Progressive multi-focus lens design method based on free-form surface expression

Abstract

The invention provides a progressive addition lens design method based on free-form surface expression, is mainly used for vision correction of presbyopia groups, and belongs to the field of eye vision optical design. According to the design method, focal power matching and astigmatism minimization of the progressive multi-focus lens are achieved by optimizing the free-form surface, the design method constructs a nonlinear functional minimization problem by means of free-form surface expression, and meanwhile an optimized initial solution is obtained by solving a linear fourth-order partial differential equation related to a spherical base disturbance term; according to the method, astigmatism in an effective area of the lens can be minimized on the premise of realizing focal power matching as far as possible, high astigmatism areas on two sides of the progressive channel are transferred to an edge area of the lens, and various customization requirements including specific light adding degree, channel length, effective visual area range and the like can be met for a wearer.

Description

Technical Field

[0001] The invention belongs to the field of ophthalmic optics design, and specifically relates to a progressive multifocal lens design method based on free surface expression, which is mainly used for vision correction of presbyopic groups. Background Art

[0002] Progressive multifocal lenses have become the main way to correct refractive errors such as presbyopia because of their continuous change in optical power between far and near vision distances. The continuous and smooth change in optical power means that the surface of progressive multifocal lenses is not a traditional spherical or aspherical surface, but a free-form surface with high design freedom and optical performance. Compared with conventional rotationally symmetric surfaces, free-form surfaces can adapt to more complex optical tasks and are widely used in the design of various imaging systems and lighting systems. They also play an important role in enhancing the performance of ophthalmic optical design and help optimize the quality and performance of vision-related products.

[0003] At present, there are still some key problems in the design methods of progressive multifocal lenses. The direct design method solves the lens surface shape by assigning a specified optical power distribution, which cannot control the astigmatism on both sides of the progressive channel, increases the wearer's adaptation time and has poor wearing comfort. The indirect design method achieves target optical power matching and astigmatism minimization by optimizing the evaluation function, but these methods are all based on linear approximation, which will result in the solved vector height distribution usually not being the optimal solution. At the same time, the above optimization methods often have difficulty in realizing grid generation with irregular boundaries, and the calculation accuracy of partial derivatives is also affected by truncation error and rounding error, resulting in low design efficiency and great limitations.

[0004] In the actual design process of progressive multifocal lenses, the wearer's personalized needs also need to be considered. Current design methods often directly obtain the lens's sagittal height distribution by solving partial differential equations, which cannot be iterated and optimized, that is, the design results are poorly adjustable. The high astigmatism areas of existing design methods are usually concentrated on both sides of the progressive channel, which greatly increases the wearer's adaptation time and poor comfort. Summary of the invention

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

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

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

[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 focal power according to the lens area divided in step 1, recorded as matrix P 0 ;

[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 multifocal 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)-P 0 (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 focal power distribution of the current lens, P 0 (x, y) is the target optical power distribution, which is the matrix P 0 elements in the equation; according to the linear approximation method, the progressive multifocal lens is set to be a combination of the spherical basis s(x,y) and the disturbance term o(x,y), that is, the actual lens sagittal height can be expressed as z(x,y)=s(x,y)+o(x,y); then the linear fourth-order partial differential equation about the disturbance term o(x,y) is obtained by variational processing of the linearly approximated equation (2):

[0014]

[0015] In the formula: o 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 about (s(x,y),α(x,y),β(x,y),P 0 (x,y)), the 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 radius of curvature R of the spherical basis s , weight distribution matrix α, β corresponding to astigmatism and optical power, target optical power distribution matrix P 0 , using the finite difference method to solve equation (3) to obtain the disturbance distribution o(x,y), then the initial vector height distribution z 0 (x,y)=s(x,y)+o(x,y);

[0018] Step 5: The discrete data points z obtained in step 4 0 (x, y) is fitted using the free surface expression, and the fitted free surface parameters are obtained by minimizing the evaluation function M(θ):

[0019]

[0020] Where: θ is the free-form surface parameter vector 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) becomes stable, and then the final optimized free-form surface f(x, y) is obtained, completing the design of the lens.

[0025] Preferably, step six includes:

[0026] Step 6.1: Based on the design parameters of the current lens, firstly, a series of discrete points (x, y) are collected in the lens design area Ω, and the corresponding vector height value z=f(θ; x, y) can be obtained by substituting these discrete points into the free surface function to be optimized;

[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: k 1 , k 2 The two principal curvatures corresponding to a point on any surface; at the same time, the difference between the optical power and astigmatism and the mean curvature and the principal curvature δ = |k 2 -k 1 The relationship between | 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)-μ 0 (x,y) 2}dxdy (10)

[0035] Where: μ 0(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 for the back propagation of the free surface parameters.

