CN101877147B - Simplified algorithm of three-dimensional triangular mesh model - Google Patents

Simplified algorithm of three-dimensional triangular mesh model Download PDF

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CN101877147B
CN101877147B CN201010215500A CN201010215500A CN101877147B CN 101877147 B CN101877147 B CN 101877147B CN 201010215500 A CN201010215500 A CN 201010215500A CN 201010215500 A CN201010215500 A CN 201010215500A CN 101877147 B CN101877147 B CN 101877147B
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CN101877147A (en
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吴庆标
金勇�
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Zhejiang University ZJU
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Abstract

The invention discloses a simplified algorithm of a three-dimensional triangular mesh model, which comprises the following steps: 1) giving the three-dimensional triangular mesh model which needs to be simplified and the number NC of edges of a polygon of a target simplified model, and dividing a triangle on the surface of the model into NC clusters; 2) approximating the error metric according to the defined plane, and iteratively updating the boundaries of all the clusters and the triangles contained in all the clusters, thereby reducing the error metric of all the clusters and the final simplified polygon; and 3) taking an intersection point of more than three clusters as a vertex of a polygon mesh after obtaining the final optimized cluster division, and obtaining the simplified polygon mesh model. The algorithm is intuitive and effective, and the simplified mesh model can effectively keep details of the original mesh; as the algorithm does not relate to the modification of mesh topology, the algorithm is simple, has better stability and can be effectively applied in a geometric modeling system.

Description

The shortcut calculation of three-dimensional triangular mesh model
Technical field
The present invention relates to a kind of shortcut calculation of three-dimensional triangular mesh model.
Background technology
Three-dimensional triangular mesh has number of applications in fields such as Computer-aided Geometric Design, computer animation, virtual reality, computer game and medical images.Development along with the 3-D scanning technology; Number of vertex is that tens thousand of model is very common; It is hundreds thousand of even more that the model vertices number that has reaches; So in the multi-resolution display of grid, the acceleration grid is painted, need fast and can keep the Mesh simplification algorithm of grid model details in the processes such as grid compression.Existing several kinds of main polygonal meshs are simplified technology as follows:
1, deletes (Decimation): Garland etc. [1]Propose a kind of Mesh simplification algorithm based on the limit reduction, the secondary range that its correlation plane is arrived on the summit of this method employing grid is as tolerance, and the iteration of carrying out the limit reduction is till reaching target gridding limit number.Similarly, Hoppe etc. [2], Klein etc. [3], Garland etc. [4]Grid elements is proposed a kind of error metrics, and its error metrics is set up based on apex coordinate, color or texture coordinate, then grid elements is carried out the iteration reduction to simplify grid.Though the method for deleting can obtain effective lattice simplified effect, these class methods obtain grid and often have the very high summit of the limit number of degrees, and more consuming time owing to the topological structure of operation grid.
2, grid refining (Mesh Refinement): Eck etc. [5], Delingette etc. [6], Lee etc. [7]All propose to approach expression original mesh model, carry out refining iteratively in the part with separately strategy then and encrypt coarse grid being able to accurately represent the precision of original mesh model, but the types of models that this type side is suitable for is very limited with a coarse grid.
3, grid reconstruction (Remeshing): Alliez etc. [8,9], Gu etc. [10]Proposition can controlled target grid vertex number grid reconstruction method, but these methods all are confined to the parametrization of grid, comprise a large amount of calculating and numerical instability.Valette etc. [11,12]Through the part the approximate Centroidal Voronoi Diagrams of greedy algorithm structure; Construct the dual graph of CVD then and simplify grid, but this method is limited by the degree of model simplification, poor effect when the model simplification ratio is very large as target.
