CN111856747A - Optimal transmission-based reflector design method - Google Patents
Optimal transmission-based reflector design method Download PDFInfo
- Publication number
- CN111856747A CN111856747A CN202010766414.0A CN202010766414A CN111856747A CN 111856747 A CN111856747 A CN 111856747A CN 202010766414 A CN202010766414 A CN 202010766414A CN 111856747 A CN111856747 A CN 111856747A
- Authority
- CN
- China
- Prior art keywords
- area
- target
- spherical
- cell
- point
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Granted
Links
Images
Classifications
-
- G—PHYSICS
- G02—OPTICS
- G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
- G02B27/00—Optical systems or apparatus not provided for by any of the groups G02B1/00 - G02B26/00, G02B30/00
- G02B27/0012—Optical design, e.g. procedures, algorithms, optimisation routines
-
- G—PHYSICS
- G02—OPTICS
- G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
- G02B5/00—Optical elements other than lenses
- G02B5/08—Mirrors
Landscapes
- Physics & Mathematics (AREA)
- General Physics & Mathematics (AREA)
- Optics & Photonics (AREA)
Abstract
The invention relates to the field of reflector design, and provides a reflector design method based on optimal transmission, which comprises the following steps of 1, taking a vertex on a spherical crown mesh as a target point, enabling each target point to have two quantities of a target area and a weight, and initializing the target area of the target point by using an interpolation method according to a target picture; step 2, establishing a paraboloid equation for each target point according to the weight, and taking the central projection of the intersection line of the paraboloid on the spherical surface as a weighted graph; calculating the area of each cell in the weighted graph, calculating the error between the area of each cell and the target area, calculating the iteration step length according to the error, and iterating to complete the calculation of optimal transmission; and 3, calculating the central envelope formed by the paraboloids by using the obtained weights of the top points of the spherical crown grids, modeling the lens according to the envelope, and printing the model by using a 3D printing technology to obtain the reflecting lens capable of reflecting the target image. The present invention enables the configuration of the lens to be finally obtained.
Description
Technical Field
The invention relates to the field of reflector design, in particular to a reflector design method based on optimal transmission.
Background
The problem of Reflector Design (Reflector Design) is a classical problem in the field of engineering, which explores how to Design the surface structure of a Reflector such that the reflected light of the Reflector constitutes a given image. This problem has proven to be equivalent to solving a single-Ampere problem, i.e., the optimal transmission problem. Although there have been many previous theoretical studies on the mirror design problem, due to the extreme nonlinearity of its numerical solution formula, it has not been converted into a mature algorithm that can actually implement the assumption.
The Optimal Transport (Optimal Transport) theory is well suited for use in solving the mapping problem between two distributions, which was originally proposed by french mathematician Monge in 1791. Given two metrics X, Y and corresponding spatial distributions μ, v, it is desirable to find a transmission transformation T X → Y that transforms the random variables corresponding to distribution-compliant μ into random variables that are distribution-compliant v while minimizing the expectation of the transmission cost c (X, T (X)).
The Brenier theorem (Brenier Theory) gives a solution to optimal transmission, which states that if the domain X is convex, and the cost function is defined as the two-norm of the transmission distance:
c(x,y)＝|x-y|2
then, there is such a convex function
Its gradient mapping is an optimal transmission mapping
Such a function is called the Brenier potential energy function. In the discrete case, the Brenier potential energy function is a piece-wise linear function determined by discrete points in the value domain, and then the piece-wise linear function forms an upper envelope in space, the projection of which on a plane is a Cell Decomposition (Cell Decomposition). By continuously changing the intercept of the linear function, the form of the cellular decomposition is changed correspondingly, and the only group exists, so that the area of each cellular is equal to the target area of the corresponding target point, and an optimal transmission mapping is obtained. Thus, solving the optimal transmission problem is translated intoAn optimization problem is solved.
