CN101071514A - Method for directly transferring three-dimensional model attitude - Google Patents

Abstract

This invention is a three-dimensional model of direct transfer method gesture involving computer graphics technology, the 3D model directly to the source posture passed to target three-dimensional model. First, the user specified source / target model features corresponding anchor, and then the corresponding algorithm automatically establish the source and destination of the triangular relationship; Then, the triangular surface film formation rotation and translation to a temporary mesh, the network from a reference targets Gatti Grid rigid elements, while retaining a possible source grid posture information; then the provisional grid in the face of the film in accordance with the predefined Laplace differential binding re-arranged, splicing generate meaningful three-dimensional model . This invention solution to a single three-dimensional model of transmission between posture, both the overall structure of the skeleton or fine skin deformation; help efficient 3D animation division has been rapid use of 3D model library to produce complex three-dimensional model of posture, and no need to start from scratch or repeated time-consuming manual adjustment.

Description

A kind of method of direct transmission three-dimensional model attitude

Technical field

The present invention relates to the computer graphics techniques field, is a kind of method of direct transmission three-dimensional model attitude---model transduction can be directly passes to the target three-dimensional model with the attitude of source three-dimensional model.

Background technology

Grid deformation is all to have at microcomputer modelling or in the computer animation field widely to use.Three-dimensional animation teacher and artists make in three-dimensional and use various deformation instruments to come the countenance and the body shape of edit virtual role in the software.These methods not only need a large amount of artistry to be used to regulate deformation, and are difficult to the result is reused on the new three-dimensional model.In order to control the deformation of other three-dimensional models with the three-dimensional model of adjusted good parameter, specific deformation parameter must match in shape new.In many cases, to such an extent as to set these parameters time-consuming in the extreme not as restarting.

People such as Noh have proposed expression clone's notion on SIGGRAPH ' 2001, countenance is delivered to object module from source model.In this method, each expression is decoded into top displacement, and it has characterized the difference with reference to face and expression face.Direction and size that expression clone comes adaptive motion vector with heuristic are to solve the different proportion problem of source and target model.This statement and adaptation technique are only applicable to countenance.

The deformation transmission method that Sumner and Popovic propose on SIGGRAPH ' 2004 is the expansion of expression concept of clone, and its deformation with the source triangular mesh is applied in the reference target triangular mesh.In order to generate the object module after the deformation, no matter being to use still deformation transmission method of expression clone, all source model and the reference target model after given reference source model, the deformation simultaneously.

Expression clone is that the source and target grid of reference must have identical athletic posture with another limitation that deformation is transmitted, because these two kinds of methods are duplicated the alteration of form that is caused by distortion.In addition, if source deformation shortage authenticity itself is obviously also necessarily lively inadequately by the deformation object module that source deformation is generated, and unfortunately can't provide effective means to improve the result.

Summary of the invention

The method that the purpose of this invention is to provide a kind of direct transmission three-dimensional model attitude---model transduction is a kind of method based on grid, can be directly the attitude of source three-dimensional model be passed to the target three-dimensional model.

Another purpose of the present invention provides a kind of method of direct transmission three-dimensional model attitude---and the model transduction, this method is general, does not need source model and reference target model to share the summit and the triangle of the same number.

For achieving the above object, technical solution of the present invention provides a kind of method of direct transmission three-dimensional model attitude---and model transduction can be directly passes to the target three-dimensional model with the attitude of source three-dimensional model; The model transduction mainly realizes by following steps: the first step, and the user specified source corresponding anchor is set up the source and target triangular relationship automatically by corresponding algorithm then; Second step, triangle surface be rotated with translation to generate a provisional grid, this model has extracted the rigidity composition of reference target grid, and has kept simultaneously the attitude information of source grid; The 3rd step reconfigured the dough sheet in the provisional grid according to Laplce's differential constraint, obtain final three-dimensional model by finding the solution global optimization deformation.

