Design method of progressive addition lenses based on free-form surface expression

Abstract

The present invention provides a design method for progressive multifocal lenses based on free-form surface expression, which is mainly used for vision correction of presbyopic people and belongs to the field of ophthalmic optical design. The present invention realizes optical power matching and astigmatism minimization of progressive multifocal lenses by optimizing free-form surfaces. The design method constructs a nonlinear functional minimization problem using free-form surface expression, and obtains an optimized initial solution by solving a linear fourth-order partial differential equation related to a perturbation term based on a spherical basis. The method can minimize the astigmatism within the effective area of ​​the lens while achieving optical power matching as much as possible, and at the same time transfer the high astigmatism areas on both sides of the progressive channel to the edge area of ​​the lens, and can realize a variety of customized requirements for the wearer, including specific addition power, channel length, effective visual area range, etc.

Description

Technical Field

[0001] The present invention belongs to the field of ophthalmic optical design, and in particular relates to a design method of progressive multifocal lenses based on free-form surface expression, which is mainly used for vision correction of presbyopic people. Background Art

[0002] Progressive addition lenses, due to their continuous change in optical power between near and far vision distances, have become the primary method for correcting refractive errors such as presbyopia. This continuous and smooth change in optical power allows progressive addition lenses to adopt free-form surfaces, rather than traditional spherical or aspherical surfaces, with high design freedom and optical performance. Compared to conventional rotationally symmetric surfaces, free-form surfaces can adapt to more complex optical tasks and are widely used in the design of various imaging and lighting systems. They also play a vital role in enhancing the performance of ophthalmic optical design, helping to optimize the quality and performance of vision-related products.

[0003] Current design methods for progressive addition lenses still face several key challenges. Direct design methods solve the lens profile by assigning a specified power distribution, but they are unable to control the astigmatism on either side of the progressive channel, increasing the wearer's adaptation time and resulting in poor wearing comfort. Indirect design methods optimize an evaluation function to achieve target power matching and minimize astigmatism, but these methods are based on linear approximations, resulting in a suboptimal sag distribution. Furthermore, these optimization methods often struggle to generate meshes with irregular boundaries, and the computational accuracy of partial derivatives is affected by truncation and rounding errors, resulting in low design efficiency and significant limitations.

[0004] The actual design of progressive addition lenses also requires consideration of the wearer's individual needs. Current design methods often directly determine the lens' sagittal distribution by solving partial differential equations, making it impossible to iterate and optimize the design. This results in poor adjustability. High astigmatism areas in existing design methods are often concentrated on either side of the progressive channel, significantly increasing wearer adaptation time and compromising comfort. Summary of the Invention

[0005] In view of this, the purpose of the present invention is to provide a progressive addition lens design method based on free-form surface expression, which can solve the problems of existing design methods such as weak ability to approximate nonlinear functionals, low design efficiency, inability to perform effective iterative optimization, and difficulty in shifting high astigmatism areas to both sides of the lens. The progressive addition lens design method based on free-form surface expression includes:

[0006] Step 1: Divide the lens design area Ω of a specified size into an effective visual area and a peripheral astigmatism area; at the same time, divide the design area Ω into an actual design area Λ and an area outside Λ;

[0007] Step 2: Based on the lens areas divided in step 1, the weights of astigmatism and optical power of each area are allocated respectively.

[0008] The astigmatism weight and the power weight distribution are recorded as matrix α and matrix β; wherein the elements in matrix α are recorded as α(x, y), and the elements in matrix β are recorded as β(x, y), which respectively represent the weight values ​​at the coordinate point (x, y);

[0009] Step 3: Distribute the optical power according to the lens area divided in step 1, which is recorded as matrix P0;

[0010] Step 4: Obtain the initial solution of free-form surface optimization by solving the linear fourth-order partial differential equation;

[0011] The indirect method for designing progressive addition lenses requires the following functional minimization:

[0012] I(z(x,y))=∫ Ω {α(x,y)P ast (x,y) 2 +β(x,y)(P m (x,y)-P0(x,y)) 2}dxdy (2)