Weighted Voronoi maps (Power Voronoi Diagram) are referred to simply as weighted maps, which represent cellular decomposition in space, a dual map of a triangular mesh. Each vertex in the triangular mesh is in dual with one cell in the weighted graph, any point in the cell is taken, the weighted distance from the point in the triangular mesh to the cell dual point is smaller than the weighted distance from the point in the mesh to any other vertex, each triangular face in the triangular mesh is in dual with one vertex in the weighted graph, and each edge is in dual with one edge in the weighted graph.
In order to solve the optimal transmission in the reflector design problem, the piecewise linear function in the discrete Brenier theorem is expanded into a piecewise paraboloid parameter equation which is geometrically expressed as a central envelope, and the central projection on the sphere is a sphere weighted graph.
Disclosure of Invention
The invention mainly solves the problem that in the design of the reflecting mirror in the prior art, the existing numerical solving method is difficult to solve, so that the theory cannot be converted into a practical algorithm, and the structure of the lens can be finally obtained.
The invention provides a reflector design method based on optimal transmission, which comprises the following steps:
step 2, establishing a paraboloid equation for each target point according to the weight, and taking the central projection of the intersection line of the paraboloid on the spherical surface as a weighted graph; calculating the area of each cell in the weighted graph, calculating the error between the area of each cell and the target area, calculating an iteration step according to the error, and then iterating to complete the calculation of optimal transmission to obtain a weighted value which enables the area of each cell to be equal to the target area; the method comprises the following substeps:
step 2-1, establishing a paraboloid equation for each target point according to the weight, calculating the intersection point of the paraboloid determined by the vertex of each triangular surface on the spherical crown grid, and taking the central projection of the intersection point on the spherical surface as the dual point of the surface;
step 2-2, calculating a plane where the intersection line of the paraboloids determined by the vertexes of the two ends of each edge on the spherical crown grid is located, wherein the plane is used as a dual surface of the edge; the section of the intersection line of the dual surface and the spherical surface is added with the dual points in the step 2-1 to form a spherical surface weighted graph of the spherical crown grid;
step 2-3, carrying out legalization treatment on the spherical crown grids;
step 2-4, the area of each cell in the weighted graph is calculated.
Step 2-5, determining the updating step length of the weight of each target point according to the difference between the target area of the target point and the actual area of the cell, updating the weight of the target point, completing optimal transmission calculation, and finally iterating to obtain a weight value which enables the cell area to be equal to the target area;
and 3, calculating the central envelope formed by the paraboloids by using the obtained weights of the top points of the spherical cap meshes, modeling the lens according to the envelope, and printing the model by using a 3D printing technology to obtain the reflecting lens capable of reflecting the target image.
Further, in step 2-1, a parabolic equation is established for each target point according to the weight, and the parabolic parameter equation is used as follows:
where x denotes the polar angle, ρ (x) denotes the polar diameter, which is the distance from a point on the paraboloid in the x direction to the focal point, piIs the coordinate of the target point, representing the direction of the optical axis, and is also a unit vector,
is the parabolic opening size;
for each triangular face [ p ] in the spherical cap meshi,pj,pk]And simultaneously establishing a parabolic parameter equation determined by three vertexes:
solving the above equation system to obtain the intersection point of the three paraboloids, and projecting the center of the intersection point on the spherical surface as a cell vertex of the spherical weighted graph.
Further, in step 2-2, the equation for the dual surface is expressed as follows:
the intersection of the dual surface and the spherical surface is used as the cell boundary, and is the side [ p ]i,pj]The dual edge of (2).
Further, in step 2-4, the calculation of the cell area includes:
for a cell [ q ]i,qj,…,qn,qi]It is a spherical non-geodesic polygon with its spherical center
Use of
Combining each boundary of the cell cavity to form a plurality of spherical non-geodetic triangles; the area of the cell is the sum of the areas of the spherical non-geodetic triangles; the area of the non-geodetic triangle is calculated by adding the area of the geodetic triangle contained in the triangle to the area of the remaining non-geodetic part.
