CN106096132A - A kind of emulation mode of the different materials clothing fold based on differential domain - Google Patents

Abstract

A kind of emulation mode of the different materials clothing fold based on differential domain, original three Cartesian coordinates is changed, set up the clothes coordinate under differential domain coordinate system, by adding different constraint control to differential domain, while ensure that rotational invariance, simulate the analogy method of the clothing of the different materials clothing fold under identical edit operation, its step is that (1) selects suitable differential domain framework, Laplce's coordinate and gradient field attribute of triangle gridding is resolved and contrast, selects suitable differential domain attribute；(2) selected differential domain method is optimized, keeps the consistency of the rotation of coordinate so that it is can apply in the deformation manipulation of larger range of clothes；(3) considering that different clothings is different to the display form of fold under identical manipulation, with the addition of the constraint of deformation targetedly, different fold under identical operation for the simulation different materials clothes produces form.

Description

A kind of emulation mode of the different materials clothing fold based on differential domain

Technical field

The present invention relates to the emulation mode of a kind of different materials clothing fold based on differential domain, belong to the emulation of virtual clothing Technical field.

Background technology

Sum up the research method of clothing fold emulation in recent years, can be largely classified into following four classes: based on geometrical property meter Calculate method, the method based on physical force and model analysis, the method based on multi-resolution grid, drive based on machine learning data Dynamic method.(1) method based on geometry.Hadap et al. with the addition of texture synthesis method in the emulation mode of geometry, innovation The fold simulating clothes of property.Culter et al. finds crest line in three-dimensional expansion model according to certain rule, by mark The smooth crest line of method generation fixed, that connect, smooth, then on crest line, regulate the geometric parameters such as fold width and height Generate difform fold.(2) method based on physics.Mainly can be attributed to two classes based on the emulation of the method for physics: Physical simulation based on energy and the physical simulation based on mechanical model.First fabric is constructed based on the physical simulation method of energy At difference energy function after the match, and in addition special constraints, then obtain non-binding error side by penalty function method Journey.Finally error equation is linearized and is solved by least square method the end-state of fabric.Feynman proposes grid matter Point model, is mapped one by one to two and three dimensions space fabric, the pendency effect according to minimum energy principle emulating fabric Really.Terzopoulos et al. obtains energy equation according to continuum Model, and utilizes calculus of finite differences to solve energy equation.Based on The physical simulation method of mechanics continues development on dynamic elasticity, sets up the motion rail of particle first with the differential equation Mark, the movement locus in then determining certain time by numerical method, existing frequently-used physical model is mainly mass spring mould Type and FEM model.(3) method based on multi-resolution grid.Russell et al. proposes a kind of dynamic based on coarse grid This algorithm of real-time simulation clothing.The method uses the reliable apparent motion pattern in image Segmentation Technology extraction time and space, Detection compression continuously, stable and stretching fold part, then utilize the motor pattern detecting to calculate each triangle Adaptive shape and the stretch tensor on the basis of this.This algorithm follows the trail of room and time conforming fold path, takes one Planting is divided into two steps to be estimated process regimes to stretch tensor, overcomes and is produced discontinuous and noise by low resolution grid surface The shortcoming of phenomenon.(4) the fold analogy method based on data-driven.The garment shape that Xu et al. calculates on the estimation provides a kind of height The solution of Folding Deformation details is generated in real time under resolution grid.The method introduces skeleton, and idiographic flow is: first First, provide an input skeleton model, obtain body area network lattice model by handling bone change.Then human body is divided into different Region, the clothing draping effect in each region carries out approximate match from lane database, is then tied to master mould.Finally, pass through Merge and obtain overall clothing draping effect, repairing is carried out to the problem producing in process and obtains target gridding.

Content of the invention

The technical problem to be solved in the present invention is: utilize differential domain method to carry out gridding edition operation, this edit methods speed Degree is fast, and effect is good.But, gridding edition operation is applied and can be made simulated effect distortion on the clothing of beformable body, in order to inherit differential The advantage of territory gridding edition, overcomes the deficiencies in the prior art, emulates this concrete application for virtual clothing and is given a kind of brand-new Based on the different clothing fold emulation mode of differential domain.

The technical solution used in the present invention: the emulation mode of a kind of different materials clothing fold based on differential domain, passes through Following steps realize:

(1) it based on Laplce's coordinate: first read original clothing grid data, is original clothing according to clothing grid data The cartesian coordinate of thing grid can obtain the Laplce's coordinate set up；Then keep Laplce's coordinate constant, to original Grid enters edlin, obtains setting up the energy error equation of original clothing grid and editor's grid, and solving equation obtains reconstruction Clothing grid；Based on gradient field coordinate: the first cartesian coordinate according to original clothing grid and gradient field, it is achieved gradient field is sat Target represents, enters edlin to original mesh, obtains editing grid, by the gradient field coordinate obtaining and editor's grid, sets up pool The energy error equation of loose measure journey, solving equation obtains the clothing grid rebuild；The clothing grid of two kinds of different coordinates reconstructions enters Row Experimental comparison and analysis, finally give be more suitable for clothing emulation differential domain solve scheme；