4, global optimization (Global optimization): Hoppe etc. [13]Proposition is regarded lattice simplified problem as the global optimization problem.Measure original mesh with an energy function, through the number of vertex of control mesh, apex coordinate and topological connection relation are to optimize the energy function of definition simultaneously, and original mesh model detail and the curvature of can being maintained is simplified effect.Cohen-Steiner etc. [14]Variation grid approach method is proposed; This method is divided into original mesh the almost plane bunch collection of destination number; Use the energy relevant to measure the bundle of planes collection with normal direction; With the grid clustering that the Lloyd algorithm is optimized, each sheet sub-clustering is at last represented to obtain final simplification grid with a polygon.This method is intuitively effective, and the grid model after the simplification can effectively keep the details of original mesh.Yet this method speed is confined to the iterations of Lloyd algorithm, and the Lloyd algorithm also can't guarantee to obtain the result of a global optimum, and needs post-processed (act of union is to almost consistent bundle of planes collection) to reach desirable effect.
List of references
[1]Garland?M,Heckbert?PS.Surface?Simplification?Using?Quadric?Error?Metrics.In:Whitted?T,ed.Proceedings?of?ACM?SIGGRAPH.Los?Angeles:ACM?Press,1997.209-216.
[2]Hoppe?H.Progressive?Meshes.In:Rushmeier?H,ed.Proceedings?of?ACM?SIGGRAPH.New?Orleans:Addison-Wesley?Professional,1996.99-108.
[3]Klein?R,Liebich?G,Straβer?W.Mesh?Reduction?with?Error?Control.In:Yagel?R,Nielson?GM,eds.Proceedings?of?IEEE?Visualization.San?Francisco:IEEE?Computer?Society?Press,1996.311-318.
[4]Garland?M,Heckbert?PS.Simplifying?Surfaces?with?Color?and?Texture?using?Quadric?Error?Metrics.In:Ebert?DS,Rushmeier?H,Hagen?H,eds.Proceedings?of?IEEE?Visulaization.Washington:IEEE?Computer?Society?Press,1998.263-269.
[5]Eck?M,DeRose?T,Duchamp?T,Hoppe?H,Lounsbery?M,Stuetzle?W.Multiresolution?analysis?of?arbitrary?meshes.In:Mair?SG,Cook?R,eds.Proceedings?of?ACM?SIGGRAPH.Los?Angeles:ACM?Press,1995.173-182.
[6]Delingette?H,Herbert?M,Ikeuchi?K.Shape?representation?and?image?segmentation?using?deformable?surfaces.Image?and?Vision?Computing,1992,10(3):132-144.
[7]Lee?AWF,Sweldens?W,
Figure BSA00000188270300031
Cowsar?L,Dobkin?D.Maps:Multiresolution?adaptive?parameterization?of?surfaces.In:Machover C,ed.Proceedings of?ACM?SIGGRAPH.Orlando:ACM?Press,1998.95-104.
[8]Alliez?P,Meyer?M,Desbrun?M.Interactive?Geometry?Remeshing.In:Appolloni?T,ed.Proceedings?of?ACM?SIGGRAPH.San?Antonio:ACM?Press,2002.355-361.
[9]Alliez?P,Cohen-Steiner?D,Devillers?O,Levy?B,Desbrun?M.Anisotropic?Polygonal?Remeshing.In:RockWood?AP,ed.Proceedings?of?ACM?SIGGRAPH.San?Diego:ACM?Press,2003.485-493.
[10]Gu?X.,Gortler?S,Hoppe?H.Geometry?Images.In:Appolloni?T,ed.Proceedings?of?ACM?SIGGRAPH.San?Antonio:ACM?Press,2002.363-374.
[11]Valette?S,Chassery?J-M.Approximated?Centroidal?Voronoi?Diagrams?for?Uniform?Polygonal?Mesh?Coarsening.Computer?Graphics?Forum(Proc.Eurographics),2004,23(3):381-389.
[12]Valette?S,Kompatsiaris?I,Chassery?J-M.Adaptive?Polygonal?Mesh?Simplification?With?Discrete?Centroidal?Voronoi?Diagrams.In:Lazzari?G,Pianesi?F,Crowley?JL,Mase?Kenji,Oviatt?SL,eds.Proceedings?ofICMI.Trento:ACM?Press,2005.655-662.