Described method, in its described first step, do not need source model and reference target model to share the summit and the triangle of the same number, if promptly source and reference target grid have the summit and the triangle of different numbers, the user can be at first by the features corresponding anchor of specified source, method with iterative closest point is deflected into the source grid as far as possible hand to hand with the reference target grid, and try to achieve simultaneously the corresponding relation of all triangle surfaces of source/target automatically, this corresponding process with the second order error function representation is:

$E\left(V^{\&prime;}\right)=w_i\underset{i=1}{\overset{\left|T\right|}{\mathrm{\&Sigma;}}}{\left|\right|Q_i-I\left|\right|}_{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="A20061007821500131.png"\; state="\{\{state\}\}">F</figure-callout>}}^2+w_m\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{\left|\right|Q_i{v}_i-m_i\left|\right|}^2$

$+\underset{s=1}{\overset{\left|T\right|}{\mathrm{\&Sigma;}}}{w}_s\left(\underset{j\&Element;\mathrm{adj}\left(s\right)}{\mathrm{\&Sigma;}}{\left|\right|Q_s-Q_j\left|\right|}_F^2\right)+w_c\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\left|\right|v_i^{\&prime;}-c_i\left|\right|}^2.$

Described method, its described second step promptly for every pair of source/target triangle, finds an interim triangle { v for generating interim rigid body grid ₁ ^r, v ₂ ^r, v ₃ ^r, it is by rotation, translation reference target triangle and obtaining; For the attitude information of coding source grid, interim triangle obtains by minimizing following second order error function:

Satisfy simultaneously: $\left|\right|v_i^r{v}_j^r\left|\right|=\left|\right|v_i^0{v}_j^0\left|\right|$ (i，j)∈{(1，2)，(2，3)，(3，1)}

Wherein, v _i ⁰For the reference target vertex of a triangle,

Be source vertex of a triangle, i=1,2,3,4.

Described method, its described third step are the translation triangle and satisfy the details differential constraint, translation item E _tWith the second order error function representation be:

$E_t\left(V^{\&prime;}\right)=\underset{t=1}{\overset{\left|T\right|}{\mathrm{\&Sigma;}}}{w}_t\left(\underset{(i,j)\&Element;\{\left(\mathrm{1,2}\right),\left(\mathrm{2,3}\right)\}}{\mathrm{\&Sigma;}}{\left|\right|v_i^{\&prime;}{v}_j^{\&prime;}-v_i^r{v}_j^r\left|\right|}^2\right)$

Differential constraint item E _lWith the second order error function representation be:

$E_1\left(V^{\&prime;}\right)=\underset{k=1}{\overset{\left|T\right|}{\mathrm{\&Sigma;}}}{w}_k\left(\underset{i=1}{\overset{3}{\mathrm{\&Sigma;}}}{\left|\right|Q_k{\mathrm{\&delta;}}_i-\mathrm{\&xi;}\left(v_i^{\&prime;}\right)\left|\right|}^2\right)+\underset{s=1}{\overset{\left|T\right|}{\mathrm{\&Sigma;}}}{w}_s\left(\underset{j\&Element;\mathrm{adj}\left(s\right)}{\mathrm{\&Sigma;}}{\left|\right|Q_s-Q_j\left|\right|}_F^2\right)$

Complete objective function E is above two second order error function sums:

E(V′)＝E _t+E _l。

The present invention has considered a very different and rich problem of challenging: if reference source model non-availability, promptly have only source model and reference target model after the deformation, can we not only be possessed the attitude of source model but also be kept the deformation object module of reference target model geometric minutia simultaneously so? the inventive method is referred to as the model transduction, the grid that this method can be applied to different structure (is different number of vertex, different triangle number, and different network annexation), by calculating the global optimization deformation of target shape, can transmit non-linear arbitrarily deformation.

Description of drawings

The method of a kind of direct transmission three-dimensional model attitude of Fig. 1 the present invention---model transduction synoptic diagram;

Generate interim rigid body grid synoptic diagram among Fig. 2 the present invention;

Fig. 3 (a) uses existing deformation transmission method, and the deformation of lion is delivered to the synoptic diagram on the cat; Fig. 3 (b) uses model transduction method of the present invention, and cat directly imitates the synoptic diagram of the attitude of lion;

Fig. 4 (a) uses existing deformation transmission method, and the deformation of cat is delivered to the synoptic diagram on the lion; Fig. 4 (b) uses model transduction method of the present invention, and lion directly imitates the synoptic diagram of the attitude of cat;

Fig. 5 uses model transduction method of the present invention, and an old man directly imitates the synoptic diagram of lad's expression.

Embodiment

In computer graphics, three-dimensional surface adopts the overall situation or local coordinate system to represent.World coordinates system explicitly specifies out the overall Eulerian coordinates of geometric data to characterize specific shape.By contrast, the local coordinate system coding surperficial inherent geometric parameter.The world coordinates system can be advantageously used in geometric transformation, texture, collision detection and play up.And on the other hand, the local coordinate system is applicable to the gridding edition operation that requires to keep the surface local minutia.