(2), on the basis of Laplce's coordinate in step (1), use improved Laplace method, maintain clothing The local geometric details of thing triangle gridding, it is ensured that clothing keeps correctness and the authenticity of deformation under rotation process；

(3) in step (2) maintain clothing triangle gridding local geometric details in the case of, for different materials Clothing to recover, stretching with bending stressing conditions different, foundation based on point, limit, the grid bound energy equation of dihedral angle, And simulate different folds under same operation for the clothing of different materials by adjusting the weight parameter in bound energy equation Effect.

The improved Laplace method detailed process of described step (2) is as follows:

(21) it is one approximate transform of calculating on the new grid in the edited of each summit, ensure scaling and rotate not Denaturation；

(22) according to the approximate transform obtaining and energy error equation, set up new energy error equation, and be converted to square Formation formula, for sparse matrix solve rebuild after optimize grid.

Described step (3) is set up based on point, limit, the grid bound energy equation of dihedral angle, and by adjusting constraint energy Different draping effect simulation under same operation for the clothing of amount equation weight parameter simulation different materials is as follows:

(31) energy equation, the recovery characteristics after control clothes compression are set up by the vertex position of triangle gridding；

(32) set up energy equation by the length of side of triangle gridding, control clothes characteristic after the stretch；

(33) energy equation, the characteristic of control clothes bending are set up by the dihedral angle of the adjacent surface of triangle gridding；

(34) arranging different weights to above three energy equation, simulation different materials clothing is under identical active force Different clothing folds tell on.

The cardinal principle of the present invention:

(1) The present invention gives expression and the reconstruction model of grid energy field under different differential domain attributes.First against Laplce's coordinate is studied, and the cartesian coordinate of original mesh Laplce's coordinate representation, edits original mesh, builds Vertical original mesh and the energy error equation of editor's grid, utilize least square method to rebuild grid, then to grid ladder Degree Domain Properties is studied, and is the manipulation to gradient field attribute the manipulation transforms of grid vertex, obtains weight by Poisson's equation Networking lattice；Finally by the contrast of two kinds of differential domain methods and analysis, have selected the La Pula being more suitable for the emulation of clothing fold This coordinate is as the basis studied further.

(2) give differential coordinate innovatory algorithm and based on Laplce's coordinate different materials clothing fold emulation Method.The operation that the present invention is directed to Laplce's coordinate at translation, rotation and scaling is studied, and discovery Laplce's coordinate is not There is rotation and scaling consistency.Therefore, give holding Laplce rotate and scale the solution of consistency, and pass through Carrying out contrast experiment with former method, demonstrating versatility and the authenticity of improved method, the present invention substantially increases clothing fold Simulation velocity, optimize the grid representation method based on differential domain according to the feature of flexible clothing.

(3) in order to simulate the draping effect of different materials clothes more really, for stretching, bending, three kinds of bases of compression This stressing conditions is set up based on point, limit, the grid bound energy equation of dihedral angle.By regulating the power of three energy constraint equations Weight parameter, generates different fold simulation effects, thus simulates different fold effect under same operation for the multiple material clothes Really.

Present invention advantage compared with prior art is:

(1) present invention substantially increases the simulation velocity of clothing fold, optimizes based on drawing according to the feature of flexible clothing The grid representation method of this coordinate of pula.

(2) characteristic according to garment material, with the addition of different energy equations, solves clothes fold under same operation Simulate an identical difficult problem, improve the vivid effect of emulation.

Brief description

Fig. 1 is the method flowchart of the present invention；

Fig. 2 is the gradient operator expression figure of the present invention；

Fig. 3 is the multi-angle sense of reality view of former grid and warp mesh；A () is original mesh, (b) is editor's grid；

Fig. 4 is for contrasting view based on Laplce's coordinate with based on gradient field editor；A () is based on Laplce's coordinate Multi-angle deformation view, (b) is the multi-angle deformation view based on gradient field；Laplce's coordinate method；

Fig. 5 is longuette initial mesh view and longuette editor's grid view；

Reconstructed view contrast under original differential coordinate for Fig. 6 longuette；

Fig. 7 is the class Laplce's energy function schematic diagram based on limit and angle；

Fig. 8 is the sense of reality view of original mesh, and (a) is right side, and (b) is bottom surface, and (c) is left side；

Fig. 9 is the grid view after editor, and (a) is right side, and (b) is bottom surface, and (c) is left side；

Figure 10 is parameter ω, the size of λ and μ and corresponding effect.