[13]Hoppe?H,DeRose?T,Duchamp?T,McDonald?J,Stuetzle?W.Mesh?Optimization.In:James?TK,ed.Proceedings?of?ACM?SIGGRAP?H.Anaheim:ACM?Press,1993.19-26.
[14]Cohen-Steiner?D,Alliez?P,Desbrun?M.Variational?Shape?Approximation.In:Marks?J,ed.Proceedings?of?ACM?SIGGRAPH.Los?Angeles:ACM?Press,2004.905-914.
Summary of the invention
The object of the invention be to propose a kind of fast, effectively and the shortcut calculation of the three-dimensional triangular mesh model of robust, it can solve the simplification of three-dimensional triangle model and the problem of multi-resolution display more effectively, more efficiently than prior art.
The shortcut calculation of three-dimensional triangular mesh model of the present invention is earlier to the plane sub-clustering of grid model type of carrying out, and sets up the polygonal mesh after the simplification according to the sub-clustering result then.Its concrete steps are:
1) N is counted on the three-dimensional triangular form grid model and the target simplified model polygon limit of given needs simplification C, be N with the model surface tessellation CBunch collection;
2) based on the plane approximate error tolerance ε of definition, the border that iteration is upgraded each bunch collection collects the triangle that is comprised with each bunch, and to reduce each bunch collection and the final polygonal error metrics of simplification, a bunch collection that finally is optimized is divided;
3) after obtaining final optimization bunch collection division, the intersection point more than three bunches of collection is regarded as the polygonal mesh summit, just can obtains a polygonal grid model, notice that polygon is not necessarily plane polygon here.Can further use classical polygon trigonometric ratio method the polygonal mesh of simplifying to be converted into the triangular mesh of simplification.
In above-mentioned steps 1) in, the initial division of said model cluster collection is based on that the model Discrete Curvature Estimation sets up, and its Curvature Estimation formula is:
Figure BSA00000188270300041
θ wherein jBe the drift angle around the vertex v, A vBe the regional area of the Voronoi of vertex v.To set up N as estimation based on the model discrete curvature CIndividual bunch of collection.
In step (2), approximate error tolerance in plane is defined as
Figure BSA00000188270300042
Wherein
Figure BSA00000188270300043
Expression bunch collection C jThe characteristic plane normal direction: T wherein jFor belonging to a bunch collection C jAll triangular plates, n jBe the unit normal vector of this triangular plate, ρ jArea for this triangular plate.
In the actual computation, with error metrics ε (F C) discretely is:
ϵ ( F , C ) = ∫ v ∈ F | | n ( v ) - n C i ( v ) | | 2 dv
= ϵ d ( F , C ) = Σ i = 0 N C - 1 ( Σ T j ∈ C i ρ j | | n j - Σ T j ∈ C i ρ j n j | | Σ T j ∈ C i ρ j n j | | | | 2 ) , Simplifying discrete form obtains:
ϵ d ( F , C ) = Σ i = 0 N C - 1 ( 2 Σ T j ∈ C i ρ j - 2 | | Σ T j ∈ C i ρ j n j | | ) .
In step (2), we have designed a kind of based on the algorithm of each bunch of iteration renewal collection border with the optimization error metrics, and in the iterative process in each step, we need consider that a bunch control collection border changes to reduce error metrics ε d(F, possibility C).For this reason, we adopt greedy algorithm to optimize error metrics: upgrade each bunch collection border iteratively, do not stop until there being a bunch collection limit that change takes place.
Above method is compared traditional method, it is advantageous that this method is intuitively effective, and the grid model after the simplification can effectively keep the details of original mesh, and this algorithm makes that algorithm is succinct and stability is preferably arranged owing to do not relate to the modification of network topology.By a large number of experiments show that, this method is easy to realization, and has geometric meaning intuitively, can effectively be applied in the geometric modeling system.
