributed numbers. The uniformly distributed initial particles is beneficial to improve the search effect of the optimization algorithm.

Fig. 3. initialization distribution by (a) Gaussian sequence, (b) Sobol sequence. (c) adaptive value of the factor.

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After initialization, the algorithm determines the crossover operation v_i and mutation operation u_i of the i-th particle and selects the better individual x_i for next iteration based on a selection criterion derived from the fitness function e that incorporates the optimization function and constraint conditions [28].The vector X_i of the i-th particle in (t + 1)-th iteration is updated by comparing the fitness function value of initialization and mutation factor. The individual optimal position P_i and population optimal position P_g are then obtained. They are calculated according to Eq. (14).

(14)$$\begin{array}{l} \begin{array}{cc} {{v_i}(t) = {x_{{r_1}}}(t) + F(t) \times ({x_{{r_2}}}(t) - {x_{{r_3}}}(t))}\\ {{u_i}(t) = \left\{ {\begin{array}{cc} {{v_i}(t)}&{{r_j}(0,1) < CR}\\ {{x_i}(t)}&{else} \end{array}} \right.} \end{array}\\ {x_i}(t + 1) = \left\{ {\begin{array}{cc} {{u_i}(t)}&{e({u_i}(t)) < e({x_i}(t))}\\ {{x_i}(t)}&{else} \end{array}} \right. \end{array}$$

where t denotes the number of generations. (x_r1j, x_r2j, x_r3j) are randomly selected from vector X_i. F is a mutation scale factor whose range is (0,1). At the start of the optimization process, a higher mutation factor enhances search effectiveness. Towards the end, a lower value facilitates faster convergence. The adaptive value of the factor (Fig. 3(c)) facilitates a compromise between the search precision and computational efficiency [30]. CR is the predetermined constant in (0,1). e is the defined fitness function. After the next generation of individuals has been determined, the algorithm enters the next cycle until the number of iterations reaches the maximum value t_max and finally output optimal solution P_g.

(2) Levy flight algorithm

The IDE algorithm possesses robust global search capability. However, it lacks an efficient strategy to escape the local optimal solution and may converge to a local extremum. To address this, the classical Levy flight method [31,32] is incorporated to jump out of the local extremum and search in a larger space thus improving the possibility to find the optimal solution fast and accurately. The steps are as follows:

Step 1 judgment: If population optimal position P_g does not change in m consecutive cycles, or its variation ΔP_g(m) is less than a set threshold T(Δg), the algorithm moves to step2 for local search.

Step 2 search: Search locally according to the levy flight algorithm, updating the mutation vector in Eq. (14).

(15)$${v_{iLevy}}(t) = {v_i}(t) + Levy(\gamma )$$

Levy(γ) is the Levy distribution that follows the parameter γ. The value is the same as the number of parameters to be optimized.

3.2 Optimization process for optimal optical structure parameters

Based on the L-IDE method, under the theoretical guidance by the optimization merit function and the constraints of optical imaging effectiveness and sensor geometry dimensions, the defined five optical structure parameters of the fringe projection system are automatically optimized aiming at improving measurement accuracy. The optimization method involves searching the optimal values of the structural optical parameters (c, k, l, λ, α_m) in the search space R^D (D = 5). The optimization merit function is defined by Eq. (9), while the boundary constraints are given in Eq. (10) and Eq. (12). The entire optimization process is shown in Fig. 4. The optimization process consists of three main modules, which are the parameters initialization module, the IDE algorithm module, and the local search operation module.

Fig. 4. optimization process of optical structure parameters based on the L-IDE algorithm.

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Firstly, the parameters were initialized as shown in Table 1 including the constraints conditions, the parameters of the IDE algorithm and Levy flight algorithm. In which the range of values for the parameter α_m was set to be within [0°,40°] through multiple pre-tests. The values of the parameters c, k, l, λ were constrained by Eq. (10) and Eq. (12). We set the lateral geometric dimension W ≤ 300 mm and the axial dimension H ≤ 200 mm according to the geometric size requirements of the actual system construction. The parameters in the optimization algorithm were determined based on the actual convergence effect of the optimization results through multiple pre-tests.

Table 1. Parameters initialization of optical process

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Secondly, following initialization by Eq. (13), the IDE algorithm begins its optimization iteration process. When the particles are in the optimization constraint space, calculating initialization vector X_i, the crossover vector V_i and mutation vector U_i according to Eq. (14). Subsequently, the individual optimal position P_i and population optimal position P_g are obtained. The local search employing Levy flight algorithm in Eq. (15) is required if continuous variation ΔP_g(m) is less than a set threshold T(Δg). The vector X_i is updated according to Eq. (14), thereby progressing to the next cycle of the iteration process. This loop continues until iterations t reaches the maximum, finally output the optimal solution P_g.

