CN101276483B - Method for implementing parallel moving hypersensitive Laplacian gridding edition - Google Patents

Abstract

The invention provides a Laplacian grid editting method for realizing translation sensitivity. The method is usually used for model editting in computer cartoons and industrial modeling, but a Laplacian coordinate have no translation sensitivity, thereby geometrical detail characters of the model can not be held well. Thus, the invention proposes a method for rotating the Laplacian coordinate by a vertex normal, which comprises re-building a middle model based on the original Laplacian coordinate, calculating the vertex normal of the model, and rotating the original Laplacian coordinate to a parallel direction of the vertex normal, and rebuilding the model to obtain the result. According to the method of the invention, the Laplacian coordinate realizes the deformation effect of translation sensitivity, the vertex normal is automatically calculated, fussy Laplacian vector estimation is avoided, and the geometrical detail characters of the model can be held well.

Description

A kind of Laplacian gridding edition method that realizes that translation is responsive

Technical field

The invention belongs to the solid modeling technology area, particularly relate to a kind of Laplacian gridding edition method that realizes that translation is responsive.

Background technology

Along with the progress of computer animation and industrial modeling, the model editing technology grows up apace.Many scholars have proposed various gridding editions technology, and like Free Transform technology, multi-resolution grid editing technique, based on the method for geometry differential attributes etc., these methods have his own strong points, and can produce new model of all kinds by master pattern according to demand.

Since the eighties of last century the eighties, big twice technological breakthrough has taken place in three-dimensional model editor field.Breakthrough for the first time occurs in the eighties, is referred to as the Free Transform technology, is that means are carried out model editing to handle agent model.Breakthrough for the second time occurs in the nineties, is referred to as the multiresolution editing technique, is that means are carried out gridding edition to handle simplified model.This twice technological breakthrough is that 3 d geometric modeling provides very strong driving force, and especially the quite ripe Free Transform technology of development has become the indispensable ingredient of commercial modeling software.

Quick progress along with the 3-D scanning technology; The precision of geometric model and data complexity have reached new height; These three-dimensional models are comprising very abundant geometric detail, are accurately also proposing new challenge for the editing and processing of model in the portrayal local geometric features---and geometric detail is difficult to effective maintenance in editing process.To having the 3 d scan data that enriches geometric detail, proposed new geometric representation method and computation model and be ready for the challenge, these algorithms are referred to as the method based on the differential attribute.Compare in preceding twice technological breakthrough, be mainly reflected in effective maintenance aspect of minutia in the geometric editor process based on the method advantage of differential attribute, so this technology is particularly suitable for editing 3 d scan data.

Deformation technology originates from the geometric modeling field, puts forward as a kind of geometric modeling method, afterwards because its great potential aspect simulation flexible article animation has little by little obtained very big development and application widely in the computer animation field.Present many commercial animation software such as Softimage, 3DMAX, Maya etc. are similar to the function of FFD (Free-Form Deformation, Free Transform).

Grid model editor algorithm based on Laplace coordinate representation is a kind of gridding edition method based on the differential attribute, and new development is in recent years got up, and through handling the differential attribute grid model is edited.This algorithm is realized simple, respond well, easy and simple to handle, and can realize multiple gridding edition functions such as distortion, Morphing, smooth, fusion, thereby receives extensive concern.

From the Laplacian coordinate when the Cartesian coordinates reconstruction model, owing to adopt the model after the Laplacian vector calculation distortion of master mould can not produce too big error, so a lot of method is all calculated the Laplacian coordinate of distortion back grid with it.But when moderate finite deformation took place, the variation that the Laplacian vector takes place was bigger, adopted said method can cause the partial geometry loss in detail.After this have a lot of methods to revise to the problem that above-mentioned Laplacian gridding edition exists, basic thought is the integral transformation result who obtains model through the estimation partial transformation.

In order before and after distortion, to keep the minutia of grid model, need rotation Laplacian coordinate to be parallel to distortion back grid surface normal, achieve the responsive deformation effect of translation.

Summary of the invention

The objective of the invention is to solve the problem that Laplacian gridding edition method does not have translation susceptibility, make the Laplacian coordinate realize the deformation effect that translation is responsive, keep local surface geometry minutia.

Method of the present invention is:

(1) sets up its Laplacian coordinate through the Cartesian coordinates of three-dimensional grid model;

(2) model is carried out deformation operation;

(3) find the solution a mid-module after the distortion by former Laplacian coordinate;

(4) calculate the vertex normal F (V) of this mid-module;

(5) the former Laplacian vector of rotation makes it be parallel to F (V);

(6) by revised Laplacian coordinate reconstruction model, obtain net result.

