CN106683169A - Sparse local decomposition and restructuring algorithm of joint motion sensing - Google Patents

Abstract

The invention discloses a sparse local decomposition and restructuring algorithm of joint motion sensing; the method includes steps of 1), calculating length of sides in a triangular mesh and size of dihedral angles corresponding to sides (the two values are recorded as LA); 2), calculating residual difference of LA vector in every triangular mesh; 3), performing sparse local decomposition on the residual difference of the LA vector and acquiring an LA base; 4), linearly mixing the LA base and acquiring a new LA vector; 5), recovering a connection mapping; 6), estimating the bid price of local orthogonality; 7), restructuring the triangular mesh. The algorithm mainly solves the problem how to perform the sparse local decomposition on a given movement mesh sequence, thereby capturing deformation of a local zone of a movement sequence; moreover, by taking it as the basis, an integrated treatment framework capable of capturing the movement zone and performing gesture edition of the triangular mesh model is structured. The sparse local decomposition and restructuring algorithm is better in locality, applicable to large-scale rotating movement and deformation, overall rotation, and good in robustness.

Description

Sparse exploded and restructing algorithm that a kind of joint motions are perceived

Technical field

The present invention relates to computer graphics and three-dimensional grid editor field, a kind of the dilute of joint motions perception is referred in particular to Dredge exploded and restructing algorithm.

Background technology

Dynamic geometry or animated Mesh sequence are the triangular mesh with identical annexation, usual these triangles Grid have recorded different attitudes of the object in a time series.This solid all right and wrong in animation and 3D game It is often universal.A succinct representation to this triangular mesh sequence is obtained, and supports to exist the editor of solid It is a very important problem in dynamic geometry treatment.

Traditional mode generally catches given triangle using succinct structure as such as skeleton or cage (Cage) The operation information of shape grid sequence, and triangular mesh reconstruct is carried out by using covering technology or broad sense quality coordinate. Although these methods can well compress triangular mesh sequence, want by operate these skeletons or cage come Build the triangular mesh with new attitude extremely difficult.

Each frame of triangular mesh sequence is considered as sparse local shape based method the change of average shape triangular mesh Shape, and extract one group of local shape base that can capture posture deforming.By this group of parameter value of shape base of editor control To constitute the combination of different shape base, so as to reconstruct the triangular mesh model with different attitudes.This method is raising eyebrow Hair, improves the facial expression aspect such as corners of the mouth and is demonstrated by powerful ability.However, what is utilized due to this group of shape base is non-locality Changes in coordinates, this method reconstruct rotating operation on a large scale when, easy bad student's artifact.Huang et al. is attempted by triangle The deformation gradient (DG) of shape patch grids spatially carries out sparse exploded and represents and joint motions filling local linear The breach of nonlinear characteristic.Although this mode can to a certain extent represent rotary motion, due to DG change it is non- Locality, its edit capability is limited by very large.The change of DG has non-locality, because firm when triangular mesh When property rotates partially around joint, all of DG can have similar change on rigid element.Additionally, special as non-intrinsically safe Levy, the conversion of DG depends on global rotation transformation.Therefore, this method can not process the animation sequence with global rotation.Most Afterwards, each deformation gradient is defined by 9 parameters, when a triangular mesh has N_F9N is accomplished by during individual face_FIndividual parameter carrys out table Show an attitude.This just substantially increases the process time of local Its Sparse Decomposition.

The above method major defect has：(1) succinct mode can be compared to represent motion sequence triangular mesh Model, but cannot but provide more succinct mode edits the attitude of triangular mesh；(2) edlin can be entered, but Rotating range cannot be processed than larger athletic posture；(3) represent that the data volume of grid sequence is big, calculate complicated.

The content of the invention

Shortcoming and deficiency it is an object of the invention to overcome prior art, there is provided it is sparse that a kind of joint motions are perceived Exploded and restructing algorithm, can exactly capture the position that rotates motion and reconstruct and provide rotation fortune on a large scale Dynamic triangular mesh attitude mode, can easily carry out attitude editor and with more preferable effect and computational efficiency.

To achieve the above object, technical scheme provided by the present invention is：The sparse part point that a kind of joint motions are perceived Solution and restructing algorithm, comprise the following steps：

1) calculate triangle gridding in while length and while corresponding dihedral angle size

Giving a series of dynamic these models of triangular mesh sequence with different attitudes all has identical topology knot Structure, first calculate respectively each bar on each triangle grid model while length and while corresponding to dihedral angle size (the two values It is designated as LA)；

2) residual error of each triangle grid model LA vectors is calculated

To step 1) on a triangle grid model obtaining each bar while corresponding LA according to while order be arranged to make up One LA vector, calculates the LA vectors of each triangle grid model, is made the difference with the LA vectors with reference to triangle grid model respectively, When corresponding LA vector residual errors；

3) LA bases are extracted

To step 2) in obtain each model LA vector residual error carry out sparse exploded, obtaining one group can catch To the LA bases of local joint motion, each LA base controls one piece of region shape on triangle grid model；

4) linear hybrid LA bases obtain new LA vectors

To step 3) in the LA bases that can capture local joint motion that obtain, edit corresponding parameter, obtain new LA vector residual errors, then make and calculate with the LA vectors with reference to triangle grid model, obtain new LA vectors；

5) connection mapping is recovered

To step 4) in the new LA vectors that obtain carry out the local coordinate value on summit on each tri patch and calculate, obtain The local coordinate value on each summit in local coordinate system lower panel, and then recover the company of each tri patch on triangle grid model Connect mapping；

6) local orthogonal marked price is estimated

To step 5) in the connection mapping relations of triangle surface that obtain, recycle and minimize marked price reconstruction of function and estimate Calculate the local orthogonal frame of triangle grid model；

7) triangle gridding is reconstructed

To step 6) in the orthogonal marked price of office that obtains, each is estimated in triangle grid model using reconstruction of function is minimized The position on summit, obtains the triangle grid model with new attitude after the meeting.

In step 1) in, it is described calculate in triangle gridding while length and while corresponding dihedral angle size (the two values It is designated as LA), it is specific as follows：

Assuming thatIt is the triangle grid model data set with different attitudes for giving, N_PTable Registration according to concentrate triangle grid model number,Represent k-th triangle grid model.Each triangle grid model surface Be by closely coupled triangular into.E and F represent line set and tri patch set in triangle grid model respectively, and And the number on side is N_E, the number of tri patch is N_F.These triangle grid models have identical annexation, i.e., all of Triangle grid model has identical line set and tri patch set.Travel through tri patch on summit when, we be with Direction counterclockwise is carried out.