In cases where the IDE algorithm converges to a local extremum, the process transitions to a local search operation module. This module applies the Levy flight algorithm to search in a larger space and improve the probability of identifying the optimal value. Then the crossover vector is updated according to Eq. (15), and this update is then integrated into the optimization processing flow of the IDE algorithm.

3.3 Design example for optimal parameters

Based on the proposed optimization process, designers can conveniently find candidate results for optical structure parameters (c, k, l, λ, α_m) prior to system construction. Algorithm parameters were initialized according to Table 1, tailored to meet practical measurement. The L-IDE algorithm, guided by a fitness function that integrates merit and constraints functions, facilitates the acquisition of optimal solutions for the optical structure parameters. Five sets of candidate solutions were selected, as shown in Table 2. The merit function in the fifth row reached its maximum value, with the corresponding parameters representing the optimal solutions identified by the optimization process.

Table 2. Optimal candidate solutions of optical structure parameters

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4. Experiment verification

4.1 Experiment setup and design

To evaluate the measurement accuracy improvement through the optimized structural parameter constructions in the fringe projection system, an optical experimental platform was built. The projector is a DLP4500 with a resolution of 912 × 1140. It utilizes Texas Instruments’ DMD chip as its projection core. The camera is a Manta G-223B with a resolution of 1088 × 2048 and a CCD with a pixel size of 5.5 μm × 5.5 μm. The model of the CCD is 2/3” CMV2000 Global Shutter CMOS. The focal length of the camera lens is 12 mm. The camera and projector were securely positioned on three-axis displacement adjustment stage. The adjustment range for the X and Y axes is 25 mm, with a resolution of 0.01 mm, a minimum scale of 0.01 mm, and an adjustment precision of 0.003 mm, using a screw rod for adjustments. The rotation has an adjustment range of 360 degrees, a resolution of 2 minutes, a minimum scale of 1 degree, and an adjustment precision of 2 minutes, adjusted via a rotary turntable. They were mounted on the optical platform, with a distance of 25 mm between two adjacent positioning holes on the platform.

Two different test objects were tested (Fig. 5(b)-(c)). The first test object, a standard 70 mm diameter sphere, has a diameter precision of 5μm, verified by a metrology institution following production. The second object is a certified 100 mm ±0.5μm ceramic block, with a flatness accuracy of 0.2μm, also calibrated post-production by the metrology institution.

Fig. 5. (a) Optical structure verification platform. (b) spherical object. (c) ceramic block.

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The uniform design method was employed to compare the measurement accuracy between the optimized structure parameters which were obtained through the optimization process, and other parameter configurations. The uniform design method is highly suitable for multifactorial and multilevel experiments, as it ensures the homogeneity of samples in terms of representativeness and data dispersion, thereby reducing the large workload of the experiment [33]. The number of defined structure parameters in this work is five. Accordingly, we selected a uniform design table with five factors and nine levels (Table 3) [33]. This means that each structure parameter takes on nine values, arranged as per the uniform table’s level sequence in each experiment. In selecting specific parameter levels, the optimal value of each parameter served as the foundation, with appropriate step lengths set for each factor. We set the range of each parameter as: c ∈[120:2:136], k∈[290:5:330], l∈[235:5:275], λ∈[0.25:0.05:0.65], α_m∈[0:5:40]. The value of each parameter in each group of experiment were determined based on the uniform design table. We conducted a total of ten group experiments, the first nine followed the uniform design's structure parameters arrangement, while the tenth used the optimized parameters. Then we proceeded to the measurement step. Firstly, the defined parameters of the measurement system in each experiment were adjusted to those values specified in Table 4. The measured object was placed in the projector's clear projection space for each experiment. Subsequently, the object was reconstructed by fringe projection system with the phase reconstruction and system calibration, leading to the reconstruction of shape point cloud, and yielding results through point-cloud fitting.

Table 3. Uniform design table with 5 factors and 9 levels

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Table 4. The structural parameters arrangement for ten group experiments

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4.2 Optimal parameters experimental verification

The point cloud of the measured sphere fitted to the standard sphere model was computed using Geomagic software. The fitted point cloud and the error data distribution in the tenth experiment are shown in Fig. 6(a)-(b). Figure 6(c) depicts the fluctuating fitted point cloud errors for nine experimental groups, and the straight line represented the error value for the optimal parameter configuration group. It was evident that the fitted point cloud error was significantly reduced using the optimal parameters compared to other configurations. The maximum value of fitted point cloud error among the nine experiment groups was 0.106 mm, while the value under the optimal structural parameters was reduced to 0.034 mm, which represents a 32.08% improvement. Enhanced measurement accuracy was achieved through the theoretical optimization process of optical structure parameters.

