CN104268936A - Construction method of barycentric coordinates - Google Patents

Abstract

The invention discloses a construction method of barycentric coordinates. The construction method of the barycentric coordinates comprises the steps that a plane polygonal mesh or a three-dimensional triangular mesh is input; for the plane polygonal mesh, triangularization is carried out on the internal area of a plane polygon to generate a plane triangular mesh inside the polygon; for the three-dimensional triangular mesh, the internal area of the three-dimensional triangular mesh is made to be tetrahedral, so that a three-dimensional tetrahedral mesh inside the triangular mesh is generated; a weighted value of each sampling point to each control point in an area Omega is calculated; according to the input plane polygonal mesh or the three-dimensional triangular mesh and the generated plane triangular mesh or the three-dimensional tetrahedral mesh, an optimization model based on a total variation model is solved, and therefore the barycentric coordinates with local characteristics are obtained. By means of the construction method of the barycentric coordinates, under the restraint of the barycentric coordinates, the weighting total variation of the barycentric coordinates is optimized while serving as an objective function, smooth and local barycentric coordinate values are obtained, memory space for storing the barycentric coordinates is smaller, and the interpolation algorithm of the barycentric coordinates based on the construction method is accelerated.

Description

A kind of construction method of barycentric coordinates

Technical field

The invention relates to a kind of given plane polygon or three-dimensional triangulation shape grid, calculate the barycentric coordinates method of the interior zone that it surrounds, its innovative point is that the barycentric coordinates utilizing the present invention to build have slickness and extraordinary local property simultaneously, can be widely used in the fields such as graphics algorithm, geometric modeling and FEM (finite element) calculation.

Background technology

As everyone knows, in triangle, any point can be write as the linear convex combination form on its three summits, and wherein three coefficients are this point relative to barycenter oftriangle coordinate.Due in plane not three points of conllinear be linear independence, so barycentric coordinates are uniquely defined.Barycenter oftriangle coordinate is widely used in the field such as graphics algorithm, geometric modeling.For in plane quadrilateral a bit, the tensor product of its bilinear interpolation gained coefficient can regard this barycentric coordinates relative to quadrilateral as.When being generalized to arbitrary plane polygon, the definition mode of its barycentric coordinates is good problems to study.The barycentric coordinates of broad sense have a wide range of applications in the field such as parametrization of FEM (finite element) calculation, complex-curved structure, polygonal patch.

Before introducing existing barycentric coordinates define method, first provide the mathematical description of polygon barycentric coordinates.Note for the vertex set of polygon C, p is a bit in this polygon or border, if having one group to depend on polygon vertex with the no negative coefficient of a p meet following relation:

● linear combination

$p=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{w}_i{c}_i$

● homogeneity

$\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{w}_i=1$

● nonnegativity

w _i≥0

In actual applications, except nonnegativity, usually also require barycentric coordinates w _ibe about polygon vertex and the smooth function putting p, and there is following character in Polygonal Boundary:

● Lagrange character

If

● linear behavio(u)r: w is linear function on the border of Ω

Current barycentric coordinates have following several construction method.Floater proposes the concept of Mean Value Barycentric Coordinates (Mean Value Coordinates, MVC), which overcomes the limitation that previous methods can only be used for convex polygon, makes barycentric coordinates may be used for star polygon.The method is generalized to three-dimensional by the people such as Tao Ju, and barycentric coordinates is applied in distortion of the mesh.But average centre coordinate does not possess nonnegativity.The people such as Derose propose be in harmonious proportion barycentric coordinates (Harmonic Barycentric Coordinates, HBC), the method by solve Poisson equation obtain one can ensure be on the occasion of barycentric coordinates, overcome the shortcoming of Mean Value Barycentric Coordinates.But these barycentric coordinates methods are all overall to the control of interior zone, namely each reference mark can affect almost whole region, and this wishes to be avoided in many applications.On the other hand, the people such as Jacobson propose bounded biharmonic weights (Bounded Biharmonic Weight, BBW), abandoning linear combination, thus strictly not belonging to barycentric coordinates to reach locality.