[0037] Preferably, 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 comprises:

[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-distance activities and the continuous optical power change area between the two; 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 comprises:

[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 small; 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, 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 focal length 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 focal length 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 focal length of other areas is the average of the focal length of the near vision zone and the far vision zone, recorded as

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

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

[0048] The invention has the following beneficial effects:

[0049] Compared with the current mainstream design method, the method proposed by the present invention minimizes the astigmatism in the effective visual area of ​​the entire lens while satisfying 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 surface to express the vector height of a progressive multifocal lens, the nonlinear functional minimization problem is constructed. Compared with the quadratic functional approximation method, the method can be as close as possible to the optimal solution of the original design problem. And because the progressive multifocal lens is constructed using a certain free surface expression, its partial derivatives of all orders with respect to the spatial coordinates can be obtained through automatic differentiation technology. Compared with traditional numerical methods, it has higher accuracy and faster speed.

[0051] The minimization of nonlinear functionals is based on solving the linear fourth-order partial differential equation 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 results of each round of optimization, thereby further obtaining satisfactory design results based on actual design requirements. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 A flow chart of a progressive multifocal lens design method based on free-form surface optimization disclosed in the present invention;

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

[0055] Figure 3 The weight function is allocated in Embodiment 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 vector height 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] FIG. 7( a ) is a cylindrical distribution diagram of the linear preliminary solution in Example 1;

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

[0061] FIG. 7( c ) is a spherical degree distribution diagram of the linear preliminary solution in Example 1;

[0062] FIG. 7( 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 progressive multifocal lens design method based on free-form surface expression, comprising the following steps:

[0065] Step 1: Divide the effective visual area and the peripheral astigmatism area within the lens design area Ω of a specified size. The effective visual area includes the upper far vision area, the lower near vision area and the middle transition area, which correspond to long-distance viewing, close-range activities and the continuous optical power change area between the two respectively; the peripheral astigmatism area is mainly distributed on both sides of the middle transition area, concentrating the high astigmatism area of ​​the entire lens. In addition, considering the actual use area of ​​the lens, the design area Ω is usually divided into the actual design area Λ and the area outside Λ. The range of area Λ is usually a circular area with the center of the lens design area Ω as the origin and R as the radius.

[0066] Step 2: According to the lens areas divided in step 1, the weights of astigmatism and focal power of each area are allocated respectively. The allocation principle of the two is consistent: the weight value of each point in the effective visual area is consistent and the largest, recorded as w 1 ; The weight values ​​of each point in the peripheral astigmatism area are consistent and small, denoted as w 2 Furthermore, the weight values ​​in the actual design region Λ are kept in accordance with the above setting requirements, and the weight values ​​outside the region Λ are set to the minimum, all set to w 3 . 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 recorded as matrix α and matrix β. Among them, the elements in matrix α are recorded as α(x, y), and the elements in matrix β are recorded as β(x, y), which represent the weight values ​​at the coordinate point (x, y) respectively.

[0067] Step 3: Distribute the focal power according to the lens areas divided in step 1. The main principle of distribution is: the focal power of the far vision area is The optical power of the near vision zone is in, is the basic focal length 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:

[0068]

[0069] Where: is the optical focal length 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 optical power of the near vision zone and the far vision zone, recorded as Similarly, the focal power distribution in 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 focal power distribution, which is recorded as the matrix P 0 .

[0070] 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 multifocal lenses needs to satisfy 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)-P 0 (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 focal power respectively (corresponding to the settings in step 2), P ast (x,y) and P m (x, y) corresponds to the astigmatism distribution and focal power distribution of the current lens, P 0 (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, the linear fourth-order partial differential equation about the perturbation term o(x, y) is obtained by performing variational processing on the linearly approximated equation (2):

[0073]

[0074] In the formula: o 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 about (s(x,y),α(x,y),β(x,y),P 0 (x,y)), the 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 radius of curvature R of the spherical basis s , weight distribution matrix α, β corresponding to astigmatism and optical power, target optical power distribution matrix P 0 , using the finite difference method to solve equation (3) we can get the perturbation distribution o(x,y), then the initial vector height distribution z 0 (x,y)=s(x,y)+o(x,y).

[0077] Step 5: The discrete data points z obtained in step 4 0 (x, y) is fitted using the free surface expression, and the fitted free surface parameters are obtained by minimizing the evaluation function M(θ):

[0078]

[0079] Where: θ is the free-form surface parameter vector 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 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). These partial derivatives can be used to directly construct the nonlinear functional shown in equation (2), and then the gradient grad(θ) of the loss function value I(z) to the parameter vector θ is calculated using back propagation, and 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 optim Usually it is the SGD and Adam optimizers commonly used in the gradient descent algorithm.

[0083] Step 7: Repeat step 6 until the loss function value I(z) tends to be stable to obtain the final optimized free 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 multifocal 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 design parameters of the current lens, firstly, a series of discrete points (x, y) are collected in the lens design area Ω, and the corresponding vector height value z=f(θ; x, y) can be obtained by substituting these discrete points into the free surface function to be optimized;

[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: k 1 , k 2The two principal curvatures corresponding to a point on any surface. At the same time, the difference between the optical power and astigmatism and the mean curvature and the principal curvature δ = |k 2 -k 1 The relationship between | is:

[0091]

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

[0093] I(z)=∫ Ω {α(x,y)(μ(x,y) 2 -κ(x,y))+β(x,y)(μ(x,y)-μ 0 (x,y) 2}dxdy (10)

[0094] Where: μ 0 (x,y) is the target mean curvature distribution.