Detailed description of the invention

As it is shown in figure 1, the present invention to implement step as follows:

First, the generation method based on differential domain energy field

1. grid representation and the reconstruction based on Laplce's coordinate

Differential coordinate has another name called Laplce (Laplacian) coordinate, is Alexa^[40]Take the lead in proposing and be applied to grid to become It in shape and gridding edition, is used for describing the relation between certain summit and its first order neighbors point.

Assuming that initial clothing grid is M, note M=(V, E, F), wherein V is the set of grid vertex, and E is Grid Edge Set, F is the set of grid surface.Wherein vertex set can be expressed asv_i=[v_ix, v_iy, v_iz]^T ∈R³.Assume that clothing grid all summits topology is connected, then claim and vertex v_iThe collection of the some composition being joined directly together is combined into vertex v_i's First order neighbors, is designated as N_i=j | (i, j) ∈ E}.According to vertex v_iWith first order neighbors N_i, v can be obtained_iDifferential coordinate δ_i:

${\mathrm{\δ}}_i=({{\mathrm{\δ}}_i}^{\left(x\right)},{{\mathrm{\δ}}_i}^{\left(y\right)},{{\mathrm{\δ}}_i}^{\left(z\right)})=v_i-{\mathrm{\Σ}}_{j\∈N_i}{\mathrm{\ω}}_{ij}{v}_j---\left(1.1\right)$

Wherein ω_ijIt is and limit e_ijThe weight on related limit, generally, ω_ijIt is added and be equal to 1.Retouch to simplify State, Unit weight scheme can be used, noted_i=| N (i) | is v_iDegree, i.e. its adjacent vertex number.

The transformational relation of differential coordinate and cartesian coordinate can be designated as:It can thus be appreciated that the original flute of grid Karr coordinate can be converted to Laplce's coordinate by matrixing.Note A is the adjacency matrix of clothing grid, represent summit it Between topological relation:

D is the diagonal matrix of Vertex Degree: then can get cartesian coordinate to the transition matrix L=I-D of Laplce's coordinate^{- 1}A.Assume that matrix is L=(L_ij), then:

Therefore, the matrixing of Laplce's coordinate and cartesian coordinate can be expressed as: Δ=LV, wherein:

$\mathrm{\Δ}=\left(\begin{array}{c}{\mathrm{\δ}}_1\\ {\mathrm{\δ}}_2\\ .\\ .\\ .\\ {\mathrm{\δ}}_n\end{array}\right)=\left(\begin{array}{ccc}{\mathrm{\δ}}_1^{\left(x\right)}& {\mathrm{\δ}}_1^{\left(y\right)}& {\mathrm{\δ}}_1^{\left(z\right)}\\ {\mathrm{\δ}}_2^{\left(x\right)}& {\mathrm{\δ}}_2^{\left(y\right)}& {\mathrm{\δ}}_2^{\left(z\right)}\\ .& .& .\\ .& .& .\\ .& .& .\\ {\mathrm{\δ}}_n^{\left(x\right)}& {\mathrm{\δ}}_n^{\left(y\right)}& {\mathrm{\δ}}_n^{\left(z\right)}\end{array}\right),$

$V=\left(\begin{array}{c}{v}_1\\ v_2\\ .\\ .\\ .\\ v_n\end{array}\right)=\left(\begin{array}{ccc}{x}_1& y_1& z_1\\ x_2& y_2& z_2\\ .& .& .\\ .& .& .\\ .& .& .\\ x_n& y_n& z_n\end{array}\right)---\left(1.4\right)$

In order to keep local geometric details, should ensure that the Laplce's coordinate after deformation is constant, then have:

Δ=LV ' (1.5)

Just can solve the cartesian coordinate of grid after deformation according to this equation, but rank of matrix is n-k, n is clothing net The quantity on lattice summit, k is UNICOM's amount of clothing grid, and generally k is equal to 1, less than number n of equation variable, therefore side Journey is unusual, can obtain infinite multiple solution.In order to grid accurately solves after trying to achieve deformation, need at least fixed deformation clothing net One of lattice point.The fixing point coordinate of note grid is u_i, then have:

v′_i=u_i, i ∈ { m ..., n}, m ＜ n (1.6)

The energy of original mesh and editor's grid thus can be set up by the constraint of Laplce's coordinate grid and control point Amount error equation:

Above formula can utilize the method for least square solution, solves free point v '_i, i ∈ 1 ..., m-1}, after being edited Clothing grid.And then the matrix equation that reconstruction clothing grid is after determining fixing point is updated to:

$\left(\begin{array}{c}\mathrm{\Δ}\\ {\mathrm{\ωu}}_{m:n}\end{array}\right)=\left(\frac{L}{{\mathrm{\ωI}}_{n-m}|0}\right)V^{\′}---\left(1.8\right)$

Wherein ω is the weight of distortion of the mesh influence degree around dominating pair of vertices on clothing grid.Matrix Calculating for convenience Solve, note

Then formula becomes:

Δ '=L ' V ' (1.9)

Observing above formula, L ' matrix is that the adjacency matrix based on grid vertex is set up, and is therefore a huge sparse matrix. In order to accelerate solving of sparse matrix.Need to matrix normal Wishart distribution, then carry out positive definite decomposition.Therefore, in above-mentioned both members Take advantage of (L ') simultaneously^T, and remember (L ')^TL '=A, (L ')^TΔ '=B, the matrix equation after final adjustment is:

AV '=B (1.10)

The equation can utilize Cholesky decomposition method, LDLT decomposition method or LLT decomposition method to solve.