Assuming thatOn a line e=(i, j) ∈ E, share this edge tri patch for τ=(j, h, i) and υ=(g, J, i), the letter in bracket, to constitute this tri patch the Vertex Labeling, is sequentially sequence counter-clockwise.When from this common edge Vertex v_iGo to v_jWhen, triangle surface τ be located at left-hand side, tri patch υ be located at right-hand side, so τ and υ be referred to as side (i, J) left tri patch and right tri patch.When calculating the length of e, the Euclidean distance between two summits is used, i.e., l_ij=| | v_j-v_i||₂.θ represents the angle of the normal of tri patch τ and υ outward direction, i.e. θ=acos (n_τ·n_υ), wherein n_τWith n_υThe normal of direction outward direction on respectively tri patch τ and υ, then the corresponding dihedral angle of common edge be defined as

A LA values (l achieved above_ij,φ_ij)。

For data setUpper all of triangle grid modelThe LA values on its each side are calculated, and according to side Number order arrangement may be constructed a 2N_EThe vector of rowThat is LA vectors.

In step 2) in, the LA vector residual errors of described each triangle grid model of calculating, specific method is：In step 1) In obtain the LA vectors of each triangle grid model after, the LA vectors respectively with first grid model are poor, draw LA to The residual error of amount, i.e. { D_k=X_k-X₁, k=1,2 ..., N_p}。

In step 3) in, described extraction LA bases, concrete mode is as follows：

Using step 2) in obtain LA vector residual error, construct a 2N_E×N_pMatrix D=(D₁,...,D_Np).Will Matrix D resolves into D=C × W.Wherein Matrix C is a 2N_EThe LA basic matrixs of × K, each row in matrix all represent a LA Each row in base, i.e. matrix all control the shape in region in triangle grid model.Here K represents the individual of LA bases Number, can be sized, if defining K value is that triangle grid model is roughly divided into K region as needed.And matrix W is one Individual K × N_PWeight matrix, the every a line in matrix represents a hybrid weight sequence.For the ease of the attitude to triangle gridding Enter edlin, it is necessary to as much as possible keep LA bases locality, i.e., each LA base control region between mixing portion as far as possible Ground is few, and needs normalization hybrid weight to eliminate due to scaling the influence to attitude.Come here by function (1) is minimized Carry out matrix decomposition.

Wherein first energy termFor the error that constraint matrix is decomposed, and second energy Quantifier is used to adjust the locality of LA bases in Matrix C, the i.e. magnitude range in the region of each LA base control.

Second energy term is specific as follows：

Wherein Λ_ikRepresent triangle grid model on i-th while kth bar as in shape base while midpoint between geodetic Linear distance, is defined as follows：

Its detailed process is：K bars side is selected first on triangle grid model as the K center of shape base, referred to as in Heart side, is represented with Edge here, edge_kRepresent kth bar center edge.Use e_iThe side that the numbering on triangle grid model is i is represented, The then d in above formula_ikRepresent e_iAnd edge_kResult after the geodesic curve range normalization at midpoint, normalization here is actually The midpoint of triangle gridding top and the geodesic curve distance at each center edge midpoint are exactly calculated, maximum therein is taken out and is designated as d_max, the value work that the geodesic curve distance for then calculating the side on triangle grid model is finally obtained in this maximum It is the geodesic curve distance after normalization.Therefore there is d_ik≤1。r_minAnd r_maxFor support radius span for [0,1), its tool Body value sets according to demand.

In whole decomposable process, c_i,kThe data of the i-th rows of representing matrix C kth row, w_kjRepresenting matrix W row ks jth row Value；And have max { w_k,j, k=1 ..., K；J=1 ..., N_PAll elements in }=1, i.e. matrix W, maximum is 1.In reality In testing, the value of η is set to 200.

In step 4) in, described linear hybrid LA bases obtain new LA vectors, and its specific method is：In step 3) in obtain After the LA basic matrixs for taking, the numerical value of any one row, obtains vectorial w in modification W matrixes_k, then Matrix C again with w_kIt is multiplied, obtains One new LA vectors residual error, finally the LA addition of vectors with first triangle grid model (i.e. with reference to triangle grid model) be The LA vectors of the triangle grid model with new attitude can be obtained.I.e. by calculatingNew LA can be obtained Vector.

In step 5) in, described recovery connection mapping, its process mainly includes setting up a new local coordinate system, leads to The LA values for crossing side are calculated two coordinates on each summit of tri patch that a this edge is common edge, and each is then calculated again The orthogonal marked price of part under new local coordinate system of dough sheet, its company is recovered finally by the local orthogonal frame of two dough sheets Details are provided below to connect mapping relations：

From step 4) in obtain LA vectors after, we can recover its annexation to every a line.Assuming that side e= (i, j) is the common edge of left tri patch τ and right tri patch υ, and h, g are respectively the 3rd summit of left and right tri patch, I.e. the left and right tri patch of common edge e=(i, j) is respectively τ=(j, h, i), υ=(g, j, i).A coordinate system is created first (r, s, t), then calculates coordinate value of two each summits of tri patch under this coordinate system.Coordinate system is with vertex v_iMake It is the origin of coordinates and vertex v_jCoordinate value be (l_ij, 0,0), the t axles of coordinate system are same with the normal of tri patch υ outward directions To.Because tri patch υ is the right tri patch of common edge e=(i, j), so vertex v_gR coordinates be necessarily smaller than 0.So V can be calculated_gCoordinate be respectively：

In order to calculate vertex v_hCoordinate value, we first assume that vertex v_hFlattening followed by triangular facet are rotated around common edge Piece υ coplanar vertex v_h', then what is be similar to can calculate vertex v_h'Coordinate value be：

By vertex v_h'φ is rotated around r reference axis_ijVertex v can be obtained after-π_hReference axis.

After two reference axis on each summit of tri patch are obtained, local coordinate newly-built in above process is calculated The orthogonal marked price of part under system.The detailed process of local orthogonal marked price is as follows：

By taking tri patch τ as an example, the orthogonal marked price of part that we are defined on this tri patch is ζ_τ=(u_τ,v_τ,n_τ) its InRepresent on tri patch τ from first vertex v_iPoint to second vertex v_jUnit vector；n_τIt is triangular facet The outside unit normal of piece；v_τ=n_τ×u_τ.Three above unit vector just constitutes ζ_τ3 × 3 matrix structure.