Description of drawings
Fig. 1 is the entire flow figure of shortcut calculation embodiment of the present invention;
Fig. 2 is the synoptic diagram of model three states in simplification of shortcut calculation embodiment of the present invention;
Fig. 3 analyzes synoptic diagram for the part bunch collection border change of shortcut calculation embodiment of the present invention.
Embodiment
Below, come specific embodiments of the invention to elaborate in conjunction with accompanying drawing and embodiment.
1. problem summary
Be the specific descriptions of embodiment of the invention method below, this method is explained algorithm based on triangular mesh, triangular mesh M can be expressed as V, E, F}, wherein V is the set of grid vertex, E is the set of grid edge, F is the set of grid surface:
V={v i=(x i,y i,z i)∈R 3|1≤i≤N V}
E = { e i = { v i 1 , v i 2 } , i = 1 , . . . , N E } - - - ( 1 )
F = { f i = { v i 1 , v i 2 , v i 3 } , i = 1 , . . . , N F }
This method target is divided into the almost plane bunch collection that the limit is communicated with for the patch grids F with original mesh M
Figure BSA00000188270300053
N C<<N FProblem, wherein
Figure BSA00000188270300054
Figure BSA00000188270300055
N CIn order to the simplification ratio of control mesh, each bunch C iIn original mesh face f ∈ C iBe communicated with about the limit each other.After sub-clustering was accomplished, each bunch collection can be represented with a polygon, and then can obtain a polygon simplification grid that approaches original mesh.
Fig. 1 is the process flow diagram of present embodiment shortcut calculation, obtain object module after, model is carried out adaptive burst initial value divides; The possibility that change takes place on each bunch collection border is observed on each bunch of iteration collection border then, if there is a bunch collection border change to take place, then continues this step; If have no a bunch collection border to change, then obtained final sub-clustering result; The sub-clustering result is carried out polygonization or trigonometric ratio obtains final simplification grid model.
Fig. 2 has shown that with the bunny model model is in three different state in the simplification process: shown in Fig. 2 a, this original bunny model has 69451 triangular plates; Master pattern is carried out sub-clustering, obtain 100 bunches of collection, shown in Fig. 2 b; Polygonal grid model after being obtained simplifying by final sub-clustering result is shown in Fig. 2 c.
2. sub-clustering algorithm
2.1 initialization
In the present embodiment,, need an initial sub-clustering to divide, implement greedy algorithm then and optimize error metrics till convergence for designing a complete sub-clustering algorithm.If use a bunch collection initial value at random to divide, be unfavorable for the speed of algorithm convergence and final effect, propose for this reason adaptive seed choose with bunch collection growth pattern to obtain initial sub-clustering.
Use the Gaussian curvature of two-dimensional manifold triangle gridding to estimate that for grid vertex, its Gaussian curvature is:
κ G ( v ) = ( 2 π - Σ θ j ∈ Neighbour ( v ) θ j ) / A v , - - - ( 2 )
θ wherein jBe the drift angle around the vertex v, A vBe the regional area of the Voronoi of vertex v.The Gaussian curvature K of triangular plate G(T i) the approximate Gaussian curvature absolute value that is taken as its three summit and mean value, in order to react the height of this triangular plate curvature.
Considering gridding sub-clustering problem: grid M={V, E, F}, target is divided number of clusters N C,, have bunch collection triangular plate that merger is more of the triangular plate of low curvature for bunch collection triangular plate that merger is less of the triangular plate that has higher curvature.Under the sub-clustering situation of our predicted ideal, the Gaussian curvature total value of each bunch collection should approach the mean value of grid:
D = 1 N C Σ T i ∈ F ρ i κ G ( T i ) , - - - ( 3 )
T wherein iBe all triangular plates, ρ iBe T iArea.