Summary of the invention

(1) technical matters that will solve

The barycentric coordinates linear combination that must meet reference mark due to barycentric coordinates must equal the European coordinate of this point, range of influence that this constraint the makes reference mark overall situation often.In addition, the barycentric coordinates desired by us must have slickness, and this increases the difficulty of design local barycentric coordinates further.Therefore, problem to be solved by this invention is:

1, design a kind of barycentric coordinates with local property, also namely the range of influence at each reference mark as much as possible near this reference mark self,

2, these barycentric coordinates are while having Partial controll, also must have slickness;

In view of this, fundamental purpose of the present invention is the construction method providing a kind of barycentric coordinates, makes the barycentric coordinates of structure have slickness and extraordinary local property simultaneously.

(2) technical scheme

For achieving the above object, the invention provides a kind of construction method of barycentric coordinates, the method comprises:

Steps A: input plane polygonal mesh or three-dimensional triangulation shape grid;

Step B: for plane polygon grid, carries out trigonometric ratio to plane polygon interior zone, generates the plane triangle grid of polygonal internal; For three-dimensional triangulation shape grid, to three-dimensional triangulation shape grid interior zone tetrahedralization, generate the three-dimensional tetrahedral grid of triangular mesh inside;

Step C: in Calculation Plane triangular mesh or three-dimensional tetrahedral grid region Ω, each sampled point is to the weighted value at each reference mark;

Step D: according to the plane polygon grid of input or the plane triangle grid of three-dimensional triangulation shape grid and generation or three-dimensional tetrahedral grid, solve the Optimized model based on total variation model, thus obtain the barycentric coordinates with local property.

In such scheme, described in step B, trigonometric ratio being carried out to plane polygon interior zone, generate the plane triangle grid of polygonal internal, realizing by calling triangle storehouse.

In such scheme, to three-dimensional triangulation shape grid interior zone tetrahedralization described in step B, generating the three-dimensional tetrahedral grid of triangular mesh inside, realizing by calling tetgen storehouse.

In such scheme, described step C comprises:

Step C1: each sampled point x in the Ω of zoning is to each reference mark c _ibetween geodesic line distance, marking this geodesic line distance is g _i(x);

Step C2: according to geodesic line value maximum in the Ω of region, by geodesic line parasang in [0,1] interval, computing formula is as follows:

$D_i\left(x\right)=\frac{g_i\left(x\right)}{\underset{y\∈\mathrm{\Ω}}{\max }{g}_i\left(y\right)};$

Wherein, y is the sampled point in the Ω of region, D _ix geodesic line distance that () is unit;

Step C3: for unitization geodesic line distance, according to formula φ _i(x)=τ (D _i(x)) calculate weighted value, wherein function τ is continuous function, φ _i(x) weighting function value of each sampled point for calculating according to unitization geodesic line distance.When described function τ is the increasing function on [0,1] interval, the locality of barycentric coordinates strengthens.Described function

In such scheme, the Optimized model solved described in step D based on total variation model carries out under the constraint of barycentric coordinates.

In such scheme, described in step D based on the Optimized model of total variation model be:

$\underset{w_1,...,w_n}{\min }\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{\∫}_{\mathrm{\Ω}}{\mathrm{\φ}}_i\left|{\▿w}_i\right|$

$s.t.\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{w}_i\left(x\right)c_i=x,\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{w}_i=1,w_i\≥0,\∀x\∈\mathrm{\Ω}$

$w_i\left(c_j\right)={\mathrm{\δ}}_{\mathrm{ij}}\∀i,j,$

$w_i\mathrm{is\; linear\; on\; cages\; and\; faces}\∀i.$

Wherein, w _ifor the barycentric coordinates value on summit, c _ifor control vertex, φ _ifor weighting function value.

In such scheme, the barycentric coordinates that the method obtains meet linear combination, homogeneity and nonnegativity, and border meets Lagrange character and linear behavio(u)r.

