Surface shape reconstruction method and system in phase deflectometry measurement of specular reflective surfaces

Abstract

A surface reconstruction method and system for phase deflectometry of specular reflective surfaces uses two-dimensional Chebyshev polynomials and their gradient polynomials as a basis. Based on the gradient data obtained from measuring the specular surface, an ill-conditioned linear equation system is modeled. This system of equations is then subjected to robust regression calculations using iterative reweighted least squares to obtain a robust estimate of the two-dimensional Chebyshev polynomials, enabling surface reconstruction of the specular reflective surface under test. This invention significantly improves the accuracy of specular surface reconstruction while reducing the algorithm's sensitivity to noise and outliers, thereby enhancing the robustness of the reconstruction algorithm.

Description

Technical Field

[0001] The present invention relates to a technology in the field of optical mirror reconstruction, in particular to a surface shape reconstruction method and system in phase deflection measurement of a mirror reflection surface. Background Art

[0002] Phase deflectometry is a technique for measuring the topography of specular reflective surfaces based on surface gradient data. The model reconstruction method uses a polynomial basis to construct a large linear system of equations from the measured gradient field, and then solves the corresponding polynomial coefficients to fit the surface topography. This linear system of equations is superpositive and has a very large condition number, making its solution ill-conditioned, leading to numerical instability and sensitivity to noise. The Chebyshev polynomial basis is orthogonal within its domain and is suitable for discrete gradient data. Summary of the Invention

[0003] In response to the above-mentioned deficiencies in the prior art, the present invention proposes a surface reconstruction method and system for phase deflection measurement of a mirror reflection surface, which can significantly improve the accuracy of surface reconstruction of the mirror reflection surface while reducing the algorithm's sensitivity to noise and outliers and improving the robustness of the surface reconstruction algorithm.

[0004] The present invention is achieved through the following technical solutions:

[0005] The present invention relates to a surface shape reconstruction method in phase deflection measurement of a mirror reflection surface. The method uses two-dimensional Chebyshev polynomials and their gradient polynomials as a basis, models a system of ill-conditioned linear equations based on gradient data obtained by measuring the mirror surface, and then performs robust regression calculation on the system of equations based on an iterative reweighted least squares method to obtain a robust estimation solution of the Chebyshev two-dimensional polynomial, thereby realizing surface shape reconstruction of the mirror reflection surface to be measured.

[0006] The present invention relates to a system for implementing the above-mentioned method, comprising: a polynomial basis generation unit, an iterative reweighted least squares algorithm unit, and a surface reconstruction unit. The polynomial basis generation unit generates a two-dimensional Chebyshev polynomial basis and a two-dimensional Chebyshev gradient polynomial basis of corresponding order based on actual measurement settings. The iterative reweighted least squares algorithm unit performs robust regression calculations on the polynomial coefficients to obtain an optimal estimate of the polynomial coefficient vector. The surface reconstruction unit, based on the optimal estimate of the polynomial coefficient vector, completes surface reconstruction of the specular reflection surface to obtain three-dimensional point cloud data.

[0007] Technical Effects

[0008] The invention constructs an equation group based on a two-dimensional Chebyshev polynomial basis and obtains the optimal estimation solution of the polynomial coefficients through an iterative reweighted least squares algorithm, thereby improving the accuracy of surface shape reconstruction of the mirror reflection surface. BRIEF DESCRIPTION OF THE DRAWINGS

[0009] Figure 1 Flowchart of the present invention;

[0010] Figure 2 A visualization diagram of the first ten two-dimensional Chebyshev polynomials and their gradients;

[0011] In the figure: (a) Two-dimensional Chebyshev polynomial; (b) Two-dimensional Chebyshev polynomial gradient field

[0012] Figure 3 Schematic diagram of (a) spherical surface and (b) complex curved surface;

[0013] Figure 4 Schematic diagram of spherical surface reconstruction error;

[0014] In the figure: (a) Direct least squares method (b) Complex surface reconstruction error of this method (c) Direct least squares method (d) This method;

[0015] Figure 5 Schematic diagram of the root mean square error of surface reconstruction with different noise levels;

[0016] Figure 6 Schematic diagram of (a) partially missing face shape, (b) direct least squares face shape reconstruction error, and (c) face shape reconstruction error of this method;

[0017] Figure 7 Schematic diagram of the phase measurement deflection system;

[0018] Figure 8 Schematic diagram of (a) plane mirror surface reconstruction, (b) direct least squares surface reconstruction error, and (c) this method’s surface reconstruction error.

[0019] Figure 9 Schematic diagram of (a) concave mirror surface reconstruction, (b) direct least squares surface reconstruction error, and (c) surface reconstruction error of this method. DETAILED DESCRIPTION

[0020] like Figure 1 As shown, this embodiment relates to a surface shape reconstruction method in phase deflection measurement of a mirror reflection surface, specifically comprising:

[0021] Step 1) constructing an ill-conditioned linear equation system based on Chebyshev polynomials and measured gradient data, specifically including:

[0022] Step 1.1) Calculate the one-dimensional Chebyshev polynomial T according to the Chebyshev polynomial recursion formula n (x) = 2xT n-1 (x)-T n-2(x), where: T0(x) = 1, T1(x) = x, -1≤x≤1; and the derivative form of the one-dimensional Chebyshev polynomial is:

[0023] Step 1.2) Generate a two-dimensional Chebyshev polynomial basis C based on the one-dimensional Chebyshev polynomial i (x, y) = T n (x)T m (y), where: C i (x,y) is the ith two-dimensional Chebyshev polynomial, and n and m are the subscripts of the one-dimensional Chebyshev polynomial in the x and y directions, respectively.

[0024] Step 1.3) Differentiate the two-dimensional Chebyshev polynomial to obtain the derivative form, that is, the two-dimensional gradient Chebyshev polynomial basis Regarding the conversion relationship between the single subscript i and the double subscript n and m of the two-dimensional Chebyshev polynomial, the first 10 polynomials are as follows: Figure 2 As shown:

[0025]

[0026]

[0027] Step 1.4) The mirror surface shape W(x,y) is obtained from the first N two-dimensional Chebyshev polynomials:

[0028] Where: α i is the coefficient of the ith polynomial; the gradient field on the mirror surface is: Where: β i are the corresponding coefficients of the two-dimensional Chebyshev gradient polynomial, which are derived by: We can get β i =α i Therefore, by solving the corresponding coefficients of the corresponding two-dimensional Chebyshev gradient polynomial, the corresponding two-dimensional Chebyshev polynomial substitution coefficients can be directly obtained to complete the surface reconstruction.

[0029] Step 1.5) Combine the measured gradient data to complete the construction of the linear equation system: Where: Q is the total number of actual measurement points, β is the column vector representing the coefficients of the N-term two-dimensional Chebyshev gradient polynomial, S x (x,y), S y (x,y) are the measured gradients in the x and y directions respectively.

[0030] Step 2) using an iterative reweighting algorithm to solve the ill-conditioned linear equations, specifically including:

[0031] Step 2.1) Assign equal weights to each point and estimate the polynomial coefficients by the least squares method. Based on the least squares method, the initial estimate of β is obtained: β0 = (G T G) -1 G T S.

[0032] Step 2.2) In the iterative reweighted least squares method, the leverage vector and weighted residual are calculated, and the hat matrix H = G (G T G) -1 G T , lever vector l=[h 11 , h 22 ,...,h nn ] T , where: the leverage vector l consists of the diagonal elements of the hat matrix H.

[0033] Step 2.3) To remove noise points, use the adjusted residual to assign larger residuals to outliers; Where: r i is the least squares residual. Standardizing the adjusted residual yields: Where: STD is the standard deviation, K is an adjustment constant used to adjust the sensitivity of the weight to the residual; in this embodiment, K is set to 1.4.

[0034] Step 2.4) Calculate the robust weight coefficient based on the standardized residual:

[0035] Step 2.5) Calculate the robust estimate of the coefficients of the two-dimensional Chebyshev gradient polynomial; Where: W is a diagonal matrix composed of robust weights.

[0036] Step 2.6) Iteratively repeat steps 2.2-2.5 until convergence or the maximum number of iterations is reached to obtain the final robust estimate solution of the polynomial coefficients, where the residual of the iterative reweighted least squares regression is:

[0037] In this embodiment, the maximum number of iterations is set to 1000.

[0038] Step 3) After obtaining the optimal estimate of the two-dimensional Chebyshev gradient polynomial coefficients, Complete the surface reconstruction of the mirror reflection surface.

[0039] After specific practical experiments, a binocular phase deflectometry system was built, in which: the binocular camera has a resolution of 2448×2048 pixels, model BASLERacA2440-20gm; the camera lens has a focal length of 16mm, model MORITEX ML-MC16HR; the LCD display has a resolution of 2560×1440 pixels and a dot pitch of 0.206mm. The three-band fringe frequencies of the display are 225, 224, and 210 respectively. In the experiment, a 15mm side rectangular plane mirror with a manufacturing accuracy of λ / 10 and a φ25mm, 250mm focal length circular concave mirror were measured. In the experiment, the first 100 two-dimensional Chebyshev polynomials were selected for surface reconstruction. The root mean square error of the surface shape of the plane mirror with a side length of 15mm was 3.97×10 -5 mm, the root mean square error of the surface shape of the φ25mm concave mirror is 5.15×10 -5 mm, and the fitting curvature radius is 498.23 mm.

[0040] like Figure 3 As shown in the figure, the spherical surface and the complex surface are reconstructed by simulation and the reconstruction accuracy of the direct least squares method and the proposed method are compared. Figure 4 As shown in Figure 2, the accuracy of this method is higher than that of the direct least squares method.

[0041] In order to compare the sensitivity of the direct least squares method and this method to noise, a noisy data reconstruction test was conducted. The test results are as follows: Figure 5 As shown in the figure, this method (IRLS-Method) has a smaller root mean square (RMS) error for surface reconstruction than the traditional least squares method (OLS-Method), is more resistant to noise, and has better robustness.

[0042] In order to compare the algorithm's ability to reconstruct the surface shape of the partially missing data, the surface shape reconstruction simulation is performed on the partially missing surface shape, such as Figure 6 As shown in Figure 3, this method has higher accuracy in surface reconstruction for locally missing data.

[0043] In order to verify the accuracy of this method through actual measurement data, the following Figure 7 The phase deflection system shown in the figure is used to verify the surface reconstruction of high-precision plane mirror and concave mirror respectively. The reconstruction results are shown in Figure 8 and Figure 9 As shown in the figure, the root mean square (RMS) error of this method for both plane and concave mirror reconstruction is superior to that of the direct least squares method. The radius of curvature (ROC) of the concave mirror fitted by this method is closer to the nominal true value of 500 mm. This shows that this method can significantly improve the accuracy of 3D surface reconstruction of specular reflective surfaces.

[0044] The above-mentioned specific implementation can be partially adjusted in different ways by those skilled in the art without departing from the principles and purpose of the present invention. The scope of protection of the present invention shall be based on the claims and shall not be limited by the above-mentioned specific implementation. All implementation schemes within its scope shall be subject to the constraints of the present invention.

