CN106934824A - The global non-rigid registration and method for reconstructing of deformable bodies - Google Patents

Abstract

The invention belongs to computer application field, three-dimensional reconstruction, registration, to realize the deformation of the big-movement for the treatment of objective object.The technical solution adopted by the present invention is, the global non-rigid registration and method for reconstructing of deformable bodies, it is assumed that have the n model at visual angle, be designated as U¹, U²..., Uⁿ.For any one model p,N is the number of the point of model p；It is a littleHomogeneous coordinates, the model U adjacent for a pair^p, U^p+1, for last model Uⁿ, that adjacent thereto is U¹, using artificial or ask the algorithm of corresponding relation to find initial mapping relation f between them_p→p+1, i.e. then corresponding points solve the non-rigid transformation X between them^p, It is a transformation matrix of 3*4.Deformation occasion present invention is mainly applied to process the big-movement of objective object.

Description

The global non-rigid registration and method for reconstructing of deformable bodies

Technical field

The invention belongs to computer application field, three-dimensional reconstruction, registration, specifically, it is related to the overall situation based on rarefaction representation Non-rigid surface aligns and method for reconstructing.

Background technology

In computer graphics and computer vision field, dynamic three-dimensional reconstruction is a hot research problem, and it is intended to Corresponding dynamic scene is recovered by single or multiple cameras.With the appearance of commercial depth camera, such as Microsoft Kinect, reconstruct that the model and texture of scene become increasingly facilitates, and cost is also more and more lower.Nowadays, it Many fields are applied, such as 3D printing, game and film.

In order to realize dynamic three-dimensional reconstruction, some seminar devise some multicamera systems.Li Kun et al. (Li K, Dai Q,Xu W.Markerless Shape and Motion Capture From Multiview Video Sequences.[J].IEEE Transactions on Circuits&Systems for Video Technology, 2011,21(3):320-334.) a dome-type multicamera system, this system have been built in Tsing-Hua University with 20 cameras Principle be by various visual angles stereoscopic approach and based on voxel deformation method realize in synchronous acquisition and Restoration dynamics scene appoint Meaning target and its texture.Aguiar et al. (De Aguiar E, Stoll C, Theobalt C, et al.Performance capture from sparse multi-view video[J].Acm Transactions on Graphics,2008,27 (3):A system for sparse sampling 15-19.) is constructed with 8 cameras, this system is by combining the deformation skill based on surface Art and the deformation technology based on voxel realize effective acquisition and reconstruction of the form and action to objective.However, this Multicamera system can run into many limitations in actual applications, such as, this multicamera system cost is very high, it is difficult to safeguard and And it is portable.Thus, due to the low cost of Microsoft's Kinect cameras and its multisensor, make it obtained it is very wide should With.Tong et al. (Tong J, Zhou J, Liu L, et al.Scanning 3D Full Human Bodies Using Kinects[J].IEEE Transactions on Visualization&Computer Graphics,2012,18(4): A complete three-dimensional (3 D) manikin 643-650.) is reconstructed with three cameras, but it requires human body target in gatherer process Remain stationary as.Guo et al. (Guo K, Xu F, Wang Y, et al.Robust Non-rigid Motion Tracking and Surface Reconstruction Using L0Regularization[C]//IEEE International Conference on Computer Vision.2015:3083-3091.) nothing is realized with three hand-held Kinect cameras The dynamic reconstruction of the interaction of mark.In order to be able to simpler and convenient, Li et al. (Li H, Vouga E, Gudym A, et al.3D Self-Portraits[J].Acm Transactions on Graphics,2013,32(6):2504-2507.) only with single Kinect reconstructs a complete threedimensional model, but it will ask for help the holding identical posture under each visual angle.Dou et al. (Dou M,Taylor J,Fuchs H,et al.3D scanning deformable objects with a single RGBD sensor[C]//Computer Vision and Pattern Recognition.IEEE,2015:493-501.) use Single Kinect devises one and allows the target object system that can have moderate finite deformation, but this system requirements adjacent view change Shape can not be big, and it has complexity and very big amount of calculation very high, thus the used time is also very long.

