Progressive lens design method based on legendre polynomials for optimizing peripheral swimming
By optimizing the lens design using Legendre polynomials, the problem of peripheral swimming effect in progressive lenses was solved, achieving high-precision modeling and dynamic visual stability of the lens, thus adapting to the needs of different wearing scenarios.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- SUPER VISION OPTICS (SHANGHAI) CO LTD
- Filing Date
- 2026-06-11
- Publication Date
- 2026-07-21
AI Technical Summary
Existing progressive lens design technology has failed to effectively solve the swimming effect in the peripheral area of the lens, causing visual discomfort and dizziness when the wearer turns their head or scans their eyes. This is mainly due to the excessive rate of change of the prism gradient, which makes it difficult for existing methods to precisely control the local prism gradient distribution.
A lens height model is constructed using an m×n order two-dimensional Legendre polynomial. The optimization objective is to minimize the maximum prism gradient. Combined with a zone-differentiated constraint strategy, optical indices for the core visual region and the surrounding region are designed. The Legendre polynomial coefficients are optimized using a quasi-Newton method or a genetic algorithm to ensure that the prism gradient is below 0.06Δ/mm.
It significantly reduces the swimming effect in the peripheral area of the lens, improves initial wear adaptability and dynamic visual stability, and more accurately aligns the optical center of the lens with the wearer's visual axis, adapting to the needs of different wearing scenarios.
Smart Images
Figure CN122431003A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of optical lens design technology, and more specifically to a progressive lens design method based on Legendre polynomial optimization of peripheral motion. Background Technology
[0002] Progressive multifocal lenses provide wearers with continuously clear vision from far to near through continuous surface changes. However, their optical structure inevitably introduces undesirable optical effects in the peripheral area of the lens, among which the "swimming effect" is particularly prominent. The swimming effect manifests as objects in the peripheral field of vision appearing to shake, slide, or distort when the wearer turns their head or scans with their eyes, leading to visual discomfort, dizziness, or even nausea.
[0003] From an optical perspective, the essence of the swimming effect is that the spatial rate of change of prism power (i.e., prism gradient) in the peripheral area of the lens is too large. When the eyeball quickly scans this area, the rapid change of the prism causes the image position of the object on the retina to shift non-uniformly. The brain has difficulty integrating this unstable signal, thus causing perceptual impairment.
[0004] Existing progressive lens design techniques typically focus on ensuring photometric accuracy between the near and far vision zones and striving for a smooth transition in the optical path, or aiming to minimize overall astigmatism. However, these methods do not directly address the root cause of the migratory effect—prism gradients—by modeling and optimizing them. Commonly used low-order polynomials or spline functions lack sufficient modeling accuracy and are difficult to precisely control the local prism gradient distribution. Although some advanced designs can control astigmatism at a low level, the local prism gradient may still exceed the human eye's comfort threshold (e.g., 0.1Δ / mm), and the migratory effect problem remains unresolved. Therefore, there is an urgent need for a design method that can directly quantify and optimize prism gradients to suppress the migratory effect at its source. Summary of the Invention
[0005] The purpose of this invention is to overcome the shortcomings of the prior art and provide a progressive lens design method, system and lens product based on Legendre polynomial optimization of peripheral swimming motion. This method directly characterizes and optimizes the prism gradient through a high-precision mathematical model, and combined with a zoned differential constraint strategy, it significantly reduces the swimming motion effect in the peripheral area while ensuring the optical performance of the core visual area.
[0006] To achieve the above objectives, the present invention adopts the following technical solution:
[0007] A progressive lens design method based on Legendre polynomial optimization of peripheral swimming motion includes the following steps:
[0008] A lens height model is constructed using an m×n order two-dimensional Legendre polynomial, where m and n take values from 7 to 51.
[0009] With minimizing the maximum prism gradient as the core optimization objective, a direct mathematical relationship is established between the surface sag expression and the prism gradient expression;
[0010] The design area is divided into a non-overlapping core visual area and a surrounding area, and different single optical index constraints are set for different areas. The core visual area is only constrained in terms of sphericity, and the surrounding area is only constrained in terms of prism gradient.
[0011] A weighted objective function is constructed, and the Legendre polynomial coefficients are solved by optimizing the solver to obtain a progressive lens surface that satisfies low swimming effect.
[0012] Preferably, the expression formula for the height Z(x,y) is:
[0013]
[0014] This formula consists of polynomial summations, where each term is a vertical k-th order polynomial. With horizontal m-order Multiplication; where the lower-order terms (k≤5, m≤5) determine the overall distribution of spherical light, the middle-order terms (6≤k≤10, 6≤m≤10) control the prism gradient and astigmatism, and the higher-order terms (k>10, m>10) correct fine morphology.
