Surface optimization method for optical lens or reflector

Abstract

A surface optimization method for an optical lens or reflector. The method comprises: given a unit illumination direction, a target image and a receiving plane, using an original mesh to represent the surface of an optical lens or reflector to be optimized (1); on the basis of the alignment degree at each position, the variation degree of luminous flux, the smoothness of the original mesh and a barrier term, establishing an optimization problem to obtain a deformed mesh (3); recalculating in the deformed mesh the positions of all basic geometric units on the receiving plane, and performing superposition and rendering to obtain an image (4); on the basis of the image difference between a rendered image and the target image, an image gradient difference, a penalty term for exceeding an image boundary, the smoothness of the deformed mesh and the barrier term, establishing an optimization problem to obtain a twice-deformed mesh (5); and refining the mesh level by level until the last iteration at the maximum resolution terminates, so as to obtain a final optimized mesh (6). The method can optimize the surface shape of an optical lens or reflector on the basis of light source information, the position of a receiving plane, and a target image.

Description

Surface optimization method of optical lens or mirror

[0001] The present disclosure claims the priority of the Chinese patent publication No. 202411102015.9, entitled "Surface optimization method of optical lens or mirror", filed on August 12, 2024, and the entire content of which is incorporated herein by reference. TECHNICAL FIELD

[0002] The present disclosure relates to the technical field of optical lens or mirror, and particularly relates to a surface optimization method of optical lens or mirror. BACKGROUND

[0003] When the refractive (or reflective) surface of an optical device is uneven, the refracted (or reflected) light rays will form a light field with uneven brightness, which is called caustics. However, it is very difficult to control the caustics phenomenon to form a specific light field by designing the surface shape of the optical device, because a slight change in the refractive (or reflective) surface can cause a huge change in the refractive (or reflective) direction. This process usually requires a large amount of calculation and uses ray tracing or other techniques to simulate the light rays passing through the lens (or mirror), and the design of the optical lens (or mirror) has important theoretical significance and practical application value in many fields such as optoelectronic detection, optical processing, medicine, architecture, etc.

[0004] The existing design method of caustic lens (or mirror) usually needs to first obtain auxiliary features such as normal field or visibility map, and then reconstruct the surface of the optical device according to the auxiliary features. However, the current method cannot ensure that the generated light field has high accuracy, first of all, these methods do not directly consider the difference between the caustic light field generated by the reconstructed surface and the target light field, so the error will gradually accumulate when calculating the auxiliary features and the reconstructed surface. In addition, many methods use triangular meshes to represent the surface of the optical device, but use the vertex normal of the triangle to refract, which is inconsistent with the use of triangular surface normal to refract in the real physical world, which further increases the error. Finally, these methods only require the simulation result to be close to the target, ignoring the difficulties that may occur in actual processing. SUMMARY

[0005] The purpose of the present disclosure is to provide a surface optimization method of optical lens or mirror, which can control the direction and intensity distribution of the light beam in a specified area, optimize the surface shape of the optical lens or mirror according to the light source information, the position of the receiving plane and the target image, so that the light emitted from the light source can present the target image on the receiving plane after passing through the lens or mirror, and improve the accuracy of the optical device.

[0006] The purpose of the present disclosure is achieved by the following technical solution:

[0007] A method for surface optimization of an optical lens or mirror, the optimization method comprising:

[0008] Step 1, Given a unit illumination direction Target image And a receiving plane p, use an original mesh M0 to represent the surface of the optical lens or mirror to be optimized;

[0009] Step 2, Calculate the position t' of each basic geometric element t in the original mesh M0 in the receiving plane p using the refraction or reflection law, and then calculate the center position of t' The target image Is regarded as a continuous probability distribution μ, and the center position The semi-discrete optimal transport problem between μ is calculated to obtain the optimized position

[0010] Step 3, according to The alignment degree E Of align The change degree E of the basic geometric element t carrying light flux Ф flux The smoothness E of the original mesh M0 smooth And the obstacle term E barr Establish an optimization problem, and optimize to obtain the deformed mesh M1;

[0011] Step 4, recalculate the position t' of all basic geometric elements t in the deformed mesh M1 in the receiving plane p, and superimpose and render to obtain the image g;

[0012] Step 5, according to the image difference E img , image gradient difference E grad , image boundary penalty term E bdr , smoothness E of the deformed mesh M1 smooth And the obstacle term E barr Between the image g obtained in step 4 and the target image Establish an optimization problem, and optimize to obtain the second deformed mesh M2;

[0013] Step 6, replace the original mesh M0 with the second deformed mesh M2, and perform several iterations of M0-M1-M2, and gradually encrypt the mesh from coarse to fine strategy until the last iteration stops at the highest resolution, to obtain the final optimized mesh Represents the surface of the optimized optical lens or mirror.

