Aspheric optical element sensor layout method based on equality constraint Latin hypercube sampling

Abstract

The invention belongs to the technical field of sensing measurement, and relates to an aspheric optical element sensor layout method based on equality constraint Latin hypercube sampling. Comprising the steps of obtaining the size of an aspheric optical element and a surface control equation of the aspheric optical element, increasing positive and negative margins to expand the control equation into an inequality sampling space, obtaining space uniform distribution samples through inequality sampling, and performing feasibility approximation so as to obtain the layout position of a sensor. According to the method, the control equation is expanded into an inequality sampling space by increasing positive and negative margins, and the problem that Latin hypercube sampling is difficult to carry out by equality constraints can be effectively solved. Spatial uniform distribution samples are obtained through inequality sampling, space division and Latin hypercube sampling are carried out on the samples obtained after two times of Monte Carlo sampling, optimization is carried out through the constraint simulated annealing algorithm, the problem of high calculation power consumption caused by direct optimization is effectively avoided, and high fidelity and high operation efficiency of the model are considered.

Description

Technical Field

[0001] The invention belongs to the technical field of sensor measurement and relates to a non-spherical optical element sensor layout method based on equality-constrained Latin hypercube sampling. Background Art

[0002] Aspheric optical elements are widely used in high-precision imaging systems, astronomical observation equipment, and high-power laser systems due to their unique optical properties. In the processing and manufacturing process of aspheric optical elements, it is usually necessary to layout non-contact sensors to monitor and control the shape, thermal deformation, and stress state of the elements. However, traditional sensor layout methods often have difficulty in achieving true spatial uniform distribution when dealing with complex aspheric shapes. In addition, due to the highly nonlinear surface shape of aspheric optical elements, traditional layout methods have great limitations in optimization calculations.

[0003] At present, the commonly used sensor layout methods are mostly based on regular grids or empirical arrangements. For example, Quan Quan et al. proposed a sensor layout optimization and rapid deployment method for wireless visual sensor networks (202011228565.7), and Zhu Qiumin et al. proposed a sensor node layout optimization method and system for electromagnetic spectrum mapping (202310091981.4). These sensor layout methods often find it difficult to achieve true spatial uniform distribution when dealing with complex aspheric shapes. In addition, due to the highly nonlinear surface shape of aspheric optical elements, traditional layout methods have great limitations in optimization calculations. Summary of the invention

[0004] In view of the problem of uneven spatial distribution in existing sensor layout methods, the present invention provides a non-spherical optical element sensor layout method based on equality constrained Latin hypercube sampling, which is used for sensor layout of non-spherical optical elements with uniform distribution in space.

[0005] In order to achieve the above object, the technical solution adopted by the present invention is:

[0006] A non-spherical optical element sensor layout method based on equality-constrained Latin hypercube sampling comprises the following steps:

[0007] Step 1: Obtain the size of the aspheric optical element and its surface control equation;

[0008] Step 2: adding positive and negative margins to expand the surface control equation into an inequality sampling space;

[0009] Step 3: Inequality sampling to obtain spatially uniformly distributed samples;

[0010] Step 4: Feasibility Approach.

[0011] The details are as follows:

[0012] Step 1: Obtain the dimensions of the aspheric optical element and its surface governing equations.

[0013] For a given aspheric optical element, its size is first obtained, and then the surface shape of the aspheric optical element is described by the Sag equation. As the surface control equation of the aspheric optical element, the Sag equation is a mathematical model used to describe the surface profile of the aspheric optical element. It accurately describes the change of the aspheric shape by defining the relationship between the vertical offset of any point on the surface and the radial distance of the point. The general form of the Sag equation is:

[0014]

[0015] Where Z is the vertical distance from the vertex to a point on the surface (surface height), r is the radial distance from the optical axis to the point; c is the curvature at the vertex; k is the cone constant, which determines the type of aspheric surface, n is a positive integer, A 2n is a high-order aspheric coefficient used to further adjust the shape of the aspheric surface.

[0016] Step 2: Add positive and negative margins to expand the surface control equation into an inequality sampling space.

[0017] After obtaining the size of the aspheric optical element and its surface control equation, a positive and negative margin ±ε is added to the surface control equation to expand the surface control equation of the aspheric optical element into an inequality sampling space, which is recorded as The spatial governing equation describing the inequality sampling space is denoted by g.

[0018] Step 3: Inequality sampling to obtain spatially uniformly distributed samples.

