CN102306397A - Method for meshing point cloud data - Google Patents

Abstract

The invention discloses a method for meshing point cloud data. In the method, aiming at discrete point cloud data obtained by laser scanning, an automatic global parameterization method with strong robustness is provided, and a parameterization result is used for directly acquiring a meshing result which is consistent with a main direction and can reflect intrinsic geometric characteristics of a model. The meshing result has two forms, i.e., the meshing result can be wholly formed by quadrangles, or formed by triangles. By using the method, only the point cloud data of the model is used, prior triangulation is not needed to perform point cloud data meshing, the processing procedure is completely automatic, point clouds containing noises can be processed without manual intervention, and the density level of quadrangle meshing can be quickly controlled through parameters to obtain quadrilateral meshes or triangular meshes with various resolution ratios.

Description

The method of cloud data gridding

Technical field

The present invention relates to computer graphics and technical field of computer vision, particularly a kind of method of cloud data gridding.

Background technology

Because the development of laser scanner fast and accurately, cloud data is widely used in computer-aided design (CAD) and field of Computer Graphics.Usually original point cloud data does not comprise any topology information, so number of research projects concentrates on how to rebuild surface mesh from cloud data.But most of existing work only pays close attention to how to produce high-quality triangular surface patch grid model, for the shape and the control of direction shortage of triangle surface.Because quadrilateral mesh tensor product characteristic, with respect to triangular mesh, quadrilateral mesh all has advantage, for example B spline-fitting, texture or the like in a lot of fields.Especially the quadrilateral that direction is consistent with principal direction has more advantage when modeling, because they can reflect the symmetry of geometric model.

Global parameterized is the method that a kind of effective solution network of quadrilaterals is formatted.Bommes (BOMMES, D., ZIMMER; H.; AND KOBBELT, L.2009.Mixed-integer quadrangulation.ACM Transactions on Graphics 28,3; 1-10.) the leg-of-mutton network of quadrilaterals problem of formatting is converted into the double optimization problem of a MIXED INTEGER, this method can generate quadrilateral mesh completely and well keep the consistance with principal direction.

But; The applicant recognizes that there is following technological deficiency in prior art: above-mentioned global parameterized method only can be used for triangular mesh; Owing to lack topological connection relation between points, directly utilize the global parameterized method that Point Cloud Processing is had certain degree of difficulty for initial cloud data.

Summary of the invention

The technical matters that (one) will solve

For addressing the aforementioned drawbacks, the invention provides a kind of method of cloud data gridding, need not the network of quadrilaterals that prior triangle gridding carries out cloud data with the cloud data that utilizes model and format.

(2) technical scheme

A kind of method of cloud data gridding is provided according to an aspect of the present invention.The method of this cloud data gridding comprises: steps A, obtain cloud data; Step B is calculated the singular point information of principal direction, normal vector and the cloud data of every bit by cloud data; Step C; Coordinate by the cloud data every bit; The normal vector of this point, principal direction and singular point information; The scalar function parameter value θ that obtains every bit is consistent with the principal direction that

this scalar function satisfies at the gradient of every bit and this point, and wherein the scalar function parameter value at singular point place is confined to round values; Step D sets up the triangular mesh consistent with the cloud data shape by scalar function parameter value θ with

.

Preferably, in the method for cloud data gridding of the present invention, step B comprises: step B1; Initial cloud data is obtained the normal vector of every bit; Step B2 by the curvature tensor of this point of normal vector calculating, obtains this and puts initial principal direction; Step B3 maximizes through the consistent of initial principal direction that makes consecutive point, and the initial principal direction of every bit is carried out smoothly obtaining the principal direction of every bit; Step B4 according to the principal direction of every bit in the cloud data, confirms the singular point in the cloud data.

Preferably, in the method for cloud data gridding of the present invention, step B1 comprises: step B1a, and each the some p for cloud data utilizes the kd of cloud data to set n neighbour's point searching a p; Step B1b; Suppose that these neighbour's points come from same plane T; Put the absolute value of the residual error of fit Plane with these neighbours; Multiply by the long-pending and structure least square problem of weight coefficient again, confirming of power wherein is that inverse of the Euclidean distance of ordering with each point in the cloud data and neighbour is as weights; Step B1c; Utilize least square method to simulate plane T; With the normal vector on this plane as in the cloud this normal vector

preferably; N is an integer, preferably gets 15 or 30.

Preferably, in the method for cloud data gridding of the present invention, step B2 comprises: step B2a, and p sets up local coordinate system to every bit, and the normal vector of the p that sets up an office does Then this p point is exactly the initial point of local coordinate system, and three directions of the portion's coordinate system of setting a trap are respectively

Step B2b puts p for the neighbour of p _i, its normal vector does

The curvature tensor of then putting the p place must satisfy following equational constraint:

$\left[\begin{array}{ccc}{\▿}_{\stackrel{\→}{u}}\stackrel{\→}{N}\·\stackrel{\→}{u}& {\▿}_{\stackrel{\→}{v}}\stackrel{\→}{N}\·\stackrel{\→}{u}& \stackrel{\→}{N}\·\stackrel{\→}{u}\\ {\▿}_{\stackrel{\→}{u}}\stackrel{\→}{N}\·\stackrel{\→}{v}& {\▿}_{\stackrel{\→}{v}}\stackrel{\→}{N}\·\stackrel{\→}{v}& \stackrel{\→}{N}\·\stackrel{\→}{v}\\ {\▿}_{\stackrel{\→}{u}}\stackrel{\→}{N}\·\stackrel{\→}{w}& {\▿}_{\stackrel{\→}{v}}\stackrel{\→}{N}\·\stackrel{\→}{w}& \stackrel{\→}{N}\·\stackrel{\→}{w}\end{array}\right]\·\left[\begin{array}{c}{\mathrm{\Δ}}_{\stackrel{\→}{p}}\·\stackrel{\→}{u}\\ {\mathrm{\Δ}}_{\stackrel{\→}{p}}\·\stackrel{\→}{v}\\ 1\end{array}\right]=\left[\begin{array}{c}\stackrel{\→}{N_i}\·\stackrel{\→}{u}\\ \stackrel{\→}{N_i}\·\stackrel{\→}{v}\\ \stackrel{\→}{N_i}\·\stackrel{\→}{w}\end{array}\right];$

Step B2c; Method direction to m the neighbour of a p ordered is carried out match; In the above-mentioned equation of substitution; Solve left side matrix; Be the curvature tensor at this some place; Initial principal direction

and

that two pairing proper vectors of eigenwert of the maximum of matrix are corresponding some p preferably, m is 15.