[0015] Preferably, the mathematical expression for the spherical light is:
[0016]
[0017] Substituting into the sag formula and combining it with the normalized coordinate derivative conversion ( , Expanding on this, we get:
[0018]
[0019] in: , The second derivative of the Legendre polynomial reflects the curvature change.
[0020] Preferably, the mathematical expression for the prism gradient is:
[0021]
[0022] The expansion of the square of its modulus (reflecting the intensity of swimming motion) is:
[0023]
[0024] in:
[0025] 0.01 is the unit conversion factor (D / mm→Δ / mm). These are unit vectors in the x and y directions;
[0026] , It is the third derivative, reflecting the rate of change of curvature.
[0027] Preferably, the core visual region includes a far-field region P1 and a near-field region P2, and the surrounding region is a prism gradient constraint region D, wherein regions P1, P2, and D do not overlap; its optical constraint condition is expressed as:
[0028]
[0029]
[0030]
[0031] Where k far For prescription distance photometric values; k near为 Prescription photometric value; p max This is the preset prism gradient squared threshold.
[0032] Preferably, the weighted objective function is:
[0033]
[0034] in:
[0035] w1 and w2 are the overall weights of each item;
[0036] P defines the region containing the photometric target.
[0037] D defines the region that limits the prism gradient.
[0038] Preferably, the optimization solver employs a quasi-Newton method or a genetic algorithm, using the Legendre polynomial coefficient matrix as the optimization variable, and performs iterative solutions based on a pre-constructed basis function library.
[0039] A progressive lens whose optical surface is designed by any of the methods described above, wherein the maximum prism gradient of the transition channel does not exceed 0.06Δ / mm.
[0040] A progressive lens customization system, characterized in that it includes:
[0041] The parameter input module is used to receive lens prescription, size, and material refractive index;
[0042] The modeling and optimization module executes any of the design methods described above and outputs the Legendre polynomial coefficient matrix.
[0043] The lens processing module generates lens surface processing data based on the coefficient matrix.
[0044] A progressive lens fitting method includes the following steps:
[0045] The customer's pupil height and pupil distance were measured and recorded using a zoned table with locked latitude and longitude lines.
[0046] Adjust the optical center position of the lens according to the pupil height and pupillary distance;
[0047] Lenses are designed and manufactured using any of the methods described above;
[0048] We conducted trial wear verification and fine-tuned the optical parameters based on feedback.
[0049] Compared with the prior art, the present invention has the following significant advantages:
[0050] This invention, by directly optimizing the prism gradient, effectively controls the maximum prism gradient in the transition channel and surrounding area to below 0.06Δ / mm, far lower than traditional designs, greatly improving initial wear adaptability and dynamic visual stability. The high-order modeling capability of Legendre polynomials ensures the accuracy of optical parameter description, while its inherent smoothness ensures natural surface transitions without optical abrupt changes. Partition constraints and weighted objective functions allow designers to flexibly adjust the optimization focus according to different prescriptions, frame sizes, and wearing scenarios (such as driving and office work). Combined with precise pupillary height and pupillary distance measurement methods and adjustable mechanisms, the optical center of the lens is more accurately aligned with the wearer's visual axis, further leveraging the design advantages. Attached Figure Description
[0051] To more clearly illustrate the technical solutions of the embodiments of this application, the accompanying drawings used in the embodiments of this application will be briefly introduced below. It should be understood that the following drawings only show some embodiments of this application and should not be regarded as a limitation of the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort.
[0052] Figure 1 This is a schematic diagram of the Legendre polynomial of the order of the present invention;
[0053] Figure 2 This is a schematic diagram showing the distribution of the spherical light confinement region and the prism region in this invention;
[0054] Figure 3 This is a three-dimensional sagittal diagram of the progressive lens of the present invention;
[0055] Figure 4 This is a schematic diagram of the prism gradient on the lens surface of the present invention;
[0056] Figure 5 This is a simulated spherical light distribution diagram of the progressive lens of the present invention. Detailed Implementation
[0057] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of this application, and not all of the embodiments. The components of the embodiments of this application described and shown in the accompanying drawings can generally be arranged and designed in various different configurations. Therefore, the following detailed description of the embodiments of this application provided in the accompanying drawings is not intended to limit the scope of the claimed application, but merely represents selected embodiments of this application. All other embodiments obtained by those skilled in the art based on the embodiments of this application without inventive effort are within the scope of protection of this application.