(3) beneficial effect

As can be seen from technique scheme, the present invention has following beneficial effect:

1, the present invention is utilized, under the constraint of barycentric coordinates, the weighting total variation of barycentric coordinates is optimized as objective function, obtain the not only smooth but also barycentric coordinates value of local, compare existing barycentric coordinates method, the barycentric coordinates of the present invention's structure can reach the effect of Partial controll, each internal data point is only by the impact at its nearest several reference mark simultaneously, and this makes the memory consumption storing barycentric coordinates reduce, and also makes the interpolation algorithm based on barycentric coordinates of the present invention accelerate.

2, as shown in Figure 3, in the construction method of barycentric coordinates provided by the invention, when having similar slickness, other barycentric coordinates more local is compared in the range of influence at reference mark.

3, as shown in Figure 4, the local gravity center coordinate (Local Barycentric Coordinates, LBC) constructed by the present invention is adopted, the distortion more local of model.

4, as shown in table 1 below, relative to original Mean Value Barycentric Coordinates, the LBC barycentric coordinates constructed by the present invention need less memory space and are out of shape elapsed time faster.

Model	Reference mark number	Memory consumption	Deformation time
				CACTUS	27	23.23％	38.71％
ELEPHANT	36	22.02％	30.45％
				GECKO	34	23.93％	31.44％
WOODY	26	30.80％	41.99％

ARMADILLO	110	16.58％	17.57％
				HORSE	51	20.75％	30.85％

Table 1

Accompanying drawing explanation

Fig. 1 is the process flow diagram of barycentric coordinates building method provided by the invention;

Fig. 2 is plane polygon grid and the three-dimensional triangulation shape grid of input, and wherein left figure is a plane polygon, and right figure is triangle control mesh;

Fig. 3 is that under different barycentric coordinates, the range of influence at reference mark is compared, which show comparing of barycentric coordinates method of the present invention and existing barycentric coordinates method, left figure exterior portion is controlling polygon, the red control vertex for choosing, color represents this control vertex influence value to interior zone under distinct methods, the barycentric coordinates value at this place red is large, as shown in the color bar of bottom right, can find the local, range of influence of our method and smooth;

Fig. 4 is two-dimentional barycentric coordinates deformation effect and contrasts with other method, which show two-dimentional deformation effect under different barycentric coordinates methods, and left figure is former figure and the controlling polygon of band texture; In the figure of the right, upper figure is after mobile controlling polygon, the deformation effect of inner band texture maps, and figure below is the size that color on inner each summit represents amount of movement, can find that distortion under barycentric coordinates control of the present invention more locally and smooth;

Fig. 5 and Fig. 6 is the deformation effect schematic diagram of barycentric coordinates under three-dimensional adopting the present invention to build.

Embodiment

Clearly understand for making the object, technical solutions and advantages of the present invention, below in conjunction with specific embodiment, and with reference to accompanying drawing, technical scheme in the embodiment of the present invention is clearly and completely described, obviously, described embodiment is only the present invention's part embodiment, instead of whole embodiments.Based on embodiments of the invention, those of ordinary skill in the art, not making the every other embodiment obtained under creative work prerequisite, belong to protection scope of the present invention.

The invention provides and a kind of there is the not only smooth but also barycentric coordinates building method of local, according to the method, can go out by the plane polygon inputted or three-dimensional triangulation shape mess generation the barycentric coordinates that this institute comprises the not only smooth of space but also local, as shown in Figure 1, the method specifically comprises the following steps:

Steps A: input plane polygonal mesh or three-dimensional triangulation shape grid;

Step B: for plane polygon grid, carries out trigonometric ratio to plane polygon interior zone, generates the plane triangle grid of polygonal internal; For three-dimensional triangulation shape grid, to three-dimensional triangulation shape grid interior zone tetrahedralization, generate the three-dimensional tetrahedral grid of triangular mesh inside;

Step C: in Calculation Plane triangular mesh or three-dimensional tetrahedral grid region Ω, each sampled point is to the weighted value at each reference mark;

Step D: according to the plane polygon grid of input or the plane triangle grid of three-dimensional triangulation shape grid and generation or three-dimensional tetrahedral grid, solve the Optimized model based on total variation model, thus obtain the barycentric coordinates with local property.

