CN104268936B - Construction method of barycentric coordinates - Google Patents

Construction method of barycentric coordinates Download PDF

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CN104268936B
CN104268936B CN201410482229.3A CN201410482229A CN104268936B CN 104268936 B CN104268936 B CN 104268936B CN 201410482229 A CN201410482229 A CN 201410482229A CN 104268936 B CN104268936 B CN 104268936B
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barycentric coodinates
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construction method
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CN104268936A (en
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张举勇
邓柏林
刘子舜
刘利刚
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University of Science and Technology of China USTC
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    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T7/00—Image analysis
    • G06T7/60—Analysis of geometric attributes
    • G06T7/66—Analysis of geometric attributes of image moments or centre of gravity
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    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
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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 coodinates
Technical field
The present invention is, with regard to a kind of given plane polygon or three-dimensional triangulation shape grid, to calculate its inner area for being surrounded The barycentric coodinates method in domain, its innovative point is that the barycentric coodinates built using the present invention have slickness and extraordinary office simultaneously Portion's property, can be widely applied for the fields such as graphics algorithm, geometric modeling and FEM calculation.
Background technology
It is well known that any point can be write as the linear convex combination form on its three summits in triangle, wherein three Coefficient is the point relative to barycenter oftriangle coordinate.Because three points not conllinear in plane are linear independences, so Barycentric coodinates are uniquely defined.Barycenter oftriangle coordinate is widely used in the fields such as graphics algorithm, geometric modeling. For a bit, the tensor product of its bilinear interpolation gained coefficient is considered as the weight relative to tetragon in plane quadrilateral Heart coordinate.When arbitrary plane polygon is generalized to, the definition mode of its barycentric coodinates is good problem to study.The weight of broad sense Heart coordinate has a wide range of applications in fields such as FEM calculation, complex-curved construction, the parametrizations of polygonal patch.
Before existing barycentric coodinates definition method is introduced, the mathematical description of polygon barycentric coodinates is given first. NoteFor the vertex set of polygon C, p be in the polygon or on border a bit, depend on if there is one group many Side shape summitWith the no negative coefficient of point pMeet following relation：
● linear combination
● it is homogeneity
● nonnegativity
wi≥0
In actual applications, in addition to nonnegativity, barycentric coodinates w are generally also requirediIt is with regard to polygon vertex and point p Smooth function, and in Polygonal Boundary have following property：
● Lagrange properties
If
● linear behavio(u)r：W is linear function on the border of Ω
At present barycentric coodinates have following several construction methods.Floater proposes Mean Value Barycentric Coordinates (Mean Value Coordinates, MVC) concept, it overcomes the limitation that previous methods are only used for convex polygon so that barycentric coodinates can For star polygon.The method is generalized to three-dimensional by Tao Ju et al., and barycentric coodinates are applied in distortion of the mesh.But It is that average centre coordinate does not possess nonnegativity.Derose et al. proposes mediation barycentric coodinates (Harmonic Barycentric Coordinates, HBC), the method by solve Poisson's equation and obtained one and can guarantee that be on the occasion of barycentric coodinates, overcome The shortcoming of Mean Value Barycentric Coordinates.But control of these barycentric coodinates methods to interior zone is all global, i.e. each control System point can affect almost whole region, and this is desirable in many applications what is avoided.On the other hand, Jacobson et al. is proposed Bounded biharmonic weights (Bounded Biharmonic Weight, BBW), linear combination is abandoned to reach locality Property, thus do not strictly belong to barycentric coodinates.
The content of the invention
(1) technical problem to be solved
Because the barycentric coodinates linear combination that barycentric coodinates must are fulfilled for control point is necessarily equal to the European coordinate of the point, this It is often global that individual constraint causes the influence area at control point.In addition, our desired barycentric coodinates must have it is smooth Property, this further increases the difficulty of design local barycentric coodinates.Therefore, problem to be solved by this invention is：
1st, a kind of barycentric coodinates with local property are designed, namely the influence area at each control point is close as far as possible The control point itself,
2nd, the barycentric coodinates are while with Partial controll, it is necessary to slickness；
In view of this, present invention is primarily targeted at providing a kind of construction method of barycentric coodinates, the center of gravity of structure is made Coordinate has slickness and extraordinary local property simultaneously.