Preferably, in the method for cloud data gridding of the present invention, step B3 comprises: step B3a, the function of the principal direction difference of cloud data consecutive point is weighed in definition

$E_{\mathrm{smoothing}}=\underset{e_{\mathrm{ij}}}{\mathrm{\Σ}}{(({\mathrm{\β}}_i-{\mathrm{\β}}_j)+{\mathrm{\α}}_i-{\mathrm{\α}}_j+\mathrm{n\π}/2)}^2,$

Wherein, α _iBe the angle of a desired principal direction and a reference direction, this reference direction is taken as any direction on the section of an i, β _iBe the angle of original principal direction K ' with this reference direction, e _IjBe a limit among the k neighbour figure, n is that integer determines how the principal direction of two consecutive point changes; Step B3b in order to find the solution above-mentioned function, at first eliminates integer variable n through approximate the variation, introduces variable sin4 α _iWith cos4 α _iReplace original variable α _i, equation becomes:

$E_{\mathrm{smoothing}}=\underset{e_{\mathrm{ij}}}{\mathrm{\Σ}}{\{\left(\begin{array}{c}\cos 4{\mathrm{\α}}_i\\ \sin {4\mathrm{\α}}_i\end{array}\right)-\left(\begin{array}{cc}\cos \mathrm{\β}& -\sin \\ \sin \mathrm{\β}& \cos \mathrm{\β}\end{array}\right)\left(\begin{array}{c}\cos {4\mathrm{\α}}_j\\ \sin {4\mathrm{\α}}_j\end{array}\right)\}}^2;$

β=β wherein _i-β _j, to sin4 α _iWith cos4 α _iDifferentiate obtains linear equation, finds the solution to obtain sin4 α _iWith cos4 α _iSeparate, through sin4 α _iWith cos4 α _iFinally obtain original variable α _iStep B3c utilizes angle _i, obtain the principal direction of every bit in the cloud data.

Preferably, in the method for cloud data gridding of the present invention, step B4 comprises: step B4a projects to the neighbour of p point and corresponding principal direction on the section of a p; Step B4b, on the section to these the point according to counterclockwise rank order; Step B4c selects the principal direction of one of them neighbour's point on section direction as a reference, from this reference direction, obtains the angle changing of the principal direction of per two consecutive point according to the order after the above-mentioned ordering; Step B4d, if all changes angle sum is positioned at outside the interval [pi/2, pi/2], then this point is marked as singular point.

Preferably, in the method for cloud data gridding of the present invention, step C comprises: step C1, cloud data is cut, and being converted into deficiency is 0, the border is one topological structure; Step C2; Each bar limit among the k neighbour figure is projected on cut-off rule and the defined plane of normal vector thereof, and cut-off rule is to cut the cut-off rule that is produced among the step C1 here, if limit after the projection and cut-off rule intersect; Then in k neighbour figure, remove this limit, thereby obtain a new k neighbour figure; Step C3; Two scalar function θ of each point obtain θ and

occurrence at each point on the cloud with

for being defined in; Make maximum principal direction and the minimum principal direction of this point be consistent with the gradient of two scalar functions respectively, two scalar function θ are the optimum solution of following equation minimum value with

as far as possible:

Wherein,

and

is respectively two functions every gradient;

and is respectively the maximum principal direction and the minimum principal direction of this point, and ω is the parameter by the controlled variable density degree of user's appointment.

Preferably, in the method for cloud data gridding of the present invention, step C1 comprises: step C1a, and utilize cloud data and Morse function to calculate the homology base of a cloud, along the homology base cloud is cut apart; Step C2b asks for the shortest path of each singular point to the border on k neighbour figure, and according to access path a cloud is cut apart once more, obtains the path that is formed by connecting some points in the cloud.

Preferably, in the method for cloud data gridding of the present invention, step C3 comprises: step C3a, and the energy function of scalar function difference between the maximum principal direction of the gradient of every bit and this point and minimum principal direction is weighed in definition:

formula 1

Wherein, θ _i,

The value of two scalar functions that expression point i is corresponding respectively, K, K ^⊥Maximum principal direction and the minimum principal direction of representing some i place respectively, ij is the limit among the k neighbour figure, e _IjExpression tie point i, the vector of the line segment of j, w are the parameters of the controlled variable density degree of user's appointment; Step C3b, for the some P that is positioned on the cut-off rule, its neighbour's point is divided into two types by cut-off rule, and this point also is divided into two some P accordingly ⁺And P ^-, and corresponding principal direction K ⁺And K ^-In order to guarantee parameter value θ and continuity at the cut-off rule place, the parameter value at some P place must meet the following conditions:

formula 2,

Wherein, (j k) is integer variable,

Expression direction K ^-To K ⁺Needed rotation matrix; And the parameter value at singular point place also is restricted to round values; Step C3c sets up a global parameterized equation continuous at the cut-off rule place through formula 1 and formula 2; Step C3d finds the solution this global parameterized equation, the scalar function parameter value θ that obtains every bit with

Preferably; In the method for cloud data gridding of the present invention; Step D comprises: step D1, for each point and on every side k neighbour point carry out the Di Luoni trigonometric ratio according to parametrization value

;

Step D2 gets the annexation that the triangle adjacent with this point set up the corresponding point in the three dimensions;

Step D3, if there is plural triangle to have same limit, it is total by two triangles at most until each bar limit then progressively to remove unnecessary triangle; Step D4 produces the cavity if remove in the redundant leg-of-mutton process, then fill this hole through the Di Luoni trigonometric ratio is carried out in this hole.Preferably, k=20.

Preferably, in the method for cloud data gridding of the present invention, this method also comprises: step e, on the triangle gridding basis of setting up, set up quadrilateral mesh.Preferably, step e comprises: step e 1, ask for the equivalent line segment in each triangle; Step e 2 is asked for the intersection point between the equivalent line segment; Step e 3 for the intersection point between each equivalent line segment, finds the abutment points of this intersection point and connects according to the annexation between the equivalent line segment, set up shape evenly and direction meet the quadrilateral mesh of principal direction.

(3) beneficial effect

The present invention directly on point cloud model, carries out global parameterized and network of quadrilaterals is formatted; Only utilize the cloud data of model and need not the result that prior triangle gridding can obtain same quality; And processing procedure is automatic fully; Need not manual intervention, and can control the density degree that network of quadrilaterals is formatted fast, obtain the quadrilateral mesh of various resolution through parameter.