Set up each bunch collection C iThe time: get a triangular plate T who does not also belong to any one bunch collection at random SeedAs working as prevariety collection C iThe seed triangular plate, upgrade circularly this bunch collection border will also not belong to any bunch of collection and the triangular plate that links to each other with this bunch collection limit with
Figure BSA00000188270300071
Concentrate for priority is incorporated into this bunch, reach goal-setting value D up to the Gaussian curvature total value of this bunch collection, promptly The time, stop to incorporate into triangular plate.For the triangular plate that almost belongs to a plane only is included in the middle of bunch of collection; After a bunch collection stops to incorporate triangle into; Still can the triangular plate of
Figure BSA00000188270300073
be incorporated into when prevariety and concentrate; Wherein, the triangular plate normal direction and a bunch collection normal error of dTh for tolerating.Circulate the method for above-mentioned foundation bunch collection up to setting up N CIndividual bunch of collection.So far, obtain an initial cluster collection and divide, final actual bunch of collection number N RCMaybe be less than N C, revise the branch number of clusters N of user's appointment so CBe N RC
2.2 the error metrics of bunch collection is approached on the plane
Define each sheet bunch collection C iCharacteristic method vector
Figure BSA00000188270300074
For this bunch collection recently like the normal direction on plane, can calculate as follows:
n C i = Unif ( ∫ v ∈ C i n ( v ) dv ) , - - - ( 4 )
Wherein (unit normal vector of v) ordering for v, Unif () refer to the unit operation to vector to n.
Hope to provide a kind of sub-clustering strategy, make and the concentrated institute of each bunch have a few pairing with it bunch to collect proper vector the most approaching that each bunch collection all approaches most the plane separately like this.
Given grid M={V, E, F}, target is divided number of clusters N C, in clustering process, hope to minimize following tolerance to grid:
ϵ ( F , C ) = ∫ v ∈ F | | n ( v ) - n C i ( v ) | | 2 dv , - - - ( 5 )
N (unit normal vector of v) ordering, C wherein for v i(bunch collection that v) belongs to for v,
Figure BSA00000188270300077
Characteristic method vector for the definition of (2) formula.
(5) formula of observation; The normal direction on plane like characteristic method vector
Figure BSA00000188270300081
representative collects recently with this bunch; The integration of the variance of the characteristic method vector of bunch collection that the unit normal vector of every bit is belonged to it is as the margin of error; The margin of error is more little; Explain each bunch concentrated to have a few pairing with it bunch of collection proper vector approaching more, that is each bunch collection more the mesh triangles sheet that normal direction is close return together.Analyze from the drafting of figure, same bunch of concentrated mesh flake has close normal direction and then having close lighting effect, helps representing a bunch of collection with a polygon.
Given grid M={V, E, F}, target is simplified polygon model and is counted N C, (F C), hopes to find the division C of an optimum to error metrics ε Opt, make ε (F, C Opt)=min{ ε (F, C) }, wherein C divides number of clusters pairing all possible division for this target.
The characteristic method vector
Figure BSA00000188270300082
of each bunch collection can discrete representation be:
n C i = Unif ( ∫ v ∈ C i n ( v ) dv ) = Unif ( Σ T j ∈ C i ρ j n j ) = Σ T j ∈ C i ρ j n j | | Σ T j ∈ C i ρ j n j | | , - - - ( 6 )
T wherein jFor belonging to a bunch collection C iAll triangular plates, n jBe the unit normal vector of this triangular plate, ρ jArea for this triangular plate.
