CN104361632A - Triangular mesh hole-filling method based on Hermite radial basis function - Google Patents

Abstract

The invention discloses a triangular mesh hole-filling method based on a Hermite radial basis function. The method includes the following steps: after a hole boundary is extracted, on the basis of vertexes around a hole and normal information, utilizing the Hermite radial basis function to interpolate to acquire an implicit hook face; projecting the boundary to a plane, subjecting the hole boundary to limited Delaunay triangulation, and performing subdivision processing on a mesh after being triangulated; adjusting a newly-added triangle onto the implicit hook face to complete repairing. The triangular mesh hole-filling method is high in robustness and needless of manual participation and has good repairing effect on mesh holes large in curvature change.

Description

A kind of triangle gridding filling-up hole method based on Hermite radial basis function

Technical field

The invention belongs to geometric manipulations and 3-D scanning model repairing field, be specifically related to the triangle gridding filling-up hole based on Hermite radial basis function.

Background technology

Triangle gridding is the important method for expressing of one of geometric model, is widely used in the fields such as virtual reality, reverse-engineering, computer graphics.Along with the fast development of the Airborne Laser Scan, obtained applying more and more widely by the three-dimensional model of scan rebuilding.But due to the restriction of measuring principle (as insensitive on black), the blocking of the reflexive impact of body surface or scanned object self structure, scan the model obtained and often there is hole.These holes greatly govern the subsequent applications of model, and such as 3D prints and requires totally-enclosed model, and the shape analysis algorithm in a large amount of graphics also requires that model does not exist hole.Therefore, the application of repairing to model of hole is significant.

Done large quantity research for grid filling-up hole people, mainly can be divided into two class methods: the method based on surface mesh and the method based on volume data, these two kinds of methods respectively have superiority.Method based on surface mesh can keep the details of non-lack part preferably, also can utilize the character of former grid simultaneously better, be adapted in the higher scene of detail.Method based on volume data adopts Global treatment mode, is first the model of volume representation by the model conversation of grid representation, then carries out filling-up hole operation to volume data.The method is used in the occasion high to algorithmic stability requirement.

Filling-up hole based on surface mesh mainly contains two large classes: carry out surface fitting according to hole and peripheral information thereof and directly predict absent region and repair.In approximating method, Branch, after hole extracts, carries out matching by radial basis function (Radial Basis Function, RBF) to the part that hole lacks, and then carries out sampling to perforated and obtain model after filling-up hole.Due to the character of radial basis function itself, the method can not simulate the curved surface meeting grid around hole in some cases.Wang is after being extracted hole border, utilization group componential analysis (Principal Component Analysis, PCA) obtain the fit Plane of hole coboundary point, hole border is projected in fit Plane, calculate the distance of these points to fit Plane as height field.Height field is brought in mobile square law (Moving Least Square, MLS) and obtain fitting surface.Sampled in hole border in fit Plane, sampled point is brought in the determined equation of MLS and obtains last filling-up hole model.Filling-up hole algorithm based on MLS has certain filling-up hole effect to on-plane surface hole, but undesirable for highly bending hole repair effect.

Above-mentioned approximating method is undesirable for the perforated repair efficiency that Curvature varying is larger, and some method complete failure in some cases.Recently, Macedo proposes Hermite radial basis function (Hemite Radial basis Fucntion, HRBF) and applies it to a cloud to rebuild, and this also becomes the focus of present stage research.The mistake that HRBF solves choosing due to improper obligatory point of RBF existence theoretically and causes rebuilds curved surface.Normal information is considered in reconstructing system simultaneously, make to rebuild curved surface and more can reflect obligatory point character.Hole grid for higher curvature change verily can rebuild the change that grid verily can react curvature.

