CN102509339B - Method for simplifying vertex clustering of three-dimensional models with texture constraint - Google Patents

Abstract

The invention provides a method for simplifying a vertex clustering of three-dimensional models with texture constraint. The method comprises the steps of: expanding the vertexes from a geometric space to a fifth-dimensional space including texture coordinates in consideration of algorithm efficiency in the self-adaptive division processes of the vertexes; dividing the vertexes according to the positions of the vertexes in the geometric space and the texture space in the self-adaptive division processes of the vertexes; in error measurement, respectively computing geometric errors and texture errors, if the error value is more than a given threshold, splitting the current node; and in the self-adaptive division processes of the vertexes, in consideration of saving the storage resources and improving the algorithm efficiency, adopting the relatively important vertex in an original model as a clustering representative instead of computing the geometric position of a new vertex through iteration. The method for simplifying vertex clustering of three-dimensional models with texture constraint organically integrates texture error measurement and geometric simplification, realizes rapid simplification of a mass of complex three-dimensional models and supports efficient three-dimensional visualization of city virtual scenes.

Description

A kind of method for simplifying vertex clustering of three-dimensional models with texture constraint

Technical field

The present invention relates to three-dimensional geographic information visualization technique field, particularly relate to a kind of for the complex three-dimensional model vertices cluster short-cut method under texture error condition.

Background technology

Texture is an important property information of three-dimensional model, is directly connected to the authenticity that three-dimensional model is expressed in virtual reality.In model simplification process, even if the maintenance of the geometric properties of model is good again, if cannot guarantee the correctness of texture, its visual effect will be had a greatly reduced quality.

Although along with the development of virtual reality technology, current existing three-dimensional model simplifying technology is very ripe aspect model geometric simplification, and for keeping in simplification process, the research of model textural characteristics is also few.

For texture error metrics, a kind of conventional method that related documents proposes is by (X, Y, Z) expansion of three-dimensional metric algorithm accounts for (X, Y, Z, R, G, B) 6 dimension spaces in, Garland, M. Multi-Resolution Modeling:Survey & future opportunities[C]. Proceedings of the Eurographics ' 99. 1999; Shen Xukun, Zhao Xuewei, Zhao Qinping. a kind of lattice simplified and compression progressive transmission method [J] of keeping characteristics. Journal of Computer Research and Development, 2007, Vol.44 (7): 1097-1104; Feng Xiang, Zhou Mingquan. with the three-dimensional model simplifying algorithm [J] of texture. computer-aided design (CAD) and graphics journal, 2009, Vol.21 (6): 842-846.Its ultimate principle is as follows: according to the texture coordinate (u, v) on summit, find the colouring information (r, g, b) corresponding to this summit on texture image.In model simplification process, according to simplified model M ₁with master pattern M ₂summit corresponding relation, define its texture error for the distance error in its vertex color space, shown in following formula, (r wherein ₁, g ₁, b ₁), (r ₂, g ₂, b ₂) be respectively the color value of vertex v 1, v2:

Wherein vertex v 1, v2 belong to respectively model M ₁, M ₂;

This error metrics method can play certain binding effect to model simplification process, can keep to a certain extent the textural characteristics of model, but still can not be satisfied with the needs of practical application.Its weak point is mainly reflected in following two aspects:

(1) reduce model simplification efficiency.Adopt this texture error metrics method guidance model to simplify process, need to obtain vertex color corresponding on texture image according to texture coordinate.Therefore, need to frequently model data texturing be loaded into internal memory, and need to each summit, carry out the map operation of (u, v) → (r, g, b), greatly increase undoubtedly the resource overhead of shortcut calculation, increase the time of model simplification.

(2) be not suitable for the three-dimensional model of texture information complexity.In the texture of three-dimensional model is expressed, the mapping relations of model vertices and texture image are not that texture coordinate (u, v) is to the simple relation of texture color (r, g, b).To with each 3d space under triangle, can in texture, define a corresponding 2D delta-shaped region according to the texture coordinate on its summit, and by this 3D triangular element of the texture of this 2D delta-shaped region.In Fig. 1, the left side is the triangle under 3d space, take in the XYZ coordinate system that O is that initial point sets up, and summit Q ₁, Q ₂, Q ₃dividing other coordinate is (x ₁, y ₁, z ₁), (x ₂, y ₂, z ₂), (x ₃, y ₃, z ₃); The right is the 2D triangle texture region that this triangle is corresponding, in UV coordinate system, and summit Q ₁, Q ₂, Q ₃dividing other coordinate is (u ₁, v ₁), (u ₂, v ₂), (u ₃, v ₃).