Error metrics ε (F, C) can disperse and be:
ϵ ( F , C ) = ∫ v ∈ F | | n ( v ) - n C i ( v ) | | 2 dv
= ϵ d ( F , C ) = Σ i = 0 N C - 1 ( Σ T j ∈ C i ρ j | | n j - Σ T j ∈ C i ρ j n j | | Σ T j ∈ C i ρ j n j | | | | 2 ) - - - ( 7 )
ε d(F is that (F, discrete form C) design and a kind ofly upgrade each bunch collection border to optimize the algorithm of error metrics based on iteration ε, and in the iterative process in each step, we need consider that a bunch control collection border changes to reduce error metrics ε C) d(F, possibility C) for this reason, need be known error metrics ε d(F, the C) value before and after bunch collection border changes is if calculate ε with (5) formula d(F, C), its complexity is O (M+N), obviously time overhead is huge.Wherein, M and N are the triangular plate number when two bunches of adjacent collection of prevariety collection border, and these two bunches of set pairs can change owing to the variation of characteristic method vector in the contribution of error metrics, need recomputate.For this reason,
(5) formula of analysis obtains: ϵ d ( F , C ) = Σ i = 0 N C - 1 ( Σ T j ∈ C i ρ j | | | | Σ T j ∈ C i ρ j n j | | · n j - Σ T j ∈ C i ρ j n j | | 2 | | Σ T j ∈ C i ρ j n j | | 2 )
= Σ i = 0 N C - 1 ( Σ T j ∈ C i ρ j | | Σ T j ∈ C i ρ j n j | | 2 | | n j | | 2 - 2 | | Σ T j ∈ C i ρ j n j | | Σ T j ∈ C i ρ j n j · Σ T j ∈ C i ρ j n j + Σ T j ∈ C i ρ j | | Σ T j ∈ C i ρ j n j | | 2 | | Σ T j ∈ C i ρ j n j | | 2 )
= Σ i = 0 N C - 1 ( Σ T j ∈ C i ρ i | | n j | | 2 - 2 | | Σ T j ∈ C i ρ j n j | | + Σ T j ∈ C i ρ j ) - - - ( 8 )
Notice ∀ T j , | | n j | | = 1 , Then
ϵ d ( F , C ) = Σ i = 0 N C - 1 ( 2 Σ T j ∈ C i ρ j - 2 | | Σ T j ∈ C i ρ j n j | | ) - - - ( 9 )
For each bunch collection G i, only need vector of record
Figure BSA00000188270300096
With a scalar
Figure BSA00000188270300097
Just can, iteration calculate error energy ε when upgrading bunch collection border with O (1) complexity d(F, C).
According to (7) formula, we propose a kind of iteration and upgrade each bunch collection C iThe greedy algorithm on border.Hypothetical trellis limit e ∈ E, the both sides triangle of e is T m, t nIf, T m, T nBelong to different bunch collection, claim that so e is a bunch of collection border, the set on all bunches collection border is E C, obviously
Figure BSA00000188270300098
In each step iteration, consider each bar bunch collection limit e ∈ E C(F, possibility C): the both sides triangle of supposing e is T to reduce ε in change m, T n, a bunch collection that belongs to is respectively C k, C lThree kinds of change situation can be considered like Fig. 3, three different error metrics can be obtained:
A) original case ε d(F, C Init): the triangle T on bunch collection both sides, border m, T nStill belong to bunch collection C separately k, C l, i.e. T m∈ C k, T n∈ C l
B) ε d(F, C 1): bunch collection border triangle T on one side mIncorporate another side bunch collection C into k, i.e. T m∈ C k, T n∈ C k(10)
C) ε d(F, C 2): bunch collection border triangle T on one side nIncorporate another side bunch collection C into l, i.e. T m∈ C l, T n∈ C l
The situation that obtains least error tolerance is upgraded a bunch collection limit set E then as last change result CCarry out iteration for a bunch collection limit and upgrade, can reduce error metrics ε iteratively, the change judged result until all bunches collection limit is C InitStop.Because error metrics ε is all dwindled in each step change dSo this convergence can be guaranteed.In order to guarantee the connectedness of bunch collection, in actual iterative computation, if the feasible connectedness of destroying bunch collection of change on bunch collection limit is then forbidden this step change.
In actual calculation, to each bunch collection C i, only need record
Figure BSA00000188270300101
Get final product.
