CN102411794A

CN102411794A - Output method of two-dimensional (2D) projection of three-dimensional (3D) model based on spherical harmonic transform

Info

Publication number: CN102411794A
Application number: CN2011102144846A
Authority: CN
Inventors: 路通; 高荣军
Original assignee: Nanjing University
Current assignee: Nanjing University
Priority date: 2011-07-29
Filing date: 2011-07-29
Publication date: 2012-04-11
Anticipated expiration: 2031-07-29
Also published as: CN102411794B

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

Output method of two-dimensional projection of three-dimensional model based on spherical harmonic transformation

Technical Field

The invention relates to a method for projecting a three-dimensional model, in particular to a method for outputting a two-dimensional projection of the three-dimensional model by utilizing a spherical harmonic transformation method aiming at the three-dimensional model only containing geometric information.

Background

At present, three-dimensional models have been widely applied to computer animation, games, virtual reality and other aspects, and meanwhile, the projection of the three-dimensional models represents the visual images of the three-dimensional models and is also widely applied to various fields.

The existing three-dimensional model projection method is parallel projection and perspective projection, and is a calculation method of geometric projection. Projection rays of the perspective projection are all emitted from a common point, and the parallel projection is a special case of the perspective projection. The visual images obtained by the two projection modes are commonly used in a three-dimensional model visual-based retrieval method, but the two projection modes have a common defect: different projection angles result in different visual images. Therefore, in order to obtain more information of the model visual image in the three-dimensional model search, it is generally necessary to set different angles to obtain visual image information in different directions of the three-dimensional model.

Disclosure of Invention

The purpose of the invention is as follows: the invention provides an output method of two-dimensional projection of a three-dimensional model based on spherical harmonic transformation, aiming at the defects of the prior art.

In order to solve the technical problem, the invention discloses an output method of two-dimensional projection of a three-dimensional model based on spherical harmonic transformation, which comprises the following steps:

inputting a three-dimensional model to be projected, wherein the input three-dimensional model consists of a group of triangular patches, and the triangular patches comprise space point coordinates and three vertex coordinates of the triangular patches;

step two, calculating the centroid coordinate of the three-dimensional model, and then normalizing the three-dimensional model;

step three, carrying out layered sampling on the three-dimensional model, and using a Monte Carlo integral method when the spherical harmonic characteristics of the three-dimensional model are conveniently calculated;

step four, according to the idea of spherical harmonic transformation, a Monte Carlo integral method is sampled to calculate the spherical harmonic characteristics of the three-dimensional model; meanwhile, calculating a conversion factor by sampling points;

step five, calculating the two-dimensional spherical harmonic characteristics after projection according to the conversion factors and the three-dimensional spherical harmonic characteristics;

and sixthly, calculating the related information after projection according to the two-dimensional spherical harmonic characteristics.

In the second step of the invention, the method for calculating the centroid coordinate C (x, y, z) of the three-dimensional model comprises the following steps:

where K (S) is the total number of triangular patches in the three-dimensional model S, w_iIs the area of the ith triangular patch in S, C_i(x_i，y_i，z_i) Is the centroid coordinate of the triangular patch i.

In the second step of the invention, the three-dimensional model is normalized, which comprises the following steps:

translating the three-dimensional model such that a centroid C (x, y, z) of the three-dimensional model coincides with an origin of the coordinate system;

the three-dimensional model is then normalized to within a unit sphere by the radial maximum distance of the three-dimensional model.

P(x，y，z)＝σ(P(x，y，z)-C(x，y，z))，

Where σ represents the reciprocal of the maximum of the distances of all points of the three-dimensional model to the centroid of the three-dimensional model, σ is 1/max (| P (x, y, z) -C (x, y, z) |), and point P (x, y, z) is a point of the three-dimensional model surface. And then, the centroid of the three-dimensional model is the origin of coordinates.