In direct prediction and method for repairing and mending, Liepa is using angle and area as constraint, and employing dynamic programming strategy carries out triangle division to hole; Then according to the thick grid repaired of density segmentation of grid around hole, in segmentation process, using edge flip to ensure, Delaunay character is met; Umbrella smoothing operator is finally adopted to carry out smooth to grid.The method is bad to the model repairing effect that Curvature varying around hole is larger, but also may produce Self-crossover.Xie proposes a kind of filling-up hole method based on differential evolution algorithm: by minimize a little between distance, the sampled point on hole border is matched.Then generate crossing plane according to pairing result, in conjunction with the tangent line of this plane and border, foramen primum hole and one ring neighborhood intersection point, crossing plane adopts differential evolution algorithm predict possible sampled point, finally sampled point is projected to plane and carry out trigonometric ratio.

Based in the method for volume data, Ju adopts Octree to carry out spatial division to model, finds the voxel crossing with boundary edge, then adopt the thought of similar Bresenham algorithm in the structure of Octree, generate at three dimensions the plane that connects these boundary edge, hole is repaired.Finally use dual contouring reconstruction model.The model conversation of grid representation is homogeneous space voxel by Nooruddin, utilizes check number (Parity Count) and light penetration (Ray Stabbing) to process the different conditions produced in conversion process.Then opening (open) and closing (close) operator in Morphology Algorithm is applied to the hole repair that three dimensions completes space voxel.Marching Cubes algorithm is finally adopted to extract contour surface.Relative to the algorithm based on mesh amending, volume data method has the advantages such as robustness is high, guarantee stream shape.But have employed the intermediate representation of voxel, cause the model after repairing to lose the details of master mould in a large number.Good repair efficiency is not had for the model that Curvature varying is larger.

In sum, based on method good effect on process plane hole of surface mesh, but good process can not be had for the grid that Curvature varying is larger, and some method also may be collapsed.Method based on volume data can process plane hole, but can not produce filling-up hole effect intuitively for the hole grid that Curvature varying is larger.And due to volume data method adopt the overall situation processing mode, for non-lack part mesh details lose more.The present invention can produce the model meeting mesh curvature variation tendency around hole by the hole large to Curvature varying, ensures that the model after repairing is consistent with master mould in density simultaneously.

Summary of the invention

The technical problem to be solved in the present invention is: overcome the deficiency of prior art to higher curvature change hole repair, net declinable complete model, provide a kind of triangle gridding filling-up hole method for generation can reflect around hole.The method mainly utilizes the generation of Hermite radial basis function to meet the fitting surface of mesh curvature change around hole, and wakes up with a start sampling in perforated, is then mapped on fitting surface by sampled point.

The technical scheme that the present invention solves the problems of the technologies described above is: based on the triangle gridding filling-up hole method of Hermite radial basis function, comprise the steps:

1) fitting surface is generated

Step (1), in grid model, detect hole border.The feature of half of data structure is two one side of something (half-edge) the every bar limit in grid being decomposed into band direction, and these two half of directions are contrary.Every bar one side of something records the triangle and two end points that are adjacent, and its one side of something of each triangle record one composition, it is the one side of something summit (outgoing vertex) that each summit records one with it.Half of data structure saves the topology information of grid, utilize these topology informations can quick-searching to the adjacency information of each element (point, line, surface).Only adjacent with triangle limit of definition is boundary edge, travel through the limit data of grid in the structure shown here, find a seed boundary edge, other boundary edge then finding next to be adjacent counterclockwise along this seed boundary edge are until again travel through go back to seed border.The annular of these boundary edge composition is hole border.

Step (2), according to hole border and neighborhood thereof, Hermite radial basis function is utilized to obtain implicit surface.After obtaining hole border, because the summit on hole does not have a complete ring field, so method line computation accurately can not be carried out to it.Choose the three ring neighborhoods on hole border, in three ring neighborhoods, the position on summit and normal information are as the input of Hermite radial basis function, reconstruct the implicit surface meeting mesh curvature change around hole.