Further, in step 2-5, a newton method is used to calculate a target point weight update step length, and a Hessian matrix used in the newton method is constructed in the following manner:
suppose an edge [ p ] in the gridi,pj]With its dual edge [ q ]k,ql]P, then the off-diagonal elements are constructed using the following equation:
wherein d isk,dl,li,ljAre each qk,ql,pi,pjGeodesic distance to the intersection point p.
The diagonal elements are constructed using the following formula:
the invention provides a reflector design method based on optimal transmission, which constructs a curved surface consisting of a large number of paraboloids with different optical axes and opening sizes, wherein the curved surface is a central convex curved surface, and the small paraboloids can reflect light rays emitted from a focus point light source out in parallel along the respective optical axes, so that the light rays in different directions form an image on a far plane. Expanding the piecewise linear function in the Brenier problem to a parabolic function by means of the existing theoretical basis of spherical optimal transmission, and converting the optimal transmission solving in the reflector design problem into gradient mapping for solving the Brenier potential energy function. Then, a parabolic parameter equation is taken as a potential energy function, and a dual graph formed by projection of the parabolas on the spherical surface is an optimal transmission mapping. An optimization solving process suitable for the problem is constructed by taking the weight quantity in the parameter equation of the paraboloid as a unique variable. The method can obtain the reflecting lens structure capable of reflecting the specific image, and can solve the problem that the current single lens can only reflect uniform light and cannot form images.
Drawings
FIG. 1 is a flow chart of an implementation of the optimal transmission based mirror design method provided by the present invention;
FIG. 2 is a schematic diagram of an initialization method for target measurement used in the optimal transmission based mirror design method provided by the present invention;
FIG. 3 is a schematic of a sphere weighting graph used in the optimal transmission based mirror design method provided by the present invention;
FIG. 4 is a schematic diagram of a mesh legalization in the optimal transmission based mirror design method provided by the present invention;
FIG. 5 is a schematic diagram of cell areas of a weighted graph calculated in the optimal transmission-based mirror design method provided by the present invention;
fig. 6 is a schematic diagram of constructing a Hessian array in a newton method in the optimal transmission-based mirror design method provided by the present invention.
Detailed Description
In order to make the technical problems solved, technical solutions adopted and technical effects achieved by the present invention clearer, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It is to be understood that the specific embodiments described herein are merely illustrative of the invention and are not limiting of the invention. It should be further noted that, for the convenience of description, only some but not all of the relevant aspects of the present invention are shown in the drawings.
Fig. 1 is a flow chart of an implementation of the optimal transmission-based mirror design method provided by the present invention. As shown in fig. 1, the present invention provides a mirror design method based on optimal transmission, comprising:
Wherein upon initializing the target measure, the spherical cap mesh and the imaging plane are arranged as shown in FIG. 2 from the center of sphere O to a point p on the spherical capiEmitting a ray and intersecting the picture at the imaging plane at giPoint, then we will piIs set as giPixel value of a dot, giThe pixel values of the points can be found by interpolation, such as bilinear interpolation.
Step 2, establishing a paraboloid equation for each target point according to the weight, and taking the central projection of the intersection line of the paraboloids on the spherical surface as a weighted graph; calculating the area of each cell in the weighted graph, calculating the error between the area of each cell and the target area, calculating an iteration step according to the error, and then iterating to complete the calculation of optimal transmission to obtain a weighted value which enables the area of each cell to be equal to the target area; the method comprises the following substeps:
step 2-1, establishing a paraboloid equation for each target point according to the weight, wherein the used paraboloid parameter equation is as follows:
where x represents the polar angle, is an independent variable, and is a unit vector representing a direction, ρ (x) represents the polar diameter, is the distance from a point on the paraboloid in the x-direction to the focal point, and piIs the coordinate of the target point, representing the direction of the optical axis, and is also a unit vector, hiIs the weight of the target point and,
is the parabolic opening size.