In the third step of the invention, the three-dimensional model is subjected to layered sampling, and the method comprises the following steps:

in step two, after the three-dimensional model is normalized to be in a unit sphere, the polar coordinates of the sphere need to be sampled:

the polar coordinates of the sphere are defined by the azimuth angle phi and the polar angle theta,

0≤θ＜π，

0≤φ＜2π，

in unit sphere polar coordinates, layered sampling is carried out according to the following formula:

wherein i is more than or equal to 0, j is less than N, N is the number of sampling points in each direction, and theta_iWhich represents the i-th polar angle,

it represents the j-th azimuth angle,

together forming a point in polar coordinates; i and j are divided intoAnd respectively representing index values of polar coordinate down-sampling, wherein each pair (i, j) determines a sampling point, and the total number of the sampling points is N.

From the three-dimensional model centroid, i.e. origin of coordinates, in the direction (sin θ)_i cosφ_j，sinθ_i sinφ_j，cosθ_i) The intersection point of the ray and the three-dimensional model is the sampling point of the three-dimensional model for layered sampling.

In the fourth step, the three-dimensional spherical harmonic characteristics of the three-dimensional model are calculated, and the method comprises the following steps:

taking a sampling point (theta, phi), and calculating a spherical function f (theta, phi) of the sampling point: taking a radial function at a sampling point of the three-dimensional model as a spherical function

Wherein the point P represents a sampling point of the three-dimensional model surface determined by the polar coordinates (θ, φ); point C is the centroid of the three-dimensional model at that time, i.e. the origin of coordinates; d represents the euclidean distance;

secondly, the basis functions of the spherical harmonic transformation are calculated from the coordinates (theta, phi)Here, only real spherical harmonic transformation basis functions are considered

The method is carried out according to the following formula:

wherein,

in order to be a scaling factor, the scaling factor,

is a conjunctive Legendre polynomial, the value range of l is an integer which is more than or equal to 0, and m is more than or equal to l and less than or equal to l;

finally, the coefficients of the spherical harmonic transformation are calculated from the spherical function and the spherical harmonic transformation basis function

<math> <mrow> <msubsup> <mi>a</mi> <mi>l</mi> <mi>m</mi> </msubsup> <mo>=</mo> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mi>sin</mi> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>θ</mi> <mo>,</mo> <mi>φ</mi> <mo>)</mo> </mrow> <msubsup> <mi>Y</mi> <mi>l</mi> <mi>m</mi> </msubsup> <mrow> <mo>(</mo> <mi>θ</mi> <mo>,</mo> <mi>φ</mi> <mo>)</mo> </mrow> <mi>dθdφ</mi> <mo>,</mo> </mrow> </math>

The numerical calculation method using the monte carlo integral for the above equation is as follows:

and selecting a preset experiment parameter B, wherein the value of B is 7, and only calculating the condition that l is more than or equal to 0 and less than or equal to B to obtain the spherical harmonic characteristic representation of the three-dimensional model S.

In the fourth step, calculating the conversion factor of the three-dimensional model comprises the following steps:

first, because three-dimensional polar coordinates can be defined by points

Determined as r is a polar coordinateDetermining the distance from a sampling point on the surface of the three-dimensional model to the centroid of the three-dimensional model, wherein the polar coordinates of the two-dimensional plane can be determined by the point (r, phi), so that through the conversion between the Cartesian coordinate systems, the following polar coordinates are converted for the sampling point:

(θ，φ)→(cosθ，φ)，

calculating conversion factor by using three-dimensional polar coordinate system and conjoint Legendre polynomial under two-dimensional polar coordinate system

Wherein,

for the associated legendre polynomial, an iterative calculation is performed according to the following formula:

P_{m}^{m} (x) = {(- 1)}^{m} (2 m - 1)!! {(1 - x^{2})}^{m / 2},

P_{m + 1}^{m} (x) = x (2 m + 1) P_{m}^{m} (x),

(l - m) P_{l}^{m} (x) = x (2 l - 1) P_{l - 1}^{m} (x) - (l + m - 1) P_{l - 2}^{m} (x),

in the fifth step, calculating the two-dimensional spherical harmonic characteristics after projection, and performing the following steps:

calculating the two-dimensional spherical harmonic characteristics after projection by the three-dimensional spherical harmonic characteristics and the conversion factors obtained by calculation in the step four,

in the sixth step, the projected two-dimensional information is calculated according to the following steps:

firstly, a two-dimensional spherical function is calculated by using an expansion formula of spherical harmonic transformation:

wherein,is a spherical harmonic basis function in the two-dimensional case.