[0013] Where: z(x,y) is the sagittal height distribution of the lens, P ast (x,y) and P m (x, y) corresponds to the astigmatism distribution and power distribution of the current lens, and P0(x, y) is the target power distribution, which is the element in the matrix P0. According to the linear approximation method, the progressive multifocal lens is set to be based on the combination of the spherical basis s(x, y) and the perturbation term o(x, y). That is, the sagittal height of the actual lens can be expressed as z(x, y) = s(x, y) + o(x, y). Then, the linear approximation formula (2) is subjected to variational processing to obtain the linear fourth-order partial differential equation for the perturbation term o(x, y):

[0014]

[0015] Where: xx ,o xy ,o yy are the second-order partial derivatives of the function o(x,y) with respect to (x,y), (·) xx ,(·) xy ,(·) yy Respectively represent the second-order partial derivative of the function in the brackets with respect to (x, y); coefficient a i (x,y) is a function of (s(x,y),α(x,y),β(x,y),P0(x,y)), and its specific form is:

[0016]

[0017] Where: s x ,s y ,s xx ,s xy ,s yy is the partial derivative of the spherical basis function s(x,y) with respect to the (x,y) coordinates; by giving the curvature radius R of the spherical basis s , the weight distribution matrices α and β corresponding to astigmatism and optical power, the target optical power distribution matrix P0, and the perturbation distribution o(x, y) is obtained by solving equation (3) using the finite difference method. Then the initial sag distribution z0(x, y) = s(x, y) + o(x, y);

[0018] Step 5. Fit the discrete data points z0(x,y) obtained in step 4 with the free surface expression, and obtain the fitted free surface parameters by minimizing the evaluation function M(θ):

[0019]

[0020] Where: θ is the parameter vector of the free-form surface to be optimized, f(θ; x, y) is the free-form surface expression, ||·| F is the Frobenius norm of the matrix;

[0021] Step 6: Use automatic differentiation technology to obtain the partial derivatives of the free surface f(θ; x, y) obtained in step 5 with respect to the input point coordinates (x, y), that is, (f x ,f y ,f xx ,f xy ,f yy ); Use these partial derivatives to form the nonlinear functional shown in formula (2), and then use back propagation to calculate the gradient grad(θ) of the loss function value I(z) to the parameter vector θ, and use the optimizer f optim Update the free surface parameter vector θ:

[0022] θ * ←f optim [θ;grad(θ),τ] (6)

[0023] Where: θ * is the updated free surface parameter, τ is the update step size (or learning rate), f optim Typically an optimizer in a gradient descent algorithm;

[0024] Step 7: Repeat step 6 until the loss function value I(z) stabilizes to obtain the final optimized free-form surface f(x, y), completing the lens design.

[0025] Preferably, step six includes:

[0026] Step 6.1: Based on the current lens design parameters, first collect a series of discrete points (x, y) within the lens design area Ω. By substituting these discrete points into the free-form surface function to be optimized, the corresponding sag value z = f(θ; x, y) can be obtained.

[0027] Step 6.2: Obtain the first-order and second-order partial derivatives (z) of the vector height value z in step 6.1 with respect to the input space coordinates (x, y) by supporting the automatic differentiation method. x ,z y ,z xx ,z xy ,z yy ); the mean curvature μ and Gaussian curvature κ are calculated from this:

[0028]

[0029] Step 6.3: Express the mean curvature and Gaussian curvature in the form of principal curvatures, that is:

[0030]

[0031] Where k1 and k2 correspond to the two principal curvatures at a point on any surface. Furthermore, the relationship between power and astigmatism and the difference between the mean curvature and the principal curvature, δ = |k2 - k1|, is:

[0032]

[0033] Where n represents the refractive index of the lens material; Substituting equations (8) and (9) into equation (2) yields:

[0034] I(z)=∫ Ω {α(x,y)(μ(x,y) 2 -κ(x,y)) +β(x,y)(μ(x,y)-μ0(x,y)) 2}dxdy(10)

[0035] Where: μ0(x,y) is the target average curvature distribution;

[0036] Step 6.4: Combine the mean curvature and Gaussian curvature calculated in step 6.2 with the known weight distribution of astigmatism and optical power and the target mean curvature distribution to obtain the loss function value I(z) corresponding to all sampling points. By further summing, the overall loss function value is obtained, which is used for the back propagation of the free surface parameters.