(2) technical scheme
To reach above-mentioned purpose, the invention provides a kind of construction method of barycentric coodinates, the method includes：
Step A：Input plane polygonal mesh or three-dimensional triangulation shape grid；
Step B：For plane polygon grid, trigonometric ratio is carried out to plane polygon interior zone, generated in polygon The plane triangle grid in portion；For three-dimensional triangulation shape grid, to three-dimensional triangulation shape grid interior zone tetrahedralization, three are generated Three-dimensional tetrahedral grid inside hexagonal lattice；
Step C：Each sampled point is controlled to each in Calculation Plane triangular mesh or three-dimensional tetrahedral grid region Ω The weighted value of point；
Step D：According to the plane polygon grid or three-dimensional triangulation shape grid and the plane triangle net of generation of input Lattice or three-dimensional tetrahedral grid, solve based on the Optimized model of total variation model, sit so as to obtain the center of gravity with local property Mark.
In such scheme, trigonometric ratio is carried out to plane polygon interior zone described in step B, generate polygonal internal Plane triangle grid, is realized by calling triangle storehouses.
In such scheme, to three-dimensional triangulation shape grid interior zone tetrahedralization described in step B, triangular mesh is generated Internal three-dimensional tetrahedral grid, is realized by calling tetgen storehouses.
In such scheme, step C includes：
Step C1：Each sampled point x in the Ω of zoning to each control point ciBetween geodesic curve distance, labelling this Geodesic curve distance is gi(x)；
Step C2：It is according to geodesic curve value maximum in the Ω of region, geodesic curve parasang is interval interior to [0,1], count Calculate formula as follows：
Wherein, y is the sampled point in the Ω of region, DiX () is the geodesic curve distance of unit；
Step C3：For unitization geodesic curve distance, according to equation φi(x)=τ (Di(x)) weighted value is calculated, wherein Function τ is continuous function, φiX () is the weighting function value of each sampled point calculated according to unitization geodesic curve distance.When When the function τ is the increasing function on [0,1] interval, the locality of barycentric coodinates strengthens.The function
In such scheme, solution described in step D is based on the constraint that the Optimized model of total variation model is in barycentric coodinates Under carry out.
In such scheme, the Optimized model described in step D based on total variation model is：
Wherein, wiFor the barycentric coodinates value on summit, ciFor control vertex, φiFor weighting function value.
In such scheme, the barycentric coodinates that the method is obtained meet linear combination, homogeneity and nonnegativity, on border Meet Lagrange properties and linear behavio(u)r.
(3) beneficial effect
From above-mentioned technical proposal as can be seen that the invention has the advantages that：
1st, using the present invention, the weighting total variation of barycentric coodinates is carried out as object function under the constraint of barycentric coodinates Optimization, has obtained not only smooth but also local barycentric coodinates value, compares existing barycentric coodinates method, and the center of gravity of present invention construction is sat Mark can reach the effect of Partial controll, while each internal data point is only affected by its nearest several control point, this is caused The memory consumption of storage barycentric coodinates reduces, and also causes to accelerate based on the interpolation algorithm of the barycentric coodinates of the present invention.
2nd, as shown in figure 3, in the construction method of the barycentric coodinates of present invention offer, there is similar slickness Under, other barycentric coodinates more local is compared in the influence area at control point.
3rd, as shown in figure 4, using local gravity center coordinate (the Local Barycentric constructed by the present invention Coordinates, LBC), the deformation more local of model.
4th, as shown in table 1 below, relative to original Mean Value Barycentric Coordinates, the LBC barycentric coodinates constructed by the present invention need Less amount of storage and deformation elapsed time faster.
Model Control point 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
Description of the drawings
Fig. 1 is the flow chart of the barycentric coodinates building method that the present invention is provided；
Fig. 2 is the plane polygon grid and three-dimensional triangulation shape grid of input, and wherein left figure is a plane polygon, right Figure is triangle control grid；
Fig. 3 is that the influence area at control point under different barycentric coodinates is compared, and which show the barycentric coodinates side of the present invention The comparison of method and existing barycentric coodinates method, left figure exterior portion is controlling polygon, and redness is the control vertex chosen, face Color represents the control vertex under distinct methods to the influence value of interior zone, and red barycentric coodinates value at this is big, such as right Under color bar shown in, it can be found that our methods influence area local and it is smooth；
Fig. 4 be two-dimentional barycentric coodinates deformation effect and with the contrast of other methods, which show different barycentric coodinates Two-dimentional deformation effect under method, left figure is textured artwork and controlling polygon；Upper figure is that mobile control is polygon in the figure of the right After shape, the deformation effect of internal textured figure, figure below is the size that the color on internal each summit represents amount of movement, Ke Yifa Deformation now under the barycentric coodinates control of the present invention is more locally and smooth；
Fig. 5 and Fig. 6 are the deformation effect schematic diagrams using the barycentric coodinates of present invention structure under three-dimensional.