2) triangle division on hole border

Step (3), utilize PCA method to be generated as the fit Plane on summit on hole border, and projected in this plane.Directly carrying out trigonometric ratio at three dimensions is a very difficult thing, but also may introduce some other defects.And carry out in the plane trigonometric ratio be one by the problem of deep research, so the first step is the fit Plane utilizing PCA method to obtain summit on hole border, then hole border is projected in this plane.The plane that PCA method obtains is actual is the least square fitting on summit on hole border.

Step (4), if there is the border of Self-crossover, also needs the minimal eigenvalue optimizing PCA method, hole is divided into small holes.For the model with highly bending hole, the projection in step (3) may produce Self-crossover border in fit Plane, so also need segmentation.Hole border is divided into two parts, calculates the minimal eigenvalue λ of every part PCA method respectively _i3and λ _j3, suppose λ _i3>=λ _j3.Get and work as λ _i3for hole is decomposed into small holes by segmentation time minimum.Ensure that the border that each small holes projects in fit Plane can not produce Self-crossover.

Step (5), the restricted Delaunay trigonometric ratio carrying out based on sweep trace in the plane, and three dimensions is returned in result back projection.After obtaining the projection of hole in fit Plane, adopt restricted Delaunay trigonometric ratio (the Constrained Delaunay Triangulation based on sweep trace, CDT) carry out triangulation to projection polygon, CDT method is compared with traditional Delaunay method has obvious advantage in speed.Then master mould is returned in the result back projection of triangle division.

3) sampled in perforated

Step (6), newly-increased triangle is met to the sub-dividing operations of density attributes.In order to mesh-density around ensureing newly-increased leg-of-mutton density and hole is consistent, simultaneously also in order to can by the curved surface that generates in newly-generated Triangular Maps to step (2).For the borderline each summit of hole defines a density attributes: the mean value of all edge lengths that this summit is adjacent.Sub-dividing operations is carried out to the newly-increased triangle obtained in step (5).The criterion of segmentation is with newly-increased triangle core for increasing point newly, and interpolation obtains the density attributes of this newly-increased point, if this density attributes and newly-increased three some distances to original triangle shape meet certain proportionate relationship, then segments this triangle.In segmentation process, utilizing edge flip to ensure, Delaunay character is guaranteed.This completes the sampling to perforated.

4) newly-increased point is mapped on implicit surface

Step (7), utilize gradient descent method by segmentation newly-increased summit be mapped on implicit surface.The sampled point obtained in step (6) is mapped on the implicit surface of generation in step (2).Gradient descent method is a kind of method asking optimum value, and the continuous iteration adjustment step-length of the method is to arrive the optimal value of function.The gradient of HRBF and each newly-increased point are brought in the iterative formula of Gradient Descent, obtains final filling-up hole grid.

The present invention compared with prior art advantage is: the present invention changes high hole for around mesh curvature good repair efficiency, and has higher stability.The present invention mainly contains 2 contributions: the first, adopts Hermite radial basis function to carry out matching to hole, obtains meeting the fitting surface of mesh curvature change around.The second, give a kind of new perforated method of sampling, the method adopts the sub-dividing operations of plane trigonometry and density based attribute.

Accompanying drawing explanation

Fig. 1 is algorithm overall flow figure;

Fig. 2 is half of data structure schematic diagram;

Fig. 3 is hole neighborhood schematic diagram; (a) ring neighborhood, (b) two ring neighborhood;

Fig. 4 is the schematic diagram that projection produces Self-crossover hole;

Fig. 5 is CDT initialization schematic diagram;

Fig. 6 is CDT summit event schematic diagram; A () current vertex projects the limit obtained on wavefront, two of this limit end points and current vertex are connected to form triangle by (b), if (c) does not meet Delaunay character, then carry out edge flip;