Problem based on the above, we have proposed a kind of the non-firm of new overall situation of the deformable bodies based on single Kinect Property registration and method for reconstructing, our method allows the target object to have larger action to deform, and data used are less, the used time It is shorter.

The content of the invention

To overcome the deficiencies in the prior art, it is contemplated that realizing the deformation of the big-movement for the treatment of objective object.This Inventing the technical scheme for using is, the global non-rigid registration and method for reconstructing of deformable bodies, it is assumed that have the n mould at visual angle Type, is designated as U¹,U²,…,Uⁿ.For any one model p,N is the number of the point of model p；It is a littleHomogeneous coordinates, the model U adjacent for a pair^p, U^p+1, for last model Uⁿ, That adjacent thereto is U¹, using artificial or ask the algorithm of corresponding relation to find initial mapping relation f between them_p→p+1, i.e., it is right Ying Dian, then solves the non-rigid transformation X between them^p,It is a conversion square of 3*4 Battle array.The method for adding multiresolution in the algorithm, will any one model all carry out down-sampled obtaining corresponding low resolution mould Type, is designated as U^(p,s),U^(p,s-1),…,U^(p,1), U^(p,s)Represent the model of lowest resolution, U^(p,1)Represent the mould of original resolution Type, then builds constraint with the corresponding relation in high-resolution models, solves the conversion between low resolution model, will finally become Change and map back in high-resolution models, be denoted asIt is the point on s resolution models, Γ_i ^(p,s+1)Be withCentered on r It is all points on s+1 resolution models in the range of the ball of radius.

The process of the conversion between solution low resolution model is as follows：

Wherein weight a_jIt is defined as follows：

R is effective radius, is defined as in low resolution model so the average weighted twice of the length of side,RepresentWithBetween Euclidean distance, and with the increase of distance, weight constantly reduces；

Based on the corresponding relation between consecutive phantom, following energy function equation is built：

E_data(X) it is data item, represents the object function energy of data item, for is corresponded to after ensureing rotated translation transformation The distance between point summation is minimum；E_smooth(X) it is smooth item, represents the object function energy of smooth item, it is ensured that the point for closing on is gathered around There is as far as possible similar conversion；E_rig(X) it is orthogonal terms, if its constraint partial transformation is rigid；Finally, E_arapIt is Laplce Bound term, the length on the side of its restricted model keeps constant as far as possible before and after conversion, and alpha, gamma and β are respectively to weigh data The weight of item, smooth item and Laplce, this four are defined as follows:

(1) data item E_data(X)：Two neighboring visual angle model U^p,U^p+1Between it is registering as general registration problems, first The corresponding relation f put between looking for model_p, the conversion X between solving model^p, because not being model U^pOn each pointHave at it Consecutive phantom U^p+1On corresponding points, thus for each point sets a weightIf this point is in its adjacent mould There are corresponding points to be then set to 1 in type, 0 is not set to then, it is U that O (p) is denoted as^p+1Upper and U^pThe set of corresponding points, obtains data item It is defined as follows：

Formula above is write as：

Diag represents the diagonalization block matrix of input vector, is denoted asObtain following form：

Wherein H is

(2) item E is smoothed_smooth(X)：For model U^pOn pointIt is denoted asIt is the set of its point being connected, is denoted as ForIt is connected thereto a littleSide, therefore, obtain the set on sideThe smooth item of definition is such as Under：

Smooth item is write as：

It is denoted as B=diag (B¹,…,Bⁿ), obtain

(3) orthogonal terms E_rig(X)：Learn that their physical aspect is to meet local stiffness from actual target object observation Conversion, introduce local stiffness bound term to reduce the flexibility of conversion, especially, it is assumed that a conversionBe it is rigid, it Comprising a spin matrix and a translation matrix, then, orthogonal terms are defined as follows：