2. the triangle gridding energy field based on gradient field represents and rebuilds

Owing to differential operator can retain the local detail of geometry to a great extent.Therefore, important by introducing two Differential operator: gradient operator and divergence operator.

Gradient operator is defined in the Grad of piecewise function on clothing surface mesh, be on surface mesh one important Vector field.On three summits of clothing triangle gridding, one piecewise function f of each definition, makes this function meet f (v_x)=f_x, when From v_xDistance is more remote, f (v_x) value less.

As shown in fig. 2 be any one triangle Δ V on clothing grid_iV_jV_k, for any point given in figure V, can be drawn by the three of triangle vertex interpolation:

For the function on clothing triangle gridding summit, noteIt is respectivelyGradient shape Formula, then gradient profile (3.17) can be written as:

BecauseFor barycentric coodinates, then must meet character:SoReplace in (3.18)I.e. available:

WithSize be respectively equal toWithDirection is respectivelyWithFor black arrow in figure The opposite direction of direction.Above formula can be write as further:

$\▿f\left(v\right)=(f_j-f_i)\·\frac{b_j{v}_j}{{\left|b_j{v}_j\right|}^2}+(f_k-f_i)\·\frac{b_k{v}_k}{{\left|b_k{v}_k\right|}^2}---\left(1.14\right)$

f_i、f_j、f_kReplace by three apex coordinate values of clothing mesh triangles shape respectively, can obtain:

$\▿f\left(v\right)=(v_j-v_i)\·\frac{b_j{v}_j}{{\left|b_j{v}_j\right|}^2}+(v_k-v_i)\·\frac{b_k{v}_k}{{\left|b_k{v}_k\right|}^2}---\left(1.15\right)$

Thus, it is possible to obtained the gradient operator of certain tri patch on clothing grid.

On the basis of gradient operator, the divergence operator of discrete point on clothing grid can be calculated, in vertex v_iPlace dissipates Degree operator can be expressed as:

$divw\left(v_i\right)={\mathrm{\Σ}}_{T_k\∈N\left(i\right)}\▿B_{ik}\·w\·\left|T_k\right|---\left(1.16\right)$

N (i) is v_iSingle order adjacent triangular faces, | T_k| it is the area of tri patch in set,Represent triangle T_kIn Vertex v_iThe Grad of piecewise linear function.Therefore, the divergence of this point is determined by the first order neighbors geometry of surface mesh Fixed.

Poisson's equation is the core of mesh reconstruction, can be described as:

${\▿}^{2}f=\▿\·w---\left(1.17\right)$

Wherein, w is the normal vector field of guiding being defined on surface mesh,It is expressed as the divergence of this vector field, Ke Yiyou The gradient operator of a upper trifle and divergence operator computational methods obtain, and f is unknown grid scalar function to be solved,Represent clothing The Laplace operator of thing grid, calculates taking the upper one Cotangent weight saving.

By the vector field of Poisson's equation and clothing surface mesh, the energy error equation that can obtain Poisson's equation is:

Also with the method for least square solution the error diffusion of local to the overall situation, thus obtain the coordinate on free summit Position.It is represented by following sparse matrix form based on the Poisson's equation of differential domain:

The method for solving taking same Laplce's coordinate i.e. can obtain rebuilding grid.

Input is low resolution skirt sub-grid, and the manipulation chosen some number is 4, and the number of fixing point is 372 (figures Point on middle clothes).If Fig. 3 is original mesh and the view of editor's grid.

Observe Fig. 4, based on the editing area deformation transitions smooth of Laplce's coordinate grid, the portion bigger to border curvature Divisional processing relatively good, and in the bigger part of skirt net boundary curvature, gradient field occurs in that grid local distorts by a small margin. And deformation Amplitude Ratio Laplce's coordinate method that grid produces near editing area is greatly, smooth degree is not so good as Laplce's coordinate It is good that method keeps.

Two kinds of methods are respectively provided with the characteristic keeping geometric detail, owing to gridding edition is associated with net dimensional Cartesian coordinates Lattice domain neighborhood property, is set up new representation, and is solved by least square method, and the error of edit operation is expanded uniformly The all free point regions dissipating, therefore can produce comparatively ideal deformation results.But have one based on gradient field grid processing method Fixed limitation:

(1) requirement solving due to gradient field grid, this processing method can not solve degeneration grid.So, for some The grid that simplifies not deleting deteriorations will be unable to obtain correct edit model.