Fig. 7 is CDT limit event schematic diagram; A triangle that () in a disguised form hands over restriction, these triangles are deleted by (b), and become upper and lower two parts according to restriction variation, and (c) carries out trigonometric ratio to these two parts respectively;

Fig. 8 is intermediate result of the present invention and end product design sketch; The model of (a) band hole, b curved surface that () obtains according to hole and the matching of neighborhood Hemite radial basis function, the fit Plane on summit and its projection in plane on (c) hole, d () plane trigonometry completes after, three dimensions returns in back projection, e () is carried out segmentation to newly-increased grid and is obtained sampled point, sampled point is mapped on curved surface by (f);

Fig. 9 is the contrast effect figure of the present invention and traditional RBF method.From left to right: (a) rabbit model with hole, based on the model repaired after RBF method, the model after this method repairing, (b) mechanical component model with hole, the model after the repairing of RBF method, the model after this method repairing.

Embodiment

The present invention is further illustrated below in conjunction with accompanying drawing and the specific embodiment of the present invention.

Step (1) detects hole border in grid model.First the elements such as the point, line, surface in half of data structure hoc mesh are used.As shown in Figure 2, the feature of this structure the every bar limit in grid is decomposed into two contrary one side of something (half-edge) of direction.In half of data structure, the one side of something in this, as initial end points is deposited in each summit (in figure a), each triangle deposits its one side of something of composition (in figure b), every bar one side of something deposits following content: the terminating point of this one side of something (in figure c), the triangle that this one side of something belongs to (in figure d), (in figure e), half of rightabout one side of something is (in figure f) with this for lower bar one side of something of the counterclockwise traversal of this one side of something.Half of data structure saves the topology information of grid, utilize these topology informations can quick-searching to the adjacency information of each element (point, line, surface).Only adjacent with triangle limit of definition is boundary edge, the limit data of grid are traveled through in Half-edge Structure, find a seed boundary edge, other boundary edge then finding next to be adjacent counterclockwise along this seed boundary edge are until again travel through go back to seed border.The annular of these boundary edge composition is hole border.

Step (2), according to hole border and neighborhood thereof, utilizes Hermite radial basis function to obtain implicit surface.After obtaining hole border, because the summit on hole does not have a complete ring neighborhood, so method line computation accurately can not be carried out to it.The neighborhood definition of hole is as follows:

Suppose that S is the simplex set that the boundary edge of hole forms, M is the combinatorial manifold of model meshes.N (S, M), by the simplex set that at least there is the adjacent triangle in summit with S and form, claims N (S, M) to be the simplex neighborhood of S.Definition hole k rank (k-order) simplex neighborhood be N (N (... N (S, M) ...), M), M), expression formula has k-1 layer nested, is called for short k rank neighborhood, is designated as N _k(S, M).The set of hole frontier point is made to be v _h, the vertex set in the neighborhood of hole k rank is v _k, then v _k-v _hfor k ring (k-ring) summit of hole.As shown in Figure 3, a is a ring neighborhood, and b is two ring neighborhoods.Choose the three ring neighborhoods on hole border, in three ring neighborhoods, the position on summit and normal information are as the input of Hermite radial basis function, reconstruct the implicit surface meeting mesh curvature change around hole.

Hermite Interpolation Property of Radial Basis Function problem can be described below: for scattered point set given in three dimensions an Implicitly function f:R is found out at a function space H ³→ R, makes it meet f (x ⁱ)=0, in approximation theory, this problem is called as single order Hermite interpolation problem.Suppose have Hilbert space its dual space of H to be H ^*.A given data set wherein γ _icontinuous linear functional, c _ireal number, i.e. γ _i∈ H ^*, c _i∈ R.Target finds a function f ∈ H, makes r _i(f)=c _i, such function is called the General Cardinal in H space.For H space, such General Cardinal has numerous, but meet H Norm minimum only have one.This function can represent v by inner this ⁱthe linear combination of ∈ H is uniquely determined: and for each u ∈ H, γ _i(u)=<v ⁱ, u>.By f ^*and γ _isimultaneous, obtains linear system A α=c, and wherein the element of matrix A is (A) _i,j=γ _i(v ^j)=<v ⁱ, v ^j> _h.In Hermite interpolation problem wherein D ^λa differentiating operator, λ ∈ N ⁿ, λ _ja jth variable in expression u is by the number of times of differential.The method building such H space is as follows.