D_iIt is a scalar matrix of 3*4, it is used to extractSpin matrix, if additionally,Multiply Caused with -1Determinant for just；

(4) Laplce E_arap(X)：Introduce Laplce and protect bound term long, the effect of this constraint is so that the side of model It is long to keep constant as far as possible before and after conversion, it is denoted as respectivelyIt is the side that point i is connected with j,Also there is similar expression, therefore, define Laplce as follows：

Here willConstant 1 is set to,It is a spin matrix, it passes throughSingle polar decomposition is obtained, It is defined as follows：

By single polar decomposition, obtainSolve as follows：

WeightDefined by the tangent angle on side：

α_ijAnd β_ijIt is the relative angle on side (i, j).Then onMinimize E_arap(X), obtain：

Therefore, Laplce is write as

L^pThe linear combination on the above formula left side is represented, discrete Laplce-Bellamy spy's operator is also；b^pBe a n tie up to Amount, its i-th row represents the expression formula on the right of above formula, and passes throughWithAlternating iteration is solved, and is denoted as L=diag (L¹,…, Lⁿ), b=[b¹,…bⁿ]^T, Laplce can be rewritten as follows：

Boundary condition：In order that our optimization problem has unique solution, first visual angle model is set to reference viewing angle, then The conversion X of first model¹Unit matrix is set to, four with reference to more than, optimization problem is obtained as follows：

Wherein C and A are auxiliary variables.

Optimization problem is solved using Augmented Lagrange method, is comprised the following steps that：

1. be input into each visual angle and its it is down-sampled after model U^p, U^p,sCorresponding points and between consecutive phantom；

If 2. X does not restrain, corresponding relation f is looked for again^p:U^p→U^p+1.Otherwise go to 4.；

3. T is updated_i, X is solved with Augmented Lagrange method.Otherwise go to 2.；

4. X is exported.

The features of the present invention and beneficial effect are：

The present invention carries out global nonrigid registration and rebuilds for the target object with big action deformation, with Lower feature：

1. this algorithm proposes the framework of global registration, i.e., two neighboring visual angle model mutually registers, it is to avoid conventional method In closed loop test problems.Thus effectively alleviate the error accumulation problem in registration process.

2. this algorithm can process object under conditions of less data amount (general visual angle model quantity is 15~30) The big-movement deformation of body, and preferable complete model can be reconstructed.In addition, the computation complexity of this algorithm is smaller than conventional method A lot.

3. the method that this algorithm uses multiresolution, this allows that solving this optimization problem obtains unique solution, and avoids Conventional method is easy to be absorbed in the problem of Local Minimum.Additionally, this reduces the difficulty for the treatment of large-scale data, further subtracting Small computation complexity.

4. this algorithm adds Laplce and protects bound term long, alleviates in registration process after imperfect Model registration Contraction problem, also for last complete model of rebuilding provides guarantee.

Brief description of the drawings：

Series of advantages of the present invention will be apparent from description below in conjunction with the accompanying drawings to example and easily manage Solution:

Fig. 1 is that, using the data (complete threedimensional model) of standard database, (a) is 10 selected complete moulds of standard Type, they have big action to deform；B () is method 1：”Yang J,Li K,Li K,et al.Sparse non-rigid registration of 3D shapes[C]//Computer Graphics Forum.2015:The result of 89-99. "；(c) It is method 2：" Robust Non-Rigid Registration With Reweighted Dual Sparsities " and knot Really；D () is the result of this algorithm.It is error size with the part of color mark, error is normalized to 0~1.

As shown in Figure 1, method 1 is worst, there is very big error of fitting；And the fitting overall error of method 2 is 9.738, we Method overall error be 3.0604.Therefore, our method has more preferable result.

Fig. 2 is the data from standard database by extracting the data set (totally 35 that the visible part at each visual angle is obtained Visual angle model), experimental result below figure.The result of the first behavioral approach 1 in figure, the result of the second behavioral approach 2, the third line is This arithmetic result.First registration result for being classified as model 1-2, second is classified as the registration result of model 1-20, and the 3rd is classified as all The result of model, the 4th is classified as the result of Poisson reconstruction.This it appears that this algorithm has obtained more preferable registration result and weight Build result.