Next, coordinate inverse mapping is performed, and the three tuples (r, phi, g (r,φ))，

representing projected image dots

The gray value of (b) is mapped by using coordinate inverse:

x＝rcos(φ)＝cos(θ)cos(φ)

y＝rsin(φ)＝cos(θ)sin(φ)，

and obtaining a new triplet (x, y, g (r, phi)) which is the information after the three-dimensional model is converted into the two-dimensional image.

Has the advantages that: the invention relates to an output method of two-dimensional projection of a three-dimensional model based on spherical harmonic transformation, which embodies global information of the three-dimensional model due to the three-dimensional model projection based on the spherical harmonic transformation and does not need other geometric projection methods to set projection direction parameters. The visual image of the three-dimensional model output by projection is used for searching the three-dimensional model, and the information of the three-dimensional model can be more accurately and more completely expressed, so that the performance of searching the three-dimensional model is improved. Moreover, the invention adopts the spherical harmonic transformation technology, ensures the rotation invariance characteristic of the three-dimensional model retrieval, namely the three-dimensional model which rotates can obtain the same two-dimensional projection visual image, thereby having better retrieval effect.

Drawings

The foregoing and/or other advantages of the invention will become further apparent from the following detailed description of the invention when taken in conjunction with the accompanying drawings.

FIG. 1 is a flow chart of the present invention.

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

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

Fig. 4 is a schematic diagram of hierarchical sampling in three-dimensional polar coordinates.

Detailed Description

The invention discloses an output method of a two-dimensional projection of a three-dimensional model based on spherical harmonic transformation.

The invention is explained in more detail below with reference to the drawings:

as shown in fig. 1, the present invention comprises the steps of:

step 1, inputting a three-dimensional model to be projected, wherein the input three-dimensional model consists of a group of triangular patches, and the geometric information extracted from the three-dimensional model comprises which three vertexes each triangular patch consists of and the geometric coordinates of each vertex.

And 2, calculating the centroid coordinate of the three-dimensional model, and then normalizing the three-dimensional model.

In step 2, the method for calculating the centroid coordinate C (x, y, z) of the three-dimensional model is as follows:

where K (S) is the total number of triangular patches in the three-dimensional model S, w_iIs the area of the ith triangular patch in S, C_i(x_i，y_i，z_i) The centroid coordinates of a single triangular patch i.

Normalizing the three-dimensional model, comprising the steps of:

translating the three-dimensional model to make the centroid C (x, y, z) of the three-dimensional model coincide with the origin of the coordinate system, namely subtracting the calculated centroid coordinates C (x, y, z), P (x, y, z) -C (x, y, z) from the coordinates of each vertex of the three-dimensional model;

the three-dimensional model is then normalized within the unit sphere, i.e., each vertex coordinate after translation is multiplied by a normalization parameter σ, as represented by:

P(x，y，z)＝σ(P(x，y，z)-C(x，y，z))，

where σ represents the reciprocal of the maximum of the distances of all points of the three-dimensional model to the centroid of the three-dimensional model, σ is 1/max (| P (x, y, z) -C (x, y, z) |), and point P (x, y, z) is a point of the three-dimensional model surface.

And 3, carrying out layered sampling on the three-dimensional model.

In step 3, the three-dimensional model is subjected to layered sampling, and the method comprises the following steps:

first, the polar coordinates of the sphere are sampled:

the sphere polar coordinate system can be composed of an azimuth angle phi and a polar angle theta, wherein theta is more than or equal to 0 and less than pi, and phi is more than or equal to 0 and less than 2 pi;

wherein i is more than or equal to 0, j is less than N, N is the number of sampling points in each direction, and theta_iWhich represents the i-th polar angle,it represents the j-th azimuth angle,together forming a point in polar coordinates; i and j respectively represent index values of polar coordinate down-sampling, each pair (i, j) determines a sampling point, and the total number of the sampling points is N;

next, sampling coordinates in the unit sphere polar coordinate system are acquired

Then, the three-dimensional model is converted into a Cartesian coordinate system of the three-dimensional model, namely, the three-dimensional model is from the centroid along the directionAnd (4) starting from the ray, calculating the intersection point of the ray and the surface patch of the surface of the three-dimensional model, namely the sampling point of the three-dimensional model for layered sampling.