The orthogonal marked price ζ of part on tri patch υ can similarly be obtained_υ.SoAs by under other coordinate systems other Coordinate value under coordinate system is transformed into the transition matrix represented under υ local coordinate systems.

Can just be counted after the orthogonal marked price of part for calculating two tri patch with side e=(i, j) as common edge Calculate the connection mapping Q of this common edge_ij。Q_ijIt is defined as follows：

So far, the connection mapping on side e=(i, j) has been calculated, the connection that can recover all sides with similar mode is reflected Penetrate.

Since when the editor of LA bases is carried out, the length of side for being LA values, i.e., directly changing in triangle gridding for directly changing The dihedral angle size of degree triangle corresponding with side.This is possible to cause the amended length of side to be unsatisfactory for the triangle length of side Formula, i.e., amended side is possible to that tri patch can not be constituted.Need to carry out specially treated in face of these situations：Work as l_gj≥ l_ji+l_igWhen, we unify the length l on side_jiAnd l_igAmplify l_gj/(l_ji+l_ig) times, thus there is l_gj=l_ji+l_ig, at this time Because the direction of side (i, j) is consistent with the direction of side (g, j), the direction of side (g, j) is (1,0,0)；Similarly, l is worked as_ji≥ l_gj+l_igWhen, the direction of side (g, j) is (1,0,0), works as l_ig≥l_gj+l_jiWhen, the direction of side (g, j) is (- 1,0,0).

In step 6) in, described estimation locally orthogonal marked price, its main method is as follows：

Assuming thatIt is the set on the side in triangular mesh, each triangle is estimated by following minimum function (2) The local orthogonal frame of dough sheet：

Wherein the left side is：

Represent connection Mapping Energy.

And the ζ in function (2)₁WithThe orthogonal marked price of first tri patch that needs to estimate and given is represented respectively First orthogonal marked price of tri patch, in order to ensure that the triangle gridding for reconstructing the rigid transformation for coming has uniqueness, we Need the orthogonal marked price of part of given first tri patch；Subscript ' F ' represents Frobenius norms；Q_ijFor step 5) in it is extensive Multiple link mapping.

Step 7) in, described reconstruct triangle gridding is exactly mainly the position on each summit of triangle grid model for calculating new Put, detailed process is as follows：

In step 6) in get local orthogonal marked price after, calculated by majorized function (3) each in triangle gridding The position on individual summit.

ε_ER({v_i, i ∈ I })=ε_EV({v_i,i∈I})+ε_VC({v_i,i∈I}) (3)

Wherein the item on the right is used to constrain the position of given fixing point, and it is specially：

Wherein I^*To give the numbering on fixed summit,It is the position of given fixed vertices.In order to ensure to reconstruct The triangle grid model of the rigid transformation for coming has uniqueness, it would be desirable to which the position for fixing some points is constant, the number of fixing point Amount is adjusted as needed.

First in function (3) is want the vertex position for restrained split-flow with step 6) in the orthogonal marked price of part that obtains The deviation of the inside vector, specially：

Wherein e_gj=-l_gjU, e_ji=l_ji(cos β u-sin β v), e_ig=l_ig(cos α u+sin α v), α is on tri patch Side (g, j) and the angle of side (i, g), and β is the angle of side (g, j) and side (j, i)；U in formula, v are represented by step 6) estimate Corresponding tri patch on orthogonal marked price ζ=(u, v, n) of part on corresponding vector.

The institute of triangle grid model position a little is obtained with after majorized function (3), that is, is reconstructed after editing Triangle grid model with new attitude.

The present invention compared with prior art, has the following advantages that and beneficial effect：

1st, more preferable locality, this is mainly reflected in the positioning to triangle gridding rigid deformation region.Relative to Huang etc. What people proposed spatially carries out sparse exploded to obtain the method (referred to herein as DG methods) of shape base in deformation gradient, this Invention is rotated in treatment has more preferable locality in the positioning of moving region.When grid model is clicked through around a joint During row rigid rotation movement, because the corresponding LA values in the side for closing on joint area can just occur big change, and other are not being closed Less, so the present invention captures the region of this change well very much, this also may be used the LA value changes on the side near nodal region Preferably to generate corresponding LA bases so that more accurate by editing the new triangle grid model of generation, can more tackle around The deformation of joint motions.And this point DG methods can not be processed well.

2nd, deform suitable for rotary motion on a large scale.Because the calculating of deformation gradient is separate, and deformation ladder The angle of rotary motion that degree is adapted to can only be [0, π) between, so the triangular facet in it move triangle grid model Around artis rotary motion, when the angle of rotation is more than π, DG methods will be unable to carry out with the deformation of the larger anglec of rotation piece The extraction of shape base.And it is relative rotation angle between adjacent triangle that the LA values for being in the present invention are represented, and this revolves Gyration is always less than 2 π, therefore, the present invention can catch from the triangle gridding sequence with rotary motion on a large scale well Grasp the region (extracting LA yls) moved.

3rd, rotated suitable for the overall situation.Although global rotary motion is not very related with posture deforming, a sparse office Part solution should also process such case.Due to being the sparse exploded to LA value residual errors in the present invention, and LA has Locality, therefore, when triangle grid model occurs global rotary motion, for example, the whole grid model of horse is around an axle Do global rotary motion and non-self attitude is when deforming, the triangle grid model of horse will not become substantially in attitude Change, and LA vectors corresponding in triangle grid model will not also occur too big change, therefore rational model can be reconstructed. And other method, such as DG methods, because deformation gradient (DG) depends on the overall situation rotationally-varying, therefore there is global rotation in model During transformationization, the change of DG is very big, and this will cause irrational deformation.

4th, with good robustness.This mainly includes that the annexation on a summit in triangle grid model has Lu Bangxing There is robustness with the selection to the corresponding area size of LA bases.The annexation on present invention summit in tri patch is processed When, a connection mapping algorithm is devised, this algorithm can well keep the annexation between summit.Therefore, in reconstruct During triangle grid model attitude after editing on a large scale, there is good robustness in the annexation on summit.At this In the experimental result of invention, the selection to the corresponding area size of LA bases can well navigate to the region of deformation, and have Good locality.

Brief description of the drawings

Fig. 1 is algorithm flow chart of the invention.