In order to realize a kind of method of direct transmission three-dimensional model attitude---the model transduction method, we have realized a kind of mesh reconstruction system, it had both comprised that global characteristics had also comprised local feature.This method for expressing supposes that some triangles are the rigidity dough sheet of independence and invariable rotary, and it adopts world coordinates to define.On the other hand, a series of predefined constraints (as geometric detail information is encoded into Laplce's differential form) are positioned over simultaneously among triangle surface, and it is defined as local coordinate.For triangular mesh being spliced into significant model, the triangle of rigidity definition is by translation, rotation, and satisfies predefined differential constraint simultaneously.The existing world coordinates of this system has the advantage of local coordinate system again: on the one hand, the triangle of rigidity definition has explicit expression, is described as absolute Eulerian coordinates.These triangles are relatively independent, and have some regional ability of the whole grid of reconstruction; On the other hand, geometric detail is encoded into the surface mesh forms of characterization that the differential coordinate provides a kind of inherence, grid can be rebuild under the situation that keeps local appearance for how much.

Triangle gridding is one of the current main-stream representation on surface.Because arbitrary polygon can be a network of triangle by subdivision easily, so the processing of triangular mesh curved surface only is discussed here.

A kind of method of direct transmission three-dimensional model attitude---model transduction, comprise main two steps: at first extract the rigidity composition of reference target grid and it is positioned on the grid of source, Laplce's attribute according to local definition retrains then, these rigidity dough sheets are reconfigured, obtain end product by finding the solution global optimization deformation.

Following processes will be described each step in detail.

1, generate interim rigid body grid:

We are expressed as leg-of-mutton affined transformation set with deformation.Use v _i ⁰With

Represent reference target triangle and source vertex of a triangle respectively.V wherein ₄(v ₄ ⁰Or

) be defined as perpendicular to a summit on the triangle projective planum:

v ₄＝c ^r+(v ₁-c ^r)×(v ₂-c ^r) (1)

Here, c ^rBe leg-of-mutton barycenter.

Then we according to the vertex of a triangle sets definition one 3 * 3 matrix Q, it has characterized the leg-of-mutton affined transformation in source from the reference target triangle to correspondence:

In this step, our target is to obtain provisional grid (seeing Fig. 1 c, Fig. 3 b-4).At first all triangles of reference target grid are broken up and guaranteed simultaneously and they are not carried out flexural deformation, then their are rotated, move on each corresponding triangle of source grid.Below we will describe this process.

In order to obtain provisional grid, we can be at first be decomposed into rotational component R with the method for svd (SVD) with matrix Q _rWith shear tension component S two parts (people such as Shoemake proposes in Graphics Interface ' 92):

$Q=R_{\mathrm{\&alpha;}}{\mathrm{DR}}_{\mathrm{\&beta;}}=R_{\mathrm{\&alpha;}}\left(R_{\mathrm{\&beta;}}{R_{\mathrm{\&beta;}}}^T\right){\mathrm{DR}}_{\mathrm{\&beta;}}=\left(R_{\mathrm{\&alpha;}}{R}_{\mathrm{\&beta;}}\right)\left({R_{\mathrm{\&beta;}}}^T{\mathrm{DR}}_{\mathrm{\&beta;}}\right)=R_{r}S---\left(3\right)$

The R here _rEnough inaccurate, but it can be used as a good initial value of following iterative solution.Be given above-mentioned initial value, for every pair of source/target triangle, we wish to find an interim triangle { v ₁ ^r, v ₂ ^r, v ₃ ^r(referring to Fig. 2), it is by rotation, translation reference target triangle and obtaining.For the attitude information of coding source grid as much as possible, interim triangle obtains by minimizing following second order error function:

Satisfy: $\left|\right|v_i^r{v}_j^r\left|\right|=\left|\right|v_i^0{v}_j^0\left|\right|$ (i，j)∈{(1，2)，(2，3)，(3，1)}

Wherein

Be another the interim triangle in the computation process, it is similar in appearance to triangle { v ₁ ^r, v ₂ ^r, v ₃ ^r, that is:

I=1,2,3 (s is a scale factor)

2, translation triangle and satisfy the details differential constraint:

After having obtained interim triangle gridding, we are stitched together them now again to obtain net result.In second step, we come the rigid body triangle of translation provisional grid according to predefined constraint.Final triangle { v ₁', v ₂', v ₃' the coordinate on three summits can obtain by the difference of the translation distance that minimizes each Atria summit:

v ₁-v ₁′＝v ₂-v ₂′＝v ₃-v ₃′ (5)

Rewrite above-mentioned minimization problem with the quadratic term error function:

$E_t\left(V^{\&prime;}\right)=\underset{t=1}{\overset{\left|T\right|}{\mathrm{\&Sigma;}}}{w}_t\left(\underset{(i,j)\&Element;\left\{\left(\mathrm{1,2}\right)\left(\mathrm{2,3}\right)\right\}}{\mathrm{\&Sigma;}}{\left|\right|v_i^{\&prime;}{v}_j^{\&prime;}-v_i^r{v}_j^r\left|\right|}^2\right)---\left(6\right)$

As mentioned above, for triangle surface reasonably is stitched together, also need between the adjacency triangle, to satisfy simultaneously predefined constraint.Here we are encoded to geometric detail on Laplce's differential coordinate of triangular mesh: ${\mathrm{\&delta;}}_i=v_i-\frac{1}{d_i}\underset{j\&Element;N_i}{\mathrm{\&Sigma;}}{v}_j,$ It is vertex v _iLaplce's coordinate be the poor of these Eulerian coordinates and the average Eulerian coordinates of adjacent vertex.We are with predefined bound term E _lWith the second order error function representation be:

$E_I\left(V^{\&prime;}\right)=\underset{k=1}{\overset{\left|T\right|}{\mathrm{\&Sigma;}}}{w}_k\left(\underset{i=1}{\overset{3}{\mathrm{\&Sigma;}}}\right||Q_k{\mathrm{\&delta;}}_i-\mathrm{\&xi;}\left(v_i^{\&prime;}\right){\left|\right|}^2)+\underset{s=1}{\overset{\left|T\right|}{\mathrm{\&Sigma;}}}{w}_s\left(\underset{j\&Element;\mathrm{adj}\left(s\right)}{\mathrm{\&Sigma;}}{\left|\right|Q_s-Q_j\left|\right|}_F^2\right)---\left(7\right)$

Wherein, Q is the affine transformation matrix with respect to each source triangle surface, and δ is the summit

v _iLaplce's differential coordinate before conversion, ξ (v _i') be the Laplce's differential seat after the conversion

Mark, || || _FBe the Frobenius norm.

E _lFirst local shape details of pointing out three-dimensional model should be retained through after the conversion; Specified space constraint for second; Point out to seamlessly transit for the 3rd in abutting connection with the affined transformation between the triangle.

In order to keep local shape details, Q _kShould be restricted to rotation matrix.People such as Sorkine have proposed a kind of local linear method for expressing on Eurographics/ACM SIGGRAPH symposium on Geometryprocessing in 2004, but this method can only guarantee rotation matrix constraint condition under low-angle situation.And in model transduction method of the present invention, we need not additional these constraints of explicitly, because the E of front _tItem has pointed out implicitly that affined transformation should be rotational transform.

We complete objective function E is two second order error function sums like this:

E(V′)＝E _t+E _l (8)

Order $\frac{\&PartialD;E}{{\&PartialD;V}^{\&prime;}}=0$ Respectively each known variables is asked local derviation, can get normal equation Ax '=b, so this optimization problem can be converted into finding the solution a sparse linear system of equations.In addition, finding the solution on three coordinates in summit (X/Y/Z) of this system can be carried out respectively separately.We at first carry out the LU decomposition to normal equation, obtain finally to separate by back substitution then.

3, set up the corresponding relation of source and target model:

If source and reference target grid have the summit and the triangle of different numbers, we should at first generate one and reference target model and have same vertices and the interim source model of triangle number purpose before step 1 and 2.The user only needs the mutually corresponding anchor point of assigned source/target (general 50～80 unique points to), we are deflected into the source grid with regard to the method for available iterative closest point as far as possible hand to hand with the reference target grid and obtain this temporary pattern, and try to achieve the corresponding relation of all triangle surfaces of source/target simultaneously automatically.Method of the present invention is similar to people such as Allen at SIGGRAPH ' 2003, the corresponding method that Sumner and Popovic propose in SIGGRAPH ' 2004, but adopted our numerical value framework, and need not setting threshold and set up corresponding lists.The inventive method need not to set up the common parameter territory of source and target grid.

