CN104778731A - Method for solving three-dimensional similarity transformation parameter of model based on fitting plane normal vector direction - Google Patents

Abstract

The invention discloses a method for solving a three-dimensional similarity transformation parameter of a model based on a fitting plane normal vector direction. The method comprises the following steps: firstly, selecting N (N is greater than or equal to 3) pairs of corresponding points in a three-dimensional model and a real model as directional reference points, completing the coordinate centralization of the directional reference points under each model, solving the coordinates of the directional reference points after coordinate centralization, and further calculating a zoom factor lambda between the three-dimensional model and the real model; then obtaining optimal fitting planes of the directional reference points in the three-dimensional model and the real model by using an overall least square method, and performing further calculation so as to obtain the normal vectors of the two optimal fitting planes; finally calculating a three-dimensional rotary matrix R between the three-dimensional model and the real model, and calculating the translation vector t of similarity transformation. The method disclosed by the invention can realize the direct solution of the three-dimensional rotary parameters between the models without providing an initial value in advance, so that the dependence on the initial values of the parameters is reduced, and the method is suitable for the solution of three-dimensional similarity transformation parameters under various rotary conditions.

Description

The method of the three-dimensional similarity transformation parameter of model is asked based on fit Plane normal vector direction

Technical field

The present invention relates to the method asking the three-dimensional similarity transformation parameter of model, particularly ask the method for the three-dimensional similarity transformation parameter of model based on fit Plane normal vector direction, belong to mapping and computer vision field.

Background technology

Often relate between different coordinates in the processing procedure of target three-dimensional fix and changing, as the conversion between partial 3 d coordinate system and earth coordinates, conversion etc. between photogrammetric image space coordinate system and object coordinates system.Conversion between three-dimensional cartesian coordinate system meets rigid body translation, has needed solving of 7 parameters, comprises the rotation parameter of conversion between scale-up factor, coordinate system translation vector and coordinate system.Wherein rotation relationship usually with one by 3 independent parameters form 3 × 3 rotation matrix represent to there is correlativity between 9 elements of rotation matrix, can cause separating unstable according to direct linear solution method.Therefore, how fast, accurately solve between coordinate system rotation relationship become focal point.

The mode at usual employing three-dimensional rotation angle expresses rotation relationship, according to three dimensions similarity transformation relation, the multivariate function of three-dimensional similarity transformation parameter are launched into the once item of Taylor series, by iterative accurate parameters, but more accurate unknown number initial value must be provided, otherwise the situation solving and do not restrain or converge to false solution can be caused.When coordinate system rotation angle is large especially, predict that rotation angle initial value is very difficult exactly.At computer vision field, the rotation relationship method for solving based on unit quaternion is widely used, but it is indefinite to there is parameter physical significance, not easily the problem such as estimated parameter precision.For such problem, be necessary to study a kind of parametric solution method being applicable to the three-dimensional similarity transformation of high rotation angle.

Summary of the invention

Technology of the present invention is dealt with problems and is: overcome the deficiencies in the prior art, provide a kind of method asking the three-dimensional similarity transformation parameter of model based on fit Plane normal vector direction, corresponding directed reference point is chosen in stereoscopic model and true model, and utilize total least square method to simulate best fit plane, the rotation matrix between two best fit planes is tried to achieve in structure by-level face, finally determine the three-dimensional similarity transformation parameter between stereoscopic model and true model, without the need to prior given initial value, get final product the direct solution of the three-dimensional rotation parameter between implementation model, reduce the dependence to initial parameter value, under being applicable to various rotating condition, three-dimensional similarity transformation parameter solves.

Technical solution of the present invention is: the method asking the three-dimensional similarity transformation parameter of model based on fit Plane normal vector direction, and step is as follows:

(1) in stereoscopic model and true model, N number of corresponding point are chosen as directed reference point, and complete the coordinate center of gravity of directed reference point under each self model, try to achieve the coordinate after each directed reference point coordinate center of gravity, described N be more than or equal to 3 natural number;

(2) utilize the coordinate after each directed reference point coordinate center of gravity obtained in step (1), calculate the zoom factor λ between stereoscopic model and true model;

(3) according to the coordinate after each directed reference point coordinate center of gravity obtained in step (1), total least squares method is adopted to carry out plane fitting to the directed reference point in stereoscopic model and true model respectively, obtain the best-fitting plane of directed reference point in stereoscopic model and true model, calculate the normal vector of two best-fitting planes further;

(4) build by-level face, according to the normal vector of two best-fitting planes obtained in step (3), calculate the three-dimensional rotation matrix R between stereoscopic model and true model;

(5) according to the coordinate after each directed reference point coordinate center of gravity obtained in the three-dimensional rotation matrix R between the stereoscopic model determined in the zoom factor λ between the stereoscopic model solved in step (2) and true model, step (4) and true model and step (1), the translation vector t of similarity transformation is calculated.

Build by-level face in described step (4), according to the normal vector of two best-fitting planes obtained in step (3), calculate the three-dimensional rotation matrix R between stereoscopic model and true model, concrete steps are:

(4-1) a by-level face π is built _h, its normal vector is [0,0,1] ^t, make the fit Plane π of stereoscopic model ₁to by-level face π _hrotation angle be then and ω ₁specifically by formula:

ω ₁＝-arcsinq ₁

Provide, wherein, (p ₁, q ₁, r ₁) be π ₁unit normal vector;

(4-2) π is calculated ₁to π _hrotation matrix and by π ₁on each directed reference point rotate to π _h, specifically by formula:

$\left[\begin{array}{c}{X}_{\mathrm{Hi}}\\ Y_{\mathrm{Hi}}\\ Z_{\mathrm{Hi}}\end{array}\right]=R_1^H\left[\begin{array}{c}{X}_{1i}\\ Y_{1i}\\ Z_{1i}\end{array}\right]$

Provide, in formula, X _1i, Y _1iand Z _1irepresent π respectively ₁on the center of gravity coordinate in stereoscopic model of i-th directed reference point, X _hi, Y _hiand Z _hirepresent π respectively ₁on i-th directed reference point by by-level face π _hcoordinate in the three-dimensional coordinate system built, described by by-level face π _hthe XOY plane of the three-dimensional coordinate system built and by-level face π _hoverlap, Z axis positive dirction and by-level face π _hnormal vector direction consistent;

(4-3) π is made _hto the fit Plane π of true model ₂between rotation angle be then π _hto π ₂rotation matrix by formula:

Provide, wherein and ω ₂by formula:

ω ₂＝-arcsinq ₂

Provide, wherein (p ₂, q ₂, r ₂) be π ₂unit normal vector;

κ ₂pass through equation:

$\left[\begin{array}{c}{X}_{2i}\\ Y_{2i}\\ Z_{2i}\end{array}\right]=R_H^2\left[\begin{array}{c}{X}_{\mathrm{Hi}}\\ Y_{\mathrm{Hi}}\\ Z_{\mathrm{Hi}}\end{array}\right]$

Obtain, X in formula _2i, Y _2iand Z _2irepresent π respectively ₂on the center of gravity coordinate in stereoscopic model of i-th directed reference point;

(4-4) π that step (4-2) obtains is utilized ₁to π _hrotation matrix with the π that step (4-3) is determined _hto π ₂rotation matrix calculate the rotation matrix R between stereoscopic model and true model, specifically by formula:

$R=R_1^H{R}_H^2$

Provide.

π in described step (4-2) ₁to π _hrotation matrix specifically by formula:

Provide.

According to the coordinate after each directed reference point coordinate center of gravity obtained in the three-dimensional rotation matrix R between the stereoscopic model determined in the zoom factor λ between the stereoscopic model solved in step (2) and true model, step (4) and true model and step (1) in described step (5), calculate the translation vector t of similarity transformation, specifically by formula:

$t=\left[\begin{array}{c}{\stackrel{\·}{X}}_2\\ {\stackrel{\·}{Y}}_2\\ {\stackrel{\·}{Z}}_2\end{array}\right]-\mathrm{\λR}\left[\begin{array}{c}{\stackrel{\·}{X}}_1\\ {\stackrel{\·}{Y}}_1\\ {\stackrel{\·}{Z}}_1\end{array}\right]$

Provide, wherein with for the barycentric coordinates of reference point directed in stereoscopic model, with for the barycentric coordinates of reference point directed in true model.

The present invention's beneficial effect is compared with prior art:

(1) existing method is difficult to the three-dimensional rotation relation under the condition not providing rotation angle initial value between accurate coordinates computed system, also the statistics of solving precision is unfavorable for, context of methods adopts and builds fit Plane, by the rotation angle between the relation solving model (coordinate system) of planar process vector, without the need to prior given initial value, realize solving of the three-dimensional similarity transformation relation between coordinate system, reduce the dependence to initial parameter value, under being applicable to various rotating condition, three-dimensional similarity transformation parameter solves;

(2) intermediate quantity that context of methods solves has clear and definite physical significance, and its value can be good at reaction solving result and precision.

Accompanying drawing explanation

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

Fig. 2 is the schematic diagram by planar process vector calculation model rotation angle in the present invention.

Embodiment

Below in conjunction with accompanying drawing, the specific embodiment of the present invention is further described in detail.

Technical solution of the present invention can adopt computer software technology to realize automatic operational scheme, describes technical solution of the present invention in detail below in conjunction with drawings and Examples.Be illustrated in figure 1 process flow diagram of the present invention, as can be seen from Figure 1, the method asking the three-dimensional similarity transformation parameter of model based on fit Plane normal vector direction that the present invention proposes, concrete steps are as follows:

(1) in stereoscopic model and true model, N number of corresponding point are chosen as directed reference point, and complete the coordinate center of gravity of directed reference point under each self model, try to achieve the coordinate after each directed reference point coordinate center of gravity, described N be more than or equal to 3 natural number;

During concrete enforcement center of gravity, the first barycentric coordinates of computation model, the formula of employing is as follows:

${\stackrel{\·}{X}}_1=\frac{\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{X}_{1i}}{N},{\stackrel{\·}{Y}}_1=\frac{\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{Y}_{1i}}{N},{\stackrel{\·}{Z}}_1=\frac{\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{Z}_{1i}}{N}$

${\stackrel{\·}{X}}_2=\frac{\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{X}_{2i}}{N},{\stackrel{\·}{Y}}_2=\frac{\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{Y}_{2i}}{N},{\stackrel{\·}{Z}}_2=\frac{\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{Z}_{2i}}{N}$

Wherein the focus point coordinate of directed reference point under being respectively stereoscopic model and true model, (X _1i, Y _1i, Z _1i), (X _2i, Y _2i, Z _2i) be respectively the three-dimensional coordinate of stereoscopic model and lower i-th the directed reference point of true model;

The process of model points coordinate center of gravity adopts following formulae discovery:

${\stackrel{\‾}{X}}_{1i}=X_{1i}-{\stackrel{\·}{X}}_1,{\stackrel{\‾}{Y}}_{1i}=Y_{1i}-{\stackrel{\·}{Y}}_1,{\stackrel{\‾}{Z}}_{1i}=Z_{1i}-{\stackrel{\·}{Z}}_1$

${\stackrel{\‾}{X}}_{2i}=X_{2i}-{\stackrel{\·}{X}}_2,{\stackrel{\‾}{Y}}_{2i}=Y_{2i}-{\stackrel{\·}{Y}}_2,{\stackrel{\‾}{Z}}_{2i}=Z_{2i}-{\stackrel{\·}{Z}}_2$

Wherein be respectively the model orientation reference point coordinate after center of gravity process under stereoscopic model and true model.

(2) utilize the coordinate after each directed reference point coordinate center of gravity obtained in step (1), calculate the zoom factor λ between stereoscopic model and true model; Specifically by formula:

$\mathrm{\λ}=\sqrt{\frac{\underset{i=1}{\overset{n}{\mathrm{\Σ}}}({\stackrel{\‾}{X}}_{2i}^2+{\stackrel{\‾}{Y}}_{2i}^2+{\stackrel{\‾}{Z}}_{2i}^2)}{\underset{i=1}{\overset{n}{\mathrm{\Σ}}}({\stackrel{\‾}{X}}_{1i}^2+{\stackrel{\‾}{Y}}_{1i}^2+{\stackrel{\‾}{Z}}_{1i}^2)}}$

Provide.

(3) according to the coordinate after each directed reference point coordinate center of gravity obtained in step (1), total least squares method is adopted to carry out plane fitting to the directed reference point in stereoscopic model and true model respectively, obtain the best-fitting plane of directed reference point in stereoscopic model and true model, calculate the normal vector of two best-fitting planes further;

The step of reference point plane fitting is as follows:

First suppose that plane equation is:

Z＝aX+bY+d

Wherein (X, Y, Z) is reference point coordinate, and a, b, d are plane parameter, are solved and can be solved a, b, d by the least square of linear equation;

Then based on the thought of total least square TLS, the distance function of point to plane is built:

$\mathrm{dist}=\frac{\mathrm{AX}+\mathrm{BY}+\mathrm{CZ}+D}{\sqrt{A^2+B^2+C^2}}=0$

Taking A into account ²+ B ²+ C ²under the constraint condition of=1, with for unknown number initial value carries out the indirect adjustment of Problem with Some Constrained Conditions, complete the Exact Solution of fit Plane parameter, wherein $T=\sqrt{a^2+b^2+1}.$

Finally calculate normalization method vector n=[A, B, the C] of corresponding flat ^t.

(4) build by-level face, according to the normal vector of two best-fitting planes obtained in step (3), calculate the three-dimensional rotation matrix R between stereoscopic model and true model; Concrete steps are:

(4-1) a by-level face π is built _h, specifically see Fig. 2, its normal vector is [0,0,1] ^t, make the fit Plane π of stereoscopic model ₁to by-level face π _hrotation angle be then and ω ₁specifically by formula:

ω ₁＝-arcsinq ₁

Provide, wherein, (p ₁, q ₁, r ₁) be π ₁unit normal vector;

(4-2) π is calculated ₁to π _hrotation matrix and by π ₁on each directed reference point rotate to π _h, specifically by formula:

$\left[\begin{array}{c}{X}_{\mathrm{Hi}}\\ Y_{\mathrm{Hi}}\\ Z_{\mathrm{Hi}}\end{array}\right]=R_1^H\left[\begin{array}{c}{X}_{1i}\\ Y_{1i}\\ Z_{1i}\end{array}\right]$

Provide, in formula, X _1i, Y _1iand Z _1irepresent π respectively ₁on the center of gravity coordinate in stereoscopic model of i-th directed reference point, X _hi, Y _hiand Z _hirepresent π respectively ₁on i-th directed reference point by by-level face π _hcoordinate in the three-dimensional coordinate system built, described by by-level face π _hthe XOY plane of the three-dimensional coordinate system built and by-level face π _hoverlap, Z axis positive dirction and by-level face π _hnormal vector direction consistent;

(4-3) π is made _hto the fit Plane π of true model ₂between rotation angle be then π _hto π ₂rotation matrix by formula:

Provide, wherein and ω ₂by formula:

ω ₂＝-arcsinq ₂

Provide, wherein (p ₂, q ₂, r ₂) be π ₂unit normal vector;

κ ₂pass through equation:

$\left[\begin{array}{c}{X}_{2i}\\ Y_{2i}\\ Z_{2i}\end{array}\right]=R_H^2\left[\begin{array}{c}{X}_{\mathrm{Hi}}\\ Y_{\mathrm{Hi}}\\ Z_{\mathrm{Hi}}\end{array}\right]$

Obtain, X in formula _2i, Y _2iand Z _2irepresent π respectively ₂on the center of gravity coordinate in stereoscopic model of i-th directed reference point;

Solving rotation matrix time, by rotation matrix be 3 × 3 matrixes, but its independent parameter only have 3, so there is strong correlation between rotation matrix 9 elements, if direct matrix carries out linear solution, can cause and separate instability, thus cause solving failure, therefore need 3 independent parameters of trying to achieve matrix respectively, and then determine rotation matrix value.

(4-4) π that step (4-2) obtains is utilized ₁to π _hrotation matrix with the π that step (4-3) is determined _hto π ₂rotation matrix calculate the rotation matrix R between stereoscopic model and true model, specifically by formula:

$R=R_1^H{R}_H^2$

Provide.

(5) according to the coordinate after each directed reference point coordinate center of gravity obtained in the three-dimensional rotation matrix R between the stereoscopic model determined in the zoom factor λ between the stereoscopic model solved in step (2) and true model, step (4) and true model and step (1), calculate the translation vector t of similarity transformation, specifically by formula:

$t=\left[\begin{array}{c}{\stackrel{\·}{X}}_2\\ {\stackrel{\·}{Y}}_2\\ {\stackrel{\·}{Z}}_2\end{array}\right]-\mathrm{\λR}\left[\begin{array}{c}{\stackrel{\·}{X}}_1\\ {\stackrel{\·}{Y}}_1\\ {\stackrel{\·}{Z}}_1\end{array}\right]$

Provide, wherein with for the barycentric coordinates of reference point directed in stereoscopic model, with for the barycentric coordinates of reference point directed in true model.

The method asking the three-dimensional similarity transformation parameter of model based on fit Plane normal vector direction that the present invention proposes, can be applicable to the processing procedure relating to space three-dimensional conversion and ordinate transform, comprise the space coordinates conversion in geodetic surveying field, the Image model absolute orientation in photogrammetric field and the model conversion of computer vision field and local coordinate system conversion etc.

The content be not described in detail in instructions of the present invention belongs to the known technology of professional and technical personnel in the field.

Claims (4)

1. ask the method for the three-dimensional similarity transformation parameter of model based on fit Plane normal vector direction, it is characterized in that step is as follows:

(1) in stereoscopic model and true model, N is chosen to corresponding point as directed reference point, and complete the coordinate center of gravity of directed reference point under each self model, try to achieve the coordinate after each directed reference point coordinate center of gravity, described N be more than or equal to 3 natural number;

(2) utilize the coordinate after each directed reference point coordinate center of gravity obtained in step (1), calculate the zoom factor λ between stereoscopic model and true model;

(3) according to the coordinate after each directed reference point coordinate center of gravity obtained in step (1), total least squares method is adopted to carry out plane fitting to the directed reference point in stereoscopic model and true model respectively, obtain the best-fitting plane of directed reference point in stereoscopic model and true model, calculate the normal vector of two best-fitting planes further;

(4) build by-level face, according to the normal vector of two best-fitting planes obtained in step (3), calculate the three-dimensional rotation matrix R between stereoscopic model and true model;

(5) according to the coordinate after each directed reference point coordinate center of gravity obtained in the three-dimensional rotation matrix R between the stereoscopic model determined in the zoom factor λ between the stereoscopic model solved in step (2) and true model, step (4) and true model and step (1), the translation vector t of similarity transformation is calculated.

2. the computing method of the three-dimensional similarity transformation parameter of model are asked as claimed in claim 1 based on fit Plane normal vector direction, it is characterized in that: in described step (4), build by-level face, according to the normal vector of two best-fitting planes obtained in step (3), calculate the three-dimensional rotation matrix R between stereoscopic model and true model, concrete steps are:

(4-1) a by-level face π is built _h, its normal vector is [0,0,1] ^t, make the fit Plane π of stereoscopic model ₁to by-level face π _hrotation angle be then and ω ₁specifically by formula:

ω ₁＝-arcsin q ₁

Provide, wherein, (p ₁, q ₁, r ₁) be π ₁unit normal vector;

(4-2) π is calculated ₁to π _hrotation matrix and by π ₁on each directed reference point rotate to π _h, specifically by formula:

$\left[\begin{array}{c}{X}_{\mathrm{Hi}}\\ Y_{\mathrm{Hi}}\\ Z_{\mathrm{Hi}}\end{array}\right]=R_1^H\left[\begin{array}{c}{X}_{1i}\\ Y_{1i}\\ Z_{1i}\end{array}\right]$

Provide, in formula, X _1i, Y _1iand Z _1irepresent π respectively ₁on the center of gravity coordinate in stereoscopic model of i-th directed reference point, X _hi, Y _hiand Z _hirepresent π respectively ₁on i-th directed reference point by by-level face π _hcoordinate in the three-dimensional coordinate system built, described by by-level face π _hthe XOY plane of the three-dimensional coordinate system built and by-level face π _hoverlap, Z axis positive dirction and by-level face π _hnormal vector direction consistent;

(4-3) π is made _hto the fit Plane π of true model ₂between rotation angle be then π _hto π ₂rotation matrix by formula:

Provide, wherein and ω ₂by formula:

ω ₂＝-arcsin q ₂

Provide, wherein (p ₂, q ₂, r ₂) be π ₂unit normal vector;

κ ₂pass through equation:

$\left[\begin{array}{c}{X}_{2i}\\ Y_{2i}\\ Z_{2i}\end{array}\right]=R_H^2\left[\begin{array}{c}{X}_{\mathrm{Hi}}\\ Y_{\mathrm{Hi}}\\ Z_{\mathrm{Hi}}\end{array}\right]$

Obtain, X in formula _2i, Y _2iand Z _2irepresent π respectively ₂on the center of gravity coordinate in stereoscopic model of i-th directed reference point;

(4-4) π that step (4-2) obtains is utilized ₁to π _hrotation matrix with the π that step (4-3) is determined _hto π ₂rotation matrix calculate the rotation matrix R between stereoscopic model and true model, specifically by formula:

$R=R_1^H{R}_H^2$

Provide.

3. the computing method asking the three-dimensional similarity transformation parameter of model based on fit Plane normal vector direction according to claim 2, is characterized in that: π in described step (4-2) ₁to π _hrotation matrix specifically by formula:

Provide.

4. the computing method asking the three-dimensional similarity transformation parameter of model based on fit Plane normal vector direction according to claim 1, it is characterized in that: according to the coordinate after each directed reference point coordinate center of gravity obtained in the three-dimensional rotation matrix R between the stereoscopic model determined in the zoom factor λ between the stereoscopic model solved in step (2) and true model, step (4) and true model and step (1) in described step (5), calculate the translation vector t of similarity transformation, specifically by formula:

$t=\left[\begin{array}{c}{\stackrel{\·}{X}}_2\\ {\stackrel{\·}{Y}}_2\\ {\stackrel{\·}{Z}}_2\end{array}\right]-\mathrm{\λR}\left[\begin{array}{c}{\stackrel{\·}{X}}_1\\ {\stackrel{\·}{Y}}_1\\ {\stackrel{\·}{Z}}_1\end{array}\right]$

Provide, wherein with for the barycentric coordinates of reference point directed in stereoscopic model, with for the barycentric coordinates of reference point directed in true model.