As shown in Figure 2, user's input plane polygonal mesh or three-dimensional triangulation shape grid.For two-dimensional case, to the plane polygon grid of input, call triangle storehouse and trigonometric ratio is carried out to plane polygon interior zone, generate the plane triangle grid of polygonal internal.Wherein triangle storehouse is the high-quality mess generation storehouse developed by CMU (CMU), and its input is a polygonal mesh, according to the parameter of user's input, generates as far as possible close to the triangular mesh of equilateral triangle.For three-dimensional situation, to the three-dimensional triangulation shape grid of input, call tetgen storehouse to three-dimensional triangulation shape grid interior zone tetrahedralization, generate the tetrahedral grid of triangular mesh inside.Wherein tetgen storehouse is that the three-dimensional tetrahedral grid of high-quality that Hang Si develops generates storehouse, and its input is a closed triangular mesh, according to the parameter of user's input, generates as far as possible close to the tetrahedral grid of positive tetrahedron.

In the Ω of zoning, each sampled point is to the weighted value at each reference mark.First each sampled point x in the Ω of zoning is to each reference mark c _ibetween geodesic line distance, marking this geodesic line distance is g _i(x).According to geodesic line value maximum in the Ω of region, by geodesic line parasang in [0,1] interval, computing formula is as follows:

$D_i\left(x\right)=\frac{g_i\left(x\right)}{\underset{y\∈\mathrm{\Ω}}{\max }{g}_i\left(y\right)}$

Wherein, y is the sampled point in the Ω of region, D _ix geodesic line distance that () is unit;

According to unitization geodesic line distance, calculate weighted value, according to following formula:

φ _i(x)＝τ(D(x))

Wherein function τ is continuous function, φ _ix () weighting function value of each sampled point for calculating according to unitization geodesic line distance, in the present invention, preferably arranging τ is continuous increasing function, such as $\mathrm{\τ}\left(D_i\left(x\right)\right)=D_i^2\left(x\right).$

Unit set after representing trigonometric ratio with T, then objective function is discrete is:

$\underset{S\∈T}{\mathrm{\Σ}}\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{\mathrm{\φ}}_i^S{A}_S{\left|\right|{\▿}_S{w}_i\left|\right|}_2$

Wherein, w _ifor the barycentric coordinates value on summit, c _ifor control vertex, φ _ifor weighting function value, w _igradient in cell S, A _sthe area (volume) of this unit, it is the weighting function value of this unit center.For the summit generated after trigonometric ratio, if this summit is on control mesh, then the barycentric coordinates value on this summit is determined by boundary condition (Lagrange character and linear behavio(u)r).So this variable optimized is internal vertex { v _j| j=1 ..., the barycentric coordinates value { w on m} _i| i=1 ..., n}, we use matrix W ∈ R ^{m × n}mark it.Its every a line W ^jrepresent vertex v _jbarycentric coordinates value, and each row W _ithen contain reference mark c _ito the barycentric coordinates value of all internal control points.Due to about W _ilinear, so it can be written to wherein represent the contribution from border vertices.Thus the discrete form of this optimization problem is:

$\underset{W}{\min }\underset{S\∈T}{\mathrm{\Σ}}\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{\mathrm{\φ}}_i^S{A}_S{\left|\right|G_S{W}_i+d_S^i\left|\right|}_2$

s.t.WK＝B，W≥0

Wherein matrix K and B get from the constraint of barycentric coordinates:

Wherein, c _icontrol vertex, v _iit is sampling summit.Can solve this problem by convex optimization method, finally output is the barycentric coordinates value of inner each sampled point.

Adopt such scheme of the present invention, can generated by the plane polygon inputted and three-dimensional triangulation shape grid inside not only smooth but also there are the barycentric coordinates of local property.

Fig. 5 and Fig. 6 shows the deformation effect of barycentric coordinates under three-dimensional adopting the present invention to build.In Figure 5, we show the 3 D deformation effect of the barycentric coordinates based on our invention, first figure is inner for needing the three-dimensional model of distortion, outside is triangle control mesh, by the summit on the triangle control mesh of mobile outside, summit on inner three-dimensional model goes out its reposition by barycentric coordinates and the reference mark interpolation of movement, thus reaches the effect of distortion.Below three large figure are the model after distortion, and the little figure in the lower right corner is the color-coding of each summit amount of movement, and it is little that blueness represents amount of movement, and it is large that redness represents amount of movement.The displacement very local of our method can be found by this example.