In order to realize goal of the invention, the method concrete steps of employing are following:

(1) input three-dimensional grid model M ₀, definition M ₀=(V, E F) have | V| summit, and V wherein, E, F represent the set of summit, limit, face respectively;

(2) pass through M ₀Its Laplacian coordinate of Cartesian coordinates structure, the note vertex v _iThe Laplacian coordinate be δ _i

(3) user passes through interactive means to model M ₀Carry out deformation operation;

(4) according to the vertex set c after the known distortion _j, satisfy v _j=c _j, j ∈ 1,2 ..., m} as the position constraint condition, rebuilds the model M the Cartesian coordinates from the Laplacian coordinate _k, k=1;

(5) work as M _kWhen differing big, be designated as ε＞E and continue step (6), otherwise jump to step (12) with the error of the model of ideal effect;

(6) by mid-module M _kObtain vertex v _iThe normal of each adjacent surface

If the concavo-convex variation of model is more, continues step (7), otherwise jump to step (8);

(7) if

With δ _iTowards inconsistent, the adjustment

The Laplacian coordinate that makes face normal and summit is towards consistent;

(8) calculate summit average normal F (v _i);

(9) rotation Laplacian coordinate δ _i

(10) find the solution mid-module M _K+1, k=k+1;

(11) if M _K+1Satisfy threshold value with the error of the model of ideal effect, continue step (12), otherwise, jump to step (5);

(12) obtain final mask M _K+1, finish.

Vertex v in the above-mentioned steps (2) _iLaplacian coordinate δ _iBe defined as:

${\mathrm{\δ}}_i=({\mathrm{\δ}}_i^{\left(x\right)},{\mathrm{\δ}}_i^{\left(y\right)},{\mathrm{\δ}}_i^{\left(z\right)})=v_i-\frac{1}{d_i}\underset{j\∈N\left(i\right)}{\mathrm{\Σ}}{v}_j---\left(1\right)$

N (i)={ j| (i, j) ∈ E}, d wherein _i=| N (i) | the expression vertex v _iAdjacent vertex number (i.e. degree).δ _iGeometric meaning be: vertex v _iWith respect to the direction and the numerical values recited of adjacent vertex, approximate local curved surface mean curvature on the numerical value, direction approximates local surface normal.Linear transformation between cartesian coordinate system and Laplacian coordinate system can be expressed as the form of matrix, δ=LV, wherein L=I-D ^-1A, A are the adjacency matrix on the grid, if vertex v _iAnd v _jThere is the limit to connect, then a _Ij=1, otherwise a _Ij=0.D is diagonal matrix (D _Ii=d _i), L is the Laplacian operator.

Rebuild M in the step (4) _kConcrete grammar be: through finding the solution the vertex set V after equation obtains distortion,

$\left(\frac{L}{\mathrm{\ω}{I}_{m\×m}|0}\right)V=\left(\begin{array}{c}\mathrm{\δ}\\ \mathrm{\ω}{c}_{1:m}\end{array}\right)---\left(2\right)$

Weights ω＞0 is used to adjust the importance of position constraint condition.Because top equality generally do not have exact solution, then find the solution approximate solution through the least square system, there is unique solution during the system full rank:

$\tilde{V}=\underset{V}{\arg \min }({\left|\right|\mathrm{LV}-\mathrm{\δ}\left|\right|}^2+\underset{J\∈C}{\mathrm{\Σ}}{\mathrm{\ω}}^2{\left|\right|v_j-c_j\left|\right|}^2)---\left(3\right)$

Normal in the step (6)

Specifically be calculated as: suppose vertex v _i, v _j, v _kConstitute a triangular mesh, then,

$\stackrel{\→}{n_j}=(v_j-v_i)\×(v_k-v_i)---\left(4\right)$

F (v in the step (8) _i) be calculated as:

$F\left(v_i\right)=\frac{\underset{j\∈M\left(i\right)}{\mathrm{\Σ}}{\mathrm{\μ}}_j\stackrel{\→}{n_j}}{\left|\right|\underset{j\∈M\left(i\right)}{\mathrm{\Σ}}{\mathrm{\μ}}_j\stackrel{\→}{n_j}\left|\right|}---\left(5\right)$

Wherein, M (i)=j| (i, j) ∈ E},

Be face normal line vector, μ _jBe weight, its value is that each adjacent surface is to vertex v _iThe influence of Laplacian vector, can be according to the area decision of triangle gridding.

Rotation δ in the step (9) _iMethod do,

δ _i＝‖δ _i‖·F(v _i)(6)

M in the step (10) _K+1The detailed process of finding the solution be: the solving equation group,

$\left(\frac{L}{\mathrm{\ω}{I}_{m\×m}|0}\right)V=\left(\begin{array}{c}\left|\right|\mathrm{\δ}\left|\right|\·F\left(V\right)\\ \mathrm{\ω}{c}_{1:m}\end{array}\right)---\left(7\right)$

Obtain approximate solution through finding the solution the least square system, have unique solution during the system full rank:

$\tilde{V}=\underset{V}{\arg \min }({\left|\right|\mathrm{LV}-\left|\right|\mathrm{\δ}\left|\right|\·F\left(V\right)\left|\right|}^2+\underset{j\∈C}{\mathrm{\Σ}}{\mathrm{\ω}}^2{\left|\right|v_j-c_j\left|\right|}^2)---\left(8\right)$

Beneficial effect of the present invention mainly embodies as follows:

1, make the Laplacian coordinate realize the deformation effect that translation is responsive

Because the method that exists is not at present carried out corresponding distortion rotation to the Laplacian coordinate after being out of shape, and causes the Laplacian coordinate that translation is not had susceptibility, can not keep the minutia of model well.The present invention makes the Laplacian coordinate be parallel to the surface normal direction through the Laplacian coordinate behind the vertex normal rotational deformation, achieves the responsive deformation effect of translation.

2, calculate the vertex normal F (V) that is used to adjust the Laplacian vector automatically

The present invention directly calculates the normal of each adjacent surface, thereby calculates vertex normal F (V).These are different with the method that exists at present, need not estimate the Laplacian vector on each summit, thereby avoid loaded down with trivial details Laplacian vector estimation.

3, realization is simple and direct, and calculated amount is little

The present invention adopts the method for implicit expression rotation Laplacian coordinate, only need carry out less step and just can obtain desirable deformation effect, realizes simple and directly, and calculated amount is little.

Description of drawings

Fig. 1 is a method execution in step synoptic diagram of the present invention.

Fig. 2 is an instantiation design sketch of the present invention.

Embodiment

Below in conjunction with accompanying drawing the present invention is done further explanation.

Method execution in step synoptic diagram of the present invention is shown in accompanying drawing 1.Through terminal input three-dimensional grid model M ₀, this model can obtain or uses modeling tool (like Maya, 3ds max etc.) to set up model through spatial digitizer.Definition M ₀=(V, E F) have | V| summit.V wherein, E, F represent the set of summit, limit, face respectively.

Set up the Laplacian coordinate of model.At first need construct the Laplacian operator.Can use even weights, also can set different weights, like the weights of cotangent or the area of triangle gridding etc. according to different grids.

Model is carried out deformation operation.This step can be passed through user-defined ROI (Region of Interest, area-of-interest) and handle.

Rebuild the mid-module M of cartesian coordinate system then according to the Laplacian coordinate after the distortion _k, k=1.Whether the error of judging this model and desirable deformation effect model is eligible.If error is bigger, then according to M _kObtain the normal of each adjacent surface on summit, calculate vertex normal.If the concavo-convex variation of model is bigger, the Laplacian coordinate that needs adjustment face normal and summit is towards consistent.Rotate the parallel direction of Laplcian vector to vertex normal then, and by the Laplacian coordinate reconstruction model M that upgrades _K+1, k=k+1.If model M _K+1Ineligible with the error of desirable deformation effect model, then repeat above-mentioned steps, otherwise, output model.

The present invention is applied to a example schematic on the three-dimensional grid model shown in accompanying drawing 2, (a) is master pattern, (b) is former Laplacian method editor's deformation result, (c) accordingly result for adopting the inventive method to obtain.Import original three-dimensional grid model, like Fig. 2 (a), set up the Laplacian coordinate of model, (a) carries out translation transformation to model, rebuilds the model after the distortion of cartesian coordinate system, obtains Fig. 2 (b) and (c).Contrast (b) with can find out that (c) after the deformation operation, (b) curve of model horizontal direction still keeps the same with master mould, caused volume-diminished in the middle part of the model.(c) curve of model horizontal direction is along with deformation operation, and the rotoflector certain angle has kept the geometric detail characteristic of model preferably, has obtained the responsive deformation effect of translation.

Claims (1)

1. Laplacian gridding edition method that realizes that translation is responsive, it is characterized in that: this method may further comprise the steps:

(1) input three-dimensional grid model M ₀, definition M ₀=(V, E F) have | V| summit, and V wherein, E, F represent the set of summit, limit, face respectively;

(2) pass through M ₀Its Laplacian coordinate of Cartesian coordinates structure, the note vertex v _iThe Laplacian coordinate be δ _i

Wherein, described vertex v _iLaplacian coordinate δ _iBe specially:

${\mathrm{\δ}}_i=({\mathrm{\δ}}_i^{\left(x\right)},{\mathrm{\δ}}_i^{\left(y\right)},{\mathrm{\δ}}_i^{\left(z\right)})=v_i-\frac{1}{d_i}\underset{j\∈N\left(i\right)}{\mathrm{\Σ}}{v}_j;$

And in formula, N (i)={ j| (i, j) ∈ E}, d _i=| N (i) |, d _iThe expression vertex v _iThe adjacent vertex number;

(3) user passes through interactive means to model M ₀Carry out deformation operation;

(4) according to the vertex set c after the known distortion _j, satisfy v _j=c _j, j ∈ 1,2 ..., m} as the position constraint condition, rebuilds the model M the Cartesian coordinates from the Laplacian coordinate _k, k=1;

Wherein, the method for rebuilding the model M k the Cartesian coordinates from the Laplacian coordinate is specially: through finding the solution the vertex set V after equation obtains distortion,

$\left(\frac{L}{{\mathrm{\ωI}}_{m\×m}|0}\right)V=\left(\begin{array}{c}\mathrm{\δ}\\ {\mathrm{\ωc}}_{1:m}\end{array}\right)$

In the formula, weights ω＞0 is used to adjust the importance of position constraint condition, and L is the Laplacian operator, and approximate solution is found the solution by system through least square, has unique solution during the system full rank:

$\tilde{V}=\underset{V}{\arg \min }({\left|\right|\mathrm{LV}-\mathrm{\δ}\left|\right|}^2+\underset{j\∈C}{\mathrm{\Σ}}{\mathrm{\ω}}^2{\left|\right|v_j-c_j\left|\right|}^2)$

(5) work as M _kWhen differing big, be designated as mark value ε＞E and continue step (6), otherwise jump to step (12) with the error of the model of ideal effect;

(6) by mid-module M _kObtain vertex v _iThe normal of each adjacent surface

If the concavo-convex variation of model is more, continues step (7), otherwise jump to step (8);

(7) if

With δ _iTowards inconsistent, the adjustment

The Laplacian coordinate that makes face normal and summit is towards consistent;

(8) calculate summit average normal F (v _i);

Wherein, the average normal F in summit (v _i) calculate gained according to the normal of summit adjacent surface,

$F\left(v_i\right)=\frac{\underset{j\∈M\left(i\right)}{\mathrm{\Σ}}{\mathrm{\μ}}_j\stackrel{\→}{n_j}}{\left|\right|\underset{j\∈M\left(i\right)}{\mathrm{\Σ}}{\mathrm{\μ}}_j\stackrel{\→}{n_j}\left|\right|}$

Wherein, M (i)=j| (i, j) ∈ E},

Be face normal line vector, μ _jBe weight, its value is that each adjacent surface is to vertex v _iThe influence of Laplacian vector, according to the area decision of triangle gridding;

(9) rotation Laplacian coordinate δ _i

Wherein, rotate the vector of Laplacian coordinate in the following way,

δ _i＝‖δ _i‖·F(v _i)；

(10) find the solution mid-module M _K+1, k=k+1;

Wherein, M _K+1The detailed process of finding the solution be: the solving equation group,

$\left(\frac{L}{{\mathrm{\ωI}}_{m\×m}|0}\right)V=\left(\begin{array}{c}\left|\right|\mathrm{\δ}\left|\right|\·F\left(V\right)\\ {\mathrm{\ωc}}_{1:m}\end{array}\right)$

Wherein, obtain approximate solution, have unique solution during the system full rank through finding the solution the least square system:

$\tilde{V}=\underset{V}{\arg \min }({\left|\right|\mathrm{LV}-\left|\right|\mathrm{\δ}\left|\right|\·F\left(V\right)\left|\right|}^2+\underset{j\∈C}{\mathrm{\Σ}}{\mathrm{\ω}}^2{\left|\right|v_j-c_j\left|\right|}^2)$

(11) if M _K+1Satisfy threshold value with the error of the model of ideal effect, continue step (12), otherwise, jump to step (5);

(12) obtain final mask M _K+1, finish.

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