Fig. 2 is the structure of tri patch in intermediate cam grid model of the present invention, and part therein is orthogonal to mark the price and two faces Angle.

Fig. 3 is the present invention to doing after local rotary motion triangle grid model carries out LA base extractions, the area that LA bases are controlled Domain visualizes schematic diagram.Figure (a) therein is the result to carrying out the iron model extraction LA bases of flexural deformation, the LA bases of extraction The position of bending has been navigated to well, and other need not then carry out extra treatment for the rigid position of deformation；Figure (b) exists The triangle grid model that the horse of deformation is rotated to afterbody extracts the result of LA bases, and the LA bases of extraction have navigated to tail base The region for rotating；Figure (c) is the result that LA base extractions are carried out to the robot for moving, and the result of extraction has navigated to hair The ancon on hand of raw rotary motion；Figure (d) is the result that the facial model changed to facial expression carries out LA base extractions, The result of extraction also can well navigate to the position for deforming.

Fig. 4 is structural representation of the present invention in the recovery process for being attached relation, the local coordinate that figure (a) is set up The distribution situation schematic diagram on each summit in system and tri patch, figure (b), (c), (d) are that LA values are passed through after editor, three sides Length violation closes the situation schematic diagram of triangle inequality, wherein figure (b) represents l_gj≥l_ji+l_igSituation, figure (c) represent l_ji ≥l_gj+l_igSituation, figure (d) represent l_ig≥l_gj+l_jiSituation.

Fig. 5 is example of the present invention when the apex coordinate of tri patch is estimated.

Fig. 6 is the result and the edited result to attitude that the present invention is extracted to motion sequence LA bases.

Fig. 7 is extraction result and the edited result of expression of the present invention to face's triangle grid model LA bases

Result and the result of DG method of Fig. 8 present invention to global rotary motion.On the right of wherein two In row, what each row were represented is result to about similar region shape base, first result of behavior DG methods, the Two behaviors result of the present invention.

Specific embodiment

With reference to specific embodiment, the invention will be further described.

As shown in figure 1, the sparse exploded that perceives of joint motions described in the present embodiment and restructing algorithm, including it is following Step：

1) calculate triangle gridding in while length and while corresponding dihedral angle size (the two values are designated as LA)

Giving a series of dynamic these models of triangular mesh sequence with different attitudes all has identical topology knot Structure, first calculate respectively each bar on each triangle grid model while length and while corresponding to dihedral angle size；

2) residual error of each triangle grid model LA vectors is calculated

To step 1) on a triangle grid model obtaining each bar while corresponding LA according to while order be arranged to make up One LA vector, calculates the LA vectors of each triangle grid model, is made the difference with the LA vectors with reference to triangle grid model respectively, When corresponding LA vector residual errors；

3) LA bases are extracted

To step 2) in obtain each model LA vector residual error carry out sparse exploded, obtaining one group can catch To the LA bases of local joint motion, each LA base controls one piece of region shape on triangle grid model；

4) linear hybrid LA bases obtain new LA vectors

To step 3) in the LA bases that can capture local joint motion that obtain, edit corresponding parameter, obtain new LA vector residual errors, then make and calculate with the LA vectors with reference to triangle grid model, obtain new LA vectors；

5) connection mapping is recovered

To step 4) in the new LA vectors that obtain carry out the local coordinate value on summit on each tri patch and calculate, obtain The local coordinate value on each summit in local coordinate system lower panel, and then recover the company of each tri patch on triangle grid model Connect mapping；

6) local orthogonal marked price is estimated

To step 5) in the connection mapping relations of triangle surface that obtain, recycle and minimize marked price reconstruction of function and estimate Calculate the local orthogonal frame of triangle grid model；

7) triangle gridding is reconstructed

To step 6) in the orthogonal marked price of office that obtains, each is estimated in triangle grid model using reconstruction of function is minimized The position on summit, obtains the triangle grid model with new attitude after the meeting.

In step 1) in, it is described calculate in triangle gridding while length and while corresponding dihedral angle size (the two values It is designated as LA), it is specific as follows：Assuming thatIt is the triangle grid model number with different attitudes for giving According to collection, N_PThe number of data set intermediate cam grid model is represented,Represent k-th triangle grid model.Each triangle gridding mould Type surface be all by closely coupled triangular into.E and F represent line set and tri patch in triangle grid model respectively Set, and the number on side is N_E, the number of tri patch is N_F.These triangle grid models have identical annexation, i.e., All of triangle grid model has identical line set and tri patch set.Travel through tri patch on summit when, I Be to be carried out with direction counterclockwise.

Assuming thatOn a line e=(i, j) ∈ E, share this edge tri patch for τ=(j, h, i) and υ=(g, J, i), the letter in bracket, to constitute this tri patch the Vertex Labeling, is sequentially sequence counter-clockwise, as shown in Figure 2.When from this The vertex v of bar common edge_iGo to v_jWhen, triangle surface τ is located at left-hand side, and tri patch υ is located at right-hand side, so τ and υ points Have another name called is the left tri patch and right tri patch of side (i, j).When calculating the length of e, use the Europe between two summits several Reed distance, i.e. l_ij=| | v_j-v_i||₂.θ represents the angle of the normal of tri patch τ and υ outward direction, i.e. θ=acos (n_τ· n_υ), wherein n_τAnd n_υThe normal of direction outward direction on respectively tri patch τ and υ, the then corresponding dihedral angle definition of common edge For

A LA values (l achieved above_ij,φ_ij)。

For data setUpper all of triangle grid modelThe LA values on its each side are calculated, and according to side Number order arrangement may be constructed a vector for 2NE rowThat is LA vectors.

In step 2) in, the LA vector value residual errors of described each triangle grid model of calculating, specific method is：In step 1) obtained in each triangle grid model LA vector values constitute vector after, respectively with first LA value of grid model to Amount makes the difference, and draws the residual vector of LA vector values, i.e.,.

In step 3) in, described extraction LA bases, concrete mode is as follows：

Using step 2) in obtain LA vector residual error, construct a 2N_E×N_pMatrix D=(D₁,...,D_Np).Will Matrix D resolves into D=C × W.Wherein Matrix C is a 2N_EThe LA basic matrixs of × K, each row in matrix all represent LA shapes Each row bar in base, i.e. matrix all controls the shape in region in triangle grid model.Here K represents LA shapes The number of base, can be sized as needed, if defining K value is that triangle grid model is roughly divided into K region.And matrix It is the weight matrix of, the every a line in matrix represents a hybrid weight sequence.For the ease of the attitude to triangle gridding Enter edlin, it is necessary to keep the locality of LA shape bases as much as possible, i.e. mixing unit between the region of each LA shapes base control Divide few as much as possible, and need normalization hybrid weight to eliminate due to scaling the influence to attitude.Here by minimum Function (1) carries out matrix decomposition.