Fig. 3 and Fig. 4 are the results (having 30 and 15 visual angle models respectively) of the True Data gathered with Kinect v2. Equally, the more preferable registration of this algorithmic descriptions and reconstructed results.

Fig. 5 is flow chart of the present invention.

Specific embodiment

To solve the problems, such as existing method, the deformation of the big-movement of objective object can be processed.Therefore, the present invention takes Scheme be, the global non-rigid registration method based on rarefaction representation.The n model at visual angle is suppose there is, U is designated as¹,U²,…,Uⁿ。 For any one model p,N is the number of the point of model p；It is a littleHomogeneous coordinates.The model U adjacent for a pair^p, U^p+1(for last model Uⁿ, that adjacent thereto is U¹), use Algorithm artificial or that seek corresponding relation (as " Tam G K L, Martin R R, Rosin P L, et al.Diffusion pruning for rapidly and robustly selecting global correspondences using local isometry[J].Acm Transactions on Graphics,2014,33(1):57-76. ") find between them just Beginning mapping relationship f_p→p+1, i.e. then corresponding points solve the non-rigid transformation X between them^p,It is a transformation matrix of 3*4.There are 12 frees degree in view of each conversion put, then There is the m point just to have the 12m free degree for a model, this for the condition of there was only m constraint be not enough to solve it is unique Solution.Thus, the method that the present invention adds multiresolution in the algorithm, will any one model all carry out down-sampled obtaining phase The low resolution model answered, is designated as U^(p,S),U^(p,s-1),…,U^(p,1), U^(p,S)Represent the model of lowest resolution, U^(p,1)Represent original The model of beginning resolution ratio.Then constraint is built with the corresponding relation in high-resolution models, between solution low resolution model Conversion, transformed mappings are finally returned in high-resolution models (look for it in low resolution model with the point in high-resolution models The weighted average of the conversion of the point of one circle neighborhood is converted as it).It is denoted asIt is the point on s resolution models, Γ_i ^(p,s+1)Be withCentered on r for radius ball in the range of all points on s+1 resolution models.Specifically ask Solution preocess is as follows：

Wherein weight a_jIt is defined as follows：

R is effective radius, is defined as in low resolution model so the average weighted twice of the length of side.RepresentWithBetween Euclidean distance, and with the increase of distance, weight constantly reduces.

Based on the corresponding relation between consecutive phantom, following energy function equation can be built：

E_data(X) it is data item, represents the object function energy of data item, for is corresponded to after ensureing rotated translation transformation The distance between point summation is minimum；E_smooth(X) it is smooth item, represents the object function energy of smooth item, it is ensured that the point for closing on is gathered around There is as far as possible similar conversion；E_rig(X) it is orthogonal terms, if its constraint partial transformation is rigid；Finally, E_arapIt is Laplce Bound term, the length on the side of its restricted model keeps constant as far as possible before and after conversion.α, γ and β are respectively to weigh data The weight of item, smooth item and Laplce.This four are defined as follows:

(1) data item E_data(X)：Two neighboring visual angle model U^p,U^p+1Between it is registering as general registration problems, first The corresponding relation f put between looking for model_p, the conversion X between solving model^p.Because not being model U^pOn each pointHave at it Consecutive phantom U^p+1On corresponding points, thus for each point sets a weightIf this point is in its adjacent mould There are corresponding points to be then set to 1 in type, 0 is not set to then, it is U that O (p) is denoted as^p+1Upper and U^pThe set of corresponding points.Can be counted It is defined as follows according to item：

Formula above can be write as：

Diag represents the diagonalization block matrix of input vector.It is denoted asCan obtain as follows Form：

Wherein H is

(2) item E is smoothed_smooth(X)：For model U^pOn pointIt is denoted asIt is the set of its point being connected, is denoted as ForIt is connected thereto a littleSide.Therefore, it can obtain the set on sideThe smooth item of definition It is as follows：