Description of drawings

Fig. 1 is the process flow diagram of cloud data gridding method;

Fig. 2 is the process flow diagram that principal direction field calculates in the embodiment of the invention cloud data gridding method;

The process flow diagram that Fig. 3 calculates for singular point in the instance cloud data gridding method of the present invention;

Fig. 4 is a global parameterized process flow diagram in the embodiment of the invention cloud data gridding method;

Fig. 5 is the process flow diagram of formatting of network of quadrilaterals in the embodiment of the invention cloud data gridding method;

Fig. 6 a is that superimposed triangular is handled synoptic diagram in the embodiment of the invention cloud data gridding method;

Fig. 6 b is the result after the superimposed triangular among Fig. 6 a is removed;

Fig. 6 c among Fig. 6 b owing to remove hole that superimposed triangular the produced result after by polishing;

Fig. 7 a is the original point cloud data of teacup model;

Fig. 7 b is the parametrization result of teacup model;

Fig. 7 c is the network of quadrilaterals of the teacup model that obtains according to the parametrization result result that formats;

Fig. 8 a is the network of quadrilaterals that adopts the Bommes method the to obtain result that formats;

Fig. 8 b is format result's the histogram of reflection quadrilateral mesh quality of network of quadrilaterals among Fig. 7 a;

Fig. 8 c is the network of quadrilaterals that adopts the inventive method the to obtain result that formats;

Fig. 8 d is format result's the histogram of reflection quadrilateral mesh quality of network of quadrilaterals among Fig. 7 c;

Fig. 9 a is the original point cloud data of part 1 model;

Fig. 9 b is the parametrization result of part 1 model;

Fig. 9 c is the network of quadrilaterals of part 1 model that obtains according to the parametrization result result that formats;

Fig. 9 d is the original point cloud data of part 2 models;

Fig. 9 e is the parametrization result of part 2 models;

Fig. 9 f is the network of quadrilaterals of part 2 models that obtain according to the parametrization result result that formats.

Figure 10 a shows for the parametrization result's that employing Floater method obtains two dimension;

Figure 10 b is that the parametrization result's that obtains of the inventive method two dimension shows;

Figure 10 c is the triangle gridding reconstructed results of Floater method;

Figure 10 d is the triangle gridding reconstructed results of the inventive method.

Embodiment

For making the object of the invention, technical scheme and advantage clearer, below in conjunction with specific embodiment, and with reference to accompanying drawing, to further explain of the present invention.

In an exemplary embodiment of the present invention, a kind of cloud data gridding method is disclosed.This cloud data gridding comprises: steps A, obtain cloud data; Step B is calculated the singular point information in normal vector, principal direction and the cloud data of every bit by cloud data; Step C; Coordinate by the cloud data every bit; The principal direction of this point; Normal vector and singular point information; The scalar function parameter value θ that obtains every bit is consistent the optimum solution of promptly following equation minimum value with the principal direction that

this scalar function satisfies at the gradient of every bit and this point as far as possible:

Wherein,

and is respectively two functions every gradient;

and

is respectively the maximum principal direction and the minimum principal direction of this point; ω is the parameter by the controlled variable density degree of user's appointment, and wherein the scalar function parameter value at singular point place is confined to round values.Step D; Set up and the consistent triangular mesh of said cloud data shape with

by scalar function parameter value θ, the size of each all interior angle of triangle is approximate in the grid.Preferably, after above-mentioned steps D, can also comprise: step e, on the triangle gridding basis of setting up, set up quadrilateral mesh, each quadrilateral approaches square in the grid.

For step B wherein, specifically comprise: step B1; Initial cloud data is obtained the normal vector of every bit; Step B2 calculates the curvature tensor of this point by normal vector, and then obtains this and put initial principal direction; Step B3 maximizes through the consistent of initial principal direction that makes consecutive point, and the initial principal direction of every bit is carried out smoothly obtaining the principal direction of every bit; Step B4 according to the principal direction of every bit in the cloud data, confirms the singular point in the cloud data.

Wherein, step B1 specifically comprises: step B1a, and each the some p for cloud data utilizes the kd of cloud data to set n neighbour's point searching a p; Step B1b; Suppose that these neighbour's points come from same plane T; Put the absolute value of the residual error of fit Plane with these neighbours; Multiply by the long-pending and structure least square problem of weight coefficient again, confirming of power wherein is that inverse of the Euclidean distance of ordering with each point in the cloud data and neighbour is as weights; Step B1c; Utilize least square method to simulate plane T, with the normal vector on this plane as this normal vector

in the cloud.Preferably, n is an integer, can be taken as 15 or 30.

Wherein, Step B2 specifically comprises: step B2a; P sets up local coordinate system to every bit; Set up an office p normal vector for

then this p point be exactly the initial point of local coordinate system; Three directions of portion's coordinate system of setting a trap are respectively step B2b; A neighbour for p puts pi, and its normal vector is then put the p place for

curvature tensor must satisfy following equational constraint:

$\left[\begin{array}{ccc}{\▿}_{\stackrel{\→}{u}}\stackrel{\→}{N}\·\stackrel{\→}{u}& {\▿}_{\stackrel{\→}{v}}\stackrel{\→}{N}\·\stackrel{\→}{u}& \stackrel{\→}{N}\·\stackrel{\→}{u}\\ {\▿}_{\stackrel{\→}{u}}\stackrel{\→}{N}\·\stackrel{\→}{v}& {\▿}_{\stackrel{\→}{v}}\stackrel{\→}{N}\·\stackrel{\→}{v}& \stackrel{\→}{N}\·\stackrel{\→}{v}\\ {\▿}_{\stackrel{\→}{u}}\stackrel{\→}{N}\·\stackrel{\→}{w}& {\▿}_{\stackrel{\→}{v}}\stackrel{\→}{N}\·\stackrel{\→}{w}& \stackrel{\→}{N}\·\stackrel{\→}{w}\end{array}\right]\·\left[\begin{array}{c}{\mathrm{\Δ}}_{\stackrel{\→}{p}}\·\stackrel{\→}{u}\\ {\mathrm{\Δ}}_{\stackrel{\→}{p}}\·\stackrel{\→}{v}\\ 1\end{array}\right]=\left[\begin{array}{c}\stackrel{\→}{N_i}\·\stackrel{\→}{u}\\ \stackrel{\→}{N_i}\·\stackrel{\→}{v}\\ \stackrel{\→}{N_i}\·\stackrel{\→}{w}\end{array}\right];$

Step B2c; Method direction to m the neighbour of a p ordered is carried out match; In the above-mentioned equation of substitution; Solve left side matrix; Be the curvature tensor at this some place; Initial principal direction

and

that two pairing proper vectors of eigenwert of the maximum of matrix are corresponding some p preferably, m is an integer, can be taken as 15.