(2) owing to the reconstruction of gradient field is based on Poisson's equation, the equation solve the topology depending on grid vertex Relation, so cannot obtain correct solution for the grid with dynamic relationship of topology.

Contrasting two kinds of differential domain method for solving, Laplce's coordinate is only relevant with the annexation on summit, does not have geometry Correlation, and the method for solving of gradient field has geometric relativity.So, for the clothing gridding edition with the non-intellectual of geometry Operation, uses Laplce's coordinate analog simulation method without geometric relativity to be more suitable for.And Laplce's coordinate Method is relatively more flexible, can add different constraint equations and obtain different deformation results.Therefore, will use based on Laplce The basic framework of the clothing fold analog simulation of coordinate.

2nd, the improvement based on Laplce's coordinate method

In order to seek to keep the scheme of invariable rotary shape in process of reconstruction, look first at and drawn based on drawing general by a upper chapter Lars conversion and the energy error equation at control point:

Equation of n th order n will be improved by the present invention, and main solution is for each vertex v_iNew net after clean up editing An approximate transform T is calculated on lattice_i, ensure scaling and rotate indeformable.Because T_i(V ') is the function of V ', it is possible to Obtain energy error equation:

Note T_iIt is all unknown with V '.If but coefficient T_iIt is the linear function of V ', then solve V ' and mean Arrive T_i, therefore E (V ') can only have the quadratic equation of V ' as known variables.

Need now to seek a kind of T_iSolution, can be from vertex v_iObtain with its first order neighbors, then according to original mesh The error equation being defined as follows with target gridding:

${\min }_{T_i}\left(\right||T_i{v}_i-v_i^{\′}|{|}^2+{\mathrm{\Σ}}_{j\∈N_i}\left|\right|T_i{v}_j-v_j^{\′}\left|{|}^2\right)---\left(1.22\right)$

Owing to above formula is secondary, the minimum of expression formula is the linear equation with regard to V '.But, if T_iUnfettered limit System, then the natural minimum of a value of E (V ') would is that, without square solution, all of geometric detail will be lost.Therefore, it should close with one The mode of reason retrains T_i.Because T_iNeeds comprise translation, rotate, the characteristic of scaling.Especially it should be noted that a bit, it should forbid Anisotropic scaling, otherwise can eliminate the normal component of Laplce's coordinate.

T_iTranslating sections can be obtained by homogeneous coordinates.Its linear segment should meet following condition: change represents Object construction should be a linear equation, but constraint is existed to the scaling and rotation of isotropic, represent each to together The matroid of the scaling of property and rotation can be described as:

T=s exp (H) (1.23)

Wherein H is an antisymmetric matrix.In three-dimensional coordinate, the product of antisymmetric matrix and certain vector can be by vector Amass and represent, as shown in (4.20) formula:

Hx=h*x； (1.24)

In conjunction with some other attributes of 3*3 skew matrix, following exponential expression can be obtained:

S exp H=s (α I+ β H+ γ h^Th) (1.25)

In (4.21) formula of observation, discovery is for unknown quantity s and h, and only I, H and s is linear, and h^TH is secondary, is The approximation that takes to restrained deformation class, order:

$T_i=\left(\begin{array}{cccc}s& -h_3& h_2& t_x\\ h_3& s& -h_1& t_y\\ -h_2& h_1& s& t_z\\ 0& 0& 0& 1\end{array}\right)---\left(1.26\right)$

Above-mentioned matrix is the linear approximation rotationally-varying to clothing.

Thus can be explicit write out the T being defined on V '_iLinear dependence.Note (s_i, h_i, t_i) T is T_iUpper the unknown to Amount, then can be obtained by minimizing equation:

||A_i(s_i, h_i, t_i)T-b_i||² (1.27)

Wherein A_iContain original mesh vertex v_iCoordinate and the coordinate of first order neighbors, b_iContain target gridding summit v_iCoordinate and the coordinate of its first order neighbors.It is expressed as follows respectively:

$A_i=\left(\begin{array}{ccccccc}{v}_{k_x}& 0& v_{k_z}& -v_{k_y}& 1& 0& 0\\ v_{k_y}& -v_{k_z}& 0& v_{k_x}& 0& 1& 0\\ v_{k_z}& v_{k_y}& -v_{k_x}& 0& 0& 0& 1\\ .& & & & & & \\ .& & & & & & \\ .& & & & & & \end{array}\right),k\∈\left\{i\right\}\∪N_i---\left(1.28\right)$

$b_i=\left(\begin{array}{c}{v}_{k_x}^{\′}\\ v_{k_y}^{\′}\\ v_{k_z}^{\′}\\ .\\ .\\ .\end{array}\right),k\∈\left\{i\right\}\∪N_i---\left(1.29\right)$

(1.22) the linear least-squares solution described by formula can be solved by following formula:

(s_i, h_i, t_i)^T=(A_i ^TA_i)^-1A_i ^Tb_i (1.30)

It can thus be appreciated that T_iCoefficient (s_i, h_i, t_i)^TShow as b_iLinear function.