[0058] This invention constructs a complete and controllable progressive lens design method by parametrically modeling the prism gradient that causes the visual swimming effect and directly processing it using optimization methods, combined with the practical application requirements of domain constraints. This achieves a balance between low swimming control and high-precision modeling computational overhead.
[0059] Precise modeling: An m×n order two-dimensional Legendre polynomial is used to construct a lens sag model. By utilizing its orthogonality in the interval [-1,1] and the continuity of its derivatives, the subtle changes in optical parameters are accurately described. The values of m and n are in the range of [7, 51].
[0060] Direct optimization: Taking minimizing the maximum prism gradient as the core objective, by establishing a direct relationship between the surface sag expression and the prism gradient expression, and based on the parameter optimization solution method, the swimming effect is reduced from the root.
[0061] Differentiated Constraints by Region: Many currently available design methods employ multiple constraints on the entire design area, which can easily lead to over-constraint and result in uneven and unstable optimization results. This method divides the design area into a core region and a peripheral region, setting differentiated optical indicators. That is, the core visual region only meets the prescription accuracy requirements, while the non-core region only considers wearing comfort. The two regions do not overlap, and the optical targets are not repeated. This regional division of labor ensures smooth and stable design results.
[0062] Parameter definition:
[0063] Two-dimensional rectangular coordinate system on the lens surface The origin is the optical center. Lens horizontal / vertical half-width Fits 30 × 30 mm Normalized coordinates ′= {-1,1} ′= Legendre polynomials are orthogonal only in the interval [-1, 1], and normalization ensures the effectiveness of the modeling. Refractive index of lens material The value ranges from 1.49 to 1.74. Legendre polynomial vertical / horizontal order It has an order of N×M and a total of (N+1)×(M+1) coefficients. Legendre polynomial coefficients are the variables to be optimized in the design methodology. k-order Legendre polynomial Satisfying orthogonality and derivative continuity Orthogonality avoids information redundancy, and the continuity of derivatives at all orders ensures that optical parameters do not change abruptly; these are the core guarantees of smoothness.
[0064] Legendre polynomial expression of lens height
[0065] The sag Z(x,y) (unit: mm) is the height of any point on the lens surface relative to the reference plane. It is the fundamental carrier of optical parameters and is expressed as:
[0066]
[0067] This formula consists of polynomial summations, where each term is a vertical k-th order polynomial. With horizontal m-order Multiplication; where the lower-order terms (k≤5, m≤5) determine the overall distribution of spherical light, the middle-order terms (6≤k≤10, 6≤m≤10) control the prism gradient and astigmatism, and the higher-order terms (k>10, m>10) correct fine morphology.
[0068] Mathematical expression of spherical light
[0069] ball light (Unit: D) can be derived from the second partial derivative of the vector height, as shown in the formula:
[0070]
[0071] Substituting into the sag formula and combining it with the normalized coordinate derivative conversion ( , Expanding on this, we get:
[0072]
[0073] in: , The second derivative of the Legendre polynomial reflects the curvature change.
[0074] Mathematical expression for prism gradient
[0075] prism gradient (Unit: Δ / mm) is the spatial rate of change of prism diopter, which is related to the third partial derivative of the sag, and the formula is:
[0076]
[0077] The expansion of the square of its modulus (reflecting the intensity of swimming motion) is:
[0078]
[0079] in:
[0080] 0.01 is the unit conversion factor (D / mm→Δ / mm). These are unit vectors in the x and y directions;
[0081] , The third derivative reflects the rate of change of curvature and is the direct cause of swimming motion.
[0082] In one embodiment, the present invention employs a regional constraint approach, with each region using different optical indices as optimization objectives. This differs from other technical documents that apply multiple optical constraints to the same region, avoiding solvability and smoothness issues caused by over-constraint. Specifically, the region can be divided into a region P (including a far-field region and a near-field region, which can be further subdivided into P1 and P2) that defines the luminance, and a region D that defines the prism gradient. The optical indices for each region can then be expressed as:
[0083]
[0084]
[0085]
[0086] That is, for P1 and P2, only their photometric targets need to be considered; for the surrounding areas, only their prism intensity is required; P1, P2 and D regions do not overlap.
[0087] Where k far For prescription distance photometric values; k near为 Prescription photometric value; p max This is the preset prism gradient squared threshold.
[0088] Preferably, the weighted objective function is:
[0089]
[0090] in:
[0091] w1 and w2 are the overall weights of each item;
[0092] P is the region defining the photometric target (including the far-field region and the near-field region). Figure 2 (Two dashed ellipses in the middle)
[0093] D defines the region that defines the prism gradient (including the nasal and temporal sides). Figure 2 (The two shaded areas on the left and right sides).