[0037] Preferably, in step 1, the method of dividing the effective visual area and the peripheral astigmatism area within the lens design area Ω of a specified size includes:

[0038] The effective visual zone includes the upper distance vision zone, the lower near vision zone and the middle transition zone, which correspond to long-distance viewing, close-range activities and the continuous optical power change area between the two respectively; the peripheral astigmatism zone is mainly distributed on both sides of the middle transition zone.

[0039] Preferably, in step 1, the range of the area Λ is usually a circular area with a radius set with the center of the lens design area Ω as the origin.

[0040] Preferably, the step 2 includes:

[0041] The principles for allocating weights of astigmatism and optical power in each area are consistent: the weight values ​​of each point in the effective visual area are consistent and maximum; the weight values ​​of each point in the peripheral astigmatism area are consistent and smaller; and the weight values ​​in the actual design area Λ maintain the above setting requirements, and the weight values ​​outside the area Λ are set to the minimum, and are all set to the same weight value.

[0042] Preferably, in the step 2, the astigmatism and the optical power weight of each region are smoothed using a convolution operation.

[0043] Preferably, in step 3, the method of allocating optical power includes: the optical power of the far vision zone is The optical power of the near vision zone is in, is the basic optical power of the lens, is the added power of the lens; the focal power of the intermediate transition area is between the near vision area and the far vision area. A trigonometric function is used to achieve continuous and smooth changes in the focal power of this area, which is recorded as:

[0044]

[0045] Where: is the optical power value at the (x, y) position, l x,y is the distance from this point to the far vision reference point, l is the total length of the intermediate transition zone; the optical power value of other areas is the average optical power of the near vision zone and the far vision zone, recorded as

[0046] Preferably, in step 3, the optical power within the entire lens design area Ω is smoothed using a convolution operation.

[0047] Preferably, during the iterative process, the optimization results are flexibly adjusted by modifying the parameters in steps one to three to adapt to the personalized needs of different wearers.

[0048] The invention has the following beneficial effects:

[0049] Compared with the current mainstream design methods, the method proposed in this invention minimizes the astigmatism in the effective visual area of ​​the entire lens while meeting the target optical power matching as much as possible; at the same time, the high astigmatism area of ​​the entire lens is transferred from both sides of the progressive channel to the edge of the lens.

[0050] By using a free-form surface to represent the sag of a progressive addition lens, a nonlinear functional minimization problem is constructed. Compared to quadratic functional approximation methods, this method approaches the optimal solution to the original design problem as closely as possible. Furthermore, because the progressive addition lens is constructed using a defined free-form surface expression, all-order partial derivatives with respect to spatial coordinates can be obtained using automatic differentiation techniques. Compared to traditional numerical methods, this method offers higher accuracy and greater speed.

[0051] The minimization of nonlinear functionals is based on solving linear fourth-order partial differential equations to obtain the initial solution, which can speed up the subsequent optimization steps and ensure the stability of the optimization process.

[0052] Compared with obtaining the sagittal height of progressive multifocal lenses by directly solving partial differential equations, the iterative optimization method can adjust fixed parameter constants through the optimization results of each round, thereby further obtaining satisfactory design results based on actual design requirements. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 This is a flow chart of the method for designing progressive addition lenses based on free-form surface optimization disclosed in the present invention;

[0054] Figure 2 The division of the lens area in Example 1 of the present invention;

[0055] Figure 3 The weight function is allocated in Example 1 of the present invention;

[0056] Figure 4 is the distribution of the optical power function in Example 1 of the present invention;

[0057] Figure 5 is the distribution of disturbance terms based on spherical sag in Example 1 of the present invention;

[0058] Figure 6 is a curve showing the variation of the error functional with the number of free-form surface optimizations in Example 1 of the present invention;

[0059] FIG7( a ) is a cylindrical degree distribution diagram of the linear preliminary solution in Example 1;

[0060] FIG7( b ) is a cylindrical surface distribution diagram of the free-form surface optimization solution in Example 1;

[0061] FIG7( c ) is a spherical degree distribution diagram of the linear preliminary solution in Example 1;

[0062] FIG7( d ) is a sphericity distribution diagram of the free-form surface optimization solution in Example 1. FIG. DETAILED DESCRIPTION

[0063] The present invention is described in detail below with reference to the accompanying drawings and embodiments.