Wherein, step B3 specifically comprises: step B3a, and the function of the principal direction difference of cloud data consecutive point is weighed in definition:

$E_{\mathrm{smoothing}}=\underset{e_{\mathrm{ij}}}{\mathrm{\Σ}}{(({\mathrm{\β}}_i-{\mathrm{\β}}_j)+{\mathrm{\α}}_i-{\mathrm{\α}}_j+\mathrm{n\π}/2)}^2,$

Wherein, α _iBe the angle of a desired principal direction and a reference direction, this reference direction can be taken as any direction on the section of an i, β _iBe the angle of original principal direction K ' with this reference direction, e _IjBe a limit among the k neighbour figure, n is that integer determines how the principal direction of two consecutive point changes; Step B3b in order to find the solution above-mentioned function, at first eliminates integer variable n through approximate the variation, introduces variable sin4 α _iWith cos4 α _iReplace original variable α _i, equation becomes

$E_{\mathrm{smoothing}}=\underset{e_{\mathrm{ij}}}{\mathrm{\Σ}}{\{\left(\begin{array}{c}\cos 4{\mathrm{\α}}_i\\ \sin {4\mathrm{\α}}_i\end{array}\right)-\left(\begin{array}{cc}\cos \mathrm{\β}& -\sin \\ \sin \mathrm{\β}& \cos \mathrm{\β}\end{array}\right)\left(\begin{array}{c}\cos {4\mathrm{\α}}_j\\ \sin {4\mathrm{\α}}_j\end{array}\right)\}}^2$

Wherein, β=β _i-β _j, above-mentioned equation is the double optimization problem, to sin 4 α _iWith cos 4 α _iDifferentiate can obtain linear equation, finds the solution to obtain sin4 α _iWith cos4 α _iSeparate, through sin4 α _iWith cos4 α _iCan finally obtain original variable α _iStep B3c, utilizing optimal direction is angle _i, obtain the principal direction of every bit in the cloud data.

Wherein, step B4 specifically comprises: step B4a projects to the neighbour of p point and corresponding principal direction on the section of a p; Step B4b, on the section to these the point according to counterclockwise rank order; Step B4c selects the principal direction of one of them neighbour's point on section direction as a reference, from this reference direction, obtains the angle changing of the principal direction of per two consecutive point according to the order after the above-mentioned ordering; Step B4d, if all changes angle sum is positioned at outside the interval [pi/2, pi/2], then this point is marked as singular point.

For step C wherein, specifically comprise: step C1, cloud data is cut, being converted into deficiency is 0, the border is one topological structure; Step C2; Each bar limit among the k neighbour figure is projected on cut-off rule and the defined plane of normal vector thereof, and cut-off rule is to cut the cut-off rule that is produced among the step C1 here, if limit after the projection and cut-off rule intersect; Then in k neighbour figure, remove this limit, thereby obtain a new k neighbour figure; Step C3; For two scalar function θ that are defined in each point on the cloud obtain with

θ with

at the occurrence of each point, make maximum principal direction and the minimum principal direction of this point be consistent with the gradient of two scalar functions respectively as far as possible.

Wherein, step C1 specifically comprises: step C1a, and utilize cloud data and Morse function to calculate the homology base of a cloud, along the homology base cloud is cut apart; Step C2b asks for the shortest path of each singular point to the border on k neighbour figure, and according to access path a cloud is cut apart once more, obtains the path that is formed by connecting some points in the cloud;

Wherein, step C3 specifically comprises: step C3a, and the energy function of scalar function difference between the maximum principal direction of the gradient of every bit and this point and minimum principal direction is weighed in definition:

Wherein, θ _i,

The value of two scalar functions that expression point i is corresponding respectively, K, K ^⊥Maximum principal direction and the minimum principal direction of representing some i place respectively, ij is the limit among the k neighbour figure, e _IjExpression tie point i, the vector of the line segment of j, w are the parameters of the controlled variable density degree of user's appointment;

Step C3b, for the some P that is positioned on the cut-off rule, its neighbour's point is divided into two types by cut-off rule, and this point also is divided into two some P accordingly ⁺And P ^-, and corresponding principal direction K ⁺And K ^-In order to guarantee parameter value θ and

continuity at the cut-off rule place, the parameter value at some P place must meet the following conditions:

Wherein, (j k) is integer variable, Expression direction K ^-To K ⁺Needed rotation matrix; In addition, the parameter value at singular point place also is restricted to round values; Step C3c can set up a global parameterized equation continuous at the cut-off rule place through formula (2) and formula (3); Step C3d finds the solution this global parameterized equation, the scalar function parameter value θ that obtains every bit with

For step D wherein, specifically comprise: step D1, for each point and on every side k neighbour's point (k is an integer, can get k=20) according to the parametrization value

Carry out the Di Luoni trigonometric ratio; Step D2 gets the annexation that the triangle adjacent with this point set up the corresponding point in the three dimensions; Step D3, if there is plural triangle to have same limit, it is total by two triangles at most until each bar limit then progressively to remove unnecessary triangle; Step D4 produces the cavity if remove in the redundant leg-of-mutton process, then fill this hole through the Di Luoni trigonometric ratio is carried out in this hole; Step D5 obtains quadrilateral mesh by triangular mesh.

Wherein, step D5 specifically comprises: step D5a, ask for the equivalent line segment in each triangle; Step D5b asks for the intersection point between the equivalent line segment; Step D5c for the intersection point between each equivalent line segment, finds the abutment points of this intersection point and connects according to the annexation between the equivalent line segment, set up shape evenly and direction meet the quadrilateral mesh of principal direction.

Below will on the basis of the foregoing description, provide optimum embodiment of the present invention.Need explain that this optimum embodiment only is used to understand the present invention, is not limited to protection scope of the present invention.And the characteristic among the optimum embodiment is not having under the situation about indicating especially, all is applicable to method embodiment and device embodiment simultaneously, and the technical characterictic that in identical or different embodiment, occurs can make up use under not conflicting situation.