Wherein first energy termFor the error that constraint matrix is decomposed, and second energy Quantifier is used to adjust the locality of shape base in Matrix C, the i.e. magnitude range of each shape base.Second energy term is specific such as Under：

Wherein Λ_ikRepresent triangle grid model on i-th while kth bar as in shape base while midpoint between geodetic Linear distance, is defined as follows：

Its detailed process is：K bars side is selected first on triangle grid model as the K center of shape base, referred to as in Heart side, is represented with Edge here, edge_kRepresent kth bar center edge.Use e_iThe side that the numbering on triangle grid model is i is represented, The then d in above formula_ikRepresent e_iAnd edge_kResult after the geodesic curve range normalization at midpoint, normalization here is actually The midpoint of triangle gridding top and the geodesic curve distance at each center edge midpoint are exactly calculated, maximum therein is taken out and is designated as d_max, the value work that the geodesic curve distance for then calculating the side on triangle grid model is finally obtained in this maximum It is the geodesic curve distance after normalization.Therefore there is d_ik≤1。r_minAnd r_maxFor support radius span for [0,1), its tool Body value sets according to demand.

In whole decomposable process, c_i,kThe data of the i-th rows of representing matrix C kth row, w_kjRepresenting matrix W row ks jth row Value；And have max { w_k,j, k=1 ..., K；J=1 ..., N_PAll elements in }=1, i.e. matrix W, maximum is 1.In reality In testing, the value of η is set to 200.The visualization result of the shape base obtained after Its Sparse Decomposition is as shown in Figure 3.

In step 4) in, described linear hybrid LA bases obtain new LA vector values, and its specific method is：In step 3) in After the LA shape bases matrix of acquisition, modification W matrixes in any one row numerical value, obtain vector, then Matrix C again with phase Multiply, obtain a new LA vector value residual vector, finally with first triangle grid model (, i.e., with reference to triangle grid model) LA vector values be added can obtain with new attitude triangle grid model LA vector values.I.e. by calculating and can obtain New LA vector values.

In step 5) in, described recovery connection mapping, its process mainly includes setting up a new local coordinate system, leads to The LA values for crossing side are calculated two coordinates on each summit of tri patch that a this edge is common edge, and each is then calculated again The orthogonal marked price of part under new local coordinate system of dough sheet, its company is recovered finally by the local orthogonal frame of two dough sheets Details are provided below to connect mapping relations：

From step 4) in obtain LA vectors after, we can recover its annexation to every a line.Assuming that side e= (i, j) is the common edge of left tri patch τ and right tri patch υ, and h, g are respectively the 3rd summit of left and right tri patch, I.e. the left and right tri patch of common edge e=(i, j) is respectively τ=(j, h, i), and υ=(g, j, i) is as shown in Figure 4.One is created first Individual coordinate system (r, s, t), then calculates coordinate value of two each summits of tri patch under this coordinate system.Coordinate system with Vertex v_iAs the origin of coordinates and vertex v_jCoordinate value be (l_ij, 0,0), t axles and the tri patch υ outward directions of coordinate system Normal is in the same direction.Because tri patch υ is the right tri patch of common edge e=(i, j), so vertex v_gR coordinates be necessarily smaller than 0.So v can be calculated_gCoordinate be respectively：

In order to calculate vertex v_hCoordinate value, we first assume that vertex v_hFlattening followed by triangular facet are rotated around common edge Piece υ coplanar vertex v_h', then what is be similar to can calculate vertex v_h'Coordinate value be：

By vertex v_h'φ is rotated around r reference axis_ijVertex v can be obtained after-π_hReference axis.

After two reference axis on each summit of tri patch are obtained, local coordinate newly-built in above process is calculated The orthogonal marked price of part under system.The detailed process of local orthogonal marked price is as follows：

By taking tri patch τ as an example, the orthogonal marked price of part that we are defined on this tri patch is ζ_τ=(u_τ,v_τ,n_τ) its InRepresent on tri patch τ from first vertex v_iPoint to second vertex v_jUnit vector；n_τIt is triangle The outside unit normal of dough sheet；v_τ=n_τ×u_τ.Three above unit vector just constitutes ζ_τ3 × 3 matrix structure.

The orthogonal marked price ζ of part on tri patch υ can similarly be obtained_υ.SoAs by under other coordinate systems other Coordinate value under coordinate system is transformed into the transition matrix represented under υ local coordinate systems.The orthogonal marked price of part on tri patch is such as Shown in Fig. 2.

Can just be counted after the orthogonal marked price of part for calculating two tri patch with side e=(i, j) as common edge Calculate the connection mapping Q of this common edge_ij。Q_ijIt is defined as follows：

So far, the connection mapping on side e=(i, j) has been calculated, the connection that can recover all sides with similar mode is reflected Penetrate.

Since when the editor of LA bases is carried out, the length of side for being LA values, i.e., directly changing in triangle gridding for directly changing The dihedral angle size of degree triangle corresponding with side.This is possible to cause the amended length of side to be unsatisfactory for the triangle length of side Formula, i.e., amended side is possible to that tri patch can not be constituted.Need to carry out specially treated in face of these situations：Work as l_gj≥ l_ji+l_igWhen (Fig. 4 (b) shown in), we unify the length l on side_jiAnd l_igAmplify l_gj/(l_ji+l_ig) times, thus there is l_gj= l_ji+l_ig, at this time because the direction on side (i, j) is consistent with the direction of side (g, j), so the direction of side (g, j) is (1,0,0)； Similarly, l is worked as_ji≥l_gj+l_igWhen (Fig. 4 (c) shown in), the direction of side (g, j) is (1,0,0), works as l_ig≥l_gj+l_jiWhen (Fig. 4 (d) It is shown), the direction of side (g, j) is (- 1,0,0).

In step 6) in, described estimation locally orthogonal marked price, its main method is as follows：

Assuming thatIt is the set on the side in triangular mesh, each triangle is estimated by following minimum function (2) The local orthogonal frame of dough sheet：

Wherein the left side is：

Represent connection Mapping Energy.