In fact, when considering e ∈ E CThe change possibility time, have only its relevant bunch collection C k, C lThe error metrics of being contributed part possibly change.By (7) formula,
E C k ∪ C l = 2 ( Σ T j ∈ C k ∪ C l ρ j - | | Σ T j ∈ C k ρ j n j | | - | | Σ T j ∈ C l ρ j n j | | ) - - - ( 11 )
In three kinds of bunches of collection limit e change situation in (8);
Figure BSA00000188270300103
is constant; So, only need to consider
L C k ∪ C l = | | Σ T j ∈ C k ρ j n j | | + | | Σ T j ∈ C l ρ j n j | | = 1 2 ( E C k ∪ C l - Σ T j ∈ C k ∪ C l ρ j ) - - - ( 12 )
Calculate C respectively Init, C 1, C 2Pairing L Init, L 1, L 2Value, get the maximum change situation of L (being that corresponding error metrics is minimum) as last change result.Pairing error metrics ε in the big more correspondence of L (7) formula d(F, C) more little, observe (10) formula, L for each bunch collection triangular plate with the area be power normal vector and mould, by the character of vector sum: the vector that the cluster identical molds is long, identical its vector sum of direction is big more; (7) formula is the geometric meaning of (10) formula so also can be optimized thus: the triangular plate that will have close normal vector belongs to same bunch of collection, thereby belongs to a certain characteristic plane with the triangular plate of cluster collection is approximate, and this is sub-clustering purpose just also.
After obtaining initial sub-clustering division, can realize the local greedy algorithm of this joint: in each step iteration, travel through all bunch collection border e ∈ E CIf one of two triangular plates that e is adjacent also do not belong to any bunch of collection, so it is belonged to bunch collection that another triangular plate belongs to; If two triangular plates belong to different bunch collection, carry out the change that proposes in the last joint so and judge, upgrade a bunch collection border E C, be C up to the change judged result on all bunches collection limit InitStop.
2.3 polygonization and trigonometric ratio
After obtaining final optimization bunch collection division, the intersection point more than three bunches of collection is regarded as the polygonal mesh summit, just can obtain a polygonal grid model, notice that polygon is not necessarily plane polygon here.We can also be triangular mesh with the polygonal mesh burst, can use Cohen-Steiner etc. [15]The bearing calibration optimization of polygon limit and this polygonal mesh of trigonometric ratio of proposing, or with the polygon trigonometric ratio method of classics the polygonal mesh of simplification is converted into the triangular mesh of simplification.

Claims (7)

1. the shortcut calculation to the three-dimensional triangular mesh model of medical image is characterized in that comprising the steps:
1) three-dimensional triangular mesh model and the polygon figurate number N of target simplified model of the medical image of given needs simplification C, estimate triangular plate is divided into N based on the Gaussian curvature of grid model triangular plate CIndividual initial cluster collection;
2) definition plane approximate error tolerance
Figure FDA0000153026420000011
Be the error of the characteristic method vector of the normal vector of each concentrated triangular plate of each bunch and its place bunch collection, wherein F representes the set of all triangular plates, and C representes the set that all bunches collection is divided, n (unit normal vector of v) representing vertex v,
Figure FDA0000153026420000012
Represent bunch collection C under each summit iCharacteristic method vector; To reduce error metrics ε is target, and iteration is upgraded the border and the triangle that each bunch collection is comprised of each bunch collection, and a bunch collection that finally is optimized is divided;
3) after obtaining final optimization bunch collection division, the intersection point more than three bunches of collection is regarded as the polygonal mesh summit, the polygonal grid model after obtaining simplifying.
2. the shortcut calculation of the three-dimensional triangular mesh model to medical image as claimed in claim 1, it is characterized in that: in the step 1), grid model vertex curvature estimation formulas is:
θ wherein jBe the drift angle around the vertex v, A vBe the regional area of the Voronoi of vertex v, Neighbour (v) is the set on summit around the vertex v; The Gaussian curvature κ of grid model triangular plate T G(T) the approximate Gaussian curvature absolute value that is taken as its three summit and mean value.
