CN108871284A - Three-dimensional space similarity transformation model parameter based on line feature constraint without initial value method for solving - Google Patents
Three-dimensional space similarity transformation model parameter based on line feature constraint without initial value method for solving Download PDFInfo
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01C—MEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
- G01C11/00—Photogrammetry or videogrammetry, e.g. stereogrammetry; Photographic surveying
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- G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
- G01B11/00—Measuring arrangements characterised by the use of optical techniques
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Abstract
A kind of three-dimensional space similarity transformation model parameter based on line feature constraint without initial value method for solving, be suitable for three-dimensional space imaging field.There are when three groups or three groups or more of homonymous line feature between two coordinate systems in known three-dimensional space, select one of coordinate system as reference, it is under the jurisdiction of in two coordinate systems the seamless fusion of all space three-dimensional data and Unified Expression as target to realize, pass through model inference, it obtains the specific method and step of parametric solution, and then realizes correct, Efficient Solution of another coordinate system relative to the optimum translation parameter between selected reference system.Its step is simple, it is high to convert analyzing efficiency, it can be under the premise of not needing to provide the approximation of conversion parameter of three-dimensional space similarity transformation, the conversion parameter of three-dimensional space similarity transformation is calculated, relieve dependence of the classical iterative method for the conversion parameter approximation of three-dimensional space similarity transformation, high reliablity.
Description
Technical field
The three-dimensional space similarity transformation model parameter based on line feature constraint that the present invention relates to a kind of is solved without initial value
Method is particularly suitable for using in coordinate system transformation in three-dimensional space imaging, belongs to survey field.
Background technique
Currently, three-dimensional space similarity transformation photogrammetric photo absolute orientation, the coordinate system of geodesic survey transformation,
The fields such as the camera Attitude estimation of computer vision and the registration of LiDAR point cloud play an important role.Three-dimensional space phase
It is the feature corresponding relationship sought and established between two coordinate systems like the essence of transformation, is based on spacial similarity transformation model,
If the relationship of relative position two coordinate basis is described in Bursa model realization, resolve between two coordinate basis of description
Seven parameters such as rotation, translation and scaling of relative positional relationship.The selection of feature of the same name can be point feature, line feature and face
Feature, it is existing major part algorithm select the similarity transformation model based on point-like character based on, as in real world more
For common one of feature, effect of the linear feature in spacial similarity transformation fails for a long time with status by due heavy
Depending on being brought not to the subsequent calculating processes such as compared with to find out its cause, being that the space expression form of linear feature is not unique
Just with difficulty.In addition, the common solution of current spatial similarity transformation model parameter is spacial similarity transformation formula using polynary
The Taylor's formula of function is unfolded, and implements linearization process to it, the solution of transformation parameter is realized based on the mode of iteration.So
And its own presence of iterative method is following insufficient：1) it needs to be determined in advance the initial value of transformation parameter, and realizes nonlinear transformation accordingly
The linearisation of model, Initial value choice is improper, then is likely to result in not restraining for result；2) such by the constraint of linearization procedure
Method is substantially only applicable to the case where small angle tower coordinate transform, matches in such as photogrammetric absolute orientation, LiDAR point cloud
Standard etc., if the approximate transform parameter that cannot be provided between Two coordinate system system is not ideal enough as initial value, or the initial value of offer,
The reliability of calculated result will be severely impacted.
Summary of the invention
Goal of the invention：In view of the above technical problems, it provides that a kind of step is simple, and calculation amount is small, resolves Two coordinate system and unite phase
Optimum translation parameter between mutually, realization be under the jurisdiction of in two coordinate systems the seamless fusion of all space three-dimensional data with uniformly
The problem of expression, overcome convergence problem of conventional method during parametric solution and parametric solution process to initial value according to
Rely property problem the three-dimensional space similarity transformation model parameter based on line feature constraint without initial value method for solving.