Corresponding system also adopts similar in appearance to the used second order error function of solving model transduction:

$E\left(V^{\&prime;}\right)=w_i\underset{i=1}{\overset{\left|T\right|}{\mathrm{\&Sigma;}}}{\left|\right|Q_i-I\left|\right|}_{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="A20061007821500131.png"\; state="\{\{state\}\}">F</figure-callout>}}^2+w_m\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{\left|\right|Q_i{v}_i-m_i\left|\right|}^2$

$+\underset{s=1}{\overset{\left|T\right|}{\mathrm{\&Sigma;}}}{w}_s\left(\underset{j\&Element;\mathrm{adj}\left(s\right)}{\mathrm{\&Sigma;}}\right||Q_s-Q_j{\left|\right|}_F^2)+w_c\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\left|\right|v_i^{\&prime;}-c_i\left|\right|}^2---\left(9\right)$

Wherein first is the unit matrix item, and effect is to prevent to produce in solution procedure violent grid deformation, this coefficient w _lGenerally getting less coefficient, is 0.001 here; Second is anchor point bound term, and purpose is to use the summit of family appointment to be deflected into the anchor point position as much as possible, the w here _mGenerally getting bigger numerical value, is 10000.0 here; The 3rd is level and smooth of deformation, and it be level and smooth as far as possible promptly changing in abutting connection with the conversion between the triangle, w _sGenerally get 1.0; The 4th is the closest approach item, points out that each vertex position on the interim source model that each iteration obtains should be a closest approach corresponding on the grid of source.

We get w above corresponding process _cBe any number from 1.0 to 1000.0, iteration can obtain separating of a satisfaction four times altogether.Like this, we have just obtained a temporary pattern (compatible grid), its similar with the source grid but and the reference target grid have the same summit and triangle number, the just direct application model technology of transduceing then.Noticing does not need to generate very fine correspondence in this step, because should just characterize by the compatibility grid as attitude, and differential constraint item E _lThe geometric detail feature of grid will be maintained.

Describe the operating process of model transduction in detail below in conjunction with accompanying drawing.Cat is a source model among Fig. 1 (a), and lion is the reference target model among Fig. 1 (b), and method of the present invention does not also require the topological relation unanimity of initial source, object module, promptly identical number of vertex, triangle number and annexation.We at first use the corresponding system of introducing previously that the reference target model is deflected into source model as far as possible hand to hand and generate an interim source model, and try to achieve the corresponding relation of source/target simultaneously automatically.After the corresponding relation that obtains each triangle surface of source/target gridding, the model transduction is directly transmitted the attitude of cat (a) to lion (b), to generate the lion (d) that has identical body posture with cat (a).Fig. 1 (c) model is an interim result, and it is mapped to it cat (a) then and goes up acquisition by extracting the rigidity composition of lion (b) earlier.Use certain constraint,, the dough sheet of grid among Fig. 1 (c) is rearranged, generate the satisfactory result shown in Fig. 1 (d) at last such as the local differential attribute of the Laplce among Fig. 1 (b).We as can be seen, no matter model transduction is overall skeleton structure or fine skin deformation if successfully having been transmitted.More the results are shown in Figure 3, Fig. 4, Fig. 5.Fig. 3 (a) is for using existing deformation transmission method, and the deformation of lion is delivered on the cat; Fig. 3 (b) is for using model transduction method proposed by the invention, and cat directly imitates the attitude of lion.Fig. 4 (a) is for using existing deformation transmission method, and the deformation of cat is delivered on the lion; Fig. 4 (b) is for using model transduction method proposed by the invention, and lion directly imitates the attitude of cat.Fig. 5 is for using model transduction method proposed by the invention, and an old man directly imitates lad's expression.Wherein, Fig. 3, Fig. 4 are the inventive method---the comparison of the deformation transmission method that model transduction and Sumner and Popovic propose on SIGGRAPH ' 2004.As can be seen from the results, the inventive method still can obtain deformation result true to nature under the situation that lacks the reference source grid.