If summit Q ₁, Q ₂, Q ₃corresponding texture color value is identical, and three summits distance in color space is 0.Suppose summit Q in simplification process ₁q ₂, Q ₃substitute summit, if while adopting so formula to carry out error metrics, gained texture error amount is also 0.Yet under actual conditions, this visual impact that brings estimated value much larger than error metrics that simplifies the operation.

Therefore, adopting texture color to carry out error metrics is only suitable in the comparatively uniform three-dimensional model of texture color distribution, for the comparatively complicated three-dimensional model (as city complex three-dimensional BUILDINGS MODELS) of texture improper.

For the Vertex Clustering method of three-dimensional model simplifying visible document: Rossignac the earliest, J., Borrel, P. Multi-Resolution 3D Approximations for Rendering [J]. Modeling in Computer Graphics, June-July 1993:455-465.The method key step is: according to factors such as the size of model meshes, complexities, by the shared spatial division of three-dimensional model, be a plurality of cubic units; Calculate the representative point in each cubic units, each summit in cubic units is replaced with this representative point; The degeneration triangle producing after replacing removes.

Weak point for above-mentioned Vertex Clustering method, related documents: Zhou Kun, Pan Zhigeng, Shi Jiaoying. a kind of new Mesh simplification algorithm based on Vertex Clustering [J]. robotization journal, 1999,1, Vol.25 (1): 1-8, provide a kind of improved clustering method, by Octree, carried out self-adaptation division, and by second order error, measured to control the generation of representative point.The method is very effective for keeping the geometric properties of model in simplification process.But in this method, do not have the attribute information of Consideration of Three-dimensional model, in simplification process, cannot keep the attributive character of three-dimensional model.

Summary of the invention

The object of the invention is to simplify deficiency for existing cluster, propose a kind of method for simplifying vertex clustering of three-dimensional models with texture constraint.

Technical scheme of the present invention is a kind of method for simplifying vertex clustering of three-dimensional models with texture constraint, it is characterized in that, comprises the following steps:

Step 1, calculates the weights on each summit in three-dimensional model;

Step 2, big or small according to default geometric units lattice size and texture cell lattice, by three-dimensional model according to (X, Y, Z, U, V) five dimensions are divided, first according to geometric units lattice, divide and obtain some leaf nodes, according to texture cell lattice, above-mentioned some leaf node Further Divisions are obtained to more leaf node, wherein (X on this basis, Y, Z) represent geometric space coordinate, (U, V) represents texture space coordinates;

Step 3, checks each leaf node inner vertex number,

If number of vertices is empty or is 1, do not do any operation;

If number of vertices is greater than default number threshold value, leaf node is carried out to node split operation;

If number of vertices is greater than 1 and be less than or equal to default number threshold value, according to the weights on step 1 gained summit, in leaf node, select the summit of weights maximum as cluster representative point, calculate the geometric error producing after cluster

with texture error

, work as geometric error

be greater than default geometric error threshold value

or texture error

be greater than default texture error threshold

shi Jinhang splitting operation;

Step 4, the leaf node that splitting operation is produced returns to execution step 3, until the number of vertices in all leaf nodes is for empty or be 1, or geometric error

with texture error

all in threshold range;

Step 5, for all geometric errors

with texture error

all the leaf node in threshold range, replaces the summit in leaf node, and upgrades index data with cluster representative point, then the triangle of degeneration is removed from the triangle list of three-dimensional model.

And the weights account form on summit is as follows,

If summit weights scope w ∈( 0,1), weights more represent that this summit is more important, for the arbitrary summit that is not positioned at boundary in three-dimensional model q, have relevant triangle set t _s = t ₀ , t ₁ ... t _m , suppose triangle set t _s intermediate cam shape t _i with t _j adjacent, definition

as follows

Wherein n _i , n _j be respectively triangle t _i , t _j normal vector, dotfor vector dot computing, w _ij value scope is (0,1), w _ij be worth greatlyr, represent triangle t _i with t _j angle less, summit qweights wbe expressed as:

For the summit that is positioned at boundary, its weights are directly given maximal value 1.