Fig. 1 is the process flow diagram of cloud data gridding method.As shown in Figure 1, the cloud data gridding comprises following three basic steps: the calculating of principal direction field promptly calculates principal direction to each point in the cloud; Global parameterized; Network of quadrilaterals is formatted.To specifying below the specific algorithm of each step.

Fig. 2 is the process flow diagram that principal direction field calculates in the embodiment of the invention cloud data gridding method.As shown in Figure 2, the calculating of principal direction field at first need be asked for a normal vector of cloud every bit.Because the coordinate information that three dimensional point cloud is generally only had a few.For asking for the curvature tensor information of a cloud, and a cloud carry out local triangleization, the method direction that obtains each point in the cloud data is necessary.At first, set up the kd tree.In computational geometry, kd tree is one of data structure the most efficiently of the certified neighbour of searching.The kd tree is divided three dimensions based on the spatial positional information of point through the dichotomy iteration, realizes optimal storage.On the kd tree, the time complexity that carries out the k neighbor searching is O (log2n), and n is the number of the point of cloud data here.

For asking for the normal vector of each point; For each some p of cloud data, utilize the kd tree of cloud data to search n neighbour's point (n is an integer, can be taken as 15 or 30); Suppose that these neighbour's points come from same plane T; So can use these neighbours to put the absolute value of the residual error of fit Plane, multiply by the long-pending and structure least square problem of weight coefficient again, confirming of power wherein is that inverse of the Euclidean distance of ordering with each point in the cloud data and neighbour is as weights.Utilize least square method to simulate plane T, with the normal vector on this plane as this normal vector

in the cloud

Because the global parameterized of present embodiment method is the principal direction constraint that receives cloud data, therefore obtain smoothly and accurately principal direction field be necessary.The foundation of principal direction field comprises two steps, and shown in two steps behind Fig. 2, the overall situation of the curvature tensor of promptly calculating each point and principal direction field smoothly.

For calculating the curvature tensor of each point, at first every bit is set up local coordinate system, the normal vector of the p that sets up an office does

Then this p point is exactly the initial point of local coordinate system, and three directions of the portion's coordinate system of setting a trap are respectively

A neighbour for p puts p _i, above-mentioned steps earning approach vector does The curvature tensor of then putting the p place must satisfy following equational constraint:

$\left[\begin{array}{ccc}{\▿}_{\stackrel{\→}{u}}\stackrel{\→}{N}\·\stackrel{\→}{u}& {\▿}_{\stackrel{\→}{v}}\stackrel{\→}{N}\·\stackrel{\→}{u}& \stackrel{\→}{N}\·\stackrel{\→}{u}\\ {\▿}_{\stackrel{\→}{u}}\stackrel{\→}{N}\·\stackrel{\→}{v}& {\▿}_{\stackrel{\→}{v}}\stackrel{\→}{N}\·\stackrel{\→}{v}& \stackrel{\→}{N}\·\stackrel{\→}{v}\\ {\▿}_{\stackrel{\→}{u}}\stackrel{\→}{N}\·\stackrel{\→}{w}& {\▿}_{\stackrel{\→}{v}}\stackrel{\→}{N}\·\stackrel{\→}{w}& \stackrel{\→}{N}\·\stackrel{\→}{w}\end{array}\right]\·\left[\begin{array}{c}{\mathrm{\Δ}}_{\stackrel{\→}{p}}\·\stackrel{\→}{u}\\ {\mathrm{\Δ}}_{\stackrel{\→}{p}}\·\stackrel{\→}{v}\\ 1\end{array}\right]=\left[\begin{array}{c}\stackrel{\→}{N_i}\·\stackrel{\→}{u}\\ \stackrel{\→}{N_i}\·\stackrel{\→}{v}\\ \stackrel{\→}{N_i}\·\stackrel{\→}{w}\end{array}\right]$

We carry out match at method direction that several of p (15) neighbour is ordered in the method, in the above-mentioned equation of substitution, solve the tensor item in the matrix of the left side, can obtain the initial principal direction K ' of corresponding some p.Principal direction plays effect of contraction in the global parameterized process.

The overall situation of principal direction field is level and smooth.Smoothing process makes the principal direction of consecutive point consistent as far as possible.This method has defined a function of weighing principal direction difference between the cloud data consecutive point, through finding the solution this minimum of a function value, can obtain the principal direction after level and smooth.

The definition of this function is following:

$E_{\mathrm{smoothing}}=\underset{e_{\mathrm{ij}}}{\mathrm{\Σ}}{(({\mathrm{\β}}_i-{\mathrm{\β}}_j)+{\mathrm{\α}}_i-{\mathrm{\α}}_j+\mathrm{n\π}/2)}^2$

Wherein, α _iBe the angle of a desired principal direction and a reference direction, this reference direction can be taken as any direction on the section of an i, β _iBe the angle of original principal direction K ' with this reference direction, e _IjBe a limit among the k neighbour figure.This equation can be converted into a double optimization problem, uses method of steepest descent to solve optimization problem, and the principal direction after optimum solution is promptly level and smooth and the angle of reference direction utilize the principal direction K after this angle can be obtained smoothly.

Obtaining principal direction field needs to confirm the position of singular point afterwards.Fig. 3 is the process flow diagram that singular point calculates.The neighbour point that at first will put p projects on the section of a p with corresponding principal direction, then on the section to these according to counterclockwise rank order.Select the principal direction of one of them neighbour's point on section direction as a reference; From this reference direction; Obtain the angle changing of the principal direction of per two consecutive point according to the order after the above-mentioned ordering; This point is marked as singular point if all changes angle sum is positioned at outside the interval [pi/2, pi/2] then.In the process of global parameterized, the parametrization value at singular point place is confined to round values.

For the some cloud to the arbitrary topology shape carries out global parameterized, need cut a cloud, being converted into deficiency is 0, the border is one topological structure.At first utilize Morse function to calculate the homology base of a cloud, a cloud is cut apart, on k neighbour figure, ask for the shortest path of each singular point then, and a cloud is cut apart once more according to access path to the border along the homology base.The input of this step be the original point cloud, the output data path that some points are formed by connecting in the cloud of serving as reasons need scheme upgrade to the k neighbour before global parameterized according to this path.