Summary of the invention：To realize the above-mentioned technical purpose, the similar change of the three-dimensional space of the invention based on line feature constraint
Change the mold shape parameter without initial value method for solving, it is characterized in that：For same tested region, different visual angles is selected to arrange two phases
Adjacent survey station, the surface characteristics LiDAR point cloud of measured target, two adjacent survey stations are collected using two adjacent survey stations
There is overlapping between point cloud, and there are three groups or three groups or more homonymous line features for overlapping region, and one of survey station is selected to make
On the basis of stand, station subject to registration is solved using being completely coincident as constraint condition for feature of the same name using another survey station as station subject to registration
Conversion parameter of the coordinate system relative to base station coordinate system, and the space three-dimensional data in station subject to registration are turned accordingly
Change, thus realize the seamless spliced of two neighboring survey station LiDAR point cloud with merge；
Using the homonymous line feature of extraction as constraint, by the origin coordinate system transform of survey station subject to registration to base station coordinate system
Specific step is as follows for system：
S-1 chooses two adjacent survey stations, there is overlapping between the LiDAR point cloud of two neighboring stations, according to two surveys
It stands the homonymous line feature extracted respectively, i.e. the same straight line line in the LiDAR point cloud information that two survey stations detect
Item, the condition premised on the stringent coincidence for ensuring homonymous line feature that two survey stations extract, building three-dimensional space are similar
The similarity measure of transformation, and indicated in the form of matrix construction；
S-2 constructs optimal turn of solution described based on dual quaterion in the similarity measure of three-dimensional space similarity transformation
Change the mathematical model of parameter；
The homonymous line feature for extracting from two neighboring survey station is substituted into the mathematical model that S-2 is obtained by S-3, is calculated three
The parameter values of dimension space similarity transformation, and three-dimensional space similarity transformation is implemented to the LiDAR point cloud in station subject to registration accordingly, it is real
The unification at existing station and base station coordinate system subject to registration, and then realize base station LiDAR point cloud and station LiDAR point cloud subject to registration
It is seamless spliced with merge.
The building of the similarity measure of the three-dimensional space similarity transformation, steps are as follows：
S-1-1：It is carried out using all linear features that regularization Pl ü cker coordinate pair extracts from base station and station subject to registration
Expression；
S-1-2：Regularization Pl ü cker straight line is sat before and after mode based on dual quaterion description provides spacial similarity transformation
Target mathematic(al) representation；
S-1-3：Based on criterion of least squares, using being completely coincident as item between the feature of the same name of spacial similarity transformation front and back
Part defines and constructs the three-dimensional space similarity transformation similitude that rule-basedization Pl ü cker coordinate describes under line feature constraint
Estimate.
Steps are as follows for the mathematical expression of the regularization Pl ü cker coordinate of any straight line in three dimensions：
S-1-1-1：Any selection is located at the straight line in three-dimensional space, two points on the optional straight lineAnd pointObtain the direction vector of the straight line
S-1-1-2：Calculate the square of the straight line For any one point on straight line,For direction vector；
S-1-1-3：Based on the definition of Pl ü cker rectilinear coordinates, in the form of hexa-atomic groupTo the straight line into
Row expression；
S-1-1-4：Regularization processing is carried out to the Pl ü cker rectilinear coordinates in S-1-1-3, i.e.,：Pl ü cker after regularization
The direction vector of rectilinear coordinates is respectively with squareWithBy the Pl ü cker rectilinear coordinates after regularization
8 dimensions are extended to, i.e.,：In formulaRespectively the quaternary number of the square of the direction vector and straight line of straight line expresses shape
Formula, i.e.,：
It is fixed using being completely coincident as condition between the feature of the same name of spacial similarity transformation front and back based on criterion of least squares
Justice and the similarity measure for constructing three-dimensional space similarity transformation under line feature constraint, its step are as follows：
S-1-3-1：According to the relationship between unit quaternion and spin matrix, the Pl ü of straight line feature in space
Cker coordinateBecome after spacial similarity transformationThe direction vector of the straight lineWithDirectly
The square of lineWithBetween corresponding relationship it is as follows：
It enables：Formula (1) can be further rewritten as：
In formula：For the original orientation vector of linear feature before spacial similarity transformation,For straight line before spacial similarity transformation
The original square of feature,To indicate the unit quaternion rotated,For unit quaternary numberConjugation, μ is zoom factor,
For the quaternary number for indicating translation；
S-1-3-2：Formula (2) is expressed with a matrix type：
Wherein,
Formula (3) is expressed with matrix, can be obtained：
S-1-3-3：Due to the presence of error, and it is based on criterion of least squares, three-dimensional space is similar under line feature constraint
The essence of the similarity measure of transformation, be after spacial similarity transformation homonymous line feature and between difference
Quadratic sum reaches minimum, and matrix construction is：
The mathematical model of the solution optimum translation parameter based on dual quaterion description includes matrix A, rotation quaternary
NumberTranslate quaternary numberWith zoom factor μ；
Mode based on dual quaterion description solves the mathematical model of optimum translation parameter and step is：Utilize extreme value point
The solution of quaternary number corresponding with rotation transformation in spacial similarity transformation is realized in analysis；
To expression formula ∑ f1 2It is decomposed：And it enables：Then：
It is knownAny straight line is in base station coordinate system, station subject to registration respectively in three-dimensional space
Direction vector and square in coordinate system,For dual quaterion corresponding with the linear feature, phase is realized
The registration of adjacent survey station LiDAR point cloud, steps are as follows：
Optimal rotation quaternary numberMeet formulaThe condition for obtaining minimum value, meets simultaneouslyCondition, in summary two factors, building objective function is：λ in formula1It is
Lagrange's multiplier；
Take objective functionAbout variablePartial derivative：It is found that quaternary numberIt is
Corresponding to matrixA feature vector, λ1It is characteristic value corresponding with feature vector；
It considers：Work as λ1When selecting the corresponding feature vector of maximum eigenvalue, become
Change the difference ∑ f between the homonymous line direction vector of front and back1 2=Cl1-λ1Minimum value is obtained, i.e.,：Maximum eigenvalue λ1Corresponding spy
Levy vectorAs required rotation quaternary number.
Construct matrixAnd the corresponding feature vector of maximum eigenvalue of solution matrix AAs
For expressing the rotation quaternary number of rotation transformation；
Wherein,
The solution of quaternary number corresponding with translation transformation and scaling in spacial similarity transformation are realized based on extreme value analysis
The solution of coefficient：
To expression formula ∑ f2 2It is decomposed, can be obtained：
It enables：Cm1=2 ∑ I,
Then ∑ f2 2It can further be expressed as：
Optimal translation quaternary numberMeet formula ∑ f2 2The condition for obtaining minimum value, meets simultaneouslyCondition, it is comprehensive
Above-mentioned two factor is closed, objective function is constructed
λ in formula2For Lagrange's multiplier；
It takes respectivelyAbout quaternary numberWith the partial derivative of zoom factor μ, obtain：
The expression formula of zoom factor μ can be obtained：
Utilize formula：Solve translation quaternary numberIn formula,
It considers：It can obtain
Lagrange coefficient lambda2Expression formula：In turn, it can obtain and translation
Convert corresponding quaternary numberExpression formula：
Beneficial effect：
The scheme that the application uses keeps the expression-form unification of any linear feature in space, describing mode more succinct,
And can be constructed based on Optimum Theory and derive spacial similarity transformation model parameter without initial value method for solving, compared to warp
The iterative method of allusion quotation avoids the linearization procedure of the function of many variables, relieves dependence of the parametric results for iterative initial value, overcomes
Iterative method algorithm instability problem when solving big corner similarity transformation parameter；
In the registration of neighboring stations LiDAR point cloud, using linear feature as the constraint condition being registrated, compared to simple
Based on the LiDAR point cloud registration Algorithm that point-like character matches, effectively enhances the constraint of LiDAR point cloud registration process, reach
Improve the purpose of quality of registration；
This programme is analyzed and the Conversion Relations and mould between Pl ü cker rectilinear coordinates and dual quaterion has been determined
Type describes method；It is sat with the Pl ü cker straight line before and after spacial similarity transformation from the homonymous line feature of two coordinate systems
Mark the equal spacial similarity transformation model for being used as constraint condition, constructing based on the description of Pl ü cker rectilinear coordinates under linear feature；
Using least square basic theories, line feature constraint down space similarity transformation ginseng is realized by the extreme valueization analysis of objective function
Several direct solutions；Basic scheme realizes spacial similarity transformation parametric solution, avoids initial value calculating and thus bring changes
For not convergence problem, regarded for solving such as photogrammetric photo absolute orientation, the transformation of the coordinate system of geodesic survey, computer
Camera Attitude estimation and the registration of LiDAR point cloud of feel etc. have important theory and realistic meaning.
Figure of description
Fig. 1 is flow diagram of the invention.
Fig. 2 is the straight line expression in three-dimensional space of the invention based on Pl ü cker coordinate.
Specific embodiment
The embodiment of the present invention is described further with reference to the accompanying drawing：
As shown in Figure 1, the three-dimensional space similarity transformation model parameter of the invention based on line feature constraint without initial value
Method for solving selects different visual angles to arrange two adjacent survey stations, utilizes two adjacent survey stations for same tested region
The surface characteristics LiDAR point cloud of measured target is collected, there is overlapping, and overlapping region between two adjacent survey site clouds
There are three groups or three groups or more homonymous line features, select one of survey station as the base station of reference, another survey station is made
Station coordinates system subject to registration is solved relative to base station using being completely coincident as constraint condition for feature of the same name for station subject to registration
The conversion parameter of coordinate system, and the space three-dimensional data in station subject to registration are converted accordingly, to realize two neighboring
Survey station LiDAR point cloud it is seamless spliced with merge；
Using the homonymous line feature of extraction as constraint, by the origin coordinate system transform of survey station subject to registration to base station coordinate system
Specific step is as follows for system：
S-1 chooses two adjacent survey stations, there is overlapping between the LiDAR point cloud of two neighboring stations, according to two surveys
It stands the homonymous line feature extracted respectively, i.e. the same straight line line in the LiDAR point cloud information that two survey stations detect
Item, the condition premised on the stringent coincidence for ensuring homonymous line feature that two survey stations extract, building three-dimensional space are similar
The similarity measure of transformation, and indicated in the form of matrix construction；
The building of the similarity measure of three-dimensional space similarity transformation, steps are as follows：
S-1-1：All linear feature carry out tables of two neighboring survey station are extracted from using regularization Pl ü cker coordinate pair
It reaches：
S-1-1-1：Any selection is located at the straight line in three-dimensional space, two points on the optional straight lineAnd pointObtain the direction vector of the straight lineAs shown in Fig. 2,With
S-1-1-2：Utilize expression formulaCalculate separately the direction vector of the linear feature
With square
S-1-1-3：Based on the definition of Pl ü cker rectilinear coordinates, in the form of hexa-atomic groupTo the straight line into
Row expression；For straight line, orthogonality relation is met between direction vector and the square of straight line, i.e.,：
S-1-1-4：Regularization processing is carried out to the Pl ü cker rectilinear coordinates in S-1-1-3, i.e.,：Pl ü cker after regularization
The direction vector of rectilinear coordinates is respectively with squareWithPl ü cker rectilinear coordinates after regularization are expanded
Exhibition is 8 dimensions, i.e.,：In formulaThe respectively quaternary number expression-form of the square of the direction vector and straight line of straight line,
I.e.：
S-1-2：Regularization Pl ü cker is straight before and after providing spacial similarity transformation in the way of describing based on dual quaterion
The mathematic(al) representation of line coordinates：
S-1-3：Based on criterion of least squares, using being completely coincident as item between the feature of the same name of spacial similarity transformation front and back
Part defines and constructs the three-dimensional space similarity transformation similitude that rule-basedization Pl ü cker coordinate describes under line feature constraint
Estimate；Based on criterion of least squares, using being completely coincident as condition between the feature of the same name of spacial similarity transformation front and back, definition is simultaneously
The similarity measure of three-dimensional space similarity transformation under line feature constraint is constructed, its step are as follows：
S-1-3-1：According to the relationship between unit quaternion and spin matrix, the Pl ü of straight line feature in space
Cker coordinateBecome after spacial similarity transformationThe direction vector of the straight lineWithDirectly
The square of lineWithBetween corresponding relationship it is as follows：
It enables：Formula (1) can be further rewritten as：
In formula：For the original orientation vector of linear feature before spacial similarity transformation,For straight line before spacial similarity transformation
The original square of feature,To indicate the unit quaternion rotated,For unit quaternary numberConjugation, μ is zoom factor,
For the quaternary number for indicating translation；
S-1-3-2：With a matrix type to formulaIt is expressed：
Wherein,
S-1-3-3：Due to the presence of error, and it is based on criterion of least squares, three-dimensional space is similar under line feature constraint
The essence of the similarity measure of transformation is homonymous line feature after spacial similarity transformationWithBetween difference
Quadratic sum reaches minimum, and matrix construction is： As f=∑ f1 2+∑f2 2It is minimum
When, the registration parameter between two neighboring LiDAR point cloud survey station coordinate system can be obtained；
S-2 constructs optimal turn of solution described based on dual quaterion in the similarity measure of three-dimensional space similarity transformation
Change the mathematical model of parameter；The mathematical model step of solution optimum translation parameter based on dual quaterion description is：
The solution procedure of the corresponding quaternary number of rotation transformation is：
It is knownAny straight line is in base station coordinate system, station subject to registration respectively in three-dimensional space
Direction vector and square in coordinate system,For dual quaterion corresponding with the linear feature, phase is realized
The registration of adjacent survey station LiDAR point cloud, steps are as follows：
S-2-1：Construct matrixAnd the corresponding feature vector of maximum eigenvalue of solution matrix AAs for expressing the unit quaternion of rotation transformation；
Wherein,
S-2-2：Solve zoom factor μ：
In formula,
S-2-3：Based on extreme value analysis realize spacial similarity transformation in quaternary number corresponding with translation transformation solution with
And the solution of zoom factor；
S-2-4：The quaternary number for expressing translation transformation (SVD) is solved according to formula
Wherein,
By above-mentioned calculating, the coordinate system at station subject to registration can be calculated relative to the coordinate system between base station
Spacial similarity transformation parameter, and then realize the registration of neighboring stations LiDAR point cloud.
In detail, realize that corresponding with rotation transformation quaternary number is asked in spacial similarity transformation based on extreme value analysis
Solution；
To expression formula ∑ f1 2It is decomposed：And it enables：Then：
Optimal dual quaterionMeet formulaThe condition for obtaining minimum value, meets simultaneouslyCondition, in summary two factors, building objective function is：λ in formula1It is
Lagrange's multiplier；
Take objective functionAbout variablePartial derivative：It is found that quaternary numberIt is
Corresponding to matrixA feature vector, λ1It is characteristic value corresponding with feature vector；
It considers：Work as λ1When selecting the corresponding feature vector of maximum eigenvalue, become
Change the difference ∑ f between the homonymous line direction vector of front and back1 2=Cl1-λ1Minimum value is obtained, i.e.,：Maximum eigenvalue λ1Corresponding spy
Levy vectorAs required rotation quaternary number.
Wherein, to expression formula ∑ f2 2Decompose can obtain：
It enables：Cm1=2 ∑ I,
Then ∑ f2 2It can further be expressed as：
Optimal dual quaterionMeet formula ∑ f2 2The condition for obtaining minimum value, meets simultaneouslyCondition, it is comprehensive
Above-mentioned two factor is closed, objective function is constructed
λ in formula2For Lagrange's multiplier.
It takes respectivelyAboutWith the partial derivative of μ, obtain：
The expression formula of zoom factor μ can be obtained：
And seek solution's expression：Its
In,
It considers：It can obtain
Lagrange coefficient lambda2Expression formula：In turn, ginseng must can be solved
NumberExpression formula：
The homonymous line feature for extracting from two neighboring survey station is substituted into the mathematical model that S-2 is obtained by S-3, is calculated three
The supplemental characteristic of dimension space similarity transformation, and three-dimensional space similarity transformation is implemented to the LiDAR point cloud in station subject to registration accordingly, it is real
The unification at existing station and base station coordinate system subject to registration, and then realize base station LiDAR point cloud and station LiDAR point cloud subject to registration
It is seamless spliced with merge.
Claims (6)
1. a kind of three-dimensional space similarity transformation model parameter based on line feature constraint without initial value method for solving, feature
It is：For same tested region, selects different visual angles to arrange two adjacent survey stations, acquired using two adjacent survey stations
To the surface characteristics LiDAR point cloud of measured target, there is overlapping between the point cloud of two neighboring stations, and there are three for overlapping region
Group or three groups or more homonymous line feature select one of survey station as base station, using another survey station as station subject to registration, with
Feature of the same name is completely coincident the conversion that station coordinates system subject to registration is solved as constraint condition relative to base station coordinate system
Parameter, and the space three-dimensional data in station subject to registration are converted accordingly, to realize two neighboring survey station LiDAR point cloud
It is seamless spliced with merge；
Using the homonymous line feature of extraction as constraint, by the origin coordinate system transform of survey station subject to registration to base station coordinate system
Specific step is as follows：
S-1 chooses two adjacent survey stations, there is overlapping between the LiDAR point cloud of two neighboring stations, according to two survey stations point
The same rectilinear strip in the LiDAR point cloud information that the homonymous line feature indescribably obtained, i.e. two survey stations detect, with
Ensure condition premised on the stringent coincidence of homonymous line feature that two survey stations extract, building three-dimensional space similarity transformation
Similarity measure, and indicated in the form of matrix construction；
S-2 constructs the solution optimum translation ginseng described based on dual quaterion in the similarity measure of three-dimensional space similarity transformation
Several mathematical models；
The homonymous line feature for extracting from two neighboring survey station is substituted into the mathematical model that S-2 is obtained by S-3, and three-dimensional space is calculated
Between similarity transformation parameter values, and accordingly in survey station subject to registration LiDAR point cloud implement three-dimensional space similarity transformation, realize
The unification at station and base station coordinate system subject to registration, and then realize the nothing of base station LiDAR point cloud and station LiDAR point cloud subject to registration
Seam splices and merges.
2. the three-dimensional space similarity transformation model parameter according to claim 1 based on line feature constraint is asked without initial value
Solution method, it is characterised in that the building of the similarity measure of three-dimensional space similarity transformation, steps are as follows：
S-1-1：All linear feature carry out tables of base station Yu station subject to registration are extracted from using regularization Pl ü cker coordinate pair
It reaches；
S-1-2：Mode based on dual quaterion description provides spacial similarity transformation front and back regularization Pl ü cker rectilinear coordinates
Mathematic(al) representation；
S-1-3：Based on criterion of least squares, using spacial similarity transformation before and after being completely coincident as condition between feature of the same name,
It defines and constructs the three-dimensional space similarity transformation similitude that rule-basedization Pl ü cker coordinate describes under line feature constraint and survey
Degree.
3. the three-dimensional space similarity transformation model parameter according to claim 2 based on line feature constraint is asked without initial value
Solution method, it is characterised in that the mathematical expression step of the regularization Pl ü cker coordinate of any straight line is such as in three dimensions
Under：
S-1-1-1：Any selection is located at the straight line in three-dimensional space, two points on the optional straight lineAnd pointIt obtains
Obtain the direction vector of the straight line
S-1-1-2：Calculate the square of the straight line For any one point on straight line,For direction vector；
S-1-1-3：Based on the definition of Pl ü cker rectilinear coordinates, in the form of hexa-atomic groupTable is carried out to the straight line
It reaches；
S-1-1-4：Regularization processing is carried out to the Pl ü cker rectilinear coordinates in S-1-1-3, i.e.,：Pl ü cker straight line after regularization
The direction vector of coordinate is respectively with squareWithPl ü cker rectilinear coordinates after regularization are extended to
8 dimensions, i.e.,：In formulaThe respectively quaternary number expression-form of the square of the direction vector and straight line of straight line, i.e.,：
4. the three-dimensional space similarity transformation according to claim 2 based on line feature constraint without initial value method for solving,
It is characterized in that：It is fixed using being completely coincident as condition between the feature of the same name of spacial similarity transformation front and back based on criterion of least squares
Justice and the similarity measure for constructing three-dimensional space similarity transformation under line feature constraint, its step are as follows：
S-1-3-1：According to the relationship between unit quaternion and spin matrix, the Pl ü cker of straight line feature is sat in space
MarkBecome after spacial similarity transformationThe direction vector of the straight lineWithThe square of straight lineWithBetween corresponding relationship it is as follows：
It enables：Formula (1) can be further rewritten as：
In formula：For the original orientation vector of linear feature before spacial similarity transformation,For linear feature before spacial similarity transformation
Original square,To indicate the unit quaternion rotated,For unit quaternary numberConjugation, μ is zoom factor,To indicate
The quaternary number of translation；
S-1-3-2：Formula (2) is expressed with a matrix type：
Wherein,
Formula (3) is expressed with matrix, can be obtained：
S-1-3-3：Due to the presence of error, and it is based on criterion of least squares, three-dimensional space similarity transformation under line feature constraint
Similarity measure essence, be after spacial similarity transformation homonymous line feature and between squared difference
Minimum with reaching, matrix construction is：
5. the three-dimensional space similarity transformation according to claim 1 based on line feature constraint without initial value method for solving,
It is characterized in that：The mathematical model of the solution optimum translation parameter based on dual quaterion description includes matrix A, rotation quaternary
NumberTranslate quaternary numberWith zoom factor μ.
6. according to claim 1, the three-dimensional space similarity transformation described in 4 or 5 based on line feature constraint without initial value solve
Method, it is characterized in that the mode based on dual quaterion description solves the mathematical model of optimum translation parameter and step is：
S-2-1：The solution of quaternary number corresponding with rotation transformation in spacial similarity transformation is realized using extreme value analysis：
To expression formula ∑ f1 2It is decomposed：And it enables：Then：
It is knownAny straight line is in base station coordinate system, station coordinates subject to registration respectively in three-dimensional space
Direction vector and square in system,For dual quaterion corresponding with the linear feature, adjacent survey is realized
The registration for LiDAR point cloud of standing, steps are as follows：
Optimal rotation quaternary numberMeet formulaThe condition for obtaining minimum value, meets simultaneouslyItem
Part, in summary two factors, building objective function are：λ in formula1It is that Lagrange multiplies
Number；
Take objective functionAbout variablePartial derivative：It is found that quaternary numberIt corresponds to
MatrixA feature vector, λ1It is characteristic value corresponding with feature vector；
It considers：Work as λ1When selecting the corresponding feature vector of maximum eigenvalue, before transformation
Difference ∑ f between secondary homonym rectilinear direction vector1 2=Cl1-λ1Minimum value is obtained, i.e.,：Maximum eigenvalue λ1Corresponding feature to
AmountAs required rotation quaternary number.
Construct matrixAnd the corresponding feature vector of maximum eigenvalue of solution matrix A As it is used for
Express the rotation quaternary number of rotation transformation；
Wherein,
S-2-2：The solution and contracting of quaternary number corresponding with translation transformation in spacial similarity transformation are realized based on extreme value analysis
Put the solution of coefficient：
To expression formula ∑ f2 2It is decomposed, can be obtained：
It enables：Cm1=2 ∑ I,
Then ∑ f2 2It can further be expressed as：
Optimal dual quaterionMeet formula ∑ f2 2The condition for obtaining minimum value, meets simultaneouslyCondition, in summary
Two factors construct objective function
λ in formula2For Lagrange's multiplier；
It takes respectivelyAbout quaternary numberWith the partial derivative of zoom factor μ, obtain：
The expression formula of zoom factor μ can be obtained：
Utilize formula：Solve dual quaterionFormula
In,
It considers：It can obtain
Lagrange coefficient lambda2Expression formula：In turn, it can obtain and become with translation
The corresponding quaternary number of commutationExpression formula：
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| CN113837326A (en) * | 2021-11-30 | 2021-12-24 | 自然资源部第一海洋研究所 | Airborne laser sounding data registration method based on characteristic curve |
| CN113837326B (en) * | 2021-11-30 | 2022-03-25 | 自然资源部第一海洋研究所 | Airborne laser sounding data registration method based on characteristic curve |
| CN116883469A (en) * | 2023-07-20 | 2023-10-13 | 中国矿业大学 | Point cloud registration method based on EIV model description under plane feature constraint |
| CN116883469B (en) * | 2023-07-20 | 2024-01-19 | 中国矿业大学 | Point cloud registration method based on EIV model description under plane feature constraint |
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