Fig. 6 is another 3 D deformation example, is also by the summit on the triangle control mesh of mobile outside, reaches the effect of 3 D deformation.

Above-described specific embodiment; object of the present invention, technical scheme and beneficial effect are further described; be understood that; the foregoing is only specific embodiments of the invention; be not limited to the present invention; within the spirit and principles in the present invention all, any amendment made, equivalent replacement, improvement etc., all should be included within protection scope of the present invention.

Claims (8)

1. a construction method for barycentric coordinates, its step is as follows:

Steps A: input plane polygonal mesh or three-dimensional triangulation shape grid;

Step B: for plane polygon grid, carries out trigonometric ratio to plane polygon interior zone, generates the plane triangle grid of polygonal internal; For three-dimensional triangulation shape grid, to three-dimensional triangulation shape grid interior zone tetrahedralization, generate the three-dimensional tetrahedral grid of triangular mesh inside;

Step C: in Calculation Plane triangular mesh or three-dimensional tetrahedral grid region Ω, each sampled point is to the weighted value at each reference mark;

Step D: according to the plane polygon grid of input or the plane triangle grid of three-dimensional triangulation shape grid and generation or three-dimensional tetrahedral grid, solve the Optimized model based on total variation model, thus obtain the barycentric coordinates with local property.

2. the construction method of barycentric coordinates according to claim 1, is characterized in that, carries out trigonometric ratio described in step B to plane polygon interior zone, generates the plane triangle grid of polygonal internal, realizes by calling triangle storehouse.

3. the construction method of barycentric coordinates according to claim 1, is characterized in that, to three-dimensional triangulation shape grid interior zone tetrahedralization described in step B, generates the three-dimensional tetrahedral grid of triangular mesh inside, realizes by calling tetgen storehouse.

4. the construction method of barycentric coordinates according to claim 1, is characterized in that, described step C comprises:

Step C1: each sampled point x in the Ω of zoning is to each reference mark c _ibetween geodesic line distance, marking this geodesic line distance is g _i(x);

Step C2: according to geodesic line value maximum in the Ω of region, by geodesic line parasang in [0,1] interval, computing formula is as follows:

$D_i\left(x\right)=\frac{g_i\left(x\right)}{\underset{y\∈\mathrm{\Ω}}{\max }{g}_i\left(y\right)};$

Wherein, y is the sampled point in the Ω of region, D _ix geodesic line distance that () is unit;

Step C3: for unitization geodesic line distance, according to formula φ _i(x)=τ (D _i(x)) calculate weighted value, wherein function τ is continuous function, φ _i(x) weighting function value of each sampled point for calculating according to unitization geodesic line distance.

5. the construction method of barycentric coordinates according to claim 4, is characterized in that, when described function τ is the increasing function on [0,1] interval, the locality of barycentric coordinates strengthens.

6. the construction method of barycentric coordinates according to claim 5, is characterized in that, described function $\mathrm{\τ}\left(D_i\left(x\right)\right)=D_i^2\left(x\right).$

7. the construction method of barycentric coordinates according to claim 1, is characterized in that, described in step D based on the Optimized model of total variation model is:

$\underset{w_1,...,w_n}{\min }\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{\∫}_{\mathrm{\Ω}}{\mathrm{\φ}}_i\left|{\▿w}_i\right|$

$s.t.\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{w}_i\left(x\right)c_i=x,\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{w}_i=1,w_i\≥0,\∀x\∈\mathrm{\Ω}$

$w_i\left(c_j\right)={\mathrm{\δ}}_{\mathrm{ij}}\∀i,j,$

$w_i\mathrm{is\; linear\; on\; cages\; and\; faces}\∀i.$

Wherein, w _ifor the barycentric coordinates value on summit, c _ifor control vertex, φ _ifor weighting function value.

8. the construction method of barycentric coordinates according to claim 1, is characterized in that, the barycentric coordinates that the method obtains meet linear combination, homogeneity and nonnegativity, and border meets Lagrange character and linear behavio(u)r.