CN109671152A - A kind of line point cloud geometrical property evaluation method weighting discrete derivative based on Gauss - Google Patents
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Abstract
The present invention relates to three dimensional point cloud processing and 3 D scene rebuilding technical field, a kind of line point cloud geometrical property evaluation method weighting discrete derivative based on Gauss, the following steps are included: (1) obtains line point cloud data, (2) line point cloud parameter is calculated, (3) it defines Gauss and weights discrete derivative, (4) parametric curve computation of geometric property formula, the estimation of (5) line point cloud geometrical property are derived.The invention has the following advantages that first is that, using Gauss weighted least-squares, discrete tangent is calculated, Gauss is defined by tangent slope and weights discrete derivative, the Gauss weight of introducing adjusts its contribution to set point discrete derivative according to the distance of adjoint point to set point, reduces the influence of noise；Second is that deriving the geometrical property formula of General Parameters curve using interspace analytic geometry theory, discrete derivative is weighted in conjunction with Gauss, by classical differential geometry discretization, the unit tangent vector and curvature of line point cloud is estimated, improves the accuracy of estimation.
Description
Technical field
The present invention relates to a kind of line point cloud geometrical property evaluation methods that discrete derivative is weighted based on Gauss, belong to three-dimensional point
Cloud data processing and 3 D scene rebuilding technical field.
Background technique
With the continuous development of 3-D scanning ranging technology, three dimensional point cloud in reverse-engineering, industrial detection, from leading
The application in the fields such as boat, historical relic's protection, virtual reality is more and more extensive.Three dimensional point cloud processing technique is above-mentioned as realizing
The basis of application has played vital effect.In three dimensional point cloud processing technique, the geometry of three dimensional point cloud is special
Property estimation be a very crucial technology, be the basis of the subsequent processings such as more view registrations, region segmentation, Model Reconstruction, to three
The application effect of dimension point cloud data has significant impact.Therefore, how research accurately estimates from three dimensional point cloud
Geometrical property, the application level for improving three dimensional point cloud have great importance.
From geometrically, three dimensional point cloud can be divided into two kinds: line point cloud data and face point cloud data.Utilize two dimension
Laser scanning and ranging instrument scans object scene, can obtain the line point cloud data of object scene, is distributed in body surface for one group
With the orderly discrete point on the intersection of the plane of scanning motion.Using 3 D laser scanning rangefinder, object scene is scanned, obtains scene object
The face point cloud data of body is the discrete point of one group of distribution on a surface of an, can extract outlet point cloud from the point cloud data of face
Data.The estimation of line point cloud geometrical property refers to the coordinate information using discrete point in line point cloud data, to estimate at discrete point
The geometrical properties such as tangent vector and curvature.
It is inventor's proposition based on the discrete of discrete derivative with the point cloud geometrical property evaluation method of line similar in the present invention
Curve geometrical property evaluation method, principle are the definition according to discrete tangent, with the method for solving for having constrained optimization problem,
The discrete tangent for calculating set point is estimated the discrete derivative of discrete curve using the slope of discrete tangent, and is led by discrete
Count the geometrical property to estimate set point.The different place of the present invention is: the present invention fully considers data noise and adjoint point
The influence that distance estimates line point cloud geometrical property, define Gauss weight discrete derivative, and by its by classical differential geometry from
Dispersion realizes the geometrical property estimation of line point cloud data, improves the accuracy and robustness of estimation.
Summary of the invention
In order to solve the deficiencies in the prior art, it is an object of the present invention to provide one kind to weight discrete derivative based on Gauss
Line point cloud geometrical property evaluation method.This method is directed to actual scene, first with two dimensional laser scanning rangefinder, scanning field
Scenery body obtains the line point cloud data of object scene, carries out parameterized treatment to line point cloud data, defines Gauss and weights discrete lead
Number, and the discrete derivative of line point cloud is calculated according to the definition that Gauss weights discrete derivative, discrete derivative and general is weighted in conjunction with Gauss
The geometrical property of logical parametric curve, estimates the unit tangent vector and curvature of line point cloud.This method has fully considered data noise
The influence estimated with adjoint point distance line point cloud geometrical property defines Gauss and weights discrete derivative, and the unit for calculating line point cloud is cut
The Gauss weight of vector sum curvature, introducing adjusts its contribution to set point discrete derivative according to the distance of adjoint point to set point,
The influence of noise bring is reduced, the accuracy of unit tangent vector and Curvature Estimate is improved.