[0064] The present invention provides a method for designing a progressive addition lens based on free-form surface expression, comprising the following steps:

[0065] Step 1: Within the lens design area Ω of a specified size, the effective visual zone and the peripheral astigmatism zone are divided. The effective visual zone consists of the upper distance zone, the lower near zone, and the central transition zone, corresponding to distance viewing, near-distance activities, and the continuous optical power variation between the two. The peripheral astigmatism zone is primarily located on either side of the transition zone, concentrating the high astigmatism areas of the entire lens. Furthermore, considering the actual lens usage area, the design area Ω is typically divided into the actual design area Λ and the area outside Λ. The area Λ typically encompasses a circular region with radius R, centered at the center of the lens design area Ω.

[0066] Step 2: Based on the lens areas divided in step 1, the weights of astigmatism and optical power of each area are allocated separately. The allocation principles of the two are consistent: the weight values ​​of each point in the effective visual area are consistent and maximum, denoted as w1; the weight values ​​of each point in the peripheral astigmatism area are consistent and smaller, denoted as w2. Furthermore, the weight values ​​in the actual design area Λ maintain the above setting requirements, and the weight values ​​outside the area Λ are set to the minimum, all set to w3. Since the weight distribution in the entire lens design area Ω is in the form of a constant discrete function, it is necessary to further use a convolution operation to smooth the weight distribution to facilitate the solution search in the subsequent optimization steps. The smoothed astigmatism weight and optical power weight distribution are denoted as matrices α and β. Among them, the elements in matrix α are denoted as α(x, y), and the elements in matrix β are denoted as β(x, y), which respectively represent the weight values ​​at the coordinate point (x, y).

[0067] Step 3: Allocate the focal power according to the lens area divided in step 1. The main principle of allocation is: the focal power of the far vision area is The optical power of the near vision zone is in, is the basic optical power of the lens, is the added power of the lens. The focal power of the intermediate transition area is between the near vision area and the far vision area. A trigonometric function is used to achieve a continuous and smooth change in the focal power of this area, which is expressed as:

[0068]

[0069] Where: is the optical power value at the (x, y) position, l x,y is the distance from the point to the far vision reference point (usually set at the center of the boundary between the far vision zone and the intermediate transition zone, and the same applies to the near vision reference point), and l is the total length of the intermediate transition zone. The optical power value of other areas is the average of the optical power of the near vision zone and the far vision zone, recorded as Similarly, the optical power distribution within the entire lens design area Ω is in the form of a constant discrete function, and a convolution operation needs to be further used to smooth the optical power distribution, which is recorded as matrix P0.

[0070] Step 4: Obtain the initial solution for free-form surface optimization by solving the linear fourth-order partial differential equation. The indirect method for designing progressive addition lenses requires the following functional minimization:

[0071] I(z(x,y))=∫ Ω {α(x,y)P ast (x,y) 2 +β(x,y)(P m (x,y)-P0(x,y)) 2}dxdy (2)

[0072] Where: z(x,y) is the sagittal height distribution of the lens, α(x,y) and β(x,y) are the weight distribution functions of astigmatism and optical power respectively (corresponding to the settings in step 2), P ast (x,y) and P m (x, y) corresponds to the astigmatism and power distribution of the current lens, and P0(x, y) is the target power distribution (corresponding to the setting in step 3). According to the linear approximation method, the progressive multifocal lens is set to be based on the combination of the spherical base s(x, y) and the perturbation term o(x, y). That is, the actual lens's sagittal height can be expressed as z(x, y) = s(x, y) + o(x, y). Then, after performing variational processing on the linearly approximated equation (2), a linear fourth-order partial differential equation for the perturbation term o(x, y) is obtained:

[0073]

[0074] Where: xx ,o xy ,o yy are the second-order partial derivatives of the function o(x,y) with respect to (x,y), (·) xx ,(·) xy ,(·) yy They represent the second-order partial derivatives of the function in the brackets with respect to (x, y). i(x,y) is a function of (s(x,y),α(x,y),β(x,y),P0(x,y)), and its specific form is:

[0075]

[0076] Where: s x ,s y ,s xx ,s xy ,s yy is the partial derivative of the spherical basis function s(x,y) with respect to the (x,y) coordinates. By giving the curvature radius R of the spherical basis s , the weight distribution matrices α and β corresponding to astigmatism and optical power, and the target optical power distribution matrix P0, can be solved by the finite difference method to obtain the disturbance distribution o(x, y), then the initial sag distribution z0(x, y) = s(x, y) + o(x, y).