According to rotating and the control matrix of scaling, just can be able to keep in a process of reconstruction scaling of clothing grid with Rotational invariance.

The input grid of the present invention is longuette sub-grid model, and this grid has 1249 summits, 3627 limits and 2386 Face.It is original mesh and editor's grid of this grid shown in Fig. 5.

It is former Laplce's coordinate method and the improvement Laplce's coordinate keeping scaling and rotational invariance respectively such as Fig. 6 Method, both calculates under Cotangent weight.The former shows the lofty deformation knot of comparison at editing area Really, the lower end of train of dress occurs in that obvious contraction, although can keep the geometric detail of local, but and real clothing shape Becoming has very big difference.And improve after Laplce's coordinate method represent at train of dress better truly, editing area Soft excessively.And in small angle rotation, illustrate good clothing fold edit effect.Therefore, keep rotating and scaling not The computational methods of denaturation have had very big improvement, make lower place mat for further research.

3rd, the research of clothing fold emulation mode and the realization for different materials

Geometric detail can well be kept based on the energy field of differential coordinate, but for the same once editor behaviour of user Making, the deformation effect drawing is unique.In order to different draping effects can be produced for same edit operation, i.e. emulate The draping effect of different materials clothes, will add the constraint equation on summit successively, the constraint equation on limit and the pact of adjacent dihedral angle The characteristic that Shu Fangcheng control clothes recovers, stretches and bend.

(1) the energy constraint equation based on point

According to the requirement of clothing grid deformation, the geometry tolerance of clothing recovery characteristics upon compression and the vertex position of grid Closely related, i.e. the position on each summit on mesh triangles dough sheet there occurs change.Grid vertex position after deformation should Approach the vertex position of initial mesh, therefore, after optimizing, need between the clothing target gridding deforming upon and initial mesh Define an energy constraint, react the recovery characteristics that true clothing is had.Point constraint mainly has two purposes, and one is Keeping the positional information of original mesh fixed vertices, another is to keep the movement to control point of user to become in each deformation Gesture.Therefore the energy constraint equation based on summit can be obtained:

$E_v={\mathrm{\Σ}}_{i=0}^n\left|\right|v_i^{\′}-v_i|{|}^2---\left(1.31\right)$

This formula controls current grid vertex v_i(x_i, y_i, z_i) move closer to target gridding vertex v '_i(x′_i, y '_i, z′_i), the summit gap of original mesh and target gridding can be calculated by Euler's formula and obtain.

If only add the energy constraint on summit to differential domain grid method, skirt draping effect can be obtained, and do not meet True effect after grid skirt editor.Accordingly, it is considered to add the energy constraint to the grid length of side and dihedral angle on this basis Equation.

(2) the class Laplce's characteristic energies constraint equation based on the length of side and dihedral angle

If the only constraint equation on summit, without the constraint equation of edge lengths and angle, deformation result only can produce Class film effect.It according to clothes and two most basic characteristics of cloth is: stretching and bending.Therefore the energy of the length of side can be set up Constraint equation controls clothing level of stretch, sets up the degree of the energy constraint equation control clothing bending of dihedral angle.Analogy below The derivative characteristic of some constraint provides the class Laplace's equation of the length of side and dihedral angle restraints.

As shown in Figure 7, it is assumed that given limit e_ij, be l and m be common edge e respectively_ijThe 3rd point of two triangles Limit e can be calculated_ijNormal direction:

$n_{ij}=\frac{A_l{n}_l+A_m{n}_m}{\left|\right|A_l{n}_l+A_m{n}_m\left|\right|}---\left(1.32\right)$

Wherein A_l, A_mAnd n_l, n_mIt is area and the direction of the triangle on two shared same limits respectively.Finally, for limit e_ijObtain class Laplce's character:

Wherein p_ij=α v_i+(1-α)v_j, q_ij=β v_l+(1-β)v_m.Factor alpha and choosing of β make σ_ijIt is parallel to the normal on limit n_ij.In equation (1.33), ρ_ijDesign be the length in order to control limit, σ_ijIt is to control the two of adjacent two triangles Face angle.