Specific embodiment
To make the object, technical solutions and advantages of the present invention become more apparent, below in conjunction with specific embodiment, and reference Accompanying drawing, is clearly and completely described to the technical scheme in the embodiment of the present invention, it is clear that described embodiment is only The a part of embodiment of the present invention, rather than the embodiment of whole.Based on embodiments of the invention, those of ordinary skill in the art exist The every other embodiment obtained under the premise of creative work is not made, protection scope of the present invention is belonged to.
The present invention provides a kind of with not only smooth but also local barycentric coodinates building method, according to the method, can be by defeated The plane polygon or three-dimensional triangulation shape mess generation for entering goes out not only smooth but also local barycentric coodinates of the institute comprising space, such as schemes Shown in 1, the method specifically includes following steps：
Step A：Input plane polygonal mesh or three-dimensional triangulation shape grid；
Step B：For plane polygon grid, trigonometric ratio is carried out to plane polygon interior zone, generated in polygon The plane triangle grid in portion；For three-dimensional triangulation shape grid, to three-dimensional triangulation shape grid interior zone tetrahedralization, three are generated Three-dimensional tetrahedral grid inside hexagonal lattice；
Step C：Each sampled point is controlled to each in Calculation Plane triangular mesh or three-dimensional tetrahedral grid region Ω The weighted value of point；
Step D：According to the plane polygon grid or three-dimensional triangulation shape grid and the plane triangle net of generation of input Lattice or three-dimensional tetrahedral grid, solve based on the Optimized model of total variation model, sit so as to obtain the center of gravity with local property Mark.
As shown in Fig. 2 user input plane polygon grid or three-dimensional triangulation shape grid.For two-dimensional case, to input Plane polygon grid, call triangle storehouses to carry out trigonometric ratio to plane polygon interior zone, generate polygonal internal Plane triangle grid.Wherein triangle storehouses are the high-quality mess generation storehouses developed by CMU (CMU), Its input is a polygonal mesh, according to the parameter of user input, generates the network of triangle as close possible to equilateral triangle Lattice.For three-dimensional situation, to the three-dimensional triangulation shape grid being input into, tetgen storehouses are called to three-dimensional triangulation shape grid interior zone four Face body, generates the tetrahedral grid inside triangular mesh.Wherein tetgen storehouses are the high-quality three-dimensionals four of Hang Si exploitations Face volume mesh generates storehouse, and its input is the triangular mesh of a closing, and according to the parameter of user input, generation connects as far as possible The tetrahedral grid of near regular tetrahedron.
Weighted value of each sampled point to each control point in the Ω of zoning.Each sampling in zoning Ω first Point x to each control point ciBetween geodesic curve distance, labelling this geodesic curve distance is gi(x).According to survey maximum in the Ω of region Ground wire value, geodesic curve parasang is interval interior to [0,1], and computing formula is as follows：
Wherein, y is the sampled point in the Ω of region, DiX () is the geodesic curve distance of unit；
According to unitization geodesic curve distance, weighted value is calculated, according to equation below：
φi(x)=τ (D (x))
Wherein function τ is continuous function, φiX () is each sampled point calculated according to unitization geodesic curve distance Weighting function value, in the present invention, it is preferred to it is continuous increasing function to arrange τ, for example
With T to represent trigonometric ratio after unit set, then object function is discrete is：
Wherein, wiFor the barycentric coodinates value on summit, ciFor control vertex, φiFor weighting function value,It is wiIn unit Gradient in S, ASIt is the area (volume) of the unit,It is the weighting function value of the unit center.It is raw for after trigonometric ratio Into summit, if the summit control grid on, the barycentric coodinates value on the summit is via boundary condition (Lagrange Property and linear behavio(u)r) determine.So, the variable of this optimization is internal vertex { vj| j=1 ..., m } on barycentric coodinates value { wi | i=1 ..., n }, we use matrix W ∈ Rm×nLabelling it.Its every a line WjRepresent vertex vjBarycentric coodinates value, it is and each Row WiThen contain control point ciBarycentric coodinates value to all internal control points.Due toIt is with regard to WiLinear, so It can be written toWhereinRepresent the contribution from border vertices.So as to this optimization problem from Scattered form is：
S.t.WK=B, W >=0
Wherein matrix K and B are got from the constraint of barycentric coodinates：
Wherein, ciIt is control vertex, viIt is sampling summit.This problem can be solved by convex optimization method, finally output is interior The barycentric coodinates value of each sampled point of portion.