[0077] Step 5. Fit the discrete data points z0(x,y) obtained in step 4 with the free surface expression, and obtain the fitted free surface parameters by minimizing the evaluation function M(θ):

[0078]

[0079] Where: θ is the parameter vector of the free-form surface to be optimized, f(θ; x, y) is the free-form surface expression, ||·|| F is the Frobenius norm of the matrix.

[0080] Step 6: Using the automatic differentiation technology, we can directly obtain the partial derivatives of the free surface f(θ; x, y) obtained in step 5 with respect to the input point coordinates (x, y), that is, (f x ,f y ,f xx ,f xy ,f yy ). Using these partial derivatives, we can directly construct the nonlinear functional shown in Equation (2), and then use backpropagation to calculate the gradient grad(θ) of the loss function value I(z) with respect to the parameter vector θ, and use the optimizer f optim Update the free surface parameter vector θ:

[0081] θ * ←f optim [θ;grad(θ),τ] (6)

[0082] Where: θ * is the updated free surface parameter, τ is the update step size (or learning rate), f optimUsually it is the SGD and Adam optimizers commonly used in gradient descent algorithms.

[0083] Step 7: Repeat step 6 until the loss function value I(z) stabilizes, obtaining the final optimized freeform surface f(x,y). During the iteration process, the optimization results can be flexibly adjusted by modifying the parameters in steps 1 to 3 to meet the personalized needs of different wearers.

[0084] In the method for designing a progressive addition lens based on free-form surface optimization of the present invention, the specific steps of step six are as follows:

[0085] Step 6.1: Based on the current lens design parameters, first collect a series of discrete points (x, y) within the lens design area Ω. By substituting these discrete points into the free-form surface function to be optimized, the corresponding sag value z = f(θ; x, y) can be obtained.

[0086] Step 6.2: Use a framework that supports automatic differentiation (such as Pytorch, Tensorflow) to obtain the first-order and second-order partial derivatives (z) of the vector height value z in step 6.1 with respect to the input space coordinates (x, y). x ,z y ,z xx ,z xy ,z yy ); The mean curvature μ and Gaussian curvature κ can be directly calculated from these partial derivatives:

[0087]

[0088] Step 6.3: Express the mean curvature and Gaussian curvature in the form of principal curvatures, that is:

[0089]

[0090] Where: k1 and k2 correspond to the two principal curvatures at a point on any surface. At the same time, the relationship between the optical power and astigmatism and the difference between the mean curvature and the principal curvature δ = |k2-k1| is:

[0091]

[0092] Where n represents the refractive index of the lens material. Substituting equations (8) and (9) into equation (2) yields:

[0093] I(z)=∫ Ω {α(x,y)(μ(x,y) 2 -κ(x,y))+β(x,y)(μ(x,y)-μ0(x,y)) 2}dxdy (10)

[0094] Where: μ0(x,y) is the target average curvature distribution.

[0095] Step 6.4: Combine the mean curvature and Gaussian curvature calculated in step 6.2 with the known weight distribution of astigmatism and optical power and the target mean curvature distribution to obtain the loss function value I(z) corresponding to all sampling points. By further summing, the overall loss function value is obtained, which is used for the backpropagation of the free surface parameters.

[0096] Example 1:

[0097] This example discloses a design method for progressive multifocal lenses based on free-form surface expression. The design process is as follows: Figure 1 As shown in the figure, the target design area Ω is a square area with a side length of 80mm, the actual design area Λ is a circular area with a radius of 30mm, and the refractive index of the lens material is set to 1.530. The parameters of the lens to be designed are: 5D of focal length for distance vision and 7D of focal length for near vision (with a 2D addition). The specific design steps are as follows:

[0098] Step 1: Follow Figure 2 The target design area Ω is divided into seven regions, including the effective visual area (far vision area, near vision area, intermediate transition area) and the peripheral astigmatism area. To embed the near vision area, it is usually necessary to linearly move it a certain distance toward the side of the nose.