Being similar to the method based on summit, newly-increased two energy equations, are utilized respectively the length of side and dihedral angle controls local Stretching and degree of crook:

Here ρ '_ijWith σ '_ijBy matrix T_ijConvert, i.e. ρ '_ij=T_ijρ_ij, σ '_ij=T_ijσ_ij.In order to generate deformation Grid M '=(V ', E), need to solve the optimal solution of equation below with the least square thought:

min_V′λE_p+μE_σ (1.35)

In above formula, λ and μ is the length of side and the weight of dihedral angle control.It is hereby achieved that with regard to the total energy in point, limit and face Amount constraint equation:

min_V′ωE_v+λE_p+μE_σ (1.36)

Owing to introducing the energy constraint equation on limit and dihedral angle, overall clothing energy equation is quadratic nonlinearity side Journey, generally uses iterative scheme to solve.In formula (1.36), it is assumed that total n free summit, then there is 3n unknown change Amount, corresponding apex coordinate is (x_i, y_i, z_i), then these variablees can pass through a column vector X=[x₁, y₁, z₁, x₂, y₂, z₂..., x_n, y_n, z_n]^TRepresent, X contains x, the coordinate information of y, z.Therefore, three energy equations can be described as closing Following geometrical constraint in X column vector:

$\left\{\begin{array}{cc}\mathrm{\μ}\left(v_i\right)=v_i^{\′}-v_i=0& i\∈\[1,n\]\\ \mathrm{\ρ}\left(e\right)=l\left(e_{ij}^{\′}\right)-l\left(e_{ij}\right)=0& i,j\∈\[1,n\]\\ \mathrm{\σ}\left(v_i{v}_j\right)=\mathrm{\θ}\left(e_{ij}^{\′}\right)-\mathrm{\θ}\left(e_{ij}\right)=0& i,j\∈\[1,n\]\end{array}\right.---\left(1.37\right)$

(1.37) equation group levoform is all the function of X.Assume X₀For the solution under current iteration, each for reducing integral energy In iteration, the difference of variable change is designated as Δ X, then have X '=X₀+ Δ X, X ' are for reducing the optimization solution of integral energy.Repeat this mistake Journey, until obtaining the final solution meeting condition.

Owing to (1.37) formula containing nonlinear terms, will utilize Taylor expansion that nonlinear terms are linearized.Due in order to Making iteration result converge to fixed value, Δ X should keep less change in each iterative process, does not use Taylor expansion And directly launch to non-linear, the reduction of solution efficiency can be caused.Below the linearisation of opposite side energy constraint equation is entered The detailed elaboration of row.

If f (x) is the functional value under current iteration, needs to reduce error variance ε to a little variable, optimize letter Number f (x₀+ ε), then can be calculated by Taylor's formula:

f(x₀+ ε)=f (x₀)+εf′(x₀)+O(ε²) (1.38)

Assuming a length of l (e) of limit e in grid, its length is by two vertex v_i1=[x_i1, y_i1, z_i1] and v_i2=[x_i2, y_i2, z_i2] determine, l (e)=| | v_i1-v_i2||.Method owing to processing three reference axis is similar to, it is contemplated that the situation of Z axis.False If variable z_i1And z_i2It is updated to z respectively_i1+Δz_i1And z_i2+Δz_i2.The amount of the change at Z axis for so ρ (e) can be expressed as Under Taylor expansion:

$\mathrm{\ρ}({\mathrm{\Δz}}_{i1}-{\mathrm{\Δz}}_{i2})=\mathrm{\ρ}\left(0\right)+\frac{(z_{i1}-z_{i2})}{l\left(e\right)}({\mathrm{\Δz}}_{i1}-{\mathrm{\Δz}}_{i2})+O\left({\left|{\mathrm{\Δz}}_{i1}-{\mathrm{\Δz}}_{i2}\right|}^2\right)---\left(1.39\right)$

In the ordinary course of things, (Δ z_i1-Δz_i2) value be much smaller than Δ z_i1With Δ z_i2Value.So (Δ z can be utilized_i1- Δz_i2) replace Δ z_i, and ensure that the correctness of algorithm.Therefore, the equation of linear expansion in z-direction can be expressed For:

$({\mathrm{\Δz}}_{i1}-{\mathrm{\Δz}}_{i2})\frac{(z_{i1}-z_{i2})}{l\left(e\right)}=l_0\left(e\right)-l\left(e\right)---\left(1.40\right)$

So (Δ z_i1-Δz_i2) can be expressed as:

$({\mathrm{\Δz}}_{i1}-{\mathrm{\Δz}}_{i2})=\frac{\left(l_0\right(e)-l(e\left)\right)l\left(e\right)}{(z_{i1}-z_{i2})}---\left(1.41\right)$

(1.41) formula is that to remove after quadratic term be with regard to Δ z_i1With Δ z_i2Linear equation.X-axis and the processing mode of Y-axis Identical with Z axis, again by Taylor expansion, nonlinear terms are linearized.After obtaining linear equation, three equation group are set Value be 0, then constant term is moved to the right-hand member of equation, can obtain following sparse matrix:

$AX=\left(\begin{array}{c}\mathrm{\ω}\[P\]\\ \mathrm{\λ}\[L\]\\ \mathrm{\μ}\[A\]\end{array}\right)X=b---\left(1.42\right)$

Wherein X=[Δ x₁, Δ x₂, Δ x₃..., Δ x_n], P is n₁* the matrix of 3n, n₁Being the number on summit, L is n₂*3n Matrix, n₂Being the number on limit, A is n₃* the matrix of 3n, n₃It is the number of required calculating dihedral angle.B is (n₁+n₂+n₃) * 1 row Vector.Three weight factors ω, λ, μ are then specified by user.