[0014] According to the technical scheme provided in the present disclosure, the above method can control the light beam direction and light intensity distribution of the specified area, and optimize the surface shape of the optical lens or mirror according to the light source information, the receiving plane position and the target image, so that the light emitted from the light source can present the target image on the receiving plane through the lens or mirror, and the accuracy of the optical device is improved. BRIEF DESCRIPTION OF DRAWINGS

[0015] In order to more clearly illustrate the technical scheme of the embodiments of the present disclosure, the drawings required to be used in the following embodiment description will be briefly introduced. Obviously, the drawings in the following description are only some embodiments of the present disclosure, and other drawings can also be obtained by those skilled in the art without creative labor.

[0016] FIG. 1 is a surface optimization method flowchart of an optical lens or mirror provided by the embodiments of the present disclosure;

[0017] FIG. 2 is a light path schematic diagram of a lens system of the embodiments of the present disclosure;

[0018] FIG. 3 is a light path schematic diagram of a mirror system of the embodiments of the present disclosure. DETAILED DESCRIPTION

[0019] The technical scheme in the embodiments of the present disclosure will be described clearly and completely in combination with the drawings in the embodiments of the present disclosure. Obviously, the described embodiments are only some embodiments of the present disclosure, but not all the embodiments, which do not constitute a limitation to the present disclosure. Based on the embodiments of the present disclosure, all other embodiments obtained by those skilled in the art without creative labor are within the protection scope of the present disclosure.

[0020] FIG. 1 is a surface optimization method flowchart of an optical lens or mirror provided by the embodiments of the present disclosure, and the method comprises:

[0021] Step 1, giving a unit light direction target image and a receiving plane p, using an original grid M0 to represent the surface of the optical lens or mirror to be optimized;

[0022] Step 2, calculating the position t' of each basic geometric unit t in the original grid M0 on the receiving plane p by using the refraction or reflection law, and further calculating the center position of t' target image as a continuous probability distribution μ, calculating the center position and the semi-discrete optimal transport problem between μ, obtaining the optimized position

[0023] In this step, Figure 2 shows a schematic diagram of the optical path of the lens system according to an embodiment of the present disclosure, and Figure 3 shows a schematic diagram of the optical path of the mirror system according to an embodiment of the present disclosure. Figure 2 includes a light source with known light emission characteristics, a lens, and a receiving plane; Figure 3 includes a light source with known light emission characteristics, a mirror, and a receiving plane. The light emitted from the light source is refracted by the lens or reflected by the mirror in sequence, and finally reaches the receiving plane, where a pattern is formed.

[0024] First, the direction of the emitted light is determined using the laws of refraction or reflection, as shown by the following formula.

[0025] Here, η is the refractive index of the lens; if it is a reflecting mirror, then η = 1. It is the unit surface normal of the basic geometric unit t; It refers to the direction of unit illumination;

[0026] The intersection of the outgoing ray and the receiving plane p gives the position t′ of the intersection point. The average value of all vertices in t′ is the center position.

[0027] Central position The semi-discrete optimal transport problem between μ and μ is equivalent to solving its dual problem, i.e., optimizing the energy function:

[0028] here, Represents the energy function; This represents the weight vector to be optimized, which is the variable in the optimization problem; These are the coordinates of the point being integrated; This represents the integral over this position; Indicating the target image Luminous flux (pixel brightness) at the location; n t Represents the number of basic geometric units t; Indicates the center position of the i-th cell; w i Representing vectors The i-th component; Ф i This represents the luminous flux carried by the i-th basic geometric unit; It is the i-th cell in the weighted Vinograph, that is:

[0029] After optimization, with For each cell, calculate the weights. The weighted center is denoted as

[0030] Step 3, according to the alignment degree E of the light flux Φ carried by the basic geometric unit t align the smoothness degree E of the original mesh M0 flux the smoothness degree E of the original mesh M0 smooth and the obstacle term E barr an optimization problem is established, and the deformed mesh M1 is obtained by optimization;

[0031] In this step, the alignment degree E of the light flux Φ carried by the basic geometric unit t align is determined by the following formula:

[0032] n t represents the number of the basic geometric unit t;

[0033] the smoothness degree E of the original mesh M0 flux is determined by the following formula:

[0034] represents the light flux carried by the basic geometric unit t before optimization;

[0035] the obstacle term E barr is composed of the following two parts:

[0036] Here P(t i ) represents the projection of t i to the receiving plane;

[0037] represents the directed area;

[0038] is the first piecewise function, which is used to prevent the triangle from turning over;

[0039] ε2>ε1>0, ε2 and ε1 are two constant thresholds;

[0040] f tir (t i ) is the second piecewise function, which is used to prevent the total reflection phenomenon, f tir (t i ) is determined by the following formula:

[0041] Here η is the refractive index of the lens, and if it is a mirror, then η=1; is the unit surface normal of the i-th basic geometric unit t i ; is the unit light direction;

[0042] Smoothness of the original mesh M0 smooth is composed of three parts:

[0043] E smooth = E face + τ1E edge + τ2E lap

[0044] where E face is the average curvature of the face; E edge is the curvature of the edge; E lap is the Laplacian smoothing term of the vertex; τ1, τ2 are constant parameters.

[0045] Firstly, let the geometric center of the basic geometric element t i be Construct an auxiliary variable which represents its second fundamental form, and the average curvature is represented as H i = (a i + b i ) / 2, a i , b i , c i are introduced optimization variables; define:

[0046] Here A i represents the area of t i ;

[0047] Next, let be a set of orthogonal basis of t i , where are two orthogonal basis functions; for each adjacent face t j , let then the Weingarten matrix should map to Consider:

[0048] δ(t i , t j ) is used to measure the continuity between t i and t j under the second fundamental form M i ;

[0049] For each adjacent edge e ij , define h(e ij ) = δ(t i , t j ) + δ(t j , t i ), and further define E edge as:

[0050] ε I Represents all adjacent edges e ij The set of; here Ψ v It is the Welsch function, defined as:

[0051] v is a user-defined constant parameter, and x is the argument of the Welsch function;

[0052] Finally E lap Determined by the following formula:

[0053] Here P U This represents the projection of the domain U of the height field; Let j represent the set of all internal vertices j; Let represent the set of adjacent points of vertex j;

[0054] The final optimization problem is expressed as:

[0055] min γ1E align +γ2E flux +γ3E smooth +γ4E barr ,

[0056] Where γ1,...,γ4 are user-defined constant parameters.

[0057] Step 4: Recalculate the positions t′ of all basic geometric elements t on the receiving plane p in the deformed mesh M1, and then overlay and render them to obtain the image g;

[0058] In this step, the intersection area of ​​t′ with each pixel grid is calculated, and the luminous flux carried by t′ is distributed into each pixel grid according to the intersection area. Finally, the luminous flux of each pixel grid is superimposed, and then gamma inverse correction is performed to obtain the superimposed rendered image g, represented as:

[0059] The Area(·) function represents area, pixel j G represents the j-th pixel. j Let t' represent the pixel value of the j-th pixel. i Represents the i-th basic geometric unit t i At position p on the receiving plane.

[0060] Step 5: Compare the image g obtained in Step 4 with the target image. Image differences E img Image gradient difference E grad Penalty term E for exceeding image boundariesbdr the smoothness of the deformed mesh M1 smooth and the obstacle term E barr An optimization problem is established to optimize the deformed mesh M2 after secondary deformation;

[0061] In this step, the smoothness of the deformed mesh M1 smooth and the obstacle term E barr is the same as defined in step 3;

[0062] The image difference E img between the rendered image g and the target image is determined by the following formula:

[0063] n p is the number of images; denotes the target pixel value of the jth pixel;

[0064] The image gradient difference E grad is determined by the following formula:

[0065] where G x denotes the horizontal gradient matrix of the rendered image g, G y denotes the vertical gradient matrix of the rendered image g, denotes the horizontal gradient matrix of the target image , denotes the vertical gradient matrix of the target image ; the subscript F denotes the Frobenius norm of the matrix, referred to as the F-norm;

[0066] The image boundary penalty term E bdr is determined by the following formula:

[0067] Here is the number of vertices of t′ i , t′ i is the position of the ith basic geometric element on the receiving plane p; is the kth vertex of the basic geometric element; denotes the point closest to inside the imaging area;

[0068] The final optimization problem is represented as:

[0069] min λ1E img +λ2E grad +λ3E bdr +λ4E smooth +λ5E barr ,

[0070] where λ1,…λ5 are user-defined constant parameters.