Fig. 4 is a global parameterized process flow diagram in the embodiment of the invention cloud data gridding method.The purpose of global parameterized be for two scalar function θ that are defined in each point on the cloud obtain with

θ with

at the occurrence of each point, make maximum principal direction and the minimum principal direction of this point be consistent with the gradient of two scalar functions respectively as far as possible.The input data are the coordinate of some cloud every bit; The principal direction of this point; Normal vector and k neighbour figure, output data be θ with

occurrence at each point.For the energy function of calculation level cloud, this method at first defines a kind of energy function and weighs this species diversity, defines as follows:

In actual asking for, above-mentioned equation disperses and turns to following form:

Wherein, θ _i,

The value of two scalar functions that expression point i is corresponding respectively, K, K ^⊥Maximum principal direction and the minimum principal direction of representing some i place respectively, ij is the limit among the k neighbour figure, e _IjExpression tie point i, the vector of the line segment of j, w are the parameters of the controlled variable density degree of user's appointment.

Because a cloud is cut apart, therefore must remove the limit of intersecting with cut-off rule among the k neighbour figure in the gradient of the scalar function on the defining point cloud, thereby keep the continuity of global parameterized.For each point near cut-off rule; Whether detect the limit that links to each other with this point among the k neighbour figure intersects with cut-off rule; Being about to this limit projects on cut-off rule and the defined plane of normal vector thereof; If limit after the projection and cut-off rule intersect, then in k neighbour figure, remove this limit, thereby obtain a new k neighbour figure.

The annexation that the k neighbour who utilizes said method to obtain schemes is set up energy function according to formula (2).Global parameterized then is converted into the optimum solution of asking for energy function.In order to guarantee that the global parameterized result cutting apart the continuity at place, in parameterized process, need to add extra constraint condition and guarantee that parameter value is consistent when crossing over the cut-off rule both sides.For the some P that is positioned on the cut-off rule, its neighbour's point is divided into two types by cut-off rule, and this point also is divided into two some P accordingly ⁺And P ^-, and corresponding principal direction K ⁺And K ^-In order to guarantee parameter value θ and continuity at the cut-off rule place, the parameter value at some P place must meet the following conditions:

Wherein, (j k) is integer variable,

Expression direction K ^-To K ⁺Needed rotation matrix.

In addition, the parameter value at singular point place also is restricted to round values.Can set up a global parameterized equation continuous at the cut-off rule place through formula (2) and formula (3), the optimization of this equation is the optimization problem of a MIXED INTEGER, utilizes document (BOMMES; D., ZIMMER, H.; AND KOBBELT; L.2009.Mixed-integer quadrangulation.ACM Transactions on Graphics 28,3, and the solver that provides in 1-10.) can rapid and precise be found the solution.This solver at first utilizes the nuisance variable in the Gaussian elimination cancellation formula (3); Find the solution linear equation then; For be defined as integer-valued variable get with this variable solving result near round values as net result, then this variable is regarded as constant substitution full scale equation and continues to find the solution other integer variables.

Fig. 5 is the process flow diagram of formatting of network of quadrilaterals in the embodiment of the invention cloud data gridding method.The parametrization result of some cloud can be used for setting up triangular mesh and quadrilateral mesh.In order to set up triangular mesh; For each point and on every side k neighbour's point (k=20) carry out the Di Luoni trigonometric ratio according to parametrization value

, and get the annexation that the triangle adjacent with this point set up the corresponding point in the three dimensions.Near cut-off rule, there is overlapping triangle to exist, then removes unnecessary triangle and keep the stream shape characteristic of triangle gridding through following steps:

If there is plural triangle to have same limit, it is total by two triangles at most until each bar limit then progressively to remove unnecessary triangle;

Produce the cavity in the redundant leg-of-mutton process if remove, then fill this hole through the Di Luoni trigonometric ratio is carried out in this hole.Fig. 6 is that superimposed triangular is handled synoptic diagram in the embodiment of the invention cloud data gridding method.What this step was imported is that an original triangle gridding result is shown in Fig. 6 a; Fig. 6 b is for removing the result after the superimposed triangular to the grid among Fig. 6 a; Because removing of superimposed triangular can produce the cavity, Fig. 6 c is for carrying out the triangle grid data behind the polishing to the cavity that is produced.

In order to obtain the final network of quadrilaterals result that formats; This method is at first asked for the equivalent line segment in each triangle; The isoline network has constituted the basic network of quadrilaterals basis of formatting; The intersection point of isoline is the summit of quadrilateral mesh, and the annexation between the summit is then by the decision of the annexation between the isoline, for the intersection point between each isoline; Find the abutment points of this intersection point and connect according to the annexation between the equivalent line segment, can set up shape evenly and direction meet the quadrilateral mesh of principal direction.Each quadrangle form in the grid approaches square.

Realized method described in the invention with C Plus Plus, and on several different data sets, tested.All experiments all are at one

Core ^TM2Quad CPU Q6600 accomplishes on the PC of 2.40GHz 4G internal memory, and the OpenGL graph function storehouse of standard has been used in the display part.

Table 1 provides part and tests the complexity of used model (number of point) and two kinds of disposal route (MIQP and TightCocone+MIQ) institute's time spent contrasts.Wherein MIQP representes directly to use at cloud data the time of this method, and TightCocone+MIQuelon representes to adopt and traditional at first cloud data carried out mesh reconstruction and then grid is used document Bommes (BOMMES, D.; ZIMMER, H., AND KOBBELT; L.2009.Mixed-integer quadrangulation.ACM Transactions on Graphics 28; 3, the method in 1-10.) is carried out network of quadrilaterals and is formatted the needed time, and employed mesh reconstruction method is TightCocone (T.K.Dey in this experiment; S.Goswami; Tight cocone:a water-tight surface reconstructor, in:Proceedings of the eighth ACM symposium on Solid modeling and applications, pp.127-134.)

Table 1

Accompanying drawing 7 has provided the result who on laser single sweep operation data teacup model, obtains.Fig. 7 (a) is an original point cloud data, and Fig. 7 a is the original point cloud data on the teacup model, and Fig. 7 b has shown the parametrization result, and Fig. 7 c utilizes the parametrization result to carry out the result that network of quadrilaterals is formatted.Picture shows that quadrilateral is evenly distributed, and trend meets the geometric properties of object.This result all carry out on point cloud model by processing, for conveniently being presented at display result on the grid surface.