According to energy equation min_V′ωE_v+λE_p+μE_σ, by regulation weight parameter ω, λ and μ controls clothing fold not Same effect, thus simulate the draping effect of different materials.The ability that wherein ω recovers after representing clothes compression, λ represents clothes The ability of stretching, the ability of μ control clothes bending.

From general knowledge, when applying power on same dress, silk cloth can produce due to the programming structure of its softness Raw fold significantly, contrary denim fabric is then less on peripheral region impact, and draping effect is inconspicuous.And cotton cloth produces Raw draping effect between, produces the bending of a certain degree of fold.In order to simulate the draping effect of three kinds of clothes, this A region at longuette grid is carried out fold deformation by literary composition, by the weight of regulation difference constraint, reaches different materials clothing The simulation effect of fold.

Sense of reality view and gridding edition view for input as shown in Figure 8 and Figure 9.

This grid has 4886 summits, 14432 limits and 9544 dough sheets.In the editor's grid shown in Fig. 9, control point Being 15 (figure middle conductors), fixing point is 1335 (point above clothing and the centers of circle in (b) figure in figure).Then by adjusting Joint parameter, sets three groups of contrast experiments, parameter ω, and size and the corresponding effect of λ and μ refer to Figure 10.

As shown in Figure 10, by identical editor and the control of different weights, skirt creates not in the Editorial Services of train of dress Same draping effect.The weight taking summit energy equation in experiment is 1, when the weight of length of side energy equation is bigger, and clothing net The shape of lattice and initial mesh closer to, therefore stronger to the capability of influence around put, create such as the first row denim in table The fold deformation effects of material.When the weight of length of side energy equation gets over hour, clothing distortion of the mesh region is closer to editor's grid, right Around the influence power of point is more weak, creates the fold deformation effects as the third line woven dacron even silk in table.When length of side energy The weight of amount equation is between the two, and the coverage to surrounding for the fold is also between first two situation, therefore can produce Raw such as the fold deformation effects of the second row purified cotton cloth in table.Therefore, the weight by adjusting mesh error equation can emulate The Folding Deformation effect of different materials.

Claims (3)

1. the emulation mode based on the different materials clothing fold of differential domain, it is characterised in that realized by following steps:

(1) it based on Laplce's coordinate: first read original clothing grid data, is original clothing net according to clothing grid data The cartesian coordinate of lattice can obtain the Laplce's coordinate set up；Then keep Laplce's coordinate constant, to original mesh Entering edlin, obtaining setting up the energy error equation of original clothing grid and editor's grid, solving equation obtains the clothing rebuild Grid；Based on gradient field coordinate: the first cartesian coordinate according to original clothing grid and gradient field, it is achieved gradient field coordinate Represent, edlin is entered to original mesh, obtain editing grid, by the gradient field coordinate obtaining and editor's grid, set up Poisson side The energy error equation of journey, solving equation obtains the clothing grid rebuild；The clothing grid of two kinds of different coordinates reconstructions carries out reality Test contrast with analyze, finally give be more suitable for clothing emulation differential domain solve scheme；

(2), on the basis of Laplce's coordinate in step (1), use improved Laplace method, maintain clothing three The local geometric details of angle grid, it is ensured that clothing keeps correctness and the authenticity of deformation under rotation process；

(3) in step (2) maintain clothing triangle gridding local geometric details in the case of, for the clothing of different materials Thing, to recovering, stretching different with the stressing conditions of bending, is set up based on point, limit, the grid bound energy equation of dihedral angle, and Simulate different draping effects under same operation for the clothing of different materials by adjusting the weight parameter in bound energy equation.

2. the emulation mode of the different materials clothing fold based on differential domain according to claim 1, it is characterised in that: institute State the improved Laplace method detailed process of step (2) as follows:

(21) be on the new grid in the edited of each summit calculating one approximate transform, ensure scaling and rotate constant Property；

(22) according to the approximate transform obtaining and energy error equation, set up new energy error equation, and be converted to rectangular Formula, for sparse matrix solve rebuild after optimize grid.

3. the emulation mode of the different materials clothing fold based on differential domain according to claim 1, it is characterised in that: institute State step (3) to set up based on point, limit, the grid bound energy equation of dihedral angle, and by adjusting bound energy equation weight Different draping effect simulation under same operation for the clothing of parameter simulation different materials is as follows:

(31) energy equation, the recovery characteristics after control clothes compression are set up by the vertex position of triangle gridding；

(32) set up energy equation by the length of side of triangle gridding, control clothes characteristic after the stretch；

(33) energy equation, the characteristic of control clothes bending are set up by the dihedral angle of the adjacent surface of triangle gridding；

(34) different weights is set to above three energy equation, simulates difference under identical active force for the different materials clothing Clothing fold tells on.