[0071] Step 6, replace the original mesh M0 with the second deformed mesh M2, and iterate several times of M0-M1-M2, through the strategy from coarse to fine, gradually encrypt the mesh until the last iteration of the highest resolution stops, and get the final optimized mesh represents the surface of the optimized optical lens or mirror.

[0072] The application introduces the piecewise smoothness constraint E in the geometric optimization process smooth so that the designed surface is easy to process, and the optical device that can accurately reproduce the target image can be manufactured.

[0073] It is worth noting that the contents not described in detail in the embodiments of the present disclosure belong to the prior art known to those skilled in the art.

[0074] In summary, the method described in the embodiments of the present disclosure has the following advantages:

[0075] 1) The present disclosure proposes an accurate differentiable rendering model based on luminous flux, without the need for complex ray tracing sampling;

[0076] 2) The present disclosure directly drives the optimization of the surface of the optical device from the difference between the rendering result and the target image, while introducing a smoothness constraint that is easy to process and manufacture, so that the optical device that can accurately reproduce the target image can be manufactured;

[0077] 3) The present disclosure uses an iterative face-based optimal transport initialization strategy, which helps to avoid local minimum and achieve effective optimization;

[0078] 4) The present disclosure can be used to design artistic devices, liquid crystal backlights, color sorter light sources, medical equipment, lighting lamps, etc., and has important theoretical significance and practical application value in many fields such as optoelectronic detection, optical processing, medicine, architecture, etc.

[0079] In addition, those skilled in the art can understand that all or part of the steps of the methods in the above embodiments can be completed by programs instructing relevant hardware, and the corresponding programs can be stored in a computer readable storage medium. The storage medium mentioned above can be a read-only memory, a magnetic disk or an optical disk, etc.

[0080] The above description is merely that of the preferred specific embodiments of the present disclosure, but the protection scope of the present disclosure is not limited thereto, and any changes or substitutions easily conceived by those skilled in the art within the technical scope disclosed by the present disclosure shall be covered within the protection scope of the present disclosure. Therefore, the protection scope of the present disclosure shall be subject to the protection scope of the claims. The information disclosed in the background section of the present disclosure is merely intended to deepen the understanding of the general background of the present disclosure, and should not be regarded as acknowledging or implying in any form that the information constitutes the prior art known to those skilled in the art.

Claims

1. A method of surface optimization of an optical lens or mirror, wherein, The surface optimization method comprises: Step 1, Given unit light direction Target image and receiving a plane p, using the original mesh M0 to represent the surface of an optical lens or mirror to be optimized; Step 2, calculate the position t' of each basic geometric element t in the original grid M0 on the receiving plane p using the refraction or reflection law, and further calculate the center position of t' target image Consider the continuous probability distribution μ, compute the center position semi-discrete optimal transport problem with μ, resulting in an optimized position Step 3, according to With the degree of alignment E align the degree of variation E of the basic geometric elements t carrying the light flux Φ flux the degree of smoothing E of the original mesh M0 smooth and the obstacle term E barr An optimization problem is established, and the deformed mesh M1 is obtained by optimization. Step 4, re-calculate the positions t' of all basic geometric elements t in the receiving plane p in the deformed mesh M1, and superimpose rendering to obtain an image g; Step 5, the image g obtained according to step 4 is compared with the target image image difference E between img image gradient difference E grad image boundary penalty term E bdr smoothness of the deformed mesh M1 E smooth obstacle term E barr establish an optimization problem, and optimize to obtain a second deformed mesh M2; Step 6, replace the original mesh M0 with the mesh M2 after the second deformation, and perform several iterations of M0-M1-M2 to gradually encrypt the mesh from coarse to fine until the last iteration stops at the highest resolution to obtain the final optimized mesh The surface of the optimized optical lens or mirror is represented.