[0019] After adding positive and negative margins to expand the surface control equation into the inequality sampling space, Latin hypercube sampling based on inequality constraints is used to obtain spatially uniformly distributed samples; specifically, as follows:

[0020] First, in the inequality sampling space A Monte Carlo sampling is performed within, and the number of samples is Get a Monte Carlo sampling point set To ensure that the number of sampling points is higher than the expected number of sampling points α, a secondary Monte Carlo sampling is performed on the basis of the primary Monte Carlo sampling to obtain the secondary Monte Carlo sampling point set The number of samples is N M .Will The space is evenly divided into N 2 Volume elements, after division Perform Latin hypercube sampling to obtain the Latin hypercube sampling point set The sample space is obtained by describing the inequality obtained in the second step The spatial control equation g is a constraint. Perform optimization based on the constrained simulated annealing algorithm to obtain the optimal Latin hypercube sampling point set

[0021] Step 4: Feasibility Approach.

[0022] After completing the inequality sampling to obtain the spatially uniformly distributed samples, the sequential least squares programming (SLSQP) is used to perform a feasibility approximation on the spatially uniformly distributed samples to obtain the uniformly distributed samples in the equality sampling space. The details are as follows:

[0023] First, define the objective function to calculate the current sampling point and The Euclidean distance of the inner sampling points. The objective function is in the following form:

[0024] fun(x)=||xx ineq ||(2)

[0025] Among them, x is the currently optimized sampling point, x ineq for The sampling points within , ‖·‖ represents the Euclidean distance.

[0026] The objective function fun(x) is used as the optimization target. As the initial value of optimization, sequential least squares programming (SLSQP) is applied for optimization, and boundary conditions and constraints are set to ensure that the optimization process is carried out within the feasible domain.

[0027] After optimization by sequential least squares programming (SLSQP), the uniformly distributed samples in the final equal sampling space are obtained. Can effectively satisfy the constraint g, uniformly distribute samples The sampling point coordinates are used as the sensor layout position, providing accurate sampling data for the sensor layout of aspherical optical elements.

[0028] Beneficial effects of the present invention:

[0029] (1) The present invention expands the control equation into an inequality sampling space by adding positive and negative margins, which can effectively solve the problem that equality constraints are difficult to perform Latin hypercube sampling.

[0030] (2) The present invention uses inequality sampling to obtain spatially uniformly distributed samples, divides the samples after two Monte Carlo samplings into spaces and performs Latin hypercube sampling, and then uses the constrained simulated annealing algorithm for optimization, thereby effectively avoiding the high computing power consumption problem of direct optimization, and taking into account both the high fidelity and high operating efficiency of the model, which is an important innovation of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] Figure 1 It is the basic flow chart of the present invention.

[0032] Figure 2 Schematic diagram of the sampling space of equation.

[0033] Figure 3 Schematic diagram of inequality sampling space.

[0034] Figure 4 Schematic diagram of uniformly distributed samples in the inequality sampling space.

[0035] Figure 5 Schematic diagram of uniformly distributed samples in the sampling space of equation. DETAILED DESCRIPTION

[0036] The present invention is further described below in conjunction with specific embodiments.

[0037] like Figure 1 As shown, the present invention provides a non-spherical optical element sensor layout method based on equality-constrained Latin hypercube sampling. Specifically, it includes the following steps:

[0038] Step 1: Obtain the dimensions of the aspheric optical element and its surface governing equations.

[0039] Figure 2 The aspheric optical element of this embodiment is shown, which is also the equation sampling space in this embodiment. First, the size of the aspheric optical element is obtained, as shown in the following table:

[0040] c k <![CDATA[A n ]]> -0.33mm -0.82239 0

[0041] The surface shape is then described by the Sag equation. The Sag equation of the aspheric optical element is as follows:

[0042]

[0043] Step 2: Add positive and negative margins to expand the surface control equation into an inequality sampling space.

[0044] After obtaining the size of the aspheric optical element and its surface control equation, a positive and negative margin ±ε=±0.05 is added to the surface control equation, and the surface control equation of the aspheric optical element is expanded to an inequality sampling space, which is recorded as The spatial governing equation describing the inequality sampling space is denoted by g. Figure 3 This embodiment ( Figure 2 The inequality sampling space is obtained by adding positive and negative margins ±ε=±0.05 to the surface control equation (Equation (3)).

[0045] Step 3: Inequality sampling to obtain spatially uniformly distributed samples.

[0046] After adding positive and negative margins to expand the surface control equation into the inequality sampling space, Latin hypercube sampling based on inequality constraints is used to obtain spatially uniformly distributed samples. First, in the inequality sampling space A Monte Carlo sampling is performed within, and the number of samples is Get a Monte Carlo sampling point set The number of sampling points in a single Monte Carlo sampling may be lower than the expected number of sampling points α = 10000. Therefore, a secondary Monte Carlo sampling is performed on the basis of the primary Monte Carlo sampling to obtain a secondary Monte Carlo sampling point set. The number of samples is N M =53419. The space is evenly divided into N 2 = 400 volume elements, after division Perform Latin hypercube sampling to obtain the Latin hypercube sampling point set right Perform optimization based on the constrained simulated annealing algorithm to obtain the optimal Latin hypercube sampling point set Figure 4 Shows the spatially uniform samples obtained by inequality sampling, i.e. the optimal Latin hypercube sampling point set The curved surface in the figure is the aspherical optical element in the embodiment, and the sampling points in the space are the layout positions of the sensors in the inequality sampling space.