Smooth item can be write as：

It is denoted as B=diag (B¹,…,Bⁿ), can obtain

(3) orthogonal terms E_rig(X)：Observed it is known that their physics shape from actual target object (human body or animal) State meets local rigid transformation.For example, the leg of people is typically rigid motion.Therefore, this algorithm introduces local stiffness about Beam reduces the flexibility of conversion.Especially, it is assumed that a conversionIt is rigid, it (is also cried comprising a spin matrix Orthogonal matrix) and a translation matrix.Then, orthogonal terms are defined as follows：

D_iIt is a scalar matrix of 3*4, it is used to extractSpin matrix.If additionally,We - 1 is multiplied by causeDeterminant for just.

(4) Laplce E_arap(X)：In experimentation in the early stage, it has been found that the side of model has that shrinks inwards to incline To in order to alleviate this problem of dtmf distortion DTMF, this algorithm introduces Laplce and protects bound term long.The effect of this constraint is so that model The length of side keeps constant as far as possible before and after conversion.We are denoted as respectivelyIt is the side that point i is connected with j, Also there is similar expression.Therefore, it can definition Laplce as follows：

Here willIt is set to constant 1.It is a spin matrix, it passes throughSingle polar decomposition is obtained, It is defined as follows

By single polar decomposition, can obtainSolve as follows：

WeightDefined by the tangent angle on side：

α_ijAnd β_ijIt is the relative angle on side (i, j).Then onMinimize E_arap(X), can obtain：

Therefore, Laplce can be write as

L^pThe linear combination on the above formula left side is represented, discrete Laplce-Bellamy spy's operator is also；b^pBe a n tie up to Amount, its i-th row represents the expression formula on the right of above formula, and passes throughWithAlternating iteration is solved.It is denoted as L=diag (L¹,…, Lⁿ), b=[b¹,…bⁿ]^T, Laplce can be rewritten as follows：

Boundary condition：In order that our optimization problem has unique solution, first visual angle model is set to reference by this algorithm Visual angle, then the conversion X of first model¹It is set to unit matrix.Four with reference to more than, the optimization problem that can obtain this algorithm is as follows：

Wherein C and A are auxiliary variables.

Optimization problem to more than can be solved using Augmented Lagrange method, be comprised the following steps that：

1. be input into each visual angle and its it is down-sampled after model U^p, U^p,sCorresponding points and between consecutive phantom.

If 2. X does not restrain, corresponding relation f is looked for again^p:U^p→U^p+1.Otherwise go to 4..

3. T is updated_i, X is solved with Augmented Lagrange method.Otherwise go to 2..

4. X is exported.

Series of advantages of the present invention will be apparent from description below in conjunction with the accompanying drawings to example and easily manage Solution:

Fig. 2 is the data from standard database by extracting the data set (totally 35 that the visible part at each visual angle is obtained Visual angle model), experimental result below figure.The result of the first behavioral approach 1 in figure, the result of the second behavioral approach 2, the third line is This arithmetic result.First registration result for being classified as model 1-2, second is classified as the registration result of model 1-20, and the 3rd is classified as all The result of model, the 4th is classified as the result of Poisson reconstruction.This it appears that this algorithm has obtained more preferable registration result and weight Build result.

Fig. 3 and Fig. 4 are the results (having 30 and 15 visual angle models respectively) of the True Data gathered with Kinect v2. Equally, the more preferable registration of this algorithmic descriptions and reconstructed results.