2. The method of optimizing the surface of an optical lens or mirror according to claim 1, wherein, In step 2, the direction of the outgoing light is first determined from the following equation using the law of refraction or reflection Here η is the refractive index of the lens, and η = 1 if it is a mirror. is the unit face normal of the basic geometric element t; is a unit light direction; The position of the intersection point t', t' is obtained by intersecting the outgoing light rays with the receiving plane p. The average of all vertices in t' is the center position center position The semi-discrete optimal transport problem between μ is equivalent to solving its dual problem, i.e., optimizing the energy function: Here, representing an energy function; denotes the weight vector to be optimized, i.e. the variable in the optimization problem; are position coordinates of the point being integrated; represents integration over this position; representing a target image in light flux of the location; n t representing the number of basic geometric units t; represents the center position of the i-th cell; w i represents the vector the i-th component of the vector; Φ i denotes the luminous flux carried by the i-th elementary geometric unit; is the ith cell in the weighted Venn diagram, i.e. After optimization, with For the weights, compute the ith cell the weighted center of gravity of the set of points, denoted as 3. The method of optimizing the surface of an optical lens or mirror according to claim 1, wherein, In step 3, The degree of alignment E with the alignment E align is determined by the following equation: n t n represents the number of basic geometric units t; The basic geometric unit t carries the degree of change E of the light flux Φ flux is determined by the equation represents the luminous flux carried by the basic geometric element t before optimization; Disorder item E barr consists of two parts: Here P(t i ) denotes the projection of t i onto the receiving plane; represents a directed area; is a first segmented function, used to prevent triangle flipping; ε2>ε1>0, ε2 and ε1 are two constant thresholds; f tir (t i ) is a second piecewise function for preventing total reflection phenomenon, f tir (t i ) is determined by the following equation: Here η is the refractive index of the lens, and η = 1 for a mirror. is the unit face normal of the i-th basic geometric element t i is the unit face normal of the i-th basic geometric element t is a unit light direction; Smoothness E of the original mesh M0 smooth consists of three parts: E smooth = E face + τ1E edge + τ2E lap where E face is the average curvature of the face; E edge is the curvature of the edge; E lap is the Laplacian smoothing term of the vertex; τ1, τ2 are constant parameters; First, a basic geometric unit t is defined i the geometric center of which is Constructing auxiliary variables represents its second fundamental form, mean curvature H i represents H i = (a i + b i ) / 2, a i , b i , c i are introduced optimization variables; defined as: Here A i represents the area of t i ; Next, set is t i a set of orthogonal basis, wherein are two orthogonal basis functions; for each adjacent surface t j , let Then the Weingarten matrix should be mapped to Consider: δ(t i , t j ) is used to measure the second basic type M i Below, t i and t j Continuity between them; For adjacent edges e ij , define h(e ij ) = δ(t i , t j ) + δ(t j , t i ), and further define E edge as: ε I denotes all sets of adjacent edges; here Ψ v is the Welsch function, defined as: v is a constant parameter defined by the user, and x is the independent variable of the Welsch function; Finally E lap is determined by the equation: Here P U denotes the projection of the height field domain U; v I denotes the set of all interior vertices j; represents a set of adjacent points of the vertex j; The finally established optimization problem is represented as: min γ1E align + γ2E flux + γ3E smooth + γ4E barr , wherein γ1, …, γ4 are constant parameters defined by the user.

4. The method of optimizing the surface of an optical lens or mirror according to claim 1, wherein, In step 4, specifically, the intersection area of t' with each pixel bin is calculated, and the luminous flux carried by t' is distributed into each pixel bin according to the intersection area. Finally, all the luminous flux of each pixel bin is superimposed, and gamma inverse correction is performed, i.e., the superimposed rendered image g is obtained, which is expressed as: where the Area(·) function represents an area; pixel j represents the jth pixel grid, g j represents the pixel value of the jth pixel, t' i represents the ith basic geometric unit t i at the position of the receiving plane p.

5. The method of optimizing the surface of an optical lens or mirror according to claim 1, wherein, In step 5, the smoothness E of the deformed mesh M1 smooth and the obstacle item E barr is the same as defined in step 3; Rendered image g with target image the image difference E between img is determined by the equation: n p for the number of images, represents the target pixel value of the jth pixel; Image gradient difference E grad is determined by the following equation: where G x denotes a horizontal gradient matrix of the rendered image g, G y denotes a vertical gradient matrix of the rendered image g, representing a target image a transverse gradient matrix of the lateral gradient matrix, representing a target image is a longitudinal gradient matrix; the subscript F represents the Frobenius norm of the matrix, referred to as F-norm; an image boundary exceeding penalty term E bdr is determined by the following equation: Here is t' i the number of vertices; t' i is the position of the i-th basic geometric element at the receiving plane p; is the kth vertex of the basic geometric unit; represents the distance from the imaging region the nearest point; The finally established optimization problem is represented as: minλ1E img +λ2E grad +λ3E bdr +λ4E smooth +λ5E barr , wherein λ1, …, λ5 are constant parameters defined by the user.

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