Provide radial basis function φ: the R of a positive definite ₊→ R, defines ψ=φ (|| ||) simultaneously.The Hilbert space of existence anduniquess, in this space, this is expressed as $v^i\left(x\right)=-\left(D^{{\mathrm{\&lambda;}}^i}\mathrm{\&psi;}\right)(x-x^i),$ Inner product is ${<v^i,v^j>}_H=-\left(D^{{\mathrm{\&lambda;}}^i}{D}^{{\mathrm{\&lambda;}}^j}\mathrm{\&psi;}\right)(x^i-x^j).$ For single order Hermite interpolation, when selecting overall support function, its representation is as follows.

$f\left(x\right)=\underset{j=1}{\overset{N}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_j\mathrm{\&psi;}(x-x^j)-{\mathrm{\&Sigma;}}_{j=1}^N<{\mathrm{\&beta;}}^j,\&dtri;\mathrm{\&psi;}(x-x^j)>+P\left(x\right)---\left(1\right)$

Wherein ψ (x)=φ (|| x||), || || represent Euclidean distance, φ (x)=x is generally got for three-dimensional reconstruction ³.P (x)=a+bx+by+cz, (x, y, z) is the coordinate of an x.Interpolation condition is brought in above-mentioned equation and can obtain factor alpha _j∈ R and β ^j∈ R ³:

$\underset{j=1}{\overset{N}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_j\mathrm{\&psi;}(x-x^j)-{\mathrm{\&Sigma;}}_{j=1}^N<{\mathrm{\&beta;}}^j,\&dtri;\mathrm{\&psi;}(x-x^j)>+P\left(x\right)=0---\left(2\right)$

$\underset{j=1}{\overset{N}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_j\&dtri;\mathrm{\&psi;}(x-x^j)-{\mathrm{\&Sigma;}}_{j=1}^N\mathrm{H\&psi;}(x-x^j){\mathrm{\&beta;}}^j+\&dtri;P\left(x\right)=n^j---\left(3\right)$

H is hessian operator, is defined as above-mentioned interpolation condition is reconfigured and obtains applicable form of calculation, for each (x ⁱ, n ⁱ) have:

${\mathrm{\&Sigma;}}_{j=1}^N\left[\begin{array}{cc}\mathrm{\&psi;}(x^i-x^j)& -\&dtri;\mathrm{\&psi;}{(x^i-x^j)}^T\\ \&dtri;\mathrm{\&psi;}(x^i-x^j)& \mathrm{H\&psi;}(x^i-x^j)\end{array}\right]\left[\begin{array}{c}{\mathrm{\&alpha;}}_j\\ {\mathrm{\&beta;}}^i\end{array}\right]+\left[\begin{array}{c}p\left(x\right)\\ \&dtri;P\left(x\right)\end{array}\right]=\left[\begin{array}{c}0\\ n^j\end{array}\right]---\left(4\right)$

${\mathrm{\&Sigma;}}_{j=1}^N\left[\begin{array}{cc}p\left(x\right)& \&dtri;p{\left(x\right)}^T\end{array}\right]\left[\begin{array}{c}{\mathrm{\&alpha;}}_j\\ {\mathrm{\&beta;}}^j\end{array}\right]=0---\left(5\right)$

From above formula, for three-dimensional each (x ⁱ, n ⁱ) once can calculate gradient about ψ and hessian matrix, obtain 4 × 4 matrixes, be then added with Polynomial Terms.A little through above formula calculate after matrix combination obtain complete linear system.