Claims

1. A surface shape reconstruction method for phase deflection measurement of a mirror reflection surface, characterized in that: Using two-dimensional Chebyshev polynomials and their gradient polynomials as the basis, an ill-conditioned linear equation system is modeled based on the gradient data obtained from measuring the mirror surface. Then, a robust regression calculation based on iterative reweighted least squares is performed on the equation system to obtain a robust estimation solution of the two-dimensional Chebyshev polynomial, thereby realizing the surface shape reconstruction of the mirror reflection surface to be measured. Specifically, the following steps are involved: Step 1) constructing an ill-conditioned linear equation system based on Chebyshev polynomials and measured gradient data, specifically including: Step 1.1) Calculate the one-dimensional Chebyshev polynomial T according to the Chebyshev polynomial recursion formula n (x) = 2xT n-1 (x)-T n-2 (x), where: T0(x) = 1, T1(x) = x, -1≤x≤1; and the derivative form of the one-dimensional Chebyshev polynomial is: Step 1.2) Generate a two-dimensional Chebyshev polynomial basis C based on the one-dimensional Chebyshev polynomial i (x, y) = T n (x)T m (y), where: C i (x,y) is the i-th two-dimensional Chebyshev polynomial, n and m are the subscripts of the one-dimensional Chebyshev polynomials in the x and y directions respectively; Step 1.3) Differentiate the two-dimensional Chebyshev polynomial to obtain the derivative form, that is, the two-dimensional gradient Chebyshev polynomial basis About the conversion relationship between single subscript i and double subscript n and m of two-dimensional Chebyshev polynomials; Step 1.4) The mirror surface shape W(x,y) is obtained from the first N two-dimensional Chebyshev polynomials: Where: α u is the coefficient of the ith polynomial; the gradient field on the mirror surface is: Where: i are the corresponding coefficients of the two-dimensional Chebyshev gradient polynomial, which are derived by: We can get β i =α i Therefore, by solving the corresponding coefficients of the corresponding two-dimensional Chebyshev gradient polynomial, the corresponding two-dimensional Chebyshev polynomial substitution coefficients can be directly obtained to complete the surface reconstruction; Step 1.5) Combine the measured gradient data to complete the construction of the linear equation system: Where: Q is the total number of actual measurement points, β is the column vector representing the coefficients of the N-term two-dimensional Chebyshev gradient polynomial, S x (x,y), S y (x,y) are the measured gradients in the x and y directions respectively; Step 2) using an iterative reweighting algorithm to solve the ill-conditioned linear equations, specifically including: Step 2.1) Assign equal weights to each point and estimate the polynomial coefficients by the least squares method. Based on the least squares method, the initial estimate of β is obtained: β0 = (G T G) -1 G T S; Step 2.2) In the iterative reweighted least squares method, the leverage vector and weighted residual are calculated, and the hat matrix H = G (G T G) - 1 G T , lever vector l=[h 11 , h 22 ,...,h nn ] T , where: the leverage vector l consists of the diagonal elements of the hat matrix H; Step 2.3) To remove noise points, use the adjusted residual to assign larger residuals to outliers; Where: r i is the least squares residual; standardizing the adjusted residual yields: Where: STD is the standard deviation, K is the adjustment constant used to adjust the sensitivity of the weight to the residual; Step 2.4) Calculate the robust weight coefficient based on the standardized residual: Step 2.5) Calculate the robust estimate of the coefficients of the two-dimensional Chebyshev gradient polynomial; Where: W is a diagonal matrix composed of robust weights; Step 2.6) Iteratively repeat steps 2.2-2.5 until convergence or the maximum number of iterations is reached to obtain the final robust estimate solution of the polynomial coefficients, where the residual of the iterative reweighted least squares regression is: Step 3) After obtaining the optimal estimate of the two-dimensional Chebyshev gradient polynomial coefficients, Complete the surface reconstruction of the mirror reflection surface.

2. A system for implementing the surface shape reconstruction method in phase deflection measurement of a mirror reflection surface according to claim 1, characterized in that: include: A polynomial basis generation unit, an iterative reweighted least squares algorithm unit, and a surface reconstruction unit, wherein: the polynomial basis generation unit generates a two-dimensional Chebyshev polynomial basis and a two-dimensional Chebyshev gradient polynomial basis of corresponding order according to actual measurement settings; the iterative reweighted least squares algorithm unit performs robust regression calculation of polynomial coefficients to obtain the optimal estimation solution of the polynomial coefficient vector; the surface reconstruction unit completes the surface reconstruction of the mirror reflection surface based on the optimal estimation solution of the polynomial coefficient vector to obtain three-dimensional point cloud data.

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