Claims (4)

1, a kind of method of direct transmission three-dimensional model attitude can be directly passes to the target three-dimensional model with the attitude of source three-dimensional model; It is characterized in that the model transduction realizes by following steps: the first step, the user specified source corresponding anchor is set up the source and target triangular relationship automatically by corresponding algorithm then; Second step, triangle surface be rotated with translation to generate a provisional grid, this grid has extracted the rigidity composition of reference target grid, and has kept simultaneously the attitude information of source grid; The 3rd step reconfigured the dough sheet in the provisional grid according to Laplce's differential constraint, obtain final three-dimensional model by finding the solution global optimization deformation.

2, the method for claim 1, it is characterized in that, in the described first step, do not need source model and reference target model to share the summit and the triangle of the same number, if promptly source and reference target grid have the summit and the triangle of different numbers, the user is at first by assigned source/target gridding features corresponding anchor, method with iterative closest point is deflected into the source grid as far as possible hand to hand with the reference target grid, and try to achieve simultaneously the corresponding relation of all triangle surfaces of source/target automatically, this corresponding process with the second order error function representation is:

$E\left(V^{\&prime;}\right)=w_i\underset{i=0}{\overset{\left|T\right|}{\mathrm{\&Sigma;}}}{\left|\right|Q_i-I\left|\right|}_F^2+w_m\underset{i=1}{\overset{m}{\mathrm{\&Sigma;}}}{\left|\right|Q_i{v}_i-m_i\left|\right|}^2$

$+\underset{s=1}{\overset{\left|T\right|}{\mathrm{\&Sigma;}}}{w}_s\left(\underset{j\&Element;\mathrm{adj}\left(s\right)}{\mathrm{\&Sigma;}}{\left|\right|Q_s-Q_j\left|\right|}_F^2\right)+w_c\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\left|\right|v_i^{\&prime;}-c_i\left|\right|}^2.$

Wherein, Q is the affine transformation matrix with respect to each source triangle surface, and I is a unit matrix, v _iBe the summit in the source model, m _iBe the anchor point of user's appointment, v _i' be the summit of model after the deformation, c _iBe v _i' the closest approach of correspondence on object module, || || _FBe the Frobenius norm.

3, the method for claim 1 is characterized in that, described second step promptly for every pair of source/target triangle, finds an interim triangle { v for generating interim rigid body grid ₁ ^r, v ₂ ^r, v ₃ ^r, it is by rotation, translation reference target triangle and obtaining; For the attitude information of coding source grid, interim triangle obtains by minimizing following second order error function:

Satisfy simultaneously: $\left|\right|v_i^r{v}_j^r\left|\right|=\left|\right|v_i^0{v}_j^0\left|\right|$ (i, j) ∈ { (1,2), (2,3), (3,1) } wherein, v _i ⁰For the reference target vertex of a triangle,

Be the source vertex of a triangle, i=1,2,3,4, For similar in appearance to { v ₁ ^r, v ₂ ^r, v ₃ ^rTriangle.

4, the method for claim 1 is characterized in that, described third step is the translation triangle and satisfies the details differential constraint, translation item E _tWith the second order error function representation be:

$E_t\left(V^{\&prime;}\right)=\underset{i=1}{\overset{\left|T\right|}{\mathrm{\&Sigma;}}}{w}_t\left(\underset{(i,j)\&Element;\{\left(\mathrm{1,2}\right),\left(\mathrm{2,3}\right)\}}{\mathrm{\&Sigma;}}{\left|\right|v_i^{\&prime;}{v}_j^{\&prime;}-v_i^r{v}_j^r\left|\right|}^2\right)$

Differential constraint item E _lWith the second order error function representation be:

$E_l\left(V^{\&prime;}\right)=\underset{k=1}{\overset{\left|T\right|}{\mathrm{\&Sigma;}}}{w}_k\left(\underset{i=1}{\overset{3}{\mathrm{\&Sigma;}}}{\left|\right|Q_k{\mathrm{\&delta;}}_i-\mathrm{\&xi;}\left(v_i^{\&prime;}\right)\left|\right|}^2\right)+\underset{s=1}{\overset{\left|T\right|}{\mathrm{\&Sigma;}}}{w}_s\left(\underset{j\&Element;\mathrm{adj}\left(s\right)}{\mathrm{\&Sigma;}}{\left|\right|Q_s-Q_j\left|\right|}_F^2\right)$

Wherein, δ is a vertex v _iLaplce's differential coordinate before conversion, ξ (v _i) be the Laplce's differential coordinate after the conversion.

Complete objective function E is above two second order error function sum: E (V ')=E _t+ E _l

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