For each triangular face [ p ] in the spherical cap meshi,pj,pk]Firstly, a paraboloid parameter equation determined by three vertexes is combined:
let ρ bei(x)＝ρj(x)＝ρk(x) The intersection point of the three paraboloids can be obtained, and the central projection of the intersection point on the spherical surface is used as a cell vertex of the spherical weighted graph, namely the dual point of the surface.
Step 2-2, aiming at each edge [ p ] in the spherical crown gridi,pj]The equation of the parameters of the paraboloid determined by two end points is combined and deformed, and the equation representation form of the dual surface of the edge can be obtained:
dual surface and spherical surface of the above typeThe intersection of (a) as the cell boundary is the side [ p ]i,pj]The dual edge of (2). The dual edges and the dual points obtained in the step 2-1 form a spherical weighting graph of the spherical cap grid, and fig. 3 is a schematic diagram of the spherical weighting graph;
step 2-3, carrying out legalization treatment on the spherical crown grids;
in this step, the spherical cap mesh needs to be changed to weighted Delaunay to ensure that no problems arise in the next step of calculating the cell area. For one edge, as shown on the left side of FIG. 4, assume that its opposite vertices are p1,p2The mating points of the adjacent surfaces are q1,q2. If the following occurs
This indicates that the edge is illegal. Wherein d (q)2,p1) Representing a vertex p1And a point of parity q2Weighted distance therebetween, the weighted distance being calculated using the formula
This edge is now turned into a legal situation as shown on the right side of fig. 4 using an edge flipping operation.
Step 2-4, calculating the area of each cell in the weighted graph;
in this step, we calculate the cell area using the following method. A cell as shown in fig. 5, which is a spherical non-geodetic polygon with its spherical center
Use of
Combining each boundary of the cell to form a plurality of spherical non-geodetic triangles, and C is a cell boundary point qi,qjThe line of intersection of the dual surface and the spherical surface is G passing qi,qjN is a pole of the sphere. The area of the cell is the sum of the areas of the spherical non-geodetic triangles. The area of the non-geodetic triangle is calculated by adding the area of the geodetic triangle included in the triangle to the area of the remaining non-geodetic part, which is represented as a geodetic triangle in FIG. 5
Plus the area of the shaded portion. Area of the shaded portion, we use to measure the ground line Nqi,NqjAnd non geodesic curve
Area of spherical triangle as boundary (denoted as Q)ijN) Minus geodetic triangle
Is calculated by area of (a), QijNThe area of (A) can be obtained by the following formula:
where d is the intercept of the plane in which C lies.
Step 2-5, determining the updating step length of the weight of each target point according to the difference between the target area of the target point and the actual area of the cell, updating the weight of the target point, completing optimal transmission calculation, and finally iterating to obtain a weight value which enables the cell area to be equal to the target area;
suppose the calculated cell area is { omega }1,ω2,…,ωkThe target area of the target point is { v }1,v2,…,vkGiven a sufficiently small value, then calculate
If it is
Then the iteration is completed and step 3 is performed; otherwise, updating the weight of the target point, and then returning to the step 2-1.
The method for updating the target point weight h is as follows:
newton's method formula is
Where H is the Hessian matrix and λ is an artificially adjustable parameter, typically set to 1, then λ H-1Is the step size of the iteration. The Hessian matrix used in newton's method is constructed as follows.
Suppose an edge [ p ] in the gridi,pj]With its dual edge [ q ]k,ql]P, then the off-diagonal elements are constructed using the following equation
Wherein d isk,dl,li,ljAre each qk,ql,pi,pjGeodesic distance to the intersection point p, fig. 6.