Using the such scheme of the present invention, can be internally generated with three-dimensional triangulation shape grid by the plane polygon being input into Not only barycentric coodinates smooth but also with local property.
Fig. 5 and Fig. 6 show the deformation effect using the barycentric coodinates of present invention structure under three-dimensional.In Figure 5, we The 3 D deformation effect of the barycentric coodinates invented based on us is illustrated, first figure inside is the threedimensional model for needing deformation, Outside is triangle control grid, the summit on grid is controlled by the triangle outside movement, on internal threedimensional model Summit goes out its new position by barycentric coodinates and mobile control point interpolation, so as to reach the effect of deformation.Three below It is the model after deformation to magnify figure, and the little figure in the lower right corner is the color-coding of each summit amount of movement, and blueness is represented moves Momentum is little, and it is big that redness represents amount of movement.The displacement very local of our methods is can be found that by this example.
Fig. 6 is another 3 D deformation example, is also to control the summit on grid to reach by mobile outside triangle The effect of 3 D deformation.
Particular embodiments described above, has been carried out further in detail to the purpose of the present invention, technical scheme and beneficial effect Describe in detail it is bright, should be understood that the foregoing is only the present invention specific embodiment, be not limited to the present invention, it is all Within the spirit and principles in the present invention, any modification, equivalent substitution and improvements done etc., should be included in the guarantor of the present invention Within the scope of shield.

Claims (7)

1. a kind of construction method of barycentric coodinates, its step is as follows：
Step A：Input plane polygonal mesh or three-dimensional triangulation shape grid；
Step B：For plane polygon grid, trigonometric ratio is carried out to plane polygon interior zone, generate polygonal internal Plane triangle grid；For three-dimensional triangulation shape grid, to three-dimensional triangulation shape grid interior zone tetrahedralization, triangle is generated Three-dimensional tetrahedral grid inside grid；
Step C：Each sampled point is to each control point in Calculation Plane triangular mesh or three-dimensional tetrahedral grid region Ω Weighted value；
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, solves based on the Optimized model of total variation model, so as to obtain the barycentric coodinates with local property；
Wherein, step C includes：
Step C1:Each sampled point x in the Ω of zoning to each control point ciBetween geodesic curve distance, labelling this geodetic Linear distance is gi(x)；
Step C2:It is according to geodesic curve value maximum in the Ω of region, geodesic curve parasang is interval interior to [0,1], calculate public Formula is as follows：
D i ( x ) = g i ( x ) max y ∈ Ω g i ( y ) ;
Wherein, x, y are the sampled point in the Ω of region, DiX () is the geodesic curve distance of unit；
Step C3：For unitization geodesic curve distance, according to equation φi(x)=τ (Di(x)) calculate weighted value, wherein function τ It is continuous function, φiX () is the weighting function value of each sampled point calculated according to unitization geodesic curve distance.
2. the construction method of barycentric coodinates according to claim 1, it is characterised in that to planar polygonal described in step B Shape interior zone carries out trigonometric ratio, generates the plane triangle grid of polygonal internal, is realized by calling triangle storehouses 's.
3. the construction method of barycentric coodinates according to claim 1, it is characterised in that to three-dimensional triangulation described in step B Shape grid interior zone tetrahedralization, generates the three-dimensional tetrahedral grid inside triangular mesh, is by calling tetgen storehouses Realize.
4. the construction method of barycentric coodinates according to claim 1, it is characterised in that when the function τ is that [0,1] is interval On increasing function when, the locality of barycentric coodinates strengthens.
5. the construction method of barycentric coodinates according to claim 4, it is characterised in that the function
6. the construction method of barycentric coodinates according to claim 1, it is characterised in that total variation is based on described in step D The Optimized model of model is：
m i n w 1 , ... , w n Σ i = 1 n ∫ Ω φ i | ▿ w i |
s . t . Σ i = 1 n w i ( x ) c i = x , Σ i = 1 n w i = 1 , w i ≥ 0 , ∀ x ∈ Ω
w i ( c j ) = δ i j ∀ i , j
And weight wiIt is linear on the face that two dimensional surface controls on the side of grid with Three dimensions control grid, wherein, wiFor top The barycentric coodinates value of point, ciFor control vertex, φiFor weighting function value.
7. the construction method of barycentric coodinates according to claim 1, it is characterised in that the barycentric coodinates that the method is obtained expire Sufficient linear combination, homogeneity and nonnegativity, meet Lagrange properties and linear behavio(u)r on border.
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