[0099] Step 2: Following the division method in Step 1, set the weight distribution corresponding to the astigmatism optimization and target power matching. The weight values ​​within the effective visual area are consistent and the largest, both set to w1 = 30; the weight values ​​in the peripheral astigmatism area are consistent and smaller, both set to w2 = 1.0. Furthermore, the weight values ​​within the actual design area Λ maintain the above setting requirements, and the weight values ​​outside Λ are the smallest, both set to w3 = 10. -8 Finally, the weight distribution of the above segments is smoothed by convolution operation, as shown in Figure 3 In this design example, the initial weight distributions for astigmatism optimization and power matching are consistent.

[0100] Step 3: Allocate the optical power of the lens according to the division method in step 1. The optical power of the far vision zone is set to The optical power of the near vision zone is The optical power of the intermediate transition area is between the near vision area and the far vision area. A trigonometric function is used to achieve continuous and smooth changes in the optical power of this area, which is expressed as:

[0101]

[0102] The optical power of other areas is the average of the optical power of the near vision area and the far vision area, which is set as Similarly, the power distribution within the entire lens design area Ω is in the form of a constant discrete function, and the convolution operation needs to be further used to smooth the power distribution, such as Figure 4 shown.

[0103] Step 4: Obtain the initial solution of the free-form surface optimization by solving the linear fourth-order partial differential equation. The curvature radius R of the spherical base s =106mm. The finite difference method is used to solve equation (3) to obtain the disturbance distribution o(x,y). The number of computational grids is 81×81, the grid spacing is 1mm, and the boundary conditions are o(-40,-40)=o(40,40)=o(40,-40)=0. The final solution is as follows: Figure 5 shown.

[0104] Step 5: Fit the discrete data points o(x,y) obtained in step 4 using a free-form surface expression. The free-form surface used in this example is a non-uniform rational B-spline surface (NURBS surface), and its expression is:

[0105]

[0106] Where: P i,j =(x p ,y p ,z p ) is the coordinate of the control point, w i,j is the weight corresponding to the control point, m s 、n s N is the number of control points in two non-parallel directions. i,p (u), N j,q (v) represents the p-order and q-order B-spline basis functions defined in the u and v directions respectively, and are defined as follows:

[0107]

[0108] Here, node u i Use quasi-uniform method to divide the interval [0,1] into m s +p copies. The coordinates of the control points of the NURBS surface along the x and y directions are uniformly collected within the target design area Ω, that is, the only variable involved in the optimization is the coordinate z of the control point. p The number of control points of the NURBS surface is set to 30×30. In order to speed up the interpolation of the NURBS surface during the optimization process, an analytical second-order B-spline basis function is used. Then, the gradient descent algorithm is used to optimize the evaluation function (5) to complete the fitting of the initial discrete points.

[0109] Step 6: Since the NURBS surface obtained in step 5 has a complete function expression (or forward expression), the automatic differentiation technology can be used to obtain the first-order and second-order partial derivatives of the lens surface height z with respect to the spatial coordinates x and y (z x ,z y ,z xx ,z xy ,z yy ). By also collecting 81×81 grid points on the NURBS surface, the average curvature and Gaussian curvature at these points are calculated. Finally, the loss function shown in formula (10) can be constructed by combining the weight distribution function in step 2 and the optical power distribution function in step 3. By using the gradient descent algorithm again, the iterative optimization of the final surface shape is completed. The change curve of the loss function I(z) with the number of iterations is shown as follows Figure 6 shown.

[0110] Step 7: Evaluate the optical performance of the NURBS surface obtained in Step 6. Collect 81×81 grid points on the NURBS surface, calculate the first- and second-order partial derivatives of these grid points using finite differences, and then calculate the mean curvature and Gaussian curvature at these points. Finally, substitute Equation (9) to obtain the optical power and astigmatism distributions of the optimized surface. Here, only the optical power and astigmatism distributions within the actual design region Λ are plotted, as shown in Figure 7.