3. the shortcut calculation of the three-dimensional triangular mesh model to medical image as claimed in claim 1 is characterized in that: in the step 1), and said N cIndividual bunch of collection set up according to following steps:
1) choose a triangle and set up a burst bunch collection as seed, a merger and bunch adjacent triangular plate of collection when the Gaussian curvature total value of this bunch collection reaches the goal-setting value, stop to incorporate into triangular plate, and this bunch collection is set up and accomplished;
2) method of the above-mentioned foundation of circulation bunch collection is up to setting up N CIndividual bunch of collection.
4. the shortcut calculation of the three-dimensional triangular mesh model to medical image as claimed in claim 3; It is characterized in that: the error of definition triangulation method vector n (T) and affiliated bunch of collection current characteristic method vector
Figure FDA0000153026420000021
for
Figure FDA0000153026420000022
after a bunch collection stops to incorporate triangular plate into; The triangular plate of continuation with
Figure FDA0000153026420000023
incorporated into when prevariety concentrated, and dTh is predefined value.
5. the shortcut calculation of the three-dimensional triangular mesh model to medical image as claimed in claim 1; It is characterized in that: step 2) in, plane approximate error tolerance
Figure FDA0000153026420000024
uses following method discrete:
1) bunch collection characteristic method vector
Figure FDA0000153026420000025
discrete representation is:
n C i = Unif ( ∫ v ∈ C i n ( v ) Dv ) = Unif ( Σ T j ∈ C i ρ j n j ) = Σ T j ∈ C i ρ j n j | | Σ T j ∈ C i ρ j n j | | , T wherein jFor belonging to a bunch collection C iAll triangular plates, n jBe the unit normal vector of this triangular plate, ρ jBe the area of this triangular plate, Unif () is the unit operation of vector;
2) error metrics ε (F, C) discrete representation is:
ϵ ( F , C ) = ∫ v ∈ F | | n ( v ) - n C i | | 2 dv
= ϵ d ( F , C ) = Σ i = 0 N C - 1 ( Σ T j ∈ C i ρ j | | n j - Σ T j ∈ C i ρ j n j | | Σ T j ∈ C i ρ j n j | | | | 2 ) , Simplifying discrete form obtains:
ϵ d ( F , C ) = Σ i = 0 N C - 1 ( 2 Σ T j ∈ C i ρ j - 2 | | Σ T j ∈ C i ρ j n j | | ) .
6. the shortcut calculation of the three-dimensional triangular mesh model to medical image as claimed in claim 1; It is characterized in that: step 2) in; Adopt greedy algorithm to optimize upgrade each bunch collection border based on iteration with the triangle that each bunch collection is comprised; Said greedy algorithm optimization step is following: in each step iteration, if one of bunch two triangular plates that collection border e is adjacent also do not belong to any bunch of collection, so it is belonged to bunch collection that another triangular plate belongs to; If two triangular plates belong to different bunch collection, carry out a bunch collection border change so and judge, upgrade a bunch collection border E C, be C up to the change judged result on all bunches collection border InitStop C InitA bunch collection border change does not take place in the expression present case.
7. the shortcut calculation of the three-dimensional triangular mesh model to medical image as claimed in claim 6 is characterized in that: said bunch of collection border change determining step is following:
1) supposes that bunch both sides triangle of collection border e is T m, T n, a bunch collection that belongs to is respectively C k, C l, consider three kinds of change situation, can obtain three different error metrics:
A) original case ε d(F, C Init): the triangle T on bunch collection both sides, border m, T nStill belong to bunch collection C separately k, C l, i.e. T m∈ C k, T n∈ C l
B) ε d(F, C 1): bunch collection border triangle T on one side mIncorporate another side bunch collection C into k, i.e. T m∈ C k, T n∈ C k
C) ε d(F, C 2): bunch collection border triangle T on one side nIncorporate another side bunch collection C into l, i.e. T m∈ C l, T n∈ C l
2) situation that obtains least error tolerance is upgraded a bunch collection limit set E then as last change result C
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