Accompanying drawing 8 has been listed in the contrast that utilizes on the model of part between the result who obtains on result who obtains on our point cloud model of method at part and the grid model of method at part that utilizes Bommes.Fig. 8 a and Fig. 8 b are the format histogram of result and reflection quadrilateral mesh quality of network of quadrilaterals that the method for Bommes obtains; Fig. 8 c and Fig. 8 d are the format histogram of result and reflection quadrilateral mesh quality of network of quadrilaterals that this method obtains; As can be seen from the figure for identical model, this method is only utilized the cloud data of model and need not the result that prior triangle gridding can obtain same quality.

Accompanying drawing 9 has been listed in the result of complicated mechanical part model more.Make that owing to noise exists with some cloud situation pockety processing is more difficult, this method still can provide good result.Wherein Fig. 9 a is the original point cloud data of part 1 model; Fig. 9 b is the parametrization result of part 1 model; Fig. 9 c is the network of quadrilaterals of part 1 model that obtains according to the parametrization result result that formats; Fig. 9 d is the original point cloud data of part 2 models, and Fig. 9 e is the parametrization result of part 2 models, and Fig. 9 f is the network of quadrilaterals of part 2 models that obtain according to the parametrization result result that formats.

Accompanying drawing 10 has been listed the triangular grid result and the Floater (M.S.Floater of this method; M.Reimers; Meshless parameterization and surface reconstruction, Comput.Aided Geom.Des.18 (2001) 77-92.) method contrast.Figure 10 a and Figure 10 b are respectively the method for Floater and the parametrization result of this method.The border, method parameter territory of Floater is fixed as circle, so torsion resistance is bigger, and this method need not fixed boundary, thereby can access the littler parametrization result of distortion.Note, be out of shape littler the overlapping of field of definition subregion that also show as, see Figure 10 b, but do not influence parameterized end product at all.Figure 10 c and Figure 10 d are respectively the method for Floater and the contrast of triangle gridding reconstructed results that the inventive method obtains.Can find out that this method can access better triangle gridding result, triangle gridding is evenly distributed, and long and narrow triangle is less.And a large amount of long and narrow triangles has appearred in the method for Floater.

The present invention directly on point cloud model, carries out global parameterized and network of quadrilaterals is formatted.And this method is automatic fully, need not manual intervention, and can control the density degree that network of quadrilaterals is formatted fast through parameter, obtains the quadrilateral mesh of various resolution.The present invention can be used for each application of computer graphics, has higher actual application value.

Above-described specific embodiment; The object of the invention, technical scheme and beneficial effect have been carried out further explain, and institute it should be understood that the above is merely specific embodiment of the present invention; Be not limited to the present invention; All within spirit of the present invention and principle, any modification of being made, be equal to replacement, improvement etc., all should be included within protection scope of the present invention.

Claims (14)

1. the method for a cloud data gridding is characterized in that, the method for this cloud data gridding comprises:

Steps A is obtained cloud data;

Step B is confirmed the singular point information of normal vector, principal direction and the cloud data of every bit by said cloud data;

Step C, by the coordinate of cloud data every bit, the principal direction of this point, normal vector and singular point information are carried out the parametrization of a cloud;

Step D by a cloud parametrization result, sets up and the consistent triangular mesh of said cloud data shape.

2. the method for cloud data gridding according to claim 1 is characterized in that, said step B comprises:

Step B1; Said initial cloud data is obtained the normal vector of every bit;

Step B2 by the curvature tensor of said this point of normal vector calculating, obtains this and puts initial principal direction;

Step B3 maximizes through the consistent of initial principal direction that makes consecutive point, and the initial principal direction of every bit is carried out smoothly obtaining the principal direction of every bit;

Step B4 according to the principal direction of every bit in the cloud data, confirms the singular point in the cloud data.

3. the method for cloud data gridding according to claim 2 is characterized in that, said step B1 comprises:

Step B1a, each the some p for cloud data utilizes the kd of cloud data to set n neighbour's point searching said some p, and wherein n is an integer;

Step B1b; Suppose that these neighbour's points come from same plane T; Put the absolute value of the residual error of fit Plane with these neighbours; Multiply by the long-pending and structure least square problem of weight coefficient again, confirming of power wherein is that inverse of the Euclidean distance of ordering with each point in the cloud data and neighbour is as weights;

Step B1c; Utilize least square method to simulate plane T, with the normal vector on this plane as this normal vector

in the cloud

4. the method for cloud data gridding according to claim 3 is characterized in that, said n=15 or 30.

5. cloud data gridding method according to claim 2 is characterized in that, said step B2 comprises:

Step B2a, p sets up local coordinate system to every bit, and the normal vector of the p that sets up an office does

Then this p point is exactly the initial point of local coordinate system, and three directions of the portion's coordinate system of setting a trap are respectively

Step B2b puts p for the neighbour of p _i, its normal vector does

The curvature tensor of then putting the p place must satisfy following equational constraint:

$\left[\begin{array}{ccc}{\▿}_{\stackrel{\→}{u}}\stackrel{\→}{N}\·\stackrel{\→}{u}& {\▿}_{\stackrel{\→}{v}}\stackrel{\→}{N}\·\stackrel{\→}{u}& \stackrel{\→}{N}\·\stackrel{\→}{u}\\ {\▿}_{\stackrel{\→}{u}}\stackrel{\→}{N}\·\stackrel{\→}{v}& {\▿}_{\stackrel{\→}{v}}\stackrel{\→}{N}\·\stackrel{\→}{v}& \stackrel{\→}{N}\·\stackrel{\→}{v}\\ {\▿}_{\stackrel{\→}{u}}\stackrel{\→}{N}\·\stackrel{\→}{w}& {\▿}_{\stackrel{\→}{v}}\stackrel{\→}{N}\·\stackrel{\→}{w}& \stackrel{\→}{N}\·\stackrel{\→}{w}\end{array}\right]\·\left[\begin{array}{c}{\mathrm{\Δ}}_{\stackrel{\→}{p}}\·\stackrel{\→}{u}\\ {\mathrm{\Δ}}_{\stackrel{\→}{p}}\·\stackrel{\→}{v}\\ 1\end{array}\right]=\left[\begin{array}{c}\stackrel{\→}{N_i}\·\stackrel{\→}{u}\\ \stackrel{\→}{N_i}\·\stackrel{\→}{v}\\ \stackrel{\→}{N_i}\·\stackrel{\→}{w}\end{array}\right];$

Step B2c; Method direction to m the neighbour of a p ordered is carried out match; In the above-mentioned equation of substitution; Solve left side matrix; Be the curvature tensor at this some place, two pairing proper vectors of eigenwert of the maximum of matrix are initial principal direction

and

of corresponding some p

6. the method for cloud data gridding according to claim 5 is characterized in that, said m is 15.

