CN101276474B - Linear constrained image distortion method based on local coordinates - Google Patents

Abstract

The invention relates to a linear image deformation method based on a local coordinate. The method comprises transforming an image into a corresponding four-side-territory grid, presenting the geometrical characteristics of the grid by affine and angle linearity restriction, constructing a boundary restriction condition based on a user-appointed control culmination, solving a linear equations set, rapidly computing the deformed image. According to the method, the deformation effect with translation sensibility can be obtained by an interaction speed, the proportion of affine deformation can be controlled by adjusting the weights of two kind of linearity restriction, complex deformation effects, such as good bend, translation sensibility, anisotropy zoom, etc., are realized.

Description

A kind of linear restriction image distortion method based on local coordinate

Technical field

The present invention relates to the deformation technology of two dimensional image, specifically relate to the animation design technique that the two dimensional image of having described animating image is out of shape.

Background technology

A large amount of artistic images as case of caricatures of persons etc., is preserved with the form of image.If can generate various new elements and expression apace with simple and direct their distortion of method control, can improve cartoon making efficient significantly, reduce manufacture difficulty.This demand has following characteristics: the distortion of at first this animating image belongs to non-accurate distortion, need not use method acquisition precise results such as physical simulation, as long as keep basic geometric properties; Secondly, the user wishes by simple means controlled deformation result; In addition, algorithm should have higher robustness, can return operation result with interactively speed.The present invention proposes a kind of method, the anamorphose problem is converted to the two-dimensional grid problem on deformation handles: with image transitions is territory, four limits grid, and this grid is out of shape solves the problems referred to above.

Distortion of the mesh is a hot issue in recent years, and people have proposed multiple scheme, but it is to be solved to still have several major issues to have.The first, serious distortion might appear in the local geometric features of grid when significantly being out of shape; The second, the user only provides the coordinate translation information of minority control vertex sometimes, and does not comprise rotation information.At this moment need algorithm to produce suitable rotation information automatically, thereby obtain desirable deformation result, be i.e. " translation sensitivity " characteristic.The 3rd, because the cause of efficient, algorithm should adopt linear operation as far as possible.It's a pity that general non-rigid body translation (as bending, irregular deformation etc.) can not be shown the summit linear operation by simple table.

For solving an above-mentioned distortion difficult problem, the present invention proposes a kind of grid deforming method based on local coordinate, thereby realizes anamorphose fast.

Summary of the invention

The present invention proposes with linear affine and angle restriction and describe the deformation method of territory, four limits grid vertex local feature, and this method is used for the non-accurate distortion of image.The user can obtain deformation result apace by specifying the position of minority control vertex simply.This method has adopted the local coordinate of grid to describe geometric properties, has the characteristic of translation sensitivity, can produce suitable rotation amount automatically according to the translation information of control vertex, generates the distortion effect of nature fast.By giving different constraint weights, can control the proportion of affine deformation in the result in addition.This method can obtain have the deformation result of translation sensitive natur with interactively speed, by a step find the solution the system of linear equations computing, generate the multiple complex deformation effect of 2 D animation image quickly and easily.

For reaching above-mentioned purpose, the technical solution used in the present invention is: a kind of linear restriction image distortion method based on local coordinate, and the method includes the steps of:

At first construct territory, corresponding four limit grid, then the anamorphose problem can be converted to the two-dimensional grid problem on deformation and handle according to input picture.Piece image I (x) conduct input, and with pixel separation d structural deformation zone $D\⊆I$ Corresponding territory, four limits grid M ₀This gridding process is regarded D as grid M ₀On pinup picture, and note the corresponding pinup picture coordinate of each grid vertex.Because M ₀The summit corresponding regularly the some pixels among the I (x), and arrange regularly, so this cancellated structure process is very fast relatively, and only needs to carry out and once then can at pretreatment stage;

Then at M ₀In specify the minority summit as control vertex, and they are moved or rotate to reposition.This method will generate corresponding position constraint condition according to control vertex information, and in conjunction with M ₀Geometric properties structure corresponding linear system of equations, use the position that solves all the other summits based on the linear restriction deformation method of local coordinate then, obtain new grid M ₁

At last according to M ₀The former pinup picture coordinate on middle summit is mapped to M with D ₁In, the image result after obtaining to be out of shape.

Technical characterstic of the present invention mainly embodies as follows:

1, affine constraint in this method and angle restriction all are linear, can find the solution new vertex position rapidly according to boundary condition, obtain deformation result fast.

2, by regulating the weight of two kinds of constraints, the user is the degree of control mesh affine deformation easily, realizes the deformation effect that good bending, translation sensitivity, anisotropy convergent-divergent etc. are complicated.

3, method has the translation sensitive natur, can produce the deformation effect of nature, distribution that need not explicit appointment rotation amount.

Description of drawings

Fig. 1 is the schematic flow sheet of the inventive method;

The deformation effect figure that Fig. 2 produces under various boundary conditions for the inventive method;

Fig. 3 is the inventive method control vertex and the weight figure that influences to deformation effect.

Embodiment

The present invention is described further below in conjunction with accompanying drawing.

The inventive method is converted into corresponding grid M with the problem on deformation of figure ₀Problem on deformation.This method at first piece image I (x) as input, and with pixel separation d structural deformation zone $D\⊆I$ Corresponding territory, four limits grid M ₀For handling conveniently, this grid is that the square of d constitutes by some length of sides.D chooses more for a short time, M ₀More near the profile of D, but the also corresponding operand that increased.Because the inventive method is towards the cartoon making field, the profile details of D is also non-key, thereby d can choose bigger value, then can as long as reflect required details rightly.This gridding process is regarded D as grid M ₀On pinup picture, and note the corresponding pinup picture coordinate of each grid vertex.Because M ₀The summit corresponding regularly the some pixels among the I (x), and arrange regularly, so this cancellated structure process is very fast relatively, and only needs to carry out and once then can at pretreatment stage.

The user is subsequently at M ₀In specify the minority summit as control vertex, and they are moved or rotate to reposition.This method will generate corresponding position constraint condition according to control vertex information, and in conjunction with M ₀Geometric properties structure corresponding linear system of equations, use the position that solves all the other summits based on the linear restriction deformation method of local coordinate then, obtain new grid M ₁

At last according to M ₀The former pinup picture coordinate on middle summit is mapped to M with D ₁In, the image result after obtaining to be out of shape.

Consider that grid local changing features when large deformation is very little, this method adopts local coordinate to describe the geometric properties of grid, be that each element all uses the relative position of adjacent element to represent, so thisly all remaining unchanged when being described in rigid transformation and convergent-divergent, is that translation, rotation and convergent-divergent are irrelevant.The translation sensitivity comes down under the prerequisite that satisfies the border vertices position constraint, make the relative rotation amount between the element as far as possible little, this just local coordinate can accomplish.Adopt local coordinate to describe the geometric properties of grid, be expressed as following formula:

$V^{*}=\underset{V^{\′}}{\arg \min }\mathrm{\α}\underset{v_i\∈V}{\mathrm{\Σ}}{\left|\right|t\left({v_i}^{\′}\right)-t\left(v_i\right)\left|\right|}^2$

$+\mathrm{\β}\underset{e_m,e_n\∈E}{\mathrm{\Σ}}{\left|\right|r({e_m}^{\′},{e_n}^{\′})-r(e_m,e_n)\left|\right|}^2$

$+\mathrm{\ω}\underset{v_k\∈\mathrm{\Ω}}{\mathrm{\Σ}}{\left|\right|v_k^{\′}-u_k\left|\right|}^2$

V wherein _iBe former vertex position, v _i' be new vertex position,

The local coordinate of expression summit in adjacent frame, e _m, e _nBe adjacent edge,

The rotation amount of expression adjacent edge, α, β, ω are corresponding weights.

Investigate initial mesh M ₀Geometric properties.It is made of some adjacent squares, and the equal in length on every limit meets at right angles between the adjacent edge or the straight angle.Therefore, utilize these geometric properties, can construct the good length of side and angle restriction, make distortion target gridding M ₁Adjacent side length approximate, angle is near the right angle or the straight angle.

The present invention M ₀Widthwise edge and adjacent vertical limit constitute a partial, right angle frame f, go to represent the limit of direct neighbor then with f, provide the concrete form of first item constraint below.

(1) affine constraint

Because M ₀Only the limit by level and vertical both direction constitutes, and equal in length, therefore for each summit, is made as v, can write out following linear relationship;

$\left\{\begin{array}{c}{v}_1-v_0=v_0-v_3\\ v_2-v_0=v_0-v_4\end{array}\right.$

So the error of new summit in adjacent local coordinate can be write:

$E_{1\left\{{v_0}^{\&prime;}\right\}}={\left|\right|{v_0}^{\&prime;}-({v_1}^{\&prime;}+{{\mathrm{<figure-callout\; id="3"\; label="v"\; filenames="S2008100274203E00021.png"\; state="\{\{state\}\}">v</figure-callout>}}_3}^{\&prime;})/2\left|\right|}^2$

$+{\left|\right|{v_0}^{\&prime;}-({v_2}^{\&prime;}+{v_4}^{\&prime;})/2\left|\right|}^2$

This error comes down to the distance of the local coordinate on new summit to the expection local coordinate.Then Quan Ju affine error is

$E_1=\underset{{v_i}^{\&prime;}\&Element;V^{\&prime;}}{\mathrm{\&Sigma;}}{E}_{1\left\{{v_i}^{\&prime;}\right\}}.$

As long as the global error of making $E_1=\underset{{v_i}^{\&prime;}\&Element;V^{\&prime;}}{\mathrm{\&Sigma;}}{E}_{1\left\{{v_i}^{\&prime;}\right\}}$ Minimum then can be distributed to the error that border vertices produces on the whole grid equably, obtains the new grid of fairing.Promptly by finding the solution down the least square linear system of equations of wearing linear barrier's constraint

$V^{*}=\underset{V^{\&prime;}}{\arg \min }\mathrm{\&alpha;\&Sigma;}{\left|\right|H{V}^{\&prime;}\left|\right|}^2+\mathrm{\&omega;}\underset{{v_k}^{\&prime;}\&Element;\mathrm{\&Omega;}}{\mathrm{\&Sigma;}}{\left|\right|{v_k}^{\&prime;}-u_k\left|\right|}^2$

Can try to achieve and make global error $E_1=\underset{{v_i}^{\&prime;}\&Element;V^{\&prime;}}{\mathrm{\&Sigma;}}{E}_{1\left\{{v_i}^{\&prime;}\right\}}$ Minimum new vertex position V '.Wherein H be by $E_{1\left\{{v_0}^{\&prime;}\right\}}={\left|\right|{v_0}^{\&prime;}-({v_1}^{\&prime;}+{{\mathrm{<figure-callout\; id="3"\; label="v"\; filenames="S2008100274203E00021.png"\; state="\{\{state\}\}">v</figure-callout>}}_3}^{\&prime;})/2\left|\right|}^2+{\left|\right|{v_0}^{\&prime;}-({v_2}^{\&prime;}+{v_4}^{\&prime;})/2\left|\right|}^2$ The weight matrix that obtains is represented the neighbouring relations on each summit.

Only adopt this linear restriction can not obtain the responsive effect of translation.So we introduce the second following item constraint.

(2) angle restriction

What consider processing is two-dimentional problem on deformation, and the mutually perpendicular x of long vector and the y component of waiting can constitute following linear restriction relation in two dimensional surface:

$\left\{\begin{array}{c}{(v_1-v_0)}_x={(v_2-v_0)}_y\\ {(v_1-v_0)}_y=-{(v_2-v_0)}_x\end{array}\right..$

So the error of the local frame that new summit is constituted is:

$E_{2\left\{{v_0}^{\&prime;}\right\}}={\left|\right|{(v_1-v_0)}_x-{(v_2-v_0)}_y\left|\right|}^2$

$+{\left|\right|{(v_1-v_0)}_y+{(v_2-v_0)}_x\left|\right|}^2.$

As seen, this error in fact also is the distance of the local coordinate on new summit to the expection local coordinate.Correspondingly obtain overall angular error and be;

$E_2=\underset{{v_i}^{\&prime;}\&Element;V^{\&prime;}}{\mathrm{\&Sigma;}}{E}_{2\left\{{v_i}^{\&prime;}\right\}}.$

Add affine constraint, obtain the least square linear system of equations of complete band linear restriction:

$V^{*}=\underset{V^{\&prime;}}{\arg \min }\mathrm{\&alpha;\&Sigma;}{\left|\right|H{V}^{\&prime;}\left|\right|}^2+\mathrm{\&beta;\&Sigma;}{\left|\right|K\left[\begin{array}{c}{V_x}^{\&prime;}\\ {V_y}^{\&prime;}\end{array}\right]\left|\right|}^2$

$+\mathrm{\&omega;}\underset{{v_k}^{\&prime;}\&Element;\mathrm{\&Omega;}}{\mathrm{\&Sigma;}}{\left|\right|{v_k}^{\&prime;}-u_k\left|\right|}^2$

Wherein K be by $E_{2\left\{{v_0}^{\&prime;}\right\}}={\left|\right|{(v_1-v_0)}_x-{(v_2-v_0)}_y\left|\right|}^2+{\left|\right|{(v_1-v_0)}_y+{(v_2-v_0)}_x\left|\right|}^2$ The weight matrix that obtains.

Because $V^{*}=\underset{V^{\&prime;}}{\arg \min }\mathrm{\&alpha;\&Sigma;}{\left|\right|H{V}^{\&prime;}\left|\right|}^2+\mathrm{\&beta;\&Sigma;}{\left|\right|K\left[\begin{array}{c}{V_x}^{\&prime;}\\ {V_y}^{\&prime;}\end{array}\right]\left|\right|}^2+\mathrm{\&omega;}\underset{{v_k}^{\&prime;}\&Element;\mathrm{\&Omega;}}{\mathrm{\&Sigma;}}{\left|\right|{v_k}^{\&prime;}-u_k\left|\right|}^2$ Be linear least square journey group, can write:

Wherein $G=\left[\begin{array}{cc}H& 0\\ 0& H\end{array}\right],$ U has preserved border vertices u _kRespective value, W is the weighting matrix of border vertices.For simplicity, make β=1/ α usually.

What need spell out is: though be system of linear equations, owing to x, the split corresponding restriction relation of structure that comes of y component, so following formula is not the system of linear equations of summit V '.Owing to adopted this skill of angle restriction, thereby the translation sensitive natur is achieved, and only demand separate one step of system of linear equations then can, need not adopt interative computation.

Owing to all constructed an angle restriction for each foursquare interior angle, so the order of K is 2 ‖ V ' ‖-4.Because the summit in edge lacks length of side constraint condition than internal vertex, the perhaps cause of border vertices conllinear, the order of G must be less than the order of K.Institute thinks that system of equations is separated when making β non-vanishing, must specify 2 border vertices at least.As seen, independent use angle constraint can be found the solution vertex position.But affine constraint can realize mistake and cut the equiaffine deformation effect.The user can be by regulating two kinds of constraint conditions of ratio balance of α and β.

System of equations is the overdetermination system of linear equations of AV '=b form, can be by calculating its normal equation group A ^TAV '=A ^TB solves V '.We have adopted the TAUCS method to try to achieve A ^TThe Cholesky of A decomposes, and calculates the least square solution of V ' by back substitution.

The deformation effect that the inventive method produces when using various boundary conditions as shown in Figure 2.(a) be former grid.(b) be the deformation effect that only adopts affine constraint to produce.After increasing angle restriction, identical control vertex position has produced smooth bending effect (c).(d) be the effect of crooked 180 degree.(e) be that the mistake that only adopts affine constraint to produce is cut effect.After increasing angle restriction, can produce the responsive effect (f) of translation and (g).(h) be the unilateral stretching effect that only adopts affine constraint to generate.(i) be to have adopted whole constraints and 2 (isotropy) amplification effects that control vertex produces.Round dot is a control vertex.

Control vertex and weight are to the influence of deformation effect as shown in Figure 3 in the inventive method.(a) be former figure.(b) with two control vertexs the left side is dwindled in, and move to right a control vertex on right side is past, amplified the right side.(c) α is 10, and β is 0.1, obtains to approach the affine deformation effect of cross directional stretch; (d) α and β are 1, and the middle part occurs amplifying owing to influence each other.(e) with the object bending, contraction to a certain degree appears in the middle part.Increase a reference mark in the centre, (e) elongated the effect (f) that area is protected in simulation.

Claims (2)

1. linear restriction image distortion method based on local coordinate is characterized in that the method includes the steps of:

(1) two dimensional image to input carries out foursquare gridding operation, and adopts local coordinate to describe the geometric properties of grid, is expressed as following formula:

$V^{*}=\underset{V^{\&prime;}}{\arg \min }\mathrm{\&alpha;}\underset{v_i\&Element;V}{\mathrm{\&Sigma;}}{\left|\right|t\left({v_i}^{\&prime;}\right)-t\left(v_i\right)\left|\right|}^2$

$+\mathrm{\&beta;}\underset{e_m,e_n\&Element;E}{\mathrm{\&Sigma;}}{\left|\right|r({e_m}^{\&prime;},{e_n}^{\&prime;})-r(e_m,e_n)\left|\right|}^2$ Formula (1)

$+\mathrm{\&omega;}\underset{v_k\&Element;\mathrm{\&Omega;}}{\mathrm{\&Sigma;}}{\left|\right|v_k^{\&prime;}-u_k\left|\right|}^2$

V wherein _iBe former vertex position, v _i' be new vertex position,

The local coordinate of expression summit in adjacent frame, e _m, e _nBe adjacent edge,

The rotation amount of expression adjacent edge, α, β, ω are corresponding weights;

(2) relative position between the summit is carried out affine constraint, obtain the constraint condition about the position, its calculation procedure is:

Each summit is represented with v, represents different summits by the subscript of v, draws linear relationship:

$\left\{\begin{array}{c}{v}_1-v_0=v_0-v_3\\ v_2-v_0=v_0-v_4\end{array}\right.$ Formula (2)

The error writing of new summit in adjacent local coordinate:

$E_{1\left\{{v_0}^{\&prime;}\right\}}={\left|\right|{v_0}^{\&prime;}-({v_1}^{\&prime;}+{v_3}^{\&prime;})/2\left|\right|}^2$ Formula (3)

$+{\left|\right|{v_0}^{\&prime;}-({v_2}^{\&prime;}+{v_4}^{\&prime;})/2\left|\right|}^2$

Then Quan Ju affine error is:

$E_1=\underset{{v_i}^{\&prime;}\&Element;V^{\&prime;}}{\mathrm{\&Sigma;}}{E}_{1\left\{{v_i}^{\&prime;}\right\}}$ Formula (4)

Find the solution the least square linear system of equations of band linear barrier constraint

$V^{*}=\underset{V^{\&prime;}}{\arg \min }\mathrm{\&alpha;\&Sigma;}{\left|\right|{\mathrm{HV}}^{\&prime;}\left|\right|}^2+\mathrm{\&omega;}\underset{{v_k}^{\&prime;}\&Element;\mathrm{\&Omega;}}{\mathrm{\&Sigma;}}{\left|\right|{v_k}^{\&prime;}-u_k\left|\right|}^2$ Formula (5)

Try to achieve the new vertex position V ' that makes the global error minimum, wherein H is the weight matrix that is obtained by formula (3), represents the neighbouring relations on each summit;

(3) angle between the adjacent edge is carried out angle restriction, obtain the constraint condition about angle, its calculation procedure is:

The mutually perpendicular x of long vector and the linear restriction relation below the y component formation of waiting in two dimensional surface:

$\left\{\begin{array}{c}{(v_1-v_0)}_x={(v_2-v_0)}_y\\ {(v_1-v_0)}_y=-{(v_2-v_0)}_x\end{array}\right.$ Formula (6)

The error of the local frame that then new summit is constituted is:

$E_{2\left\{{v_0}^{\&prime;}\right\}}={\left|\right|{(v_1-v_0)}_x-{(v_2-v_0)}_y\left|\right|}^2$

$+{\left|\right|{(v_1-v_0)}_y+{(v_2-v_0)}_x\left|\right|}^2$ Formula (7)

Correspondingly obtain overall angular error and be:

$E_2=\underset{{v_i}^{\&prime;}\&Element;V^{\&prime;}}{\mathrm{\&Sigma;}}{E}_{2\left\{{v_i}^{\&prime;}\right\}}$ Formula (8)

Thereby obtain the least square linear system of equations of complete band linear restriction:

$V^{*}=\underset{V^{\&prime;}}{\arg \min }\mathrm{\&alpha;\&Sigma;}{\left|\right|{\mathrm{HV}}^{\&prime;}\left|\right|}^2+\mathrm{\&beta;\&Sigma;}{\left|\right|K\left[\begin{array}{c}{V_x}^{\&prime;}\\ {V_y}^{\&prime;}\end{array}\right]\left|\right|}^2$

$+\mathrm{\&omega;}\underset{{v_k}^{\&prime;}\&Element;\mathrm{\&Omega;}}{\mathrm{\&Sigma;}}{\left|\right|{v_k}^{\&prime;}-u_k\left|\right|}^2$ Formula (9)

Wherein K is the weight matrix that is obtained by formula (7);

(4) regulate two kinds of constraint conditions of ratio balance of α and β, and find the solution system of linear equations;

(5) the multiple complex deformation effect of generation 2 D animation image.

2. according to the described a kind of linear restriction image distortion method based on local coordinate of claim 1, it is characterized in that: two kinds of constraint conditions of ratio balance of described adjusting α and β are specially: make β=1/ α, formula (9) formula is write:

Formula (10)

Wherein

U has preserved border vertices u _kRespective value, W is the weighting matrix of border vertices;

By group of equations A ^TAV '=A ^TB solves V '; And adopted TAUCS to try to achieve A ^TThe Cholesky of A decomposes, and calculates the least square solution of V ' by back substitution.

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