And, in step 1, judge whether summit is positioned at the specific implementation of boundary as follows,

If summit qthere is relevant limit set l _s with triangle set t _s if, limit set l _s in have limit lmeet triangle set t _s in only have unique triangle tcomprise this limit, so this summit qbe frontier point; If limit set l _s in all limits do not meet this condition, this summit is not frontier point so.

And in step 3, computational geometry error adopts error of curvature measure, specific implementation is as follows,

Be set in a vertex set in leaf node q _s = q _w , Q ₀ , Q ₁ q _n , summit wherein q _w for cluster representative point, vertex set q _s other any one summits in interior q _i there is relevant triangle set ts= t ₀ , t ₁ ... t _m , by triangle set tsin all vertexs of a triangle q _i use summit q _w after replacement, obtain triangle set ts '= t ₀ ', t ₁ ' ... t _m '; If triangle set tsintermediate cam shape t _j normal vector be n _j , triangle set ts 'intermediate cam shape t _j 'normal vector be n _j ', this summit q _i geometric error be

Wherein

represent vector dot computing, jvalue is 0,1, m;

This vertex set q _s geometric error after cluster is

Wherein, max{} represents to get maximal value, ivalue is 0,1, n.

And in step 3, the specific implementation of calculating texture error is as follows,

Be set in a vertex set in leaf node q _s = q _w , Q ₀ , Q ₁ q _n , summit wherein q _w for cluster representative point, summit q _w texture coordinate be ( u _w , v _w ), vertex set q _s other any one summits in interior q _i texture coordinate be ( u _i , v _i ), by summit q _i use summit q _w texture error after replacement is

This vertex set q _s texture error after cluster is

Wherein, max{} represents to get maximal value, ivalue is 0,1, n.

The present invention, creatively using texture coordinate space length as texture error metrics, makes three-dimensional model simplifying have stronger binding character and higher efficiency; The texture error metrics method of complex three-dimensional model and how much simplification organic unities are got up, realized the quick simplification of magnanimity complex three-dimensional model, and effectively kept the textural characteristics after model simplification.The present invention can support the efficient three-dimensional visualization of city virtual scene.

Accompanying drawing explanation

Fig. 1 is three-dimensional model texture graph of a relation.

Fig. 2 is the process flow diagram of the embodiment of the present invention.

Embodiment

Embodiment adopts method of the present invention, uses C Plus Plus to write the method for simplifying vertex clustering with texture constraint, by typical 3 D complex building model, specifically implements.Consider that 3 D complex BUILDINGS MODELS is all to carry out modeling according to true ratio, the redundant data amount of model is limited, for guaranteeing good reduced mass, simplify result data amount maintain master pattern 30% ~ 50% between.

For the purpose of reference implementation, provide embodiment flow process as follows, referring to Fig. 2:

Step 1, calculates the weights on each summit.

The object of three-dimensional model simplifying is in the situation that the model and the similar as far as possible summit of deleting to greatest extent of master pattern that guarantees to simplify.Therefore in the process of Vertex Clustering, the cluster representative of selection point should be that some can reflect model principal character and the summit that keeps model silhouette.Summit weights can reflect that this summit is for the importance that keeps the aspect of model.Generally, be positioned at the variations such as model boundary, acute angle and comparatively significantly should give larger weights in the summit of part, and should give less weights for the summit that model changes level and smooth part.According to this principle, embodiment is after importing three-dimensional modeling data, and the method for calculating summit weights is as follows:

For ease of calculating, establish summit weights scope w ∈( 0,1), weights more represent that this summit is more important, for the arbitrary summit that is not positioned at boundary in three-dimensional model q, have relevant triangle set t _s = t ₀ , t ₁ ... t _m , subscript 0,1, for identifying the triangle of set.Suppose triangle set t _s intermediate cam shape t _i with t _j adjacent, definition

as follows

Wherein n _i , n _j be respectively triangle t _i , t _j normal vector, dotfor vector dot computing, w _ij value scope is (0,1), w _ij be worth greatlyr, represent triangle t _i with t _j angle less, geometric properties is more obvious.Summit qweights wbe expressed as:

For the summit that is positioned at boundary, its weights are directly given maximal value 1.