And the ζ in function (2)₁WithThe orthogonal marked price of first tri patch that needs to estimate and given is represented respectively First orthogonal marked price of tri patch, in order to ensure that the triangle gridding for reconstructing the rigid transformation for coming has uniqueness, we Need the orthogonal marked price of part of given first tri patch；Subscript ' F ' represents Frobenius norms；Q_ijFor step 5) in it is extensive Multiple link mapping.

Step 7) in, described reconstruct triangle gridding is exactly mainly the position on each summit of triangle grid model for calculating new Put, detailed process is as follows：

In step 6) in get local orthogonal marked price after, calculated by majorized function (3) each in triangle gridding The position on individual summit.

ε_ER({v_i, i ∈ I })=ε_EV({v_i,i∈I})+ε_EC({v_i,i∈I}) (3)

Wherein the item on the right is used to constrain the position of given fixing point, and it is specially：

Wherein I^*To give the numbering on fixed summit,It is the position of given fixed vertices.Come to ensure to reconstruct The triangle grid model of rigid transformation there is uniqueness, it would be desirable to the position for fixing some points is constant, the quantity of fixing point It is adjusted as needed.

First in function (3) is want the vertex position for restrained split-flow with step 6) in the orthogonal marked price of part that obtains The deviation of the inside vector, specially：

Wherein e_gj=-l_gjU, e_ji=l_ji(cos β u-sin β v), e_ig=l_ig(cos α u+sin α v), α is on tri patch Side (g, j) and the angle of side (i, g), and β is the angle (as shown in Figure 5) of side (g, j) and side (j, i)；U in formula, v represent by Step 6) estimate corresponding tri patch on orthogonal marked price ζ=(u, v, n) of part on corresponding vector.

The institute of triangle grid model position a little is obtained with after majorized function (3), that is, is reconstructed after editing Triangle grid model with new attitude.

The present invention proves its feasibility by experiment, can be widely used in various models.Fig. 6 shows the present invention to LA Base positioning and the result to being rotated editor around node.Wherein figure (a) is editor's two LA bases therein, is passed through The hybrid weight parameter for changing the LA bases for controlling the two regions has reconfigured a triangle gridding mould with new attitude Type, intuitively say be exactly will control left side model upper part of the body region LA bases hybrid weight parameter assignment with control mid-module The hybrid weight parameter combination of the LA bases in lower part of the body region is got up, and constitutes a complete hybrid weight parameter, is finally reconstructed The right has the triangle grid model of new attitude.Figure (b) is also similar result.And (c) is then by changing hybrid weight The triangle grid model with new attitude for coming is reconstructed after parameter, these attitudes are concentrated in original triangle grid data It is non-existent.Fig. 7 shows the edit effect to face triangle grid model, wherein from first triangle gridding of left side number Model display is the LA bases for controlling left eye region extracted by the present invention, and second shows control right eye The LA bases in region, the 3rd shows the LA bases for controlling left corners of the mouth region, and rightmost shows by that will control these three The hybrid weight parameter group of LA bases is combined together the three dimensional network for constructing and being reconstructed after a complete hybrid weight parameter Lattice model.The comparing phase of the shape with first model of the three-dimensional grid model left eye for coming is reconstructed as we can see from the figure Seemingly, the shape of right eye is similar with the comparing of second model, and the shape of the left side corners of the mouth is with the 3rd comparing phase of model Seemingly.Fig. 8 be with the triangle grid model of horse as example illustrate the present invention to global rotary motion treatment legitimate result and The result of DG methods treatment.The model of horse carries out running motion around a circle in experiment.The left side is not carry out originally in figure The model of deformation, the row of the right two shows the result obtained after global rotation editor is carried out to identical base, and first is The deformation of head, second is the deformation of foreleg, and the 3rd deformation of back leg, experiment shows, the present invention is in the global rotation for the treatment of During motion, the result for the treatment of is more reasonable.

The examples of implementation of the above are only the preferred embodiments of the invention, not limit implementation model of the invention with this Enclose, therefore the change that all shapes according to the present invention, principle are made, all should cover within the scope of the present invention.

Claims (2)

1. a kind of joint motions are perceived sparse exploded and restructing algorithm, it is characterised in that comprise the following steps：

1) calculate triangle gridding in while length and while corresponding dihedral angle size

Giving a series of dynamic these models of triangular mesh sequence with different attitudes all has identical topological structure, first Calculate respectively each bar on each triangle grid model while length and while corresponding to dihedral angle size, the two values are designated as LA；

2) residual error of each triangle grid model LA vector values is calculated

To step 1) on a triangle grid model obtaining each bar while corresponding LA according to while order be arranged to make up one LA vector, calculate each triangle grid model LA vector, the LA values of each triangle grid model, respectively with refer to the triangulation network The LA vector values of lattice model make the difference, when corresponding LA vector values residual error；

3) shape LA bases are extracted

To step 2) in the LA vector values residual error of each model that obtains carry out sparse exploded, obtaining one group can capture The LA bases of local joint motion, each LA base controls one piece of region shape on triangle grid model；

4) linear hybrid LA bases obtain new LA vector values

To step 3) in the LA bases that can capture local joint motion that obtain, edit corresponding parameter, obtain new LA to Value residual error, then make and calculate with the LA vector values with reference to triangle grid model, obtain new LA vector values；

5) connection mapping is recovered

To step 4) in the new LA vector values that obtain carry out the local coordinate value on summit on each tri patch and calculate, acquisition office The local coordinate value on each summit in portion's coordinate system lower panel, and then recover the connection of each tri patch on triangle grid model Mapping relations；

6) local orthogonal marked price is estimated

To step 5) in the connection mapping relations of triangle surface that obtain, recycle and minimize marked price reconstruction of function and estimate The local orthogonal frame of triangle grid model；

7) triangle gridding is reconstructed

To step 6) in the orthogonal price mark display of office that obtains, each top in triangle grid model is estimated using reconstruction of function is minimized The position of point, obtains the triangle grid model with new attitude after the meeting.