[0111] The optimization of Example 1 is divided into two parts: the first step is to use NURBS surface to fit discrete points, with a total of 1000 iterations, an initial learning rate of 0.01, and a learning rate decay coefficient of 0.975 every 50 iterations; the second step is to use NURBS surface to optimize the target optical power and astigmatism, with a total of 1300 iterations, an initial learning rate of 0.005, and a learning rate decay coefficient of 0.975 every 50 iterations. Figure 7 shows the changes from the linear initial solution to the NURBS surface optimization solution. The evaluation index of the added light is defined as the optical power difference between the reference far point and the reference near point. Figure 7 (a) shows the astigmatism distribution of the linear solution. The astigmatism value in the effective visual area fluctuates within the range of 0.5D; Figure 7 (b) shows the astigmatism distribution of the NURBS surface optimization solution. The astigmatism values ​​in the effective visual area are all within 0.25D. Although the maximum astigmatism value in the peripheral astigmatism area has increased, these high astigmatism areas have shifted to the sides of the lens. Figure 7(c) shows the power distribution of the linear solution, with an addition of 1.499D. Figure 7(d) shows the power distribution of the NURBS surface optimized solution, with an addition of 1.625D, an 8.41% improvement over the linear solution. In summary, the above are merely preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention shall be included within the scope of protection of the present invention.

Claims

1. A progressive addition lens design method based on free-form surface expression, characterized in that: include: Step 1: Divide the lens design area Ω of a specified size into an effective visual area and a peripheral astigmatism area; at the same time, divide the design area Ω into an actual design area Λ and an area outside Λ; Step 2: Based on the lens areas divided in step 1, the weights of astigmatism and optical power of each area are allocated respectively. The astigmatism weight and the power weight distribution are recorded as matrix α and matrix β; wherein the elements in matrix α are recorded as α(x, y), and the elements in matrix β are recorded as β(x, y), which respectively represent the weight values ​​at the coordinate point (x, y); Step 3: Distribute the optical power according to the lens area divided in step 1, which is recorded as matrix P0; Step 4: Obtain the initial solution of free-form surface optimization by solving the linear fourth-order partial differential equation; The indirect method for designing progressive addition lenses requires the following functional minimization: I(z(x,y))=∫ Ω {α(x,y)P ast (x,y) 2 +β(x,y)(P m (x,y)-P0(x,y)) 2 }dxdy (2) Where: z(x,y) is the sagittal height distribution of the lens, P ast (x,y) and P m (x, y) corresponds to the astigmatism distribution and power distribution of the current lens, and P0(x, y) is the target power distribution, which is the element in the matrix P0. According to the linear approximation method, the progressive multifocal lens is set to be based on the combination of the spherical basis s(x, y) and the perturbation term o(x, y). That is, the sagittal height of the actual lens can be expressed as z(x, y) = s(x, y) + o(x, y). Then, the linear approximation formula (2) is subjected to variational processing to obtain the linear fourth-order partial differential equation for the perturbation term o(x, y): Where: xx ,o xy ,o yy are the second-order partial derivatives of the function o(x,y) with respect to (x,y), (·) xx ,(·) xy ,(·) yy Respectively represent the second-order partial derivative of the function in the brackets with respect to (x, y); coefficient a i (x,y) is a function of (s(x,y),α(x,y),β(x,y),P0(x,y)), and its specific form is: Where: s x ,s y ,s xx ,s xy ,s yy is the partial derivative of the spherical basis function s(x,y) with respect to the (x,y) coordinates; by giving the curvature radius R of the spherical basis s , the weight distribution matrices α and β corresponding to astigmatism and optical power, the target optical power distribution matrix P0, and the perturbation distribution o(x, y) is obtained by solving equation (3) using the finite difference method. Then the initial sag distribution z0(x, y) = s(x, y) + o(x, y); Step 5. Fit the discrete data points z0(x,y) obtained in step 4 with the free surface expression, and obtain the fitted free surface parameters by minimizing the evaluation function M(θ): Where: θ is the parameter vector of the free-form surface to be optimized, f(θ; x, y) is the free-form surface expression, ||·|| F is the Frobenius norm of the matrix; Step 6: Use automatic differentiation technology to obtain the partial derivatives of the free surface f(θ; x, y) obtained in step 5 with respect to the input point coordinates (x, y), that is, (f x ,f y ,f xx ,f xy ,f yy ); Use these partial derivatives to form the nonlinear functional shown in formula (2), and then use back propagation to calculate the gradient grad(θ) of the loss function value I(z) to the parameter vector θ, and use the optimizer f optim Update the free surface parameter vector θ: θ * ←f optim [θ;grad(θ),τ](6) where: θ * is the updated free surface parameter, τ is the update step size or learning rate, f optim Typically an optimizer in a gradient descent algorithm; Step 7: Repeat step 6 until the loss function value I(z) stabilizes to obtain the final optimized free-form surface f(x, y), completing the lens design.