In addition,, because model boundary is the pith that keeps model silhouette, therefore for the summit that is positioned at boundary, its weights should be given maximal value 1.A kind of conventional frontier point determination methods is as follows: establish summit qthere is relevant limit set l _s with triangle set t _s if, limit set l _s in have limit l(limit quantity >=1) meets triangle set t _s in only have unique triangle tcomprise this limit, this summit is frontier point so q; If limit set l _s in all limits do not meet this condition, this summit is not frontier point so.

During concrete enforcement, can also adopt other existing weights account forms.

Step 2, big or small according to default geometric units lattice size and texture cell lattice, by three-dimensional model according to (X, Y, Z, U, V) five dimensions divide, (X wherein, Y, Z) represent geometric space coordinate, (U, V) represents texture space coordinates.

The geometric space that three-dimensional model is corresponding is its bounding box volume, texture space is a two dimensional surface, in its X and Y-direction, span is 0 to 1, geometric units lattice are geometric space volumes of user-defined least unit, texture cell lattice are texture space areas of user-defined least unit, for example defining geometric units lattice is 1 meter of square, texture cell lattice for long and wide be 0.1 texture area.During concrete enforcement, user can set voluntarily according to accuracy requirement.According to user-defined cell (comprising geometric space volume, texture space area) size, by master pattern according to (X, Y, Z, U, V) five dimensions divide, first according to geometric units lattice, divide and obtain some leaf nodes, according to texture space, divide above-mentioned some leaf node Further Divisions are obtained to more leaf node on this basis, complete initial division.The leaf node number obtaining after initial division is initial number, at subsequent step 3, may divide the leafier child node of generation.

Step 3, checks each leaf node inner vertex number,

If number of vertices is empty or is 1, do not do any operation;

If number of vertices is greater than default number threshold value, leaf node is carried out to node split operation;

If number of vertices is greater than 1 and be less than or equal to default number threshold value, according to the weights on step 1 gained summit, in leaf node, select the summit of weights maximum as cluster representative point, calculate the geometric error producing after cluster with texture error , work as geometric error

be greater than default geometric error threshold value

or texture error

be greater than default texture error threshold

shi Jinhang splitting operation.Certainly, if geometric error with texture error

all in threshold range, i.e. geometric error

be less than or equal to default geometric error threshold value

, and texture error be less than or equal to default texture error threshold

time, leaf node is not needed to carry out node split operation, in step 5, the summit in leaf node is replaced with cluster representative point.

The splitting operation that embodiment carries out leaf node is according to (X, Y, Z, U, V) five dimensions again subdivision 2 by present node ⁵child node, according to dimension X, Y, Z, U, V, each dimension is carried out the division of one-to-two.Summit in this leaf node is reentered in the leaf node that subdivision obtains, and by returning to the leaf node that step 3 obtains each subdivision judges whether continue division.Default number threshold value, error threshold

,

size is relevant with the precision of modeling, can according to actual conditions, suitable value be set by user.

The key factor of three-dimensional model summit and 2 d texture mapping is the texture coordinate (u, v) on summit.For summit (u, v) value, differ larger 3D triangle, the texture information comprising between its 2D triangle corresponding on texture is also more, deletes the texture error that this triangle brings larger; Otherwise, for summit (u, v) value, differing less 3D triangle, the texture information comprising between its 2D triangle corresponding on texture is also less, deletes the texture error that this triangle brings less.Therefore, the present invention is carrying out in three-dimensional model simplifying process, geometric space can be expanded to and comprise texture coordinate (X, Y, Z, U, V) quintuple space in, according to the summit corresponding relation of simplified model and master pattern, defining its texture error is the distance difference of its summit in texture coordinate space:

Wherein, wherein vertex v 1, v2 belong to respectively simplified model M ₁, master pattern M ₂, (u ₁, v ₁), (u ₂, v ₂) be vertex v 1, the apex coordinate of v2.

According to this texture Error Measure Rule, in carrying out the process of model simplification, preferentially delete the less summit of texture coordinate distance, retain the larger summit of texture coordinate distance, thereby make to simplify result, can retain more textural characteristics.

Adopt texture coordinate space length as the method for error metrics, with take texture color space length and compare as error metrics method, not only its binding character is stronger, and owing to not needing each summit to obtain the operation of color, and texture coordinate space than color space few a dimension, in difference calculation process runs, the complexity of algorithm also can be lower, thereby improved the efficiency of Model Simplification Algorithm.