[0094] In one embodiment, a weighted objective function is constructed with minimizing astigmatism in the core visual region as the core objective and constraining the maximum prism gradient in the surrounding area as the constraint:
[0095]
[0096] in:
[0097] w1 and w2 are the overall weights of each item;
[0098] P is the region defining the photometric target (including the far-field region and the near-field region). Figure 2 (Two dashed ellipses in the middle)
[0099] D defines the region that defines the prism gradient (including the nasal and temporal sides). Figure 2 (The two shaded areas on the left and right sides).
[0100] Because Legendre polynomials have continuous derivatives, their objective functions and constraints exhibit good smoothness, providing a foundation for efficient solutions. This solver uses the Legendre coefficient matrix as the core variable and can solve the combined objective of minimizing prism gradients and constraining spherical optics using quasi-Newton methods and genetic algorithms. During the solution process, a base function library of Legendre polynomials can be pre-built, and various objectives in the evaluation function (the size of the specified far-field and near-field regions and their corresponding photometric target values) and correction hyperparameters (the weights of the photometric and prism components) can be set until the objective function stabilizes or reaches the upper limit of the number of iterations, outputting a coefficient matrix that meets the requirements of accuracy and efficiency.
[0101] An embodiment of the present invention is described below:
[0102] Parameter settings
[0103] The lens size is 60×60mm, and the material refractive index is n=1.56; the lens prescription is: plano for distance vision, with 2.0D for downlight.
[0104] An 11×11 Legendre polynomial is used to construct the vector height model, and the optimization variable is the coefficient matrix. (k,m=0~20), the initial values are set to random values in the interval [-5, 5].
[0105] Target and constraint configuration
[0106] The core optimization objective is to minimize the gradient of the surrounding prism (weight w1=0.6), while taking into account the spherical optical error in the core region (far-use area and near-use area) (weight w2=0.4). The convergence condition is that the change in the objective function is ≤1e-6 or the maximum number of iterations is 1500.
[0107] The attached diagrams show the remote and near usage zones. The remote usage zone is defined as the area with a radius of 3.5 mm around coordinates (0, 8), while the near usage zone is defined as the area with a radius of 3.5 mm around coordinates (2, -12).
[0108] Solution method
[0109] A genetic algorithm is used for optimization. A basis function library is constructed by pre-computing Legendre polynomials of orders 0-11 and their first to third derivatives. The optimization steps of the genetic algorithm are as follows:
[0110] Initialize a population of 100 individuals, with each individual corresponding to a group. .
[0111] Begin core iteration:
[0112] The optical parameters are calculated using individual coefficients, substituted into the composite objective function, and normalized to obtain the fitness.
[0113] The next generation population is formed by selecting and retaining the top 30% of elite individuals through roulette wheel selection and randomly retaining 10 low-fit individuals. With an 80% probability, single-point crossover is performed on paired individuals in the mid-order coefficient region (6≤k≤10, 6≤m≤10) to generate offspring. Mutation is performed only on higher-order coefficient individuals with a 5% probability, and the mutation magnitude decreases with iteration.
[0114] The iteration stops when the objective function value of the optimal individual changes by ≤1e-6 for 1500 generations.
[0115] The first 3 columns and last 3 rows of the output parameter set Ck,m are as follows:
[0116] k=0 3.794725165 0.724683095 3.639863009 k=1 -0.085648303 0.037628577 -0.017954119 k=2 3.972503573 0.274018904 0.048436473 k=3 -0.039889834 0.063293577 -0.002023774 k=4 0.058933181 -0.072207901 0.012924413 k=5 0.013390578 -0.027137557 0.002440675 k=6 -0.027043511 0.039450794 0.002560989 k=7 -0.008739727 0.013210569 -0.001128331 k=8 0.012177131 -0.020433063 -0.00267779 k=9 0.004334155 -0.005877978 -0.000586982 k=10 -0.004811507 0.00943247 0.003032219
[0117] Optical performance verification
[0118] Simulations were performed on the lens model corresponding to the optimized coefficient matrix, and the results show:
[0119] The average spherical light intensity in the far-field area is 0.02D.
[0120] Near-field downlight: 1.96D
[0121] The maximum gradient of the transition channel prism is 0.054Δ / mm.