2. a kind of joint motions according to belonging to claim 1 are perceived sparse exploded and restructing algorithm, it is characterised in that In step 1) in, it is described calculate in triangle gridding while length and while corresponding dihedral angle size, it is specific as follows：

Assuming thatIt is the triangle grid model data set with different attitudes for giving, N_PRepresent number According to the number for concentrating triangle grid model,Represent k-th triangle grid model, each triangle grid model surface be by Closely coupled triangular is into E and F represents line set and tri patch set in triangle grid model, and side respectively Number be N_E, the number of tri patch is N_F, these triangle grid models have identical annexation, i.e., all of triangle Grid model has identical line set and tri patch set；Travel through tri patch on summit when, we are with the inverse time The direction of pin is carried out；

Assuming thatOn a line e=(i, j) ∈ E, share the tri patch of this edge for τ=(j, h, i) and

υ=(g, j, i), the letter in bracket, to constitute this tri patch the Vertex Labeling, is sequentially sequence counter-clockwise；When from this The vertex v of bar common edge_iGo to v_jWhen, triangle surface τ is located at left-hand side, and tri patch υ is located at right-hand side, so τ and υ points Have another name called is the left tri patch and right tri patch of side (i, j)；When calculating the length of e, use the Europe between two summits several Reed distance, i.e. l_ij=| | v_j-v_i||₂；θ represents the angle of the normal of tri patch τ and υ outward direction, i.e. θ=a cos (n_τ·n_υ), wherein n_τAnd n_υThe normal of direction outward direction on respectively tri patch τ and υ, the then corresponding dihedral angle of common edge It is defined as

A LA values (l achieved above_ij,φ_ij)；

For data setUpper all of triangle grid modelThe LA values on its each side are calculated, and according to the numbering on side Order arrangement may be constructed a 2N_EThe vector of rowThat is LA vectors；

In step 2) in, the LA vector residual errors of described each triangle grid model of calculating, specific method is：In step 1) in obtain After obtaining the vector of LA vector compositions of each triangle grid model, the LA vectors respectively with first grid model are poor, obtain Go out the residual error of LA vectors, i.e. { D_k=X_k-X₁, k=1,2 ..., N_p}；

In step 3) in, described extraction LA bases, concrete mode is as follows：

Using step 2) in obtain LA vector residual error, construct a 2N_E×N_pMatrix D=(D₁,...,D_Np), by matrix D D=C × W is resolved into, wherein Matrix C is a 2N_EThe LA basic matrixs of × K, each row in matrix all represent a LA base, i.e., Each row in matrix all control the shape in region in triangle grid model, and K here represents the number of LA bases, energy It is enough to be sized as needed, if defining K value is that triangle grid model is roughly divided into K region；And matrix W be a K × N_PWeight matrix, the every a line in matrix represents a hybrid weight sequence；Carried out for the ease of the attitude to triangle gridding Editor, it is necessary to keep the locality of LA bases, i.e., mixing portion is few as much as possible between each LA base, and needs are returned as much as possible One changes hybrid weight to eliminate due to scaling the influence to attitude；Matrix decomposition is carried out here by function (1) is minimized：

$\arg \underset{C,W}{min}{\mathrm{\ϵ}}_{MF}(C,W)+{\mathrm{\η\ϵ}}_{LC}\left(C\right)---\left(1\right)$

Wherein, first energy termFor the error that constraint matrix is decomposed, and second energy term The area size scope of the locality for adjusting LA bases in Matrix C, i.e. each LA base control；

Second energy term is specific as follows：

${\mathrm{\ϵ}}_{LC}\left(C\right)=\underset{k=1}{\overset{K}{\Σ}}\underset{i=1}{\overset{N_E}{\Σ}}{\mathrm{\Λ}}_{ik}\left|\right|(c_{2i-1,k},c_{2i,k})|{|}_2$

Wherein, Λ_ikRepresent triangle grid model on i-th while kth bar as in shape base while midpoint between geodesic curve Distance, is defined as follows：

${\mathrm{\&Lambda;}}_{ik}=\left\{\begin{array}{cc}1& d_{ik}>r_{\max }\\ \frac{d_{ik}-r_{min}}{r_{\max }-r_{min}}& r_{min}\&le;d_{ik}\&le;r_{\max }\\ 0& d_{ik}<r_{\min }\end{array}\right.$

Its detailed process is：K bars side is selected first on triangle grid model as the K center of shape base, referred to as center edge, Here represented with Edge, edge_kRepresent kth bar center edge；Use e_iThe side that the numbering on triangle grid model is i is represented, then above formula In d_ikRepresent e_iAnd edge_kResult after the geodesic curve range normalization at midpoint, normalization here is actually meter The midpoint of triangle gridding top and the geodesic curve distance at each center edge midpoint are calculated, maximum therein is taken out and is designated as d_max, then The geodesic curve that side on triangle grid model is calculated is apart from the value finally obtained in this maximum as normalization Geodesic curve distance afterwards, therefore have d_ik≤ 1, r_minAnd r_maxFor support radius span for [0,1), its occurrence according to Demand sets；

In whole decomposable process, c_i,kThe data of the i-th rows of representing matrix C kth row, w_kjThe value of representing matrix W row ks jth row； And have max { w_k,j, k=1 ..., K；J=1 ..., N_PAll elements in }=1, i.e. matrix W, maximum is 1；

In step 4) in, described linear hybrid LA bases obtain new LA vectors, and its specific method is：In step 3) middle acquisition After LA basic matrixs, the numerical value of any one row, obtains vectorial w in modification W matrixes_k, then Matrix C again with w_kIt is multiplied, obtains one New LA vector residual errors, finally can obtain the triangle with new attitude with first LA addition of vectors of triangle grid model The LA vectors of grid model, i.e., by calculatingNew LA vectors can be obtained；

In step 5) in, described recovery connection mapping, its process mainly includes setting up a new local coordinate system, by side LA values be calculated a this edge be common edge two coordinates on each summit of tri patch, each dough sheet is then calculated again The orthogonal marked price of the part under new local coordinate system, recover its connection finally by the local orthogonal frame of two dough sheets and reflect Details are provided below to penetrate relation：

From step 4) in obtain LA vectors after, its annexation is recovered to every a line, it is assumed that side e=(i, j) is left three The common edge of edged surface piece τ and right tri patch υ, and h, g are respectively the 3rd summit of left and right tri patch, i.e. common edge e= The left and right tri patch of (i, j) is respectively τ=(j, h, i), υ=(g, j, i)；A coordinate system (r, s, t) is created first, then Calculate coordinate value of two each summits of tri patch under this coordinate system；Coordinate system is with vertex v_iAs the origin of coordinates and Vertex v_jCoordinate value be (l_ij, 0,0), the t axles of coordinate system are in the same direction with the normal of tri patch υ outward directions；Due to triangular facet Piece υ is the right tri patch of common edge e=(i, j), so vertex v_gR coordinates be necessarily smaller than 0, so calculate v_gCoordinate point It is not：