The diagonal elements are constructed using the following formula
And 3, calculating the central envelope formed by the paraboloids by using the obtained weights of the top points of the spherical cap meshes, modeling the lens according to the envelope, and printing the model by using a 3D printing technology to obtain the reflecting lens capable of reflecting the target image.
The center envelope is the Brenier potential energy function, expressed as using the formula
A large number of samples are taken on ρ (p) to obtain a point cloud, from which a model of the lens can be built using mesh generation software.
According to the reflector design method based on optimal transmission, the piecewise linear function in the Brenier problem is expanded to the parabolic function by means of the existing theoretical basis of spherical optimal transmission, and therefore the optimal transmission solving in the reflector design problem is converted into gradient mapping for solving the Brenier potential energy function. Then, the parabolic parameter equation is taken as a potential energy function, and a duality graph formed by projection of the parabolic surfaces on the spherical surface is an optimal transmission mapping. An optimization solving process suitable for the problem is constructed by taking the weight quantity in the parameter equation of the paraboloid as a unique variable. The method can obtain the reflecting lens structure capable of reflecting the specific image, and can solve the problem that the current single lens can only reflect uniform light and cannot form images.
Finally, it should be noted that: the above embodiments are only used to illustrate the technical solution of the present invention, and not to limit the same; while the invention has been described in detail and with reference to the foregoing embodiments, it will be understood by those skilled in the art that: modifications of the technical solutions described in the embodiments or equivalent replacements of some or all technical features may be made without departing from the scope of the technical solutions of the embodiments of the present invention.
Claims (5)
1. A reflector design method based on optimal transmission is characterized by comprising the following steps:
step 1, using a vertex on a spherical crown mesh as a target point, wherein each target point has two quantities of a target area and a weight, the target weight is initialized to 0, and the target area of the target point is initialized by using an interpolation method according to a target picture;
step 2, establishing a paraboloid equation for each target point according to the weight, and taking the central projection of the intersection line of the paraboloid on the spherical surface as a weighted graph; calculating the area of each cell in the weighted graph, calculating the error between the area of each cell and the target area, calculating an iteration step according to the error, and then iterating to complete the calculation of optimal transmission to obtain a weighted value which enables the area of each cell to be equal to the target area; the method comprises the following substeps:
step 2-1, establishing a paraboloid equation for each target point according to the weight, calculating the intersection point of the paraboloid determined by the vertex of each triangular surface on the spherical crown grid, and taking the central projection of the intersection point on the spherical surface as the dual point of the surface;
step 2-2, calculating a plane where the intersection line of the paraboloids determined by the vertexes of the two ends of each edge on the spherical crown grid is located, wherein the plane is used as a dual surface of the edge; the section of the intersection line of the dual surface and the spherical surface is added with the dual points in the step 2-1 to form a spherical surface weighted graph of the spherical crown grid;
step 2-3, carrying out legalization treatment on the spherical crown grids;
step 2-4, calculating the area of each cell in the weighted graph;
step 2-5, determining the updating step length of the weight of each target point according to the difference between the target area of the target point and the actual area of the cell, updating the weight of the target point, completing optimal transmission calculation, and finally iterating to obtain a weight value which enables the cell area to be equal to the target area;
and 3, calculating the central envelope formed by the paraboloids by using the obtained weights of the top points of the spherical cap meshes, modeling the lens according to the envelope, and printing the model by using a 3D printing technology to obtain the reflecting lens capable of reflecting the target image.
2. The optimal transmission-based mirror design method of claim 1, wherein in step 2-1, a parabolic equation is established for each target point according to the weight, and the parabolic parametric equation is used as follows:
where x denotes the polar angle, ρ (x) denotes the polar diameter, which is the distance from a point on the paraboloid in the x direction to the focal point, piIs the coordinate of the target point, representing the direction of the optical axis, and is also a unit vector,
is the parabolic opening size;
for each triangular face [ p ] in the spherical cap meshi,pj,pk]And simultaneously establishing a parabolic parameter equation determined by three vertexes:
solving the above equation system to obtain the intersection point of the three paraboloids, and projecting the center of the intersection point on the spherical surface as a cell vertex of the spherical weighted graph.