7. the method for cloud data gridding according to claim 2 is characterized in that, said step B3 comprises:

Step B3a, the function of the principal direction difference of cloud data consecutive point is weighed in definition

$E_{\mathrm{smoothing}}=\underset{e_{\mathrm{ij}}}{\mathrm{\Σ}}{(({\mathrm{\β}}_i-{\mathrm{\β}}_j)+{\mathrm{\α}}_i-{\mathrm{\α}}_j+\mathrm{n\π}/2)}^2,$

Wherein, α _iBe the angle of a desired principal direction and a reference direction, this reference direction is taken as any direction on the section of an i, β _iBe the angle of original principal direction K ' with this reference direction, e _IjBe a limit among the k neighbour figure, n is that integer determines how the principal direction of two consecutive point changes;

Step B3b in order to find the solution above-mentioned function, at first eliminates integer variable n through approximate the variation, introduces variable sin 4 α _iWith co54 α _iReplace original variable α _i, equation becomes:

$E_{\mathrm{smoothing}}=\underset{e_{\mathrm{ij}}}{\mathrm{\Σ}}{\{\left(\begin{array}{c}\cos 4{\mathrm{\α}}_i\\ \sin {4\mathrm{\α}}_i\end{array}\right)-\left(\begin{array}{cc}\cos \mathrm{\β}& -\sin \\ \sin \mathrm{\β}& \cos \mathrm{\β}\end{array}\right)\left(\begin{array}{c}\cos {4\mathrm{\α}}_j\\ \sin {4\mathrm{\α}}_j\end{array}\right)\}}^2;$

β=β wherein _i-β _j, to sin4 α _iWith co54 α _iDifferentiate obtains linear equation, finds the solution to obtain sin 4 α _iWith co54 α _iSeparate, through sin4 α _iWith co54 α _iFinally obtain original variable α _i

Step B3c utilizes said angle _i, obtain the principal direction of every bit in the cloud data.

8. the method for cloud data gridding according to claim 2 is characterized in that, said step B4 comprises:

Step B4a projects to the neighbour of p point and corresponding principal direction on the section of a p;

Step B4b, on the section to these the point according to counterclockwise rank order;

Step B4c selects the principal direction of one of them neighbour's point on section direction as a reference, from this reference direction, obtains the angle changing of the principal direction of per two consecutive point according to the order after the above-mentioned ordering;

Step B4d, if all changes angle sum is positioned at outside the interval [pi/2, pi/2], then this point is marked as singular point.

9. the method for cloud data gridding according to claim 1 is characterized in that, said step C comprises:

Step C1 cuts said cloud data, and being converted into deficiency is 0, and the border is one topological structure;

Step C2; Each bar limit among the k neighbour figure is projected on cut-off rule and the defined plane of normal vector thereof; Cut-off rule described herein is to cut the cut-off rule that is produced among the step C1; If limit after the projection and cut-off rule intersect, then in k neighbour figure, remove this limit, thereby obtain a new k neighbour figure;

Step C3; Two scalar function θ of each point obtain θ and

occurrence at each point on the cloud with

for being defined in; Make maximum principal direction and the minimum principal direction of this point be consistent with the gradient of two scalar functions respectively, two scalar function θ are the optimum solution of following equation minimum value with

as far as possible:

Wherein, and

is respectively two functions every gradient; and is respectively the maximum principal direction and the minimum principal direction of this point, and ω is the parameter by the controlled variable density degree of user's appointment.

10. the method for cloud data gridding according to claim 9 is characterized in that, said step C1 comprises:

Step C1a utilizes cloud data and Morse function to calculate the homology base of a cloud, along the homology base cloud is cut apart;

Step C2b asks for the shortest path of each singular point to the border on k neighbour figure, and according to access path a cloud is cut apart once more, obtains the path that is formed by connecting some points in the cloud.

11. the method for cloud data gridding according to claim 9 is characterized in that, said step C3 comprises:

Step C3a, the energy function of scalar function difference between the maximum principal direction of the gradient of every bit and this point and minimum principal direction is weighed in definition:

formula 1

Wherein, θ _i,

The value of two scalar functions that expression point i is corresponding respectively, K, K ^⊥Maximum principal direction and the minimum principal direction of representing some i place respectively, ij is the limit among the k neighbour figure, e _IjExpression tie point i, the vector of the line segment of j, w are the parameters of the controlled variable density degree of user's appointment;

Step C3b, for the some P that is positioned on the cut-off rule, its neighbour's point is divided into two types by cut-off rule, and this point also is divided into two some P accordingly ⁺And P ^-, and corresponding principal direction K ⁺And K ^-In order to guarantee parameter value θ and

continuity at the cut-off rule place, the parameter value at some P place must meet the following conditions:

formula 2

Wherein, (j k) is integer variable,

Expression direction K ^-To K ⁺Needed rotation matrix; And the parameter value at singular point place also is restricted to round values;

Step C3c sets up a global parameterized equation continuous at the cut-off rule place through formula 1 and formula 2;

Step C3d; Find the solution this global parameterized equation, obtain the scalar function parameter value θ and

of every bit

12. the method for cloud data gridding according to claim 1 is characterized in that, said step D comprises:

Step D1, for each point and on every side k neighbour point carry out the Di Luoni trigonometric ratio according to parametrization value

;

Step D2 gets the annexation that the triangle adjacent with this point set up the corresponding point in the three dimensions;

Step D3, if there is plural triangle to have same limit, it is total by two triangles at most until each bar limit then progressively to remove unnecessary triangle;

Step D4 produces the cavity if remove in the redundant leg-of-mutton process, then fill this hole through the Di Luoni trigonometric ratio is carried out in this hole.

13. the method for cloud data gridding according to claim 1 is characterized in that, this method also comprises:

Step e is set up quadrilateral mesh on the triangle gridding basis of setting up.

14. the method for cloud data gridding according to claim 13 is characterized in that, said step e comprises:

Step e 1 is asked for the equivalent line segment in each triangle;

Step e 2 is asked for the intersection point between the equivalent line segment;

Step e 3 for the intersection point between each equivalent line segment, finds the abutment points of this intersection point and connects according to the annexation between the equivalent line segment, set up shape evenly and direction meet the quadrilateral mesh of principal direction.

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