And 4, calculating the spherical harmonic characteristics of the three-dimensional model by using a Monte Carlo integral method, and calculating a conversion factor by using the sampling points.

In step 4, calculating the spherical harmonic characteristics of the three-dimensional model, comprising the following steps:

according to the step 3, taking the polar coordinates (theta, phi), and calculating the spherical function f (theta, phi) at the sampling point corresponding to the polar coordinates: taking a radial function at a sampling point of the three-dimensional model as a spherical function:

f(θ，φ)＝d(C，P)；

wherein the point P represents a sampling point of the three-dimensional model surface determined by the polar coordinates (θ, φ); point C is the centroid of the three-dimensional model; d represents the euclidean distance;

calculating the basis functions of the spherical harmonic transformation from the polar coordinates (theta, phi)

Wherein the hair isThe real spherical harmonic transformation basis function is adopted and still recorded as

Calculated according to the following formula:

wherein,

in order to be a scaling factor, the scaling factor,

for the associated Legendre polynomial, l is an integer with the value range of more than or equal to 0, m is more than or equal to l, a parameter value B of experimental operation can be preset, and only the spherical harmonic basis function under the condition that l is more than or equal to 0 and less than B is calculated;

calculating coefficients of spherical harmonic transforms from spherical functions and real spherical harmonic transform basis functions

In the case of determining each pair (l, m), the corresponding spherical harmonic characteristic is calculated by the numerical calculation method of the Monte Carlo integral method using the above formula

The Monte Carlo integration method corresponding to the above equation is as follows:

accordingly, a series of spherical harmonic characteristics of the three-dimensional model can be calculatedIn total B²And (4) respectively.

In step 4, calculating the conversion factor from the sampling points comprises the following steps:

the distance from the sampling point of the three-dimensional model surface to the three-dimensional model centroid determined by the three-dimensional polar coordinates (theta, phi, r) and the two-dimensional plane polar coordinates

Through the conversion between the polar coordinates and the cartesian coordinates and the conversion between the cartesian coordinate systems, the following conversion between the polar coordinates can be performed on the sampling points:

(θ，φ)→(cosθ，φ)，

Wherein,

P_{m}^{m} (x) = {(- 1)}^{m} (2 m - 1)!! {(1 - x^{2})}^{m / 2},

P_{m + 1}^{m} (x) = x (2 m + 1) P_{m}^{m} (x),

(l - m) P_{l}^{m} (x) = x (2 l - 1) P_{l - 1}^{m} (x) - (l + m - 1) P_{l - 2}^{m} (x) .

thus, a set of values for the conversion factor for different l and m conditions is obtained.

And 5, calculating the projected two-dimensional spherical harmonic characteristics according to the conversion factors and the spherical harmonic characteristics of the three-dimensional model.

In step 5, calculating the two-dimensional spherical harmonic characteristics after projection: a set of spherical harmonic features of the three-dimensional model calculated in step 3

And the set of conversion factors obtained in step 4

To calculate a set of spherical harmonic features of the projected two-dimensional image

Each of which

Calculated from the following formula,

and 6, calculating the projected two-dimensional information according to the two-dimensional spherical harmonic characteristics.

In step 6, calculating the projected two-dimensional information according to the following steps:

in selecting two-dimensional polar coordinates

Calculating the spherical function at the projection point corresponding to the polar coordinates

Firstly, an expansion formula of spherical harmonic transformation is utilized to calculate a spherical function under the condition of two-dimensional polar coordinates:

wherein,

is a spherical harmonic basis function in two dimensions;

secondly, coordinate inverse mapping is carried out, and the obtained triad (r, phi, g (r, phi)),

representing projected images in polar coordinatesThe gray value of (b) is mapped by using coordinate inverse:

x＝rcos(φ)＝cos(θ)cos(φ)

y＝rsin(φ)＝cos(θ)sin(φ)，

As shown in fig. 2, the cartesian coordinates and polar coordinates of the three-dimensional model are schematically shown: in the whole calculation process, the mutual conversion between the Cartesian coordinates and the polar coordinates is used for multiple times.

The three-dimensional model polar coordinate is composed of an azimuth angle phi epsilon [0, 2 pi) and a polar angle theta [0, pi ]. The Cartesian coordinate system is represented by the x-, y-, and z-axes.

In fig. 2, Γ represents a curved surface in three-dimensional space, and P (x, y, z) represents a point on the curved surface. Through some relationships of trigonometric functions, a polar coordinate to cartesian coordinate conversion formula can be obtained as follows:

polar coordinates are calculated from cartesian coordinates:

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

conversely, cartesian coordinates are calculated from polar coordinates:

x＝rcosφsinθ，

y＝rsinφsinθ，

z＝rcosθ，

as shown in fig. 3, the three-dimensional model projection process is schematically illustrated as follows: converting the input of the three-dimensional model into the representation of the three-dimensional spherical harmonic characteristic, then calculating the conversion factor of the spherical harmonic in two-dimensional and three-dimensional conditions, calculating to obtain the representation of the projected two-dimensional spherical harmonic characteristic, and finally restoring the representation into the projected two-dimensional image.

Fig. 3 shows the specific experimental results (the experimental parameters are 1024 samples, i.e., N is 32, the bandwidth B is 7, and the projection size is 256 × 256 pixels.)

The three-dimensional model (ant) shown on the left side of fig. 3, as an input object, then solves its three-dimensional spherical harmonic features, and writes them in the form of a matrix, as follows:

[\begin{matrix} - 1.23448 \\ - 3.53098 e - 016 & - 0.0648568 & L \\ 32.311 & 2.49106 e - 015 & 33.4847 & L \\ 1.83729 e - 015 & 0.334174 & L \\ 4.41466 \end{matrix}]

the conversion factor is then calculated using the two-dimensional and three-dimensional spherical harmonic basis functions. In the lower graph, the upper left matrix is a spherical harmonic basis function under the two-dimensional condition, the lower left matrix is a spherical harmonic basis function under the three-dimensional condition, and the right matrix is a conversion factor obtained by calculation of the two matrixes.

And finally, calculating to obtain the two-dimensional spherical harmonic characteristic by utilizing the three-dimensional spherical harmonic characteristic and the conversion factor obtained in the previous step.

[\begin{matrix} - 2.27326 \\ - 9.30704 e - 017 & - 0.759054 & L \\ 32.311 & 72.7984 & 55.9754 & L \\ 1.77207 e - 015 & - 4.15861 & L \\ 2.77679 \end{matrix}]

By using the two-dimensional spherical harmonic feature, the projection effect diagram on the right side in fig. 3 can be calculated.

As shown in fig. 4, a schematic diagram of layered sampling in the case of three-dimensional polar coordinates: the three-dimensional polar coordinate is composed of an azimuth angle phi epsilon [0, 2 pi) and a polar angle theta epsilon [0, pi) which are respectively expressed as a vertical coordinate and a horizontal coordinate, N points are taken on each coordinate, and therefore theta is respectively arranged_iAnd

the two components together form a polar coordinate

The present invention provides a method and a system for outputting a two-dimensional projection of a three-dimensional model based on spherical harmonic transformation, and a plurality of methods and ways for implementing the technical solution are provided, and the above description is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, a plurality of improvements and modifications can be made without departing from the principle of the present invention, and these improvements and modifications should also be regarded as the protection scope of the present invention. All the components not specified in the present embodiment can be realized by the prior art.

Claims

1. An output method of a two-dimensional projection of a three-dimensional model based on spherical harmonic transformation, characterized by comprising the following steps:

inputting a three-dimensional model S to be projected, wherein the input three-dimensional model S consists of a group of triangular patches, and the triangular patches comprise three vertex coordinates;

calculating the centroid coordinate of the three-dimensional model, and then normalizing the three-dimensional model;

step three, carrying out layered sampling on the three-dimensional model;

step four, calculating the spherical harmonic characteristics of the three-dimensional model, and calculating a conversion factor by using the sampling points;

step five, calculating the two-dimensional spherical harmonic characteristics after projection;

and step six, calculating the projected two-dimensional information according to the two-dimensional spherical harmonic characteristics.

2. The output method of the two-dimensional projection of the three-dimensional model based on the spherical harmonic transformation as claimed in claim 1, wherein in the second step, the centroid coordinate C (x, y, z) of the three-dimensional model is calculated by:

where K (S) is the total number of triangular patches in the three-dimensional model S, w_iIs the area of the ith triangular patch, C, in the three-dimensional model S_i(x_i，y_i，z_i) The centroid coordinates of a single triangular patch i.

3. The method as claimed in claim 2, wherein the step two of normalizing the three-dimensional model comprises the steps of:

translating the three-dimensional model to translate so that the centroid C (x, y, z) of the three-dimensional model coincides with the origin of the coordinate system;

normalizing the three-dimensional model to be within a unit sphere by the radial maximum distance of the three-dimensional model:

P(x，y，z)＝σ(P(x，y，z)-C(x，y，z))，

4. The method as claimed in claim 3, wherein the step three of hierarchically sampling the three-dimensional model comprises the following steps:

sampling the polar coordinates of the sphere:

the polar coordinates of the sphere are represented by an azimuth angle phi and a polar angle theta, wherein theta is more than or equal to 0 and less than pi, and phi is more than or equal to 0 and less than 2 pi;

it represents the j-th azimuth angle,

together forming a point in polar coordinates; i and i respectively represent index values of polar coordinate down-sampling, each pair (i, j) determines a sampling point, and the total number of the sampling points is N;

from the three-dimensional model centroid in the direction (sin θ i cos φ)_j，sinθ_isinφ_j，cosθ_i) The intersection point of the ray and the three-dimensional model is the sampling point of the three-dimensional model for layered sampling.

5. The method as claimed in claim 4, wherein the step four of calculating the spherical harmonic characteristics of the three-dimensional model comprises the steps of:

taking the polar coordinates (theta, phi), calculating the spherical function f (theta, phi) of the sampling point: taking a radial function at a sampling point of the three-dimensional model as a spherical function:

f(θ，φ)＝d(C，P)；

calculating the basis functions of the spherical harmonic transformation from the polar coordinates (theta, phi)Wherein, real number spherical harmonic transformation basis functionCalculated according to the following formula:

wherein,

in order to be a scaling factor, the scaling factor,

is a conjunctive Legendre polynomial, l is any integer with the value range of more than or equal to 0, m is a calculation parameter, and-l is more than or equal to m and less than or equal to l;

from spherical functions and real spherical harmonic transformsCalculating coefficients of a spherical harmonic transform using basis functions

The above formula is obtained by adopting a Monte Carlo integral method:

6. the method of claim 5, wherein the step four of calculating the conversion factor from the sampling points comprises the steps of:

determining three-dimensional polar coordinates from the points (theta, phi, r), wherein r is the distance from the sampling point of the three-dimensional model surface to the mass center of the three-dimensional model determined by the polar coordinates (theta, phi)

Determining two-dimensional plane polar coordinates, and performing Cartesian coordinate system conversion on the sampling pointsThe following polar coordinates:

(θ，φ)→(cosθ，φ)，

Wherein,

P_{m}^{m} (x) = {(- 1)}^{m} (2 m - 1)!! {(1 - x^{2})}^{m / 2},

P_{m + 1}^{m} (x) = x (2 m + 1) P_{m}^{m} (x),

(l - m) P_{l}^{m} (x) = x (2 l - 1) P_{l - 1}^{m} (x) - (l + m - 1) P_{l - 2}^{m} (x) .

7. the output method of the two-dimensional projection based on the spherical harmonic transformed three-dimensional model as claimed in claim 6, wherein in the fifth step, the projected two-dimensional spherical harmonic features are calculated:

calculating the two-dimensional spherical harmonic characteristics after projection according to the conversion factor and the spherical harmonic characteristics of the three-dimensional model

8. The output method of the two-dimensional projection of the three-dimensional model based on the spherical harmonic transformation as claimed in claim 1, wherein in the sixth step, the two-dimensional information after projection is calculated according to the following steps:

calculating a two-dimensional spherical function by using an expansion formula of spherical harmonic transformation:

wherein,is a spherical harmonic basis function in two dimensions,representing projected images in polar coordinates

The gray value at the represented point, B being the parameter value;

and (3) carrying out coordinate inverse mapping to obtain a triplet (r, phi, g (r, phi)), and using the coordinate inverse mapping:

x＝rcos(φ)＝cos(θ)cos(φ)

y＝rsin(φ)＝cos(θ)sin(φ)，