The present invention first using ask corresponding points algorithm (as " Tam G K L, Martin R R, Rosin P L, et al.Diffusion pruning for rapidly and robustly selecting global correspondences using local isometry[J].Acm Transactions on Graphics,2014,33 (1):57-76. ") initial corresponding relation between consecutive phantom is found out, data item constraint is then built by corresponding relation, then by it His priori can build smooth item, orthogonal item constraint, in order to alleviate the contraction problem produced for imperfect model deformation, introduce Laplce's item constraint, the above four collectively constitutes our majorized function equation.Finally, using augmentation Lagrangian method and side Majorized function equation is solved to back-and-forth method.Specific method comprises the following steps：

1) to each initial model U^pCarry out the down-sampled model U for obtaining corresponding low resolution^(p,S),U^(p,S-1),…, U^(p,1), and be that every a pair of consecutive phantoms find out initial corresponding relation f_p。

2) it is constraint by the corresponding relation on above archetype, adds the priori features of model, optimization letter can be built Number equation is as follows：

E_data(X) it is data item, represents the object function energy of data item, for is corresponded to after ensureing rotated translation transformation The distance between point summation is minimum；E_smooth(X) it is smooth item, represents the object function energy of smooth item, it is ensured that the point for closing on is gathered around There is as far as possible similar conversion；E_rig(X) it is orthogonal terms, if its constraint partial transformation is rigid；Finally, E_arapIt is Laplce Bound term, the length on the side of its restricted model keeps constant as far as possible before and after conversion.α, γ and β are respectively to weigh data The weight of item, smooth item and Laplce.

3) sparse matrix H, B, L are defined

Build data item：Wherein

Smooth item：E_smooth(X)=∑_p‖BX‖₁, wherein B=diag (B¹,…,Bⁿ)；

Orthogonal terms：

Laplce：Wherein L=diag (L¹,…,Lⁿ), b=[b¹,…bⁿ]^T。

4) following majorized function equation is solved using Augmented Lagrange method and set direction method.

S.t C=HX

A=BX

Claims (3)

1. the global non-rigid registration and method for reconstructing of a kind of deformable bodies, it is characterized in that, it is assumed that there is the n model at visual angle, It is designated as U¹,U²,…,Uⁿ.For any one model p,N is the number of the point of model p；It is a littleHomogeneous coordinates, the model U adjacent for a pair^p, U^p+1, for last model Uⁿ, That adjacent thereto is U¹, using artificial or ask the algorithm of corresponding relation to find initial mapping relation f between them_p→p+1, i.e., it is right Ying Dian, then solves the non-rigid transformation X between them^p, It is a transformation matrix of 3*4 In the algorithm add multiresolution method, will any one model all carry out it is down-sampled obtain corresponding low resolution model, It is designated as U^(p,S),U^(p,s-1),…,U^(p,1), U^(p,s)Represent the model of lowest resolution, U^(p,1)The model of original resolution is represented, so Constraint is built with the corresponding relation in high-resolution models afterwards, the conversion between low resolution model is solved, finally conversion is reflected It is emitted back towards in high-resolution models, is denoted asIt is the point on s resolution models, Γ_i ^(p,s+1)Be withCentered on r be half All points on s+1 resolution models in the range of the ball in footpath.

2. the global non-rigid registration and method for reconstructing of deformable bodies as claimed in claim 1, it is characterized in that, solve low point The solution procedure of the conversion between resolution model is as follows：

$X_i^{(p,s)}=\underset{j\∈{\mathrm{\Γ}}_i^{(p,s+1)}}{\Σ}{a}_j(u_i^{(p,s)},u_j^{(p,s+1)},r)X_j^{(p,s+1)}---\left(1\right)$

Wherein weight a_jIt is defined as follows：

$a_j(u_i^{(p,s)},u_j^{(p,s+1)},r)=max(0,(1-\frac{d^2(u_i^{(p,s)},u_j^{(p,s+1)})}{r^2}\left)\right)---\left(2\right)$

R is effective radius, is defined as in low resolution model so the average weighted twice of the length of side,RepresentWithBetween Euclidean distance, and with the increase of distance, weight constantly reduces；

Based on the corresponding relation between consecutive phantom, following energy function equation is built：

$E(X;f)\stackrel{\mathrm{\Δ}}{=}{E}_{data}(X;f)+{\mathrm{\αE}}_{smooth}\left(X\right)+{\mathrm{\γE}}_{rig}\left(X\right)+{\mathrm{\βE}}_{arap}\left(X\right)---\left(3\right)$

E_data(X) be data item, represent the object function energy of data item, for ensure after rotated translation transformation corresponding points it Between it is minimum apart from summation；E_smooth(X) it is smooth item, represents the object function energy of smooth item, it is ensured that the point for closing on possesses to the greatest extent The similar conversion of amount；E_rig(X) it is orthogonal terms, if its constraint partial transformation is rigid；Finally, E_arapIt is Laplce's constraint Item, the length on the side of its restricted model keeps constant as far as possible before and after conversion, and alpha, gamma and β are respectively to weigh data item, put down Sliding item and the weight of Laplce, this four are defined as follows:

(1) data item E_data(X)：Two neighboring visual angle model U^p,U^p+1Between it is registering as general registration problems, first look for mould The corresponding relation f put between type_p, the conversion X between solving model^p, because not being model U^pOn each pointHave adjacent at its Model U^p+1On corresponding points, thus for each point sets a weightIf this point is on its consecutive phantom There are corresponding points to be then set to 1,0 is not set to then, it is U that O (p) is denoted as^p+1Upper and U^pThe set of corresponding points, obtains determining for data item Justice is as follows：

$E_{data}(X;f)\stackrel{\mathrm{\Δ}}{=}\underset{p}{\Σ}\underset{u_i^p\∈U^p}{\Σ}{w}_i^p\left|\right|X_i^p{u}_i^p-X_{f_p\left(i\right)}^p{u}_{f_p\left(i\right)}^{p+1}|{|}_1---\left(4\right)$

Formula above is write as：

$E_{data}(X;f)\stackrel{\mathrm{\Δ}}{=}\underset{p}{\Σ}\left|\right|W^p(U^p{X}^p-U^{p+1}{X}^{p+1})|{|}_1$

$W^p=diag(\sqrt{w_1^p},\sqrt{w_2^p},...,\sqrt{w_N^p})$

$U^p=diag(u_1^p,u_2^p,...,u_N^p)---\left(5\right)$

Diag represents the diagonalization block matrix of input vector, is denoted asObtain following form：

$E_{data}(X;f)\stackrel{\mathrm{\Δ}}{=}\left|\right|HX|{|}_1---\left(6\right)$

Wherein H is

(2) item E is smoothed_smooth(X)：For model U^pOn pointIt is denoted asIt is the set of its point being connected, is denoted asFor It is connected thereto a littleSide, therefore, obtain the set on sideThe smooth item of definition is as follows：

$E_{smooth}\left(X\right)=\underset{p}{\Σ}\underset{e_{ij}^p\∈{\mathrm{\ϵ}}^p}{\Σ}\left|\right|X_i^p{u}_j^p-X_j^p{u}_j^p|{|}_1---\left(7\right)$

Smooth item is write as：

$E_{smooth}\left(X\right)=\underset{p}{\Σ}\left|\right|B^p{X}^p|{|}_1---\left(8\right)$

It is denoted as B=diag (B¹,…,Bⁿ), obtain

$E_{smooth}\left(X\right)=\underset{p}{\Σ}\left|\right|BX|{|}_1---\left(9\right)$

(3) orthogonal terms E_rig(X)：Learn that their physical aspect is to meet local rigid transformation from actual target object observation , introduce local stiffness bound term to reduce the flexibility of conversion, especially, it is assumed that a conversionIt is rigid, it is included One spin matrix and a translation matrix, then, define orthogonal terms as follows：

$E_{rig}\left(X\right)=\underset{p}{\Σ}\underset{i}{\Σ}\left|\right|D_i{X}_i^p-R_i^p|{|}_F^2---\left(10\right)$

$\begin{array}{cc}s.t.& R_i^{p^T}{R}_i^p=I_3\end{array}$

$\det \left(R_i^p\right)>0$

D_iIt is a scalar matrix of 3*4, it is used to extractSpin matrix, if additionally,Being multiplied by -1 makes Determinant for just；

(4) Laplce E_arap(X)：Introduce Laplce and protect bound term long, the effect of this constraint is so that the length of side of model exists Keep constant as far as possible before and after conversion, be denoted as respectivelyIt is the side that point i is connected with j, Also there is similar expression, therefore, define Laplce as follows：

$E_{arap}\left(X\right)=\underset{T_i^p}{min}\underset{p}{\Σ}\underset{i}{\Σ}{w}_i^p\underset{j\∈N\left(i\right)}{\Σ}{w}_{ij}^p\left|\right|e_{ij}^{\′p}-T_i^p{e}_{ij}^p|{|}_F^2---\left(11\right)$

Here willConstant 1 is set to,It is a spin matrix, it passes throughSingle polar decomposition is obtained,Definition It is as follows：

$S_i^p=\underset{p}{\Σ}\underset{j\∈N\left(i\right)}{\Σ}{w}_{ij}^p{e}_{ij}^p{e}_{ij}^{\′p^T}---\left(12\right)$

By single polar decomposition, obtain Solve as follows：

$T_i^p=V_i^p{U}_i^{p^T}---\left(13\right)$

WeightDefined by the tangent angle on side：

$w_{ij}^p=\frac{1}{2}({\mathrm{cot\α}}_{ij}+{\mathrm{cot\β}}_{ij})---\left(14\right)$

α_ijAnd β_ijIt is the relative angle on side (i, j).Then onMinimize E_arap(X), obtain：

$\underset{j\∈N\left(i\right)}{\mathrm{\Σ}}{w}_{ij}^p\left(u_i^{\′p}-u_j^{\′p}\right)=\underset{j\∈N\left(i\right)}{\mathrm{\Σ}}\frac{w_{ij}^p}{2}\left(T_i^p+T_j^p\right)\left(u_i^p-u_j^p\right)---\left(15\right)$

Therefore, Laplce is write as

$E_{arap}\left(X\right)=\underset{p}{\Σ}\left|\right|L^p{X}^p-b^p|{|}_F^2---\left(16\right)$

L^pThe linear combination on the above formula left side is represented, discrete Laplce-Bellamy spy's operator is also；b^pIt is a n-dimensional vector, Its i-th row represents the expression formula on the right of above formula, and passes throughWithAlternating iteration is solved, and is denoted as L=diag (L¹,…,Lⁿ), B=[b¹,…bⁿ]^T, Laplce can be rewritten as follows：

$E_{arap}\left(X\right)=\left|\right|LX-b|{|}_F^2---\left(17\right)$

Boundary condition：In order that our optimization problem has unique solution, first visual angle model is set to reference viewing angle, then first The conversion X of individual model¹Unit matrix is set to, four with reference to more than, optimization problem is obtained as follows：

$\begin{array}{c}\underset{X,C,A}{\min }\left|\right|C|{|}_1+\mathrm{\α}|\left|A\right|{|}_1+\mathrm{\γ}\underset{p}{\mathrm{\Σ}}\underset{i}{\mathrm{\Σ}}\left|\right|D_i{X}_i^p-R_i^p|{|}_F^2+\mathrm{\β}||LX-b|{|}_F^2\\ \begin{array}{cc}s.t& C=HX\end{array}\\ A=BX\\ R_i^{p^T}{R}_i^p=I_3,\det \left(R_i^p\right)>0\end{array}---\left(18\right)$

Wherein C and A are auxiliary variables.

3. the global non-rigid registration and method for reconstructing of deformable bodies as claimed in claim 1, it is characterized in that, optimization is asked Topic is solved using Augmented Lagrange method, is comprised the following steps that：

1. be input into each visual angle and its it is down-sampled after model U^p, U^p,sCorresponding points and between consecutive phantom；

If 2. X does not restrain, corresponding relation f is looked for again^p:U^p→U^p+1.Otherwise go to 4.；

3. T is updated_i, X is solved with Augmented Lagrange method.Otherwise go to 2.；

4. X is exported.