2. The method for designing progressive addition lenses based on free-form surface expression according to claim 1, wherein: Step six includes: Step 6.1: Based on the current lens design parameters, first collect a series of discrete points (x, y) within the lens design area Ω. By substituting these discrete points into the free-form surface function to be optimized, the corresponding sag value z = f(θ; x, y) can be obtained. Step 6.2: Obtain the first-order and second-order partial derivatives (z) of the vector height value z in step 6.1 with respect to the input space coordinates (x, y) by supporting the automatic differentiation method. x ,z y ,z xx ,z xy ,z yy ); the mean curvature μ and Gaussian curvature κ are calculated from this: Step 6.3: Express the mean curvature and Gaussian curvature in the form of principal curvatures, that is: Where k1 and k2 correspond to the two principal curvatures at a point on any surface. Furthermore, the relationship between power and astigmatism and the difference between the mean curvature and the principal curvature, δ = |k2 - k1|, is: Where n represents the refractive index of the lens material; Substituting equations (8) and (9) into equation (2) yields: I(z)=∫ Ω {α(x,y)(μ(x,y) 2 -κ(x,y))+β(x,y)(μ(x,y)-μ0(x,y)) 2 }dxdy (10) Where: μ0(x,y) is the target average curvature distribution; Step 6.4: Combine the mean curvature and Gaussian curvature calculated in step 6.2 with the known weight distribution of astigmatism and optical power and the target mean curvature distribution to obtain the loss function value I(z) corresponding to all sampling points. By further summing, the overall loss function value is obtained, which is used for the back propagation of the free surface parameters.

3. The method for designing progressive addition lenses based on free-form surface expression according to claim 2, wherein: In step 1, the method for dividing the effective visual area and the peripheral astigmatism area within the lens design area Ω of a specified size includes: The effective visual zone includes the upper distance vision zone, the lower near vision zone and the middle transition zone, which correspond to long-distance viewing, close-range activities and the continuous optical power change area between the two respectively; the peripheral astigmatism zone is mainly distributed on both sides of the middle transition zone.

4. The method for designing progressive addition lenses based on free-form surface expression according to claim 3, wherein: In step 1, the range of the region Λ is generally a circular region with a radius set with the center of the lens design region Ω as the origin.

5. The method for designing progressive addition lenses based on free-form surface expression according to claim 4, wherein: The second step includes: The principles for allocating weights of astigmatism and optical power in each area are consistent: the weight values ​​of each point in the effective visual area are consistent and maximum; the weight values ​​of each point in the peripheral astigmatism area are consistent and smaller; and the weight values ​​in the actual design area Λ maintain the above setting requirements, and the weight values ​​outside the area Λ are set to the minimum, and are all set to the same weight value.

6. The method for designing progressive addition lenses based on free-form surface expression according to claim 5, wherein: In the second step, the astigmatism and the optical power weight of each region are smoothed using a convolution operation.

7. The method for designing a progressive addition lens based on free-form surface expression according to claim 2, 3, 4, 5 or 6, wherein: In step 3, the method for allocating optical power includes: the optical power of the far vision zone is The optical power of the near vision zone is in, is the basic optical power of the lens, is the added power of the lens; the focal power of the intermediate transition area is between the near vision area and the far vision area. A trigonometric function is used to achieve continuous and smooth changes in the focal power of this area, which is recorded as: Where: is the optical power value at the (x, y) position, l x,y is the distance from this position to the far vision reference point, l is the total length of the intermediate transition zone; the optical power value of other areas is the average of the optical power of the near vision zone and the far vision zone, recorded as 8. The method for designing progressive addition lenses based on free-form surface expression according to claim 7, wherein: In step 3, the optical power within the entire lens design area Ω is smoothed using a convolution operation.

9. The method for designing progressive addition lenses based on free-form surface expression according to claim 2, wherein: During the iterative process, the optimization results are flexibly adjusted by modifying the parameters in steps one to three to adapt to the personalized needs of different wearers.

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