3. The optimal transmission based mirror design method according to claim 2, wherein in step 2-2, the equation for the dual surface is expressed as follows:
the intersection of the dual surface and the spherical surface is used as the cell boundary, and is the side [ p ]i,pj]The dual edge of (2).
4. A mirror design method based on optimal transmission according to claim 1 or 2, wherein in step 2-4, the cell area is calculated in a manner comprising:
for a cell [ q ]i,qj,…,qn,qi]It is a spherical non-geodesic polygon with its spherical center
Use of
Each side of the combined cellA boundary forming a plurality of spherical non-geodetic triangles; the area of the cell is the sum of the areas of the spherical non-geodetic triangles; the area of the non-geodetic triangle is calculated by adding the area of the geodetic triangle contained in the triangle to the area of the remaining non-geodetic part.
5. The optimal transmission-based mirror design method according to claim 4, wherein in step 2-5, a Newton method is used to calculate the update step length of the target point weight, and the Hessian matrix used in the Newton method is constructed in a manner that:
suppose an edge [ p ] in the gridi,pj]With its dual edge [ q ]k,ql]P, then the off-diagonal elements are constructed using the following equation:
wherein d isk,dl,li,ljAre each qk,ql,pi,pjGeodesic distance to the intersection point p;
the diagonal elements are constructed using the following formula:
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202010766414.0A CN111856747B (en) | 2020-08-03 | 2020-08-03 | Optimal transmission-based reflector design method |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202010766414.0A CN111856747B (en) | 2020-08-03 | 2020-08-03 | Optimal transmission-based reflector design method |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN111856747A true CN111856747A (en) | 2020-10-30 |
| CN111856747B CN111856747B (en) | 2021-05-14 |
Family
ID=72952762
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN202010766414.0A Active CN111856747B (en) | 2020-08-03 | 2020-08-03 | Optimal transmission-based reflector design method |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN111856747B (en) |
Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN112612029A (en) * | 2020-12-24 | 2021-04-06 | 哈尔滨工业大学芜湖机器人产业技术研究院 | Grid map positioning method fusing NDT and ICP |
| CN113064272A (en) * | 2021-03-04 | 2021-07-02 | 武汉大学 | Optical free-form surface construction method and system under semi-discrete optimal transmission |
| CN113936070A (en) * | 2021-10-14 | 2022-01-14 | 厦门大学 | Two-dimensional point cloud shape regional parallel reconstruction method based on optimal transmission theory |
Citations (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| EP0661559A1 (en) * | 1993-12-31 | 1995-07-05 | AEROSPATIALE Société Nationale Industrielle | Retroreflector for a laser for geodesy, with omnidirectional correction for velocity errors |
| US5682448A (en) * | 1995-01-17 | 1997-10-28 | Remote Source Lighting International | Reflector and illumination system |
| CN101667604A (en) * | 2009-09-30 | 2010-03-10 | 中国科学技术大学 | Design method of solar energy uniform light superposition reflective condenser |
| CN106650180A (en) * | 2017-02-16 | 2017-05-10 | 中国人民解放军电子工程学院 | Optimization design method for infrared directional irradiation device |
| CN106764553A (en) * | 2016-12-21 | 2017-05-31 | 中国科学院工程热物理研究所 | A kind of secondary optics mirror design method that uniform irradiation is measured based on quantum |
-
2020
- 2020-08-03 CN CN202010766414.0A patent/CN111856747B/en active Active
Patent Citations (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| EP0661559A1 (en) * | 1993-12-31 | 1995-07-05 | AEROSPATIALE Société Nationale Industrielle | Retroreflector for a laser for geodesy, with omnidirectional correction for velocity errors |
| US5682448A (en) * | 1995-01-17 | 1997-10-28 | Remote Source Lighting International | Reflector and illumination system |