[0122] This invention, by directly optimizing the prism gradient, effectively controls the maximum prism gradient in the transition channel and surrounding area to below 0.06Δ / mm, far lower than traditional designs, greatly improving initial wear adaptability and dynamic visual stability. The high-order modeling capability of Legendre polynomials ensures the accuracy of optical parameter description, while its inherent smoothness ensures natural surface transitions without optical abrupt changes. Partition constraints and weighted objective functions allow designers to flexibly adjust the optimization focus according to different prescriptions, frame sizes, and wearing scenarios (such as driving and office work). Combined with precise pupillary height and pupillary distance measurement methods and adjustable mechanisms, the optical center of the lens is more accurately aligned with the wearer's visual axis, further leveraging the design advantages.
[0123] Finally, it should be noted that the above are merely preferred embodiments of the present invention and are not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.
Claims
1. A progressive lens design method based on Legendre polynomial optimization of peripheral swimming motion, characterized in that, Includes the following steps: A lens height model is constructed using an m×n order two-dimensional Legendre polynomial, where m and n take values from 7 to 51. With minimizing the maximum prism gradient as the core optimization objective, a direct mathematical relationship is established between the surface sag expression and the prism gradient expression; The design area is divided into a non-overlapping core visual area and a surrounding area, and different single optical index constraints are set for different areas. The core visual area is only constrained in terms of sphericity, and the surrounding area is only constrained in terms of prism gradient. A weighted objective function is constructed, and the Legendre polynomial coefficients are solved by optimizing the solver to obtain a progressive lens surface that satisfies low swimming effect.
2. The progressive lens design method based on Legendre polynomial optimization of peripheral swimming motion according to claim 1, characterized in that, The formula for the height Z(x,y) is as follows: This formula consists of polynomial summations, where each term is a vertical k-th order polynomial. With horizontal m-order Multiplication; where the lower-order terms (k≤5, m≤5) determine the overall distribution of spherical light, the middle-order terms (6≤k≤10, 6≤m≤10) control the prism gradient and astigmatism, and the higher-order terms (k>10, m>10) correct fine morphology.
3. The progressive lens design method based on Legendre polynomial optimization of peripheral swimming motion according to claim 2, characterized in that, The mathematical expression for the spherical light is: Substituting into the sag formula and combining it with the normalized coordinate derivative conversion ( , Expanding on this, we get: in: , The second derivative of the Legendre polynomial reflects the curvature change.
4. The progressive lens design method based on Legendre polynomial optimization of peripheral swimming motion according to claim 1, characterized in that, The mathematical expression for the prism gradient is: The expansion of the square of its modulus (reflecting the intensity of swimming motion) is: in: 0.01 is the unit conversion factor (D / mm→Δ / mm). These are unit vectors in the x and y directions; , It is the third derivative, reflecting the rate of change of curvature.
5. The progressive lens design method based on Legendre polynomial optimization of peripheral swimming motion according to claim 1, characterized in that, The core visual region includes a far-field region P1 and a near-field region P2. The surrounding region is a prism gradient constraint region D, and regions P1, P2, and D do not overlap. Its optical constraint conditions are expressed as follows: Where k far For prescription distance photometric values; k near为 Prescription photometric value; p max This is the preset prism gradient squared threshold.
6. The progressive lens design method based on Legendre polynomial optimization of peripheral swimming motion according to claim 1, characterized in that, The weighted objective function is: in: w1 and w2 are the overall weights of each item; P defines the region containing the photometric target. D defines the region that limits the prism gradient.
7. The progressive lens design method based on Legendre polynomial optimization of peripheral swimming motion according to claim 1, characterized in that, The optimization solver employs a quasi-Newton method or a genetic algorithm, using the Legendre polynomial coefficient matrix as the optimization variable, and performs iterative solutions based on a pre-built basis function library.
8. A progressive lens, characterized in that, Its optical surface is designed by the method described in any one of claims 1-7, wherein the maximum prism gradient of the transition channel does not exceed 0.06Δ / mm.
9. A progressive lens customization system, characterized in that, include: The parameter input module is used to receive lens prescription, size, and material refractive index; The modeling and optimization module executes the design method according to any one of claims 1-7 and outputs the Legendre polynomial coefficient matrix. The lens processing module generates lens surface processing data based on the coefficient matrix.
10. A method for fitting progressive lenses, characterized in that, Includes the following steps: The customer's pupil height and pupil distance were measured and recorded using a zoned table with locked latitude and longitude lines. Adjust the optical center position of the lens according to the pupil height and pupillary distance; The lens is designed and manufactured using the method described in any one of claims 1-7; We conducted trial wear verification and fine-tuned the optical parameters based on feedback.