[0095] Step 6.4: Combining 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, we can get the loss function value I(z) corresponding to all sampling points. By further summing, we can get the overall loss function value, which is used for the back propagation of the free surface parameters.

[0096] Embodiment 1:

[0097] This example discloses a progressive multifocal lens design method based on free-form surface expression. The design process is as follows: Figure 1 As shown. 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: the focal length of the far vision area is 5D, and the focal length of the near vision area is 7D (the addition power is 2D). The specific design steps are as follows:

[0098] Step 1: Follow Figure 2 The target design area Ω is divided into 7 areas including the effective visual area (far vision area, near vision area, intermediate transition area) and the peripheral astigmatism area. In order to embed the near vision area, it is usually necessary to linearly move it a certain distance to the nose side of the person.

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

[0100] Step 3: Allocate the focal power of the lens according to the division method in step 1. The focal power of the far vision zone is set to The optical power of the near vision zone is 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 expressed as:

[0101]

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

[0103] Step 4: Obtain the initial solution of free-form surface optimization by solving the linear fourth-order partial differential equation. The radius of curvature 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 the free surface expression. The free 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 The interval [0,1] is divided 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 region 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 speed of the NURBS surface during the optimization process, the analytical 2nd-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 vector 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 in 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, use finite differences to calculate the first-order and second-order partial derivatives of these grid points, and then calculate the average curvature and Gaussian curvature at these points. Finally, substitute into formula (9) to obtain the power distribution and astigmatism distribution of the optimized surface. Here, only the power and astigmatism distribution within the actual design area 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 times; 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 times. 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) is the astigmatism distribution of the linear solution, and the astigmatism value in the effective visual area fluctuates within the range of 0.5D; Figure 7 (b) is the astigmatism distribution of the NURBS surface optimization solution, and 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 both sides of the lens. Figure 7(c) is the focal power distribution of the linear solution, and its addition is 1.499D; Figure 7(d) is the focal power distribution of the NURBS surface optimization solution, and its addition is 1.625D, which is 8.41% higher than the linear solution. In summary, the above are only 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 should be included in the scope of protection of the present invention.

Claims

1. A progressive multifocal lens design method based on free-form surface expression, characterized in that: include: Step 1: Divide the effective visual area and the peripheral astigmatism area within the lens design area Ω of a specified size; at the same time, divide the design area Ω into an actual design area Λ and an area outside Λ; Step 2: According to the lens areas divided in step 1, the weights of astigmatism and optical power of each area are respectively allocated. 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 focal 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 multifocal 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, P0(x, y) is the target power distribution, that is, the element in the matrix P0; according to the linear approximation method, the progressive multifocal lens is set to be a combination of the spherical basis s(x, y) and the disturbance term o(x, y), that is, the actual lens's vector height 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 about the disturbance term o(x, y): In the formula: o 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 radius of curvature R of the spherical basis s , weight distribution matrices α and β corresponding to astigmatism and optical power, target optical power distribution matrix P0, use finite difference method to solve equation (3) to get perturbation distribution o(x, y), then initial vector height 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 free-form surface parameter vector 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) becomes stable, and then the final optimized free-form surface f(x, y) is obtained, completing the design of the lens.

2. The method for designing a progressive addition lens based on free-form surface expression according to claim 1, characterized in that: Step six includes: Step 6.1: Based on the design parameters of the current lens, firstly, a series of discrete points (x, y) are collected in the lens design area Ω, and the corresponding vector height value z=f(θ; x, y) can be obtained by substituting these discrete points into the free surface function to be optimized; 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 of 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: 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 for the back propagation of the free surface parameters.

3. The method for designing a progressive multifocal lens based on free-form surface expression according to claim 2, characterized in that: In the 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-distance activities and the continuous optical power change area between the two; the peripheral astigmatism zone is mainly distributed on both sides of the middle transition zone.

4. The method for designing a progressive addition lens based on free-form surface expression according to claim 3, characterized in that: In the 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.

5. The method for designing a progressive addition lens based on free-form surface expression according to claim 4, characterized in that: The second step comprises: 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 small; 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, all set to the same weight value.

6. The method for designing a progressive addition lens based on free-form surface expression according to claim 5, characterized in that: In the step 2, 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, characterized in that: 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 focal length 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 focal length 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 focal length of other areas is the average of the focal length of the near vision zone and the far vision zone, recorded as 8. The method for designing a progressive addition lens based on free-form surface expression according to claim 7, characterized in that: In the step three, the optical power in the entire lens design area Ω is smoothed using a convolution operation.

9. The method for designing a progressive addition lens based on free-form surface expression according to claim 2, characterized in that: During the iteration 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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