| CN101667604A (en) * | 2009-09-30 | 2010-03-10 | 中国科学技术大学 | Design method of solar energy uniform light superposition reflective condenser |
| CN106764553A (en) * | 2016-12-21 | 2017-05-31 | 中国科学院工程热物理研究所 | A kind of secondary optics mirror design method that uniform irradiation is measured based on quantum |
| CN106650180A (en) * | 2017-02-16 | 2017-05-10 | 中国人民解放军电子工程学院 | Optimization design method for infrared directional irradiation device |
Cited By (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN112612029A (en) * | 2020-12-24 | 2021-04-06 | 哈尔滨工业大学芜湖机器人产业技术研究院 | Grid map positioning method fusing NDT and ICP |
| CN113064272A (en) * | 2021-03-04 | 2021-07-02 | 武汉大学 | Optical free-form surface construction method and system under semi-discrete optimal transmission |
| CN113064272B (en) * | 2021-03-04 | 2022-05-17 | 武汉大学 | Optical free-form surface construction method and system under semi-discrete optimal transmission |
| CN113936070A (en) * | 2021-10-14 | 2022-01-14 | 厦门大学 | Two-dimensional point cloud shape regional parallel reconstruction method based on optimal transmission theory |
Also Published As
| Publication number | Publication date |
|---|---|
| CN111856747B (en) | 2021-05-14 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| CN111856747B (en) | Optimal transmission-based reflector design method | |
| US10176626B2 (en) | Systems and methods for computer-based visualization, rendering, and representation of surfaces using point clouds | |
| Sheffer et al. | Parameterization of faceted surfaces for meshing using angle-based flattening | |
| CN105336003B (en) | The method for drawing out three-dimensional terrain model with reference to the real-time smoothness of GPU technologies | |
| US7586489B2 (en) | Method of generating surface defined by boundary of three-dimensional point cloud | |
| US10732405B2 (en) | Refractive surface | |
| Liang et al. | A visualization-oriented 3D method for efficient computation of urban solar radiation based on 3D–2D surface mapping | |
| JP2642070B2 (en) | Method and system for generating quadrilateral mesh | |
| CN113781667B (en) | Three-dimensional structure simplified reconstruction method and device, computer equipment and storage medium | |
| WO2024067209A1 (en) | Three-dimensional model unfolding method and apparatus, device, computer-readable storage medium, and computer program product | |
| Livny et al. | A GPU persistent grid mapping for terrain rendering | |
| Yadav et al. | A least-squares method for the design of two-reflector optical systems | |
| Yao et al. | Quadratic error metric mesh simplification algorithm based on discrete curvature | |
| CN103645463B (en) | The method of synthetic aperture radar image-forming data three-dimensional display | |
| Zhou et al. | Improvement of normal estimation for point clouds via simplifying surface fitting | |
| CN114494641B (en) | Three-dimensional model light weight method and device | |
| JP6899915B2 (en) | Shadow casting for elevation data grid | |
| US8022949B2 (en) | System and method for generating curvature adapted isosurface based on delaunay triangulation | |
| CN117274527A (en) | Method for constructing three-dimensional visualization model data set of generator equipment | |
| CN113643423B (en) | Waterbomb derived paper folding structure modeling method with axisymmetric characteristic and paper folding structure | |
| CN109448106A (en) | Fruit point cloud method for reconstructing based on spherical coordinates | |
| CN109448107A (en) | Fruit point cloud method for reconstructing based on spherical coordinates | |
| KR100433947B1 (en) | Exact geometry based triangle mesh generating method | |
| CN103745492B (en) | A kind of two-dimentional shadow generation method based on vector data | |
| CN110414059A (en) | The radiant energy dfensity analogy method of planar heliostats in tower type solar thermo-power station |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant |