CN102411794B - Output method of two-dimensional (2D) projection of three-dimensional (3D) model based on spherical harmonic transform - Google Patents

Abstract

The invention discloses an output method of two-dimensional (2D) projection of a three-dimensional (3D) model based on spherical harmonic transform. The output method comprises the following steps of: 1. inputting the 3D model to be projected; 2. calculating barycentric coordinates of the 3D model, and then normalizing the 3D model; 3. carrying out layering sampling on the 3D model; 4. calculating the spherical harmonic characteristic of the 3D model by a Monte Carlo integration method, and calculating conversion factors based on sampling points; 5. calculating the 2D spherical harmonic characteristic after projection according to the conversion factors and the spherical harmonic characteristics of the 3D model; and 6. calculating the 2D information after projection according to the 2D spherical harmonic characteristic.

Description

A kind of output intent of two-dimensional projection of the three-dimensional model based on the humorous conversion of ball

Technical field

The present invention relates to a kind of method of three-dimensional model projection, a kind of three-dimensional model for only comprising geological information particularly utilizes the method for the humorous conversion of ball, the method for the two-dimensional projection of output three-dimensional model.

Background technology

Now, three-dimensional model has been widely applied to the aspects such as computer animation, game, virtual reality, and simultaneously, the projection of three-dimensional model has characterized the visual pattern of three-dimensional model, also is widely applied in each field.

The three-dimensional model projecting method that exists at present is parallel projection and perspective projection, is all the computing method of geometric projection.The projection ray of perspective projection is sent by common point, and parallel projection is special circumstances of perspective projection.The visual pattern that these two kinds of projection patterns obtain is usually used in three-dimensional model based on the search method of vision, but there is a common shortcoming in these two kinds of projection patterns: the difference of projection angle can obtain different visual patterns.So, in three-dimensional model search, in order to obtain the more information of model-based vision image, generally need to set different angles, obtain the visual pattern information on the three-dimensional model different directions.

Summary of the invention

Goal of the invention: technical matters to be solved by this invention is for the deficiencies in the prior art, and a kind of output intent of two-dimensional projection of the three-dimensional model based on the humorous conversion of ball is provided.

In order to solve the problems of the technologies described above, the invention discloses a kind of output intent of two-dimensional projection of the three-dimensional model based on the humorous conversion of ball, comprise the following steps:

The three-dimensional model of projection is treated in step 1, input, and the three-dimensional model of inputting is comprised of one group of tri patch, and the three-dimensional model of inputting is comprised of one group of tri patch, comprises three apex coordinates of spatial point coordinate, tri patch in tri patch;

Then the center-of-mass coordinate of step 2, Calculation of Three Dimensional model standardizes to three-dimensional model;

Step 3, three-dimensional model is carried out stratified sampling, use the monte carlo integration method when facilitating the humorous feature calculation of three-dimensional model ball;

Step 4, according to the thought of the humorous conversion of ball, sampling monte carlo integration method, the humorous feature of the ball of Calculation of Three Dimensional model; Simultaneously, calculate conversion factor by sampled point;

Step 5, according to conversion factor and the humorous feature of three-dimensional ball, calculate the humorous feature of two-sphere after projection;

Step 6, feature humorous according to two-sphere, the relevant information after the calculating projection.

In step 2 of the present invention, the computing method of the center-of-mass coordinate C of three-dimensional model (x, y, z) are:

$C(x,y,z)=\frac{\underset{i=1}{\overset{K\left(S\right)}{\mathrm{\Σ}}}(w_i\×C_i(x_i,y_i,z_i))}{\underset{i=1}{\overset{K\left(S\right)}{\mathrm{\Σ}}}{w}_i},$

Wherein K (S) is three-dimensional model S intermediate cam dough sheet sum, w _iThe area of i tri patch in S, C _i(x _i, y _i, z _i) be the center-of-mass coordinate of tri patch i.

In step 2 of the present invention, three-dimensional model is standardized, comprises following steps:

The translation three-dimensional model makes the barycenter C (x, y, z) of three-dimensional model overlap with the initial point of coordinate system;

Then the radially ultimate range with three-dimensional model normalizes to three-dimensional model in unit ball.

P(x，y，z)＝σ(P(x，y，z)-C(x，y，z))，

Wherein, σ represents the peaked inverse of distance of the barycenter of three-dimensional model that three-dimensional model is had a few, σ=1/max (‖ P (x, y, z)-C (x, y, z) ‖), and some P (x, y, z) is the point on three-dimensional model surface.Afterwards, the barycenter of three-dimensional model is true origin.

In step 3 of the present invention, three-dimensional model is carried out stratified sampling, comprises following steps:

In step 2, after standardizing in unit ball to three-dimensional model, need to sample to the spheroid polar coordinates:

The spheroid polar coordinates are defined by position angle φ and polar angle θ,

0≤θ＜π，

0≤φ＜2π，

In the unit ball polar coordinates, carry out as follows stratified sampling:

${\mathrm{\θ}}_i=(i+0.5)\frac{\mathrm{\π}}{N},$

${\mathrm{\φ}}_j=(j+0.5)\frac{2\mathrm{\π}}{N},$

Wherein, 0≤i, j＜N, N are the sampled point number on each direction, θ _iRepresent i polar angle,

Represent j position angle,

A common point that consists of in polar coordinates; I and j represent respectively the desired value of polar coordinates down-sampling, and often a pair of (i, j) determines a sampled point, and total sampled point number is N*N.

Be true origin from the three-dimensional model barycenter, along direction (sin θ _iCos φ _j, sin θ _iSin φ _j, cos θ _i) ray and the intersection point of three-dimensional model, be the sampled point of three-dimensional model stratified sampling.

In step 4, the humorous feature of the three-dimensional ball of Calculation of Three Dimensional model comprises following steps:

Get sampled point (θ, φ), calculate the spheric function f (θ, φ) of this sampled point: the radial function of getting the three-dimensional model sample point is spheric function

Wherein, some P represents the sampled point by the definite three-dimensional model surface of polar coordinates (θ, φ); Point C is the three-dimensional model barycenter of this moment, i.e. true origin; D represents Euclidean distance;

Secondly, calculated the basis function of the humorous conversion of ball by coordinate (θ, φ)

Only consider the humorous transform-based function of real number ball herein, Carry out according to following formula:

$Y_l^m(\mathrm{\&theta;},\mathrm{\&phi;})=\left\{\begin{array}{cc}\sqrt{2}{K}_l^m\cos \left(\mathrm{m\&phi;}\right)P_l^m\left(\cos \mathrm{\&theta;}\right),& m>0\\ \sqrt{2}{K}_l^m\sin (-\mathrm{m\&phi;})P_l^{-m}\left(\cos \mathrm{\&theta;}\right),& m<0\\ K_l^0{P}_l^0\left(\cos \mathrm{\&theta;}\right),& m=0\end{array}\right.,$

Wherein,

Be zoom factor,

Be associated Legendre polynomial, the l span is the integer more than or equal to 0 ,-l≤m≤l;

At last, calculate the coefficient of the humorous conversion of ball by spheric function and the humorous transform-based function of ball

$a_l^m={\&Integral;}_0^{2\mathrm{\&pi;}}{\&Integral;}_0^{\mathrm{\&pi;}}\sin \left(\mathrm{\&theta;}\right)f(\mathrm{\&theta;},\mathrm{\&phi;})Y_l^m(\mathrm{\&theta;},\mathrm{\&phi;})\mathrm{d\&theta;d\&phi;},$

Adopt the numerical computation method of monte carlo integration as follows to following formula:

$a_l^m=\frac{4\mathrm{\&pi;}}{N}\underset{\mathrm{\&theta;},\mathrm{\&phi;}}{\mathrm{\&Sigma;}}f(\mathrm{\&theta;},\mathrm{\&phi;})Y_l^m(\mathrm{\&theta;},\mathrm{\&phi;}),$

Choose default experiment parameter B, in the present invention, the B value is 7, only calculates the situation under 0≤l≤B, obtains the humorous character representation of ball of three-dimensional model S.

In step 4, the conversion factor of Calculation of Three Dimensional model comprises following steps:

At first, due to, three-dimensional polar coordinates can be by point

Determine, r is by polar coordinates The sampled point on the three-dimensional model surface of determining is to the distance of three-dimensional model barycenter, and the two dimensional surface polar coordinates can be definite by point (r, φ), so, by the conversion between cartesian coordinate system, carry out conversion between following polar coordinates to sampled point:

(θ，φ)→(cosθ，φ)，

Utilize the associated Legendre polynomial under three-dimensional polar coordinate system and two-dimentional polar coordinate system, calculate conversion factor

Wherein,

Be associated Legendre polynomial, carry out iterative computation according to following formula:

$P_m^m\left(x\right)={(-1)}^m(2m-1)!!{(1-x^2)}^{m/2},$

$P_{m+1}^m\left(x\right)=x(2m+1)P_m^m\left(x\right),$

$(l-m)P_l^m\left(x\right)=x(2l-1)P_{l-1}^m\left(x\right)-(l+m-1)P_{l-2}^m\left(x\right),$

In step 5, the humorous feature of two-sphere after the calculating projection, carry out according to following steps:

By the humorous feature of the three-dimensional ball that calculates in step 4 and conversion factor, calculate the humorous feature of two-sphere after projection,

$c_l^m=a_l^m\&CenterDot;{\mathrm{\&beta;}}_l^m;$

In step 6, the two-dimensional signal after the calculating projection, carry out according to following steps:

At first, utilize the expansion formula of the humorous conversion of ball, the spheric function of Two-dimensional:

$g(r,\mathrm{\&phi;})=\underset{l=0}{\overset{B}{\mathrm{\&Sigma;}}}\underset{m=-l}{\overset{m=l}{\mathrm{\&Sigma;}}}{c}_l^m{H}_l^m(r,\mathrm{\&phi;})$

Wherein,

Be the humorous basis function of the ball under two-dimensional case.

Secondly, carry out the coordinate inverse mapping, to tlv triple obtained above (r, φ, g (r, φ)),

Image after the expression projection is at point

The gray-scale value at place, utilize the coordinate inverse mapping:

x＝rcos(φ)＝cos(θ)cos(φ)

y＝rsin(φ)＝cos(θ)sin(φ)，

Obtain new tlv triple (x, y, g (r, φ)), be the information after three-dimensional model is transformed into two dimensional image.

Beneficial effect: the present invention is a kind of output intent of two-dimensional projection of the three-dimensional model based on the humorous conversion of ball, due to the three-dimensional model projection based on the humorous conversion of ball, embody a kind of global information of three-dimensional model, do not needed other geometric projection method to set the projecting direction parameter.The visual pattern of the three-dimensional model of projection output is used for the retrieval of three-dimensional model, the information of can be more accurate more complete expression three-dimensional model, thus improve the performance of three-dimensional model search.And the present invention has adopted the technology of the humorous conversion of ball, has guaranteed the feature of the rotational invariance of three-dimensional model search, and the three-dimensional model that namely rotates can obtain identical two-dimensional projection's visual pattern, thereby retrieval effectiveness is better.

Description of drawings

Below in conjunction with the drawings and specific embodiments, the present invention is done further illustrating, above-mentioned and/or otherwise advantage of the present invention will become apparent.

Fig. 1 is process flow diagram of the present invention.

Fig. 2 is three-dimensional model polar coordinates of the present invention and Cartesian coordinates schematic diagram.

Fig. 3 is three-dimensional model projection process schematic diagram.

Fig. 4 is the stratified sampling schematic diagram in three-dimensional polar coordinates.

Embodiment

The output intent of the two-dimensional projection of a kind of three-dimensional model based on the humorous conversion of ball of the present invention, its basic point of departure is by the humorous feature of the ball of three-dimensional model, under the conversion factor effect, obtain the humorous feature of ball after projection, afterwards, by the coordinate inverse mapping, obtain the information of two-dimensional projection image.

Below in conjunction with accompanying drawing, the present invention is done more detailed explanation:

As shown in Figure 1, the present invention includes following steps:

Step 1, the three-dimensional model of projection is treated in input, and the three-dimensional model of inputting is comprised of one group of tri patch, and the geological information that extracts from three-dimensional model comprises each tri patch which three summit is made of, and the geometric coordinate on each summit.

Step 2, then the center-of-mass coordinate of Calculation of Three Dimensional model standardizes to three-dimensional model.

In step 2, the computing method of the center-of-mass coordinate C of three-dimensional model (x, y, z) are:

$C(x,y,z)=\frac{\underset{i=1}{\overset{K\left(S\right)}{\mathrm{\&Sigma;}}}(w_i\&times;C_i(x_i,y_i,z_i))}{\underset{i=1}{\overset{K\left(S\right)}{\mathrm{\&Sigma;}}}{w}_i},$

Wherein K (S) is three-dimensional model S intermediate cam dough sheet sum, w _iThe area of i tri patch in S, C _i(x _i, y _i, z _i) center-of-mass coordinate of single tri patch i.

Three-dimensional model is standardized, comprises following steps:

The barycenter C (x, y, z) that three-dimensional model is moved to three-dimensional model overlaps with the initial point of coordinate system, is about to each apex coordinate of three-dimensional model and deducts the center-of-mass coordinate C (x, y, z) that has calculated, P (x, y, z)-C (x, y, z);

Then with three-dimensional model normalization in the unit spheroid, namely each apex coordinate after translation be multiply by a normalized parameter σ, as shown in the formula expression:

P(x，y，z)＝σ(P(x，y，z)-C(x，y，z))，

Wherein, σ represents the peaked inverse of distance of the barycenter of three-dimensional model that three-dimensional model is had a few, σ=1/max (‖ P (x, y, z)-C (x, y, z) ‖), and some P (x, y, z) is the point on three-dimensional model surface.

Step 3 is carried out stratified sampling to three-dimensional model.

In step 3, three-dimensional model is carried out stratified sampling, comprises following steps:

At first, the spheroid polar coordinates are sampled:

The spheroid polar coordinate system can be made of position angle φ and polar angle θ, 0≤θ＜π wherein, 0≤φ＜2 π;

In the unit ball polar coordinates, carry out as follows stratified sampling:

${\mathrm{\&theta;}}_i=(i+0.5)\frac{\mathrm{\&pi;}}{N},$

${\mathrm{\&phi;}}_j=(j+0.5)\frac{2\mathrm{\&pi;}}{N},$

Wherein, 0≤i, j＜N, N are the sampled point number on each direction, θ _iRepresent i polar angle,

Represent j position angle,

A common point that consists of in polar coordinates; I and j represent respectively the desired value of polar coordinates down-sampling, and often a pair of (i, j) determines a sampled point, and total sampled point number is N*N;

Secondly, the sample coordinate in obtaining the unit ball polar coordinate system Afterwards, it is transformed in the cartesian coordinate system at three-dimensional model place, namely from the three-dimensional model barycenter along direction

Ray set out, calculate the intersection point of the dough sheet on itself and three-dimensional model surface, be the sampled point of three-dimensional model stratified sampling.

Step 4 is used the monte carlo integration method, the humorous feature of the ball of Calculation of Three Dimensional model, and calculate conversion factor by sampled point.

In step 4, the humorous feature of the ball of Calculation of Three Dimensional model comprises following steps:

According to step 3, get polar coordinates (θ, φ), calculate the spheric function f (θ, φ) of sample point corresponding to these polar coordinates: the radial function of getting the three-dimensional model sample point is spheric function:

f(θ，φ)＝d(C，P)；

Wherein, some P represents the sampled point by the definite three-dimensional model surface of polar coordinates (θ, φ); Point C is the barycenter of three-dimensional model; D represents Euclidean distance;

Calculated the basis function of the humorous conversion of ball by polar coordinates (θ, φ)

Wherein, the present invention adopts the humorous transform-based function of real number ball, still is designated as

Calculate according to following formula:

$Y_l^m(\mathrm{\&theta;},\mathrm{\&phi;})=\left\{\begin{array}{cc}\sqrt{2}{K}_l^m\cos \left(\mathrm{m\&phi;}\right)P_l^m\left(\cos \mathrm{\&theta;}\right),& m>0\\ \sqrt{2}{K}_l^m\sin (-\mathrm{m\&phi;})P_l^{-m}\left(\cos \mathrm{\&theta;}\right),& m<0\\ K_l^0{P}_l^0\left(\cos \mathrm{\&theta;}\right),& m=0\end{array}\right.,$

Wherein,

Be zoom factor,

Be associated Legendre polynomial, the l span is the integer more than or equal to 0, and-l≤m≤l can pre-set the parameter value B of experimental implementation, only calculates the humorous basis function of ball in 0≤l＜B situation;

Calculate the coefficient of the humorous conversion of ball by spheric function and the humorous transform-based function of real number ball

$a_l^m={\&Integral;}_0^{2\mathrm{\&pi;}}{\&Integral;}_0^{\mathrm{\&pi;}}\sin \left(\mathrm{\&theta;}\right)f(\mathrm{\&theta;},\mathrm{\&phi;})Y_l^m(\mathrm{\&theta;},\mathrm{\&phi;})\mathrm{d\&theta;d\&phi;},$

In every pair (l, m) definite situation, utilize following formula, adopt the numerical computation method of monte carlo integration method to calculate the humorous feature of corresponding ball

The monte carlo integration method that following formula is corresponding is as follows:

According to this, but the humorous feature of a series of balls of Calculation of Three Dimensional model

Amount to B ²Individual.

In step 4, calculate conversion factor by sampled point and comprise following steps:

By three-dimensional polar coordinates (θ, φ, r), r is by the sampled point on polar coordinates (θ, φ) the definite three-dimensional model surfaces distance to the three-dimensional model barycenter, and the two dimensional surface polar coordinates

By the conversion between polar coordinates and Cartesian coordinates, and the conversion between cartesian coordinate system, can carry out conversion between following polar coordinates to sampled point:

(θ，φ)→(cosθ，φ)，

Utilize the associated Legendre polynomial under three-dimensional polar coordinate system and two-dimentional polar coordinate system, calculate conversion factor

Wherein,

Be associated Legendre polynomial, carry out iterative computation according to following formula:

$P_m^m\left(x\right)={(-1)}^m(2m-1)!!{(1-x^2)}^{m/2},$

$P_{m+1}^m\left(x\right)=x(2m+1)P_m^m\left(x\right),$

$(l-m)P_l^m\left(x\right)=x(2l-1)P_{l-1}^m\left(x\right)-(l+m-1)P_{l-2}^m\left(x\right).$

Like this, just obtained the value of one group of conversion factor in different l and m situation.

Step 5, feature humorous according to the ball of described conversion factor and three-dimensional model, the humorous feature of two-sphere after the calculating projection.

In step 5, the humorous feature of two-sphere after the calculating projection: by one group of humorous feature of ball of the three-dimensional model that calculates in step 3

With the one group of conversion factor that obtains in step 4

Calculate one group of humorous feature of ball of the two dimensional image after projection

Wherein each

Calculated by following formula,

$c_l^m=a_l^m\&CenterDot;{\mathrm{\&beta;}}_l^m.$

Step 6, feature humorous according to two-sphere, the two-dimensional signal after the calculating projection.

In step 6, the two-dimensional signal after the calculating projection, carry out according to following steps:

Choosing two-dimentional polar coordinates

Calculate the spheric function at subpoint place corresponding to these polar coordinates

At first, utilize the expansion formula of the humorous conversion of ball, the spheric function in Two-dimensional polar coordinates situation:

Wherein,

Be the humorous basis function of the ball under two dimension;

Secondly, carry out the coordinate inverse mapping, the tlv triple that obtains (r, φ, g (r, φ)),

The expression projected image is at polar coordinates

The gray-scale value at place, utilize the coordinate inverse mapping:

x＝rcos(φ)＝cos(θ)cos(φ)

y＝rsin(φ)＝cos(θ)sin(φ)，

Obtain new tlv triple (x, y, g (r, φ)), be the information after three-dimensional model is transformed into two dimensional image.

As shown in Figure 2, three-dimensional model Cartesian coordinates and polar coordinates schematic diagram: in whole computation process, repeatedly use the mutual conversion between Cartesian coordinates and polar coordinates.

The three-dimensional model polar coordinates by position angle φ ∈ [0,2 π) and polar angle θ ∈ [0, π) form.Cartesian coordinate system is by x-, y-, and the z-axle represents.

In Fig. 2, Γ represents curved surface in three dimensions, and P (x, y, z) represents a point on this curved surface.By some relations of trigonometric function, can obtain polar coordinates and the Cartesian coordinates conversion formula is as follows:

Calculate polar coordinates by Cartesian coordinates:

$r=\sqrt{x^2+y^2+z^2},$

$\mathrm{\&phi;}={\tan }^{-1}\left(\frac{y}{x}\right),$

$\mathrm{\&theta;}={\cos }^{-1}\left(\frac{z}{r}\right),$

Otherwise, calculate Cartesian coordinates by polar coordinates:

x＝rcosφsinθ，

y＝rsinφsinθ，

z＝rcosθ，

As shown in Figure 3, three-dimensional model projection process schematic diagram: from the input of three-dimensional model, to the expression that is converted to the humorous feature of three-dimensional ball, the humorous conversion factor of ball under Two-dimensional and three-dimensional situation afterwards, calculate the humorous character representation of two-sphere after projection, be reduced at last the two dimensional image after projection.

Fig. 3 schematic diagram is that (experiment parameter is respectively concrete experimental result: 1024 of sampled points, i.e. N=32; Bandwidth B=7; Perspective view is of a size of the 256*256 pixel.)

Three-dimensional model shown in Fig. 3 left side (ant) as input object, is then found the solution the humorous feature of its three-dimensional ball, is write as the form of matrix, and is as follows:

$\left[\begin{array}{cccc}& & -1.23448& \\ & -3.53098e-016& -0.0648568& L\\ 32.311& 2.49106e-015& 33.4847& L\\ & 1.83729e-015& 0.334174& L\\ & & 4.41466& \end{array}\right]$

Afterwards, utilize the humorous basis function of two and three dimensions ball, calculate conversion factor.In figure below, the upper left side matrix is the humorous basis function of ball under two-dimensional case, and the lower left matrix is the humorous basis function of ball under three-dimensional situation, the conversion factor that right-hand matrix is served as reasons and both calculated.

Recycle at last the humorous feature of the three-dimensional ball that obtains previously and conversion factor, can calculate the humorous feature of two-sphere.

$\left[\begin{array}{cccc}& & -2.27326& \\ & -9.30704e-017& -0.759054& L\\ 32.311& 72.7984& 55.9754& L\\ & 1.77207e-015& -4.15861& L\\ & & 2.77679& \end{array}\right]$

Utilize the humorous feature of two-sphere, can calculate the figure of drop shadow effect on the right in Fig. 3.

As shown in Figure 4, the stratified sampling schematic diagram in three-dimensional polar coordinates situation: three-dimensional polar coordinates by position angle φ ∈ [0,2 π) and polar angle θ ∈ [0, π) form, be expressed as respectively ordinate and horizontal ordinate, get N point on each coordinate, therefore θ is arranged respectively _iWith

By both jointly consisting of polar coordinates

The invention provides a kind of thinking and method of output intent of two-dimensional projection of the three-dimensional model based on the humorous conversion of ball; method and the approach of this technical scheme of specific implementation are a lot; the above is only the preferred embodiment of the present invention; should be understood that; for those skilled in the art; under the prerequisite that does not break away from the principle of the invention, can also make some improvements and modifications, these improvements and modifications also should be considered as protection scope of the present invention.In the present embodiment not clear and definite each ingredient all available prior art realized.

Claims (3)

1. the output intent based on the two-dimensional projection of the computer animation three-dimensional model of the humorous conversion of ball, is characterized in that, comprises the following steps:

Step 1, the three-dimensional model S of projection is treated in input, the three-dimensional model S that inputs is comprised of one group of tri patch, comprises three apex coordinates in tri patch;

Step 2, then the center-of-mass coordinate of Calculation of Three Dimensional model standardizes to three-dimensional model;

Step 3 is carried out stratified sampling to three-dimensional model;

Step 4, the humorous feature of the ball of Calculation of Three Dimensional model, and calculate conversion factor by sampled point;

Step 5, the humorous feature of two-sphere after the calculating projection;

Step 6, feature humorous according to two-sphere, the two dimensional image after the calculating projection;

In step 2, the computing method of the center-of-mass coordinate C of three-dimensional model (x, y, z) are:

$C(x,y,z)=\frac{\underset{i=1}{\overset{K\left(S\right)}{\mathrm{\&Sigma;}}}(w_i\&times;C_i(x_i,y_i,z_i))}{\underset{i=1}{\overset{K\left(S\right)}{\mathrm{\&Sigma;}}}{w}_i},$

Wherein K (S) is three-dimensional model S intermediate cam dough sheet sum, w _iThe area of i tri patch in three-dimensional model S, C _i(x _i, y _i, z _i) center-of-mass coordinate of single tri patch i;

In step 2, three-dimensional model is standardized, comprises following steps:

The translation three-dimensional model makes the barycenter C (x, y, z) of three-dimensional model overlap with the initial point of coordinate system;

Radially ultimate range with three-dimensional model normalizes to three-dimensional model in unit ball:

P(x,y,z)=σ(P(x,y,z)-C(x,y,z))，

Wherein, σ represents the peaked inverse of distance of the barycenter of three-dimensional model that three-dimensional model is had a few,

σ=1/max (|| P (x, y, z)-C (x, y, z) ||), some P (x, y, z) is the point on three-dimensional model surface;

In step 3, three-dimensional model is carried out stratified sampling, comprises following steps:

The spheroid polar coordinates are sampled:

The spheroid polar coordinates represent by position angle φ and polar angle θ, 0≤θ＜π wherein, 0≤φ＜2 π;

In the unit ball polar coordinates, carry out as follows stratified sampling:

${\mathrm{\&theta;}}_i=(i+0.5)\frac{\mathrm{\&pi;}}{N},$

${\mathrm{\&phi;}}_j=(j+0.5)\frac{2\mathrm{\&pi;}}{N},$

Wherein, 0≤i, j＜N, N are the sampled point number on each direction, θ _iRepresent i polar angle, φ _jRepresent j position angle, (θ _i, φ _j) jointly consist of a point in polar coordinates; I and j represent respectively the desired value of polar coordinates down-sampling, and often a pair of (i, j) determines a sampled point, and total sampled point number is N*N;

(the sin θ from the three-dimensional model barycenter along direction _iCos φ _j, sin θ _iSin φ _j, cos θ _i) ray and the intersection point of three-dimensional model, be the sampled point of three-dimensional model stratified sampling;

In step 4, the humorous feature of the ball of Calculation of Three Dimensional model comprises following steps:

Get polar coordinates (θ, φ), calculate the spheric function f (θ, φ) of this sampled point: the radial function of getting the three-dimensional model sample point is spheric function:

f(θ,φ)=d(C,P)；

Wherein, some P represents the sampled point by the definite three-dimensional model surface of polar coordinates (θ, φ); Point C is the barycenter of three-dimensional model; D represents Euclidean distance;

Calculated the basis function of the humorous conversion of ball by polar coordinates (θ, φ)

Wherein, the humorous transform-based function of real number ball

Calculate according to following formula:

$Y_l^m(\mathrm{\&theta;},\mathrm{\&phi;})=\left\{\begin{array}{cc}\sqrt{2}{K}_l^m\cos \left(\mathrm{m\&phi;}\right)P_l^m\left(\cos \mathrm{\&theta;}\right),& m>0\\ \sqrt{2}{K}_l^m\sin (-\mathrm{m\&phi;})P_l^{-m}\left(\cos \mathrm{\&theta;}\right),& m<0\\ K_l^0{P}_l^0\left(\cos \mathrm{\&theta;}\right),& m=0\end{array}\right.$

Wherein,

Be zoom factor,

Be associated Legendre polynomial, l is that span is the arbitrary integer more than or equal to 0, and m is calculating parameter ,-l≤m≤l;

Calculate the coefficient of the humorous conversion of ball according to spheric function and the humorous transform-based function of real number ball

$a_l^m={\&Integral;}_0^{2\mathrm{\&pi;}}{\&Integral;}_0^{\mathrm{\&pi;}}\sin \left(\mathrm{\&theta;}\right)f(\mathrm{\&theta;},\mathrm{\&phi;})Y_l^m(\mathrm{\&theta;},\mathrm{\&phi;})\mathrm{d\&theta;d\&phi;},$

Adopt the monte carlo integration method to obtain to following formula:

$a_l^m=\frac{4\mathrm{\&pi;}}{N}\underset{\mathrm{\&theta;},\mathrm{\&phi;}}{\mathrm{\&Sigma;}}f(\mathrm{\&theta;},\mathrm{\&phi;})Y_l^m(\mathrm{\&theta;},\mathrm{\&phi;});$

In step 4, calculate conversion factor by sampled point and comprise following steps:

By point (θ, φ, r) determine three-dimensional polar coordinates, r is by polar coordinates (θ, φ) sampled point on definite three-dimensional model surface to the distance of three-dimensional model barycenter, is determined the two dimensional surface polar coordinates by point (r, φ), change by cartesian coordinate system, sampled point carried out conversion between following polar coordinates:

(θ,φ)→(cosθ,φ)，

Utilize the associated Legendre polynomial under three-dimensional polar coordinate system and two-dimentional polar coordinate system, calculate conversion factor

${\mathrm{\&beta;}}_l^m\&ap;\frac{\underset{r,\mathrm{\&phi;}}{\mathrm{\&Sigma;}}{P}_l^m(r,\mathrm{\&phi;})}{\underset{\mathrm{\&theta;},\mathrm{\&phi;}}{\mathrm{\&Sigma;}}{P}_l^m(\mathrm{\&theta;},\mathrm{\&phi;})},$

Wherein,

Be associated Legendre polynomial, carry out iterative computation according to following formula:

$P_m^m\left(x\right)={(-1)}^m(2m-1)!!{(1-x^2)}^{m/2},$

$P_{m+1}^m\left(x\right)=x(2m+1)P_m^m\left(x\right),$

$(l-m)P_l^m\left(x\right)=x(2l-1)P_{l-1}^m\left(x\right)-(l+m-1)P_{l-2}^m\left(x\right).$

2. the output intent of the two-dimensional projection of a kind of computer animation three-dimensional model based on the humorous conversion of ball according to claim 1, is characterized in that, in step 5, calculates the humorous feature of two-sphere after projection:

Feature humorous according to the ball of described conversion factor and three-dimensional model, the humorous feature of two-sphere after the calculating projection $c_l^m=a_l^m\&CenterDot;{\mathrm{\&beta;}}_l^m.$

3. the output intent of the two-dimensional projection of a kind of computer animation three-dimensional model based on the humorous conversion of ball according to claim 2, is characterized in that, in step 6, calculates the two-dimensional signal after projection, carries out according to following steps:

Utilize the expansion formula of the humorous conversion of ball, the spheric function of Two-dimensional:

$g(r,\mathrm{\&phi;})=\underset{l=0}{\overset{B}{\mathrm{\&Sigma;}}}\underset{m=-l}{\overset{m=l}{\mathrm{\&Sigma;}}}{c}_l^m{H}_l^m(r,\mathrm{\&phi;}),$

Wherein,

Be the humorous basis function of the ball under two dimension, g (r, φ) expression projected image is at the gray-scale value at the represented some place of polar coordinates (r, φ), and B is parameter value;

Carry out the coordinate inverse mapping, the tlv triple that obtains (r, φ, g (r, φ)), utilize the coordinate inverse mapping:

x=rcos(φ)=cos(θ)cos(φ)

y=rsin(φ)=cos(θ)sin(φ)，

Obtain new tlv triple (x, y, g (r, φ)), be the information after three-dimensional model is transformed into two dimensional image.

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