The geometric error that comprises of error metrics is measured and texture error metrics two parts.Embodiment adopts error of curvature measure of the prior art for geometric error tolerance, specific as follows:

Suppose to be positioned at the vertex set of a leaf node q _s = q _w , Q ₀ , Q ₁ q _n , summit wherein q _w for cluster representative point, summit q ₀ , Q ₁ q _n be in leaf node in addition n+ 1 node.Vertex set q _s other any one summits in interior q _i there is relevant triangle set ts= t ₀ , t ₁ ... t _m , by triangle set tsin all vertexs of a triangle q _i use summit q _w after replacement, obtain triangle set ts '= t ₀ ', t ₁ ' ... t _m '; If triangle set tsintermediate cam shape t _j normal vector be n _j , triangle set ts 'intermediate cam shape t _j 'normal vector be n _j ', this summit q _i geometric error be

Wherein

represent vector dot computing, jvalue is 0,1, m;

This vertex set q _s geometric error after cluster is

Wherein, max{} represents to get maximal value, ivalue is 0,1, n.

During concrete enforcement, computational geometry error can also adopt additive method, such as distance error tolerance, and angular error tolerance etc.

The basic skills of texture error metrics is as follows:

Be set in a vertex set in leaf node q _s = q _w , Q ₀ , Q ₁ q _n , summit wherein q _w for cluster representative point, summit q _w texture coordinate be ( u _w , v _w ), vertex set q _s other any one summits in interior q _i texture coordinate be ( u _i , v _i ), by summit q _i use summit q _w texture error after replacement is

This vertex set v _s texture error formula after cluster is identical with geometric error.This vertex set q _s texture error after cluster is

Wherein, max{} represents to get maximal value, ivalue is 0,1, n.

Step 4, the leaf node that splitting operation is produced returns to execution step 3, until all leaf nodes meet one of following two conditions:

(1) number of vertices in leaf node is empty or is 1;

(2) number of vertices in leaf node is greater than 1 and be less than or equal to default number threshold value, and geometric error

with texture error

all in threshold range.In leaf node, select the summit of weights maximum as cluster representative point, calculate the geometric error producing after cluster

with texture error , geometric error

be less than or equal to default geometric error threshold value

, texture error be less than or equal to default texture error threshold

.

Step 5, for all geometric errors

with texture error

all the leaf node in threshold range, replaces the summit in leaf node, and upgrades index data with cluster representative point, then the triangle of degeneration is removed from the triangle list of three-dimensional model.

Embodiment carries out Vertex Clustering operation by all leaf nodes that meet cluster standard, and the degeneration triangle producing after cluster is removed.In prior art, in the data of three-dimensional model, with triangle set, represent object external outline shape, the definition of a plurality of triangle surfaces, consist of, the definition of each tri patch comprises the three-dimensional coordinate on each summit of triangle and the method vector of tri patch.Index data is index corresponding to summit in three-dimensional model, behind replacement summit, needs to upgrade.Triangle list is the set of three-dimensional model intermediate cam shape, with the index on triangle sequence number and corresponding three summits, expresses.The triangle of degenerating, is after simplifying, and therefore the triangle shifting out from three-dimensional model needs correspondingly from triangle list, to remove.

By above concrete enforcement, can find out, increase the simplification after texture constraint, although there is certain increase on algorithm complex, but its efficiency can meet the needs of practical application, more crucial is to keep there is obvious effect for the textural characteristics of simplifying result, is applicable to the three-dimensional building object model of some grain details complexity.In addition, because Vertex Clustering method is a kind of method that does not keep topological structure, thereby be applicable to the model simplification of some non-manifolds, and simplify speed.

Claims (4)

1. for the visual method for simplifying vertex clustering of three-dimensional models of geography information, it is characterized in that, comprise the following steps:

Step 1, calculates the weights on each summit in three-dimensional model;

Step 2, big or small according to default geometric units lattice size and texture cell lattice, by three-dimensional model according to (X, Y, Z, U, V) five dimensions are divided, first according to geometric units lattice, divide and obtain some leaf nodes, according to texture cell lattice, above-mentioned some leaf node Further Divisions are obtained to more leaf node, wherein (X on this basis, Y, Z) represent geometric space coordinate, (U, V) represents texture space coordinates;

Step 3, checks each leaf node inner vertex number,

If number of vertices is empty or is 1, do not do any operation;