$r_g=\frac{l_{gi}^2+l_{ij}^2-l_{gj}^2}{2l_{ij}},s_g=-\sqrt{l_{gi}^2-r_g^2},t_g=0$

In order to calculate vertex v_hCoordinate value, assume initially that vertex v_hFlattening is rotated around common edge followed by tri patch υ is coplanar Vertex v_h', then the calculating vertex v being similar to_h'Coordinate value be：

$r_{h^{\&prime;}}=\frac{l_{hi}^2+l_{ij}^2-l_{hj}^2}{2l_{ij}},s_{h^{\&prime;}}=-\sqrt{l_{hi}^2-r_{h^{\&prime;}}^2},t_{h^{\&prime;}}=0$

By vertex v_h'φ is rotated around r reference axis_ijVertex v can be obtained after-π_hReference axis；

After two reference axis on each summit of tri patch are obtained, calculate under local coordinate system newly-built in above process The orthogonal marked price of part；The detailed process of local orthogonal marked price is as follows：

By taking tri patch τ as an example, it is ζ to define the orthogonal marked price of the part on this tri patch_τ=(u_τ,v_τ,n_τ) whereinRepresent on tri patch τ from first vertex v_iPoint to second vertex v_jUnit vector；n_τIt is tri patch Outside unit normal；v_τ=n_τ×u_τ；Three above unit vector just constitutes ζ_τ3 × 3 matrix structure；

The orthogonal marked price ζ of part on tri patch υ can similarly be obtained_υ, thenAs by other coordinates under other coordinate systems Coordinate value under system is transformed into the transition matrix represented under υ local coordinate systems；

This can be just calculated after the orthogonal marked price of part for calculating two tri patch with side e=(i, j) as common edge The connection mapping Q of common edge_ij, Q_ijIt is defined as follows：

$Q_{ij}={\mathrm{\&zeta;}}_{\mathrm{\&upsi;}}^{-1}{\mathrm{\&zeta;}}_{\mathrm{\&tau;}}$

So far, the connection mapping on side e=(i, j) has been calculated, the connection mapping on all sides can have just been recovered with similar fashion；

Since when the editor of shape base is carried out, the edge lengths for being LA values, i.e., directly changing in triangle gridding for directly changing The dihedral angle size of triangle corresponding with side, this is possible to cause the amended length of side to be unsatisfactory for the triangle length of side Formula, i.e., amended side is possible to that tri patch can not be constituted, and needs to carry out specially treated in face of these situations：Work as l_gj≥ l_ji+l_igWhen, the unified length l by side_jiAnd l_igAmplify l_gj/(l_ji+l_ig) times, thus there is l_gj=l_ji+l_ig, at this time because The direction on side (i, j) is consistent with the direction of side (g, j), so the direction of side (g, j) is (1,0,0)；Similarly, l is worked as_ji≥l_gj+l_ig When, the direction of side (g, j) is (1,0,0), works as l_ig≥l_gj+l_jiWhen, the direction of side (g, j) is (- 1,0,0)；

In step 6) in, described estimation locally orthogonal marked price, its main method is as follows：

Assuming thatIt is the set on the side in triangular mesh, each tri patch is estimated by following minimum function (2) Local orthogonal frame：

${\mathrm{\&epsiv;}}_{FR}\left(\right\{{\mathrm{\&zeta;}}_t,t\&Element;F\left\}\right)={\mathrm{\&epsiv;}}_{CM}\left(\right\{{\mathrm{\&zeta;}}_t,t\&Element;F\left\}\right)+\left|\right|{\mathrm{\&zeta;}}_1-{\mathrm{\&zeta;}}_1^{*}|{|}_F^2---\left(2\right)$

Wherein the left side is：

Represent connection Mapping Energy；

And the ζ in function (2)₁WithThe orthogonal marked price and given first of first tri patch for needing to estimate is represented respectively The orthogonal marked price of individual tri patch, in order to ensure that the triangle gridding for reconstructing the rigid transformation for coming has uniqueness, it is necessary to given The orthogonal marked price of part of first tri patch；Subscript ' F ' represents Frobenius norms；Q_ijFor step 5) the middle link for recovering Mapping；

Step 7) in, described reconstruct triangle gridding is exactly mainly the position on each summit of triangle grid model for calculating new, tool Body process is as follows：

In step 6) in get local orthogonal marked price after, each top in triangle gridding is calculated by majorized function (3) The position of point；

ε_ER({v_i, i ∈ I })=ε_EV({v_i,i∈I})+ε_VC({v_i,i∈I}) (3)

Wherein, the item on the right is used to constrain the position of given fixing point, and it is specially：

${\mathrm{\&epsiv;}}_{VC}\left(\right\{v_i,i\&Element;I\left\}\right)=\underset{i\&Element;I^{*}}{\&Sigma;}\left|\right|v_i-v_i^{*}|{|}^2$

Wherein, I^*To give the numbering on fixed summit,It is the position of given fixed vertices, in order to ensure to reconstruct what is come The triangle grid model of rigid transformation there is uniqueness, it is necessary to fix some point position it is constant, the quantity of fixing point is according to need It is adjusted；

First in function (3) is want the vertex position for restrained split-flow with step 6) in the orthogonal marked price the inside of the part that obtains The deviation of vector, specially：

${\mathrm{\&epsiv;}}_{EV}\left(\right\{v_i,i\&Element;I\left\}\right)=\underset{(g,j,i)\&Element;F}{\&Sigma;}\&lsqb;\left|\right|v_g-v_j-e_{gj}|{|}^2+||v_j-v_i-e_{ji}|{|}^2+\left|\right|v_i-v_g-e_{ig}|{|}^2\&rsqb;$

Wherein, e_gj=-l_gjU, e_ji=l_ji(cos β u-sin β v), e_ig=l_ig(cos α u+sin α v), α is tri patch top The angle of (g, j) and side (i, g), and β is the angle of side (g, j) and side (j, i)；U in formula, v are represented by step 6) estimate Corresponding vector on orthogonal marked price ζ=(u, v, n) of part on correspondence tri patch；

The institute of triangle grid model position a little is just obtained in that after majorized function (3), that is, reconstruct has after editing The triangle grid model of new attitude.