Fig. 6. results in sphere experiment. (a) fitted point cloud. (b)the error data distribution in the tenth experiment. (c) fitted point cloud error of ten groups.

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The ceramic block was also measured under the same experiment conditions. The captured measurement scene by camera and point cloud fitted plane are shown in Fig. 7(a)-(b). We employed the multiple exposure method [34] to obtain accurate phase reconstruction results and keep a set exposure time consistent. In each set of experiments, the camera captures images with a uniform sequence of exposure times. Under optimal structure parameters, the flatness error achieved its lowest value compared to other groups. These experimental results provided empirical validation of the effectiveness of the proposed method in improving measurement accuracy.

Fig. 7. Results in plane experiment. (a) measured object captured by camera. (b) Point cloud plane fitting in Geomagic. (c) flatness error of ten groups.

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4.3 Comparison of different optimization algorithms

We compared the optimal results of optical structure parameters using different optimization algorithms. The three selected algorithms were the proposed L-IDE algorithm, the traditional differential evolution (DE) algorithm and the classical evolution algorithm known as genetic algorithm (GA). The detailed parameters setup for each algorithm were displayed in Table 5. During the optimization process, the max iteration t_max is 1000. P_m was the mutation probability and P_c was the crossover probability in the GA.

Table 5. Parameters setup of different optimization algorithms

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The convergence process of GA, DE and L-IDE algorithms was guided by the same fitness function. The optimization results during the 200 iterations are displayed in Fig. 8(a). It can be observed that both the L-IDE and IDE methods converged within 200 iterations process. The L-IDE method reached convergence in the 31st iteration comparing the 42nd iteration for DE method. The DE algorithm experienced two times of optimization stagnation during the optimization process. In contrast, the proposed L-IDE method exhibited the capability to swiftly escape local extreme, resulting in a significant improvement in convergence speed. The GA algorithm, however, has not yet converged. By extending the number of iterations to 1000 (Fig. 8(b)), the GA algorithm reached convergence in the 482nd iteration, signifying a notably slower optimization speed compared to the L-IDE method. Additionally, the GA algorithm also exhibited prolonged periods of being trapped in local extreme. Therefore, the proposed L-IDE method facilitates the rapid discovery of optimal values while avoiding trapping in local extreme, thereby enhancing the efficiency of the optimization process.

Fig. 8. Optimization results comparison (a) during 200 iterations. (b) during 1000 iterations.

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5. Discussion and conclusion

In this work, a theoretical optimization model and an automated optimization method for obtaining the multiple optimal structure parameters of fringe projection measurement system are proposed. The primary objective is to address the challenge of improving the measurement accuracy for fringe projection system, which is influenced by multiple optical structural parameters. The approach aims to determine the optimal structural parameters by transitioning from traditional reliance on operator skill to a theoretically-guided and automated optimization process. The applicability of this work focuses on the design process of the imaging sensor based on fringe projection prior to the system construction. An optimization merit function and constraints are established based on the physical model of the fringe projection system and the phase-sensitivity index reflecting accuracy. We then propose an optimization method to qualitatively calculate the optimal value of multiple structure parameters, thereby determining optimal structure parameters values and improving measurement accuracy.

Under the theoretical guidance of the proposed method, an automated assisted parameter- adjustment system can be developed for high-precision measurement scenarios. Initially, this system autonomously acquires the actual values of structural parameters through a high-precision calibration method. Subsequently, adjust the measurement system's structural parameters towards their optimal values using an automated, high-precision adjustment platform. After multiple calibration-adjustment iterations, the actual structural parameters progressively approach ideal values. Adjustment ceases when the deviation between the actual and optimal structural parameters falls within the predetermined permissible range, thereby achieving automatic control and adjustment of the optimal structural parameters. Following this conceptual framework, the experiments in this work executed manual adjustments of structural parameters: the actual values of the parameters were first calculated using the calibration method, and then the system's structural parameters were adjusted using a three-dimensional high-precision displacement adjustment platform based on the deviation between theoretical and current values. And then calibrated again to ensure the parameters adjustment and obtain the accurate values. The experiment results demonstrated a significant enhancement in measurement accuracy under the optimal structural parameters, thus validating the feasibility of developing an automatic assisted parameter adjustment system toward the proposed method.