If number of vertices is greater than default number threshold value, leaf node is carried out to node split operation;

If number of vertices is greater than 1 and be less than or equal to default number threshold value, according to the weights on step 1 gained summit, in leaf node, select the summit of weights maximum as cluster representative point, calculate the geometric error ε producing after cluster _swith texture error ε ' _s, as geometric error ε _sbe greater than default geometric error threshold epsilon _maxor texture error ε ' _sbe greater than default texture error threshold ε ' _maxshi Jinhang splitting operation; The specific implementation of calculating texture error is as follows,

Be set in a vertex set Q in leaf node _s={ Q _w, Q ₀, Q ₁q _n, summit Q wherein _wfor cluster representative point, summit Q _wtexture coordinate be (u _w, v _w), vertex set Q _sother any one summit Q in interior _itexture coordinate be (u _i, v _i), by summit Q _iuse summit Q _wtexture error after replacement is

ε' _i＝(u _i-u _w) ²+(v _i-v _w) ²

This vertex set Q _stexture error after cluster is

ε' _s＝max{ε' _i}

Wherein, max{} represents to get maximal value, and i value is 0,1 ... n;

Step 4, the leaf node that splitting operation is produced returns to execution step 3, until the number of vertices in all leaf nodes is for empty or be 1, or geometric error ε _swith texture error ε ' _sall in threshold range;

Step 5, for all geometric error ε _swith texture error ε ' _sall the leaf node in threshold range, replaces the summit in leaf node, and upgrades index data with cluster representative point, then the triangle of degeneration is removed from the triangle list of three-dimensional model.

2. as claimed in claim 1 for the visual method for simplifying vertex clustering of three-dimensional models of geography information, it is characterized in that: the weights account form on summit is as follows,

If summit weights scope w ∈ (0,1), weights more represent that this summit is more important, for the arbitrary summit Q that is not positioned at boundary in three-dimensional model, have relevant triangle set T _s={ t ₀, t ₁... t _m, suppose triangle set T _sintermediate cam shape t _iwith t _jadjacent, definition w _ijas follows

w _ij＝(1-dot(n _i,n _j))/2

N wherein _i, n _jbe respectively triangle t _i, t _jnormal vector, dot is vector dot computing, w _ijvalue scope is (0,1), w _ijbe worth greatlyr, represent triangle t _iwith t _jangle less, the weight w of summit Q is expressed as:

$\left\{\begin{array}{c}w=\max \left\{w_{\mathrm{ij}}\right\}\\ i\≠j\\ i,j\∈(0,m)\end{array}\right.$

For the summit that is positioned at boundary, its weights are directly given maximal value 1.

3. as claimed in claim 2 for the visual method for simplifying vertex clustering of three-dimensional models of geography information, it is characterized in that: in step 1, judge whether summit is positioned at the specific implementation of boundary as follows,

If summit Q has relevant limit set L _swith triangle set T _sif, limit set L _sin have limit l and meet triangle set T _sin only have unique triangle t to comprise this limit, this summit Q is frontier point so; If limit set L _sin all limits do not meet this condition, this summit is not frontier point so.

As described in claim 1 or 2 or 3 for the visual method for simplifying vertex clustering of three-dimensional models of geography information, it is characterized in that: in step 3, computational geometry error adopts error of curvature measure, and specific implementation is as follows,

Be set in a vertex set Q in leaf node _s={ Q _w, Q ₀, Q ₁q _n, summit Q wherein _wfor cluster representative point, vertex set Q _sother any one summit Q in interior _ithere is relevant triangle set Ts={t ₀, t ₁... t _m, by all vertex of a triangle Q in triangle set Ts _iuse summit Q _wafter replacement, obtain triangle set Ts '={ t ₀', t ₁' ... t _m'; If triangle set Ts intermediate cam shape t _jnormal vector be n _j, triangle set Ts ' intermediate cam shape t _j' normal vector be n _j', this summit Q _igeometric error be

${\mathrm{\ϵ}}_i=\frac{1}{m+1}\underset{j=0}{\overset{m}{\mathrm{\Σ}}}\mathrm{dot}(n_j,{n_j}^{\′})$

Wherein dot represents vector dot computing, and j value is 0,1 ... m;

This vertex set Q _sgeometric error after cluster is

ε _s＝max{ε _i}

Wherein, max{} represents to get maximal value, and i value is 0,1 ... n.

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