Hermite radial basis function is the improvement of conventional radial basis function, it solves conventional radial basis function theoretically choosing the randomness of the subjectivity on obligatory point (off-surface points), substantially increases the reconstruction robustness based on this function.In addition, due to the linear system of rebuilding using normal as interpolation condition, make the model rebuild can reflect the character of interpolation point better.The intermediate result obtained as shown in Figure 8 b.

Step (3) utilizes PCA method to generate the fit Plane of hole border vertices.Because directly carrying out trigonometric ratio at three dimensions is a very difficult thing, but also some other defects may be introduced, and carry out in the plane trigonometric ratio be one by the problem of deep research, so the first step is the fit Plane utilizing PCA method to obtain summit on hole border, then hole border is projected in this plane.The plane that PCA method obtains is the least square fitting on summit on hole border, and key step is as follows:

A) mean value on summit, hole coboundary is calculated wherein p _ibe the border vertices on hole, n is the number of border vertices, it is a point in required plane.Using this as the point of in fit Plane.

B) obtain matrix M is set up: M after point _i=(x _i-x ₀, y _i-y ₀, z _i-z ₀), wherein M _ithe i-th row of matrix M, (x _i, y _i, z _i) be p _icoordinate.M has following form:

$M=\left[\begin{array}{ccc}{x}_1-x_0& y_1-y_0& z_1-z_0\\ x_2-x_0& y_2-y_0& z_2-z_0\\ .& .& .\\ .& .& .\\ .& .& .\\ x_n-x_0& y_n-y_0& z_n-z_0\end{array}\right]---\left(6\right)$

C) M is calculated with SVD ^tthree eigenvalue λ of M ₁>=λ ₂>=λ ₃, with λ ₃characteristic of correspondence vector as the normal vector of plane.According to with determine plane, the boundary edge of hole is projected in this plane.

Step (4), if there is the border of Self-crossover, also needs the minimal eigenvalue optimizing PCA method, hole is divided into small holes.For highly folding complicated hole, as Fig. 4, above-mentioned projecting method can cause hole border to produce Self-crossover.Needing this kind of Hole partition is two parts, projects to two planes respectively and carries out subsequent operation again, and the optimum segmentation how obtaining hole in this step is key issue.In the method being obtained three-dimensional point set projection plane by PCA, the value of minimal eigenvalue determines these fitting degree to plane: λ ₃relative to λ ₁less, represent that the degree of fitting of plane and point set is higher.Therefore, minimize larger that in the minimal eigenvalue determined of latter two point set of segmentation and obtain optimum segmentation, i.e. solution optimization problem below:

$\underset{1\&le;i,j\&le;n}{\min }\max \{{\mathrm{\&lambda;}}_{3,{\mathrm{ps}}_1},{\mathrm{\&lambda;}}_{3,{\mathrm{ps}}_2}\}---\left(7\right)$

Wherein, PS ₁={ p _j... p _n..., p _i-1, PS ₂={ p _i... p ₁..., p _j-1, pS respectively ₁, PS ₂by the minimal eigenvalue that PCA method obtains.

After hole is decomposed into small holes, can ensure that the border that each small holes projects in fit Plane can not produce Self-crossover.After hole border projects to plane, the intuitive of conveniently follow-up calculating, by this Plane Rotation in the parallel position of x-y plane, and the corresponding relation of point in record space and Plane-point.The intermediate result obtained as shown in Figure 8 c.

Step (5) adopts the CDT based on sweep trace to carry out triangulation to the projection of hole in the fit Plane generated in step (3), and CDT method is compared with traditional Delaunay method has obvious advantage in speed.The step of CDT is as follows:

A) initialization.All points sort from small to large according to y value, for the point that y is identical, sort according to x value.For each interpolation additional information, indicate that whether this point is the upper end vertex of binding side.In order to avoid the discussion of too much complex situations, add two virtual point P _-1, P _-2, these two points respectively in lower left and the lower right of all input vertexs, as shown in Figure 5.By P _-1, P _-2, P ₀connect and compose initial delta, by P ₀p _-1, P ₀p _-2as preliminary sweep wavefront.