Overall, this work provides a theoretical foundation for determining the optimal values of multiple optical structure parameters at the beginning of system setup. The proposed method can guide the design of the optimal structure parameters in practical system setup and avoid the arbitrariness of parameter selection. By automatic parameters design, it holds the advantage to enhance the measurement accuracy of imaging sensor based on fringe projection. It is further expected to promote the development in fields such as high-precision imaging sensor design and manufacturing, and development of the automated system for optimal parameters adjustment.

Funding

Science and Technology Commission of Shanghai Municipality (20S31908300).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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No.	(c, k, l, λ, α_m)/ (mm, mm, mm, mm, deg)	merit function
1	(127.256, 305.187, 273.243, 0.623, 0.225)	3.647
2	(128.328, 306.493, 260.781, 0.513, 0.000)	4.413
3	(127.945, 303.741, 255.839, 0.442, 8.949)	5.363
4	(129.923, 300.625, 250.301, 0.403, 11.543)	6.052
5	(140.385, 300.135, 250.209, 0.343, 11.390)	7.141

*Number*	c(mm)	k(mm)	l(mm)	λ(mm)	α_m (deg)
1	120.50	295.20	250.30	0.55	35°
2	122.50	305.20	270.30	0.45	30°
3	124.50	315.20	245.30	0.35	25°
4	126.50	325.20	265.30	0.25	20°
5	128.50	290.20	240.30	0.60	15°
6	130.50	300.20	260.30	0.50	11.5°
7	132.50	310.20	235.30	0.40	5°
8	134.50	320.20	255.30	0.30	0°
9	136.50	330.20	275.30	0.65	40°
10(Optimal)	140.50	300.20	250.30	0.35	11.5°

Method	Details of main parameters setup
GA	t_max =1000	P_m= 0.5	P_c= 0.8
DE	t_max =1000	F = 0.5
L-IDE	t_max =1000	F(0) = 0.5	γ=5

No.	(c, k, l, λ, α_m)/ (mm, mm, mm, mm, deg)	merit function
1	(127.256, 305.187, 273.243, 0.623, 0.225)	3.647
2	(128.328, 306.493, 260.781, 0.513, 0.000)	4.413
3	(127.945, 303.741, 255.839, 0.442, 8.949)	5.363
4	(129.923, 300.625, 250.301, 0.403, 11.543)	6.052
5	(140.385, 300.135, 250.209, 0.343, 11.390)	7.141

*Number*	c(mm)	k(mm)	l(mm)	λ(mm)	α_m (deg)
1	120.50	295.20	250.30	0.55	35°
2	122.50	305.20	270.30	0.45	30°
3	124.50	315.20	245.30	0.35	25°
4	126.50	325.20	265.30	0.25	20°
5	128.50	290.20	240.30	0.60	15°
6	130.50	300.20	260.30	0.50	11.5°
7	132.50	310.20	235.30	0.40	5°
8	134.50	320.20	255.30	0.30	0°
9	136.50	330.20	275.30	0.65	40°
10(Optimal)	140.50	300.20	250.30	0.35	11.5°

Optimization of parameters for a fringe projection measurement system by use of an improved differential evolution method

Abstract

1. Introduction

2. Optimization principle

2.1 Imaging principle of fringe projection system

2.2 Optimization merit function construction

2.3 Boundary condition constraints of system

3. Parameters optimization process

3.1 Joint of improved differential evolution with levy flight algorithm (L-IDE)

3.2 Optimization process for optimal optical structure parameters

3.3 Design example for optimal parameters

4. Experiment verification

4.1 Experiment setup and design

4.2 Optimal parameters experimental verification

4.3 Comparison of different optimization algorithms

5. Discussion and conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Tables (5)

Equations (15)

Optics Express

R^D	α_m∈[0°,40°], W ≤ 300 mm, H ≤ 200 mm, c, k, l, λ
IDE	F	t_max	n	CR
IDE	0.5	300	50	0.6
Levy flight	Γ	m	T(Δg)
Levy flight	5	5	0.1

Levels/Factors	1	2	3	4	5
1	1	2	4	7	8
2	2	4	8	5	7
3	3	6	3	3	6
4	4	8	7	1	5
5	5	1	2	8	4
6	6	3	6	6	3
7	7	5	1	4	2
8	8	7	5	2	1
9	9	9	9	9	9

Levels/Factors	1	2	3	4	5
1	1	2	4	7	8
2	2	4	8	5	7
3	3	6	3	3	6
4	4	8	7	1	5
5	5	1	2	8	4
6	6	3	6	6	3
7	7	5	1	4	2
8	8	7	5	2	1
9	9	9	9	9	9