[0047] Step 4: Feasibility Approach.

[0048] After completing the inequality sampling to obtain the uniformly distributed samples in space, the feasibility approximation of the samples is performed using the sequential least squares programming (SLSQP) to obtain the uniformly distributed samples in the equality sampling space. The objective function fun(x) is used as the optimization target. As the initial value of optimization, sequential least squares programming (SLSQP) is applied for optimization, and boundary conditions and constraints g are set to ensure that the optimization process is carried out within the feasible domain.

[0049] After optimization by sequential least squares programming (SLSQP), the uniformly distributed samples in the final equal sampling space are obtained. The constraint g can be effectively satisfied. The present invention uses the uniform distribution of samples in the sampling space as The sampling point coordinates are the layout positions of the sensors, providing accurate sampling data for the sensor layout of aspherical optical elements. Figure 5 The result of using equality-constrained Latin hypercube sampling on the aspheric optical element in this embodiment is shown. The curved surface in the figure is the aspheric optical element in the embodiment, and the sampling points evenly distributed on the curved surface are the layout positions of the sensor.

Claims

1. A non-spherical optical element sensor layout method based on equality-constrained Latin hypercube sampling, characterized in that: The following steps are involved: Step 1: Obtain the size of the aspheric optical element and its surface control equation; Step 2: adding positive and negative margins to expand the surface control equation into an inequality sampling space; Step 3: Inequality sampling to obtain spatially uniformly distributed samples; Step 4: Feasibility Approach.

2. The method for sensor layout of non-spherical optical elements based on equality-constrained Latin hypercube sampling according to claim 1, characterized in that: The details are as follows: Step 1: Obtain the size of the aspheric optical element and its surface control equation; For a given aspheric optical element, its size is first obtained, and then the surface shape of the aspheric optical element is described by the Sag equation; the form of the Sag equation is: Where Z is the vertical distance from the vertex to a point on the surface, r is the radial distance from the optical axis to the point; c is the curvature at the vertex; k is the cone constant, n is a positive integer, and A 2n is the high-order aspheric coefficient; Step 2: adding positive and negative margins to expand the surface control equation into an inequality sampling space; After obtaining the size of the aspheric optical element and its surface control equation, a positive and negative margin ±ε is added to the surface control equation to expand the surface control equation of the aspheric optical element into an inequality sampling space, which is recorded as The spatial control equation describing the sampling space of the inequality is denoted as g; Step 3: Inequality sampling to obtain spatially uniformly distributed samples; After adding positive and negative margins to expand the surface control equation into the inequality sampling space, Latin hypercube sampling based on inequality constraints is used to obtain spatially uniformly distributed samples; specifically, as follows: First, in the inequality sampling space A Monte Carlo sampling is performed within, and the number of samples is Get a Monte Carlo sampling point set To ensure that the number of sampling points is higher than the expected number of sampling points α, a secondary Monte Carlo sampling is performed on the basis of the primary Monte Carlo sampling to obtain the secondary Monte Carlo sampling point set The number of samples is N M ;Will The space is evenly divided into N 2 Volume elements, after division Perform Latin hypercube sampling to obtain the Latin hypercube sampling point set The sample space is obtained by describing the inequality obtained in the second step The spatial control equation g is a constraint. Perform optimization based on the constrained simulated annealing algorithm to obtain the optimal Latin hypercube sampling point set Step 4: Feasibility Approximation; After completing the inequality sampling to obtain the spatially uniformly distributed samples, the feasibility approximation of the spatially uniformly distributed samples is performed using the sequential least squares programming to obtain the uniformly distributed samples in the equality sampling space. The details are as follows: First, define the objective function to calculate the current sampling point and The Euclidean distance of the inner sampling points; the objective function is in the following form: fun(x)=||x-x ineq ||(2) Among them, x is the currently optimized sampling point, x ineq for The sampling points within, ||·|| represents the Euclidean distance; The objective function fun(x) is used as the optimization target. As the initial value of optimization, sequential least squares programming is applied for optimization, and boundary conditions and constraints are set to ensure that the optimization process is carried out within the feasible domain; After optimization of the sequential least squares programming, the uniformly distributed samples in the final equal sampling space are obtained. Can effectively satisfy the constraint g, uniformly distribute samples The sampling point coordinates are used as the layout position of the sensor.

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