B) scan.According to y value size, from the bottom up motion scan line.Just stop whenever sweep trace runs into a summit, if this summit is not the upper end vertex of a binding side, process summit event, otherwise process limit event.Summit event: by P _ivertical projection obtains P to wavefront _j, determine P _jthe limit P of place wavefront _lp _r(Fig. 6 a), connects P _i, P _l, P _lform new triangle (Fig. 6 b), if this triangle does not meet empty round matter, then opposite side adjusts, and this step is called as edge flip (Fig. 6 c); Limit event: if during the upper end vertex on a current vertex limit, just form limit event.First determine first triangle crossing with this limit, then by (Fig. 7 a) along this traversing triangle to other triangles crossing with this limit.After having traveled through, delete and all triangles crossing with this limit, and the Π obtaining being positioned at below e that carried out sorting in their summit foundation and the position of limit e _lwith the top Π being positioned at e _u(Fig. 7 b), finally carries out the conventional CDT based on delta algorithm (Fig. 7 c) to these two parts respectively.

C) aftertreatment.By all with virtual point P _-1, P _-2the triangle that is connected is deleted, and adds edge triangles and complete trigonometric ratio.

After the polygonal trigonometric ratio of boundary edge composition completes, according to the corresponding relation in upper step, three dimensions is returned in the some back projection on two dimensional surface.The intermediate result obtained as shown in figure 8d.

Step (6) is consistent in order to mesh-density around ensureing newly-increased leg-of-mutton density and hole, simultaneously also in order to, by the curved surface that generates in newly-generated Triangular Maps to step (2), need the triangle to increasing newly to carry out sub-dividing operations.Concrete steps are as follows:

A) be each frontier point v on hole _iall increase a density attributes θ (x _i), it is defined as: with v _ithe mean value of adjacent edge lengths.

B) to each newly-increased triangle (v _m, v _n, v _l), calculate its barycenter v _cwith its density attributes θ (v _c) :=(θ (v _n)+θ (v _m)+θ (v _l))/3.To each i=m, n, l: if α || v _c-v _i|| > θ (v _c), α || v _c-v _i|| > θ (v _i), so just by triangle (v _m, v _n, v _l) be subdivided into (v _c, v _n, v _l), (v _m, v _c, v _l), (v _m, v _n, v _c), check (v simultaneously _m, v _n), (v _m, v _l), (v _l, v _n) the need of carrying out edge flip.

If c) do not have new triangle to produce in b jump to d, otherwise jump to b.

D) limit that the needs in all newly-increased limits exchange is exchanged.

In the implementation, choose the intermediate result obtained as figure 8 e shows.

Step (7), after sampling terminates, needs to be adjusted to by sampled point on the curved surface of HRBF matching.The present invention adopts gradient descent method to complete this step.The gradient calculation method of HRBF is:

$f\left(x\right)={\mathrm{\&Sigma;}}_{j=1}^N{\mathrm{\&alpha;}}_j\&dtri;\mathrm{\&psi;}(x^i-x^j)-{\mathrm{\&Sigma;}}_{j=1}^N\mathrm{H\&psi;}(x^i-x^j){\mathrm{\&beta;}}^j+\&dtri;P\left(x\right)---\left(8\right)$

H is defined as Hessian matrix as above-mentioned.After obtaining gradient, for each newly-increased summit r in repairing grid ₀, adopt iterative formula to be below mapped on curved surface:

$r_{k+1}=r_k-\frac{f\left(r_k\right)}{{\left|\right|\&dtri;f\left(r_k\right)\left|\right|}^2}\&dtri;f\left(r_k\right)---\left(9\right)$

Meet || r _k+1-r _k|| during≤ε condition, iteration stopping, ε is the error range of specifying.

Algorithm visual studio 2010 of the present invention and OpenGL realizes, and allocation of computer used is: dominant frequency 2.53GH, Xeon E5630 processor and 8G internal memory.The LU in Eigen storehouse is adopted to decompose when the determined linear system of solution HRBF.Method final effect figure as illustrated in fig. 8f.Fig. 9 is this method and traditional comparison diagram based on RBF method, is from left to right the master mould of band hole respectively, model after RBF method filling-up hole, model after HRFB method filling-up hole.

Claims (5)

1., based on a triangle gridding filling-up hole method for Hermite radial basis function, be applied to the hole repair of 3-D scanning model, it is characterized in that, comprise the steps:

(1) extract hole border, obtain hole border and its neighborhood summit, utilize Hermite radial basis function to generate and meet the implicit surface that around hole, mesh curvature changes;

(2) utilize PCA method to generate the fit Plane on summit on hole border, the hole border of extraction is projected in this plane; If there is Self-crossover border, then optimize the minimal eigenvalue of PCA method, hole is decomposed into small holes; Utilize the CDT algorithm based on sweep trace to carry out triangle division to the hole border projected in plane, then master mould is returned in the result back projection of triangle division;

(3) in order to mesh-density around ensureing newly-increased leg-of-mutton density and hole is consistent, for the borderline each summit of hole defines a density attributes, as guidance, newly-increased triangle is segmented using this density attributes, complete choosing of sampled point;

(4) gradient descent method is utilized to be mapped on the implicit surface that step (1) generates on the newly-increased summit of segmentation.

2. the triangle gridding filling-up hole method based on Hermite radial basis function according to claim 1, is characterized in that: step (1) comprising: utilize half of data structure tissue three-dimensional grid data; Limit data in traversal grid, detect only adjacent with triangle limit, are boundary edge; The closed-loop that end to end boundary edge is formed forms a hole; Extract the summit in hole border and field, utilize position and the normal information on these summits, generate the implicit surface meeting mesh curvature change around hole in conjunction with Hermite radial basis function.

3. the triangle gridding filling-up hole method based on Hermite radial basis function according to claim 2, it is characterized in that: step (2) comprising: in order to reduce the complexity of the triangle division in three dimensions, being transformed into the enterprising line operate of two dimensional surface; According to the borderline vertex information of hole, utilize PCA method to obtain the fit Plane of these points, then frontier point is projected in fit Plane; If there is Self-crossover border, then hole border is divided into two parts, calculates the minimal eigenvalue λ of every part PCA method respectively _i3and λ _j3, suppose λ _i3>=λ _j3; Get and work as λ _i3for hole is decomposed into small holes by segmentation time minimum; On this plane, the restriction Delaunay trigonometric ratio based on sweep trace is adopted to carry out trigonometric ratio to the polygon obtained that projects; Then three dimensions is returned in the result back projection of trigonometric ratio.

4. the triangle gridding filling-up hole method based on Hermite radial basis function according to claim 3, it is characterized in that: step (3) comprising: in order to mesh-density around ensureing the filling-up hole mesh-density that finally obtains and hole is consistent, need to segment the newly-increased triangle in step (3); For the borderline each summit definition density attributes of hole, this attribute is defined as the mean value putting adjacent all length of sides with this; Then segment triangle with newly-increased barycenter oftriangle coordinate for newly-increased point, Delaunay character will be kept in segmentation process; The sampled point of perforated is obtained after sub-dividing operations completes.

5. the triangle gridding filling-up hole method based on Hermite radial basis function according to claim 4, it is characterized in that: step (4) comprising: the sampled point in the implicit surface generated in integrating step (1) and step (3), utilize gradient descent method to be mapped on implicit surface by sampled point, obtain meeting around hole and net declinable filling-up hole grid.