In order to achieve the above-mentioned object of the invention, it solves the problems of in the prior art, the technical solution that the present invention takes
It is: a kind of line point cloud geometrical property evaluation method weighting discrete derivative based on Gauss, comprising the following steps:
Step 1 obtains line point cloud data, using two dimensional laser scanning rangefinder, scans object scene, obtains object scene
Line point cloud data P={ pi=(xi,yi,zi) | 1≤i≤n }, it is one group and is distributed on the plane of scanning motion and body surface intersection
Orderly discrete point, wherein pi=(xi,yi,zi) it is discrete point on line point cloud P, i is the serial number of discrete point, and n is discrete point
Number；
Step 2 calculates line point cloud parameter, using the method for accumulative Chord Length Parameterization, calculates each discrete point in line point cloud P
piChord length parameter ti, it is described by formula (1),
Parameter ti∈ S and discrete point pi=(xi,yi,zi) ∈ p-shaped is at the relationship α mapped one by oned: S → P, for each
A parameter ti∈ S has a discrete point pi=(xi,yi,zi) ∈ P is corresponding to it, it indicates are as follows: pi=αd(ti)=(xd(ti),yd
(ti),zd(ti)), ti∈ S, wherein S={ ti| 1≤i≤n } it is parameter sets, xi=xd(ti)、yi=yd(ti) and zi=zd
(ti) be parameter discrete function；
Step 3 defines Gauss weighting discrete derivative, leads geometry of numbers meaning using continuous function, defines discrete function
Derivative, calculate discrete function xi=xd(ti)、yi=yd(ti) and zi=zd(ti) derivative, specifically include following sub-step:
Step (a), on geometric meaning, continuous function is equivalent to continuous function in the tangent line of the point in the derivative of set point
Slope therefore define discrete function first in the tangent line of set point, referred to as discrete tangent；Discrete function xi=xd(ti) giving
Pinpoint (ti,xi) discrete tangent be defined as one while meeting the straight line of following three conditions
1) straight lineBy set point (ti,xi)；
2) straight lineWith set point (ti,xi) adjoint point { (tj,xj) | i-m≤j≤i+m } between along x-axis side
The quadratic sum of upward distance is minimum；
3) adjoint point is remoter apart from set point, and the influence to straight line is smaller；
Wherein, m is the radius of neighborhood, and j is the serial number of adjoint point, and a is the slope of discrete tangent, and b is discrete tangent in x-axis
Intercept；
Step (b), the definition according to discrete tangent are calculated using there is the method for solving of constrained optimization problem given
Point (ti,xi) discrete tangent, there is constrained optimization problem to be described by formula (2),
Wherein,For adjoint point (tj,xj) Gauss weight, be described by formula (3),
Wherein, σ is bandwidth, is using the slope a that method of Lagrange multipliers can solve discrete tangent,
According to the slope a of discrete tangent, discrete function x is definedi=xd(ti) in set point (ti,xi) derivative, by formula
(5) it is described,
Wherein, x 'i=x 'd(ti) it is discrete function xi=xd(ti) in set point (ti,xi) derivative；
Step (c), using with step 3 sub-step (a) and (b) identical method, discrete function y can be definedi=yd(ti)
And zi=zd(ti) in set point (ti,yi) and (ti,zi) derivative, be described respectively by formula (6) and (7),
Wherein, (tj,yj) it is set point (ti,yi) adjoint point, (tj,zj) it is set point (ti,zi) adjoint point, σ is bandwidth,
y′i=y 'd(ti) it is discrete function yi=yd(ti) in set point (ti,yi) derivative, z 'i=z 'd(ti) it is discrete function zi=
zd(ti) in set point (ti,zi) derivative,WithFor Gauss weight, it is described respectively by formula (8) and (9),
Then line point cloud vector discrete function pi=αd(ti)=(xd(ti),yd(ti),zd(ti)) in tiThe Derivative Definition at place
For,
p′i=α 'd(ti)=(x 'd(ti),y′d(ti),z′d(ti)), (10)
Wherein, p 'i=α 'd(ti) it is vector discrete function pi=αd(ti)=(xd(ti),yd(ti),zd(ti)) in tiPlace
Derivative, abbreviation discrete derivative；
Step 4 derives parametric curve computation of geometric property formula, according to interspace analytic geometry theory, is commonly joined
Numberization curve α (t)=(x (t), y (t), z (t)) unit tangent vector v (t) and curvature κ (t) be respectively,
Wherein, α ' (t) is the derivative of α (t), and v ' (t) is the derivative of v (t)；
Step 5, the estimation of line point cloud geometrical property, utilize the geometrical property of discrete derivative and General Parameters curve, estimation
Line point cloud is in tiThe unit tangent vector v at placed(ti) and curvature κd(ti) be respectively,
Wherein, v 'd(ti) it is vector discrete function vd(ti) in tiThe discrete derivative at place utilizes method identical with step 3
To calculate.
The medicine have the advantages that a kind of line point cloud geometrical property evaluation method for weighting discrete derivative based on Gauss, packet
It includes following steps: (1) obtaining line point cloud data, (2) calculate line point cloud parameter, and (3) define Gauss and weight discrete derivative, and (4) push away
Lead parametric curve computation of geometric property formula, the estimation of (5) line point cloud geometrical property.Compared with the prior art, the present invention has
Following advantages: first is that, the method that the present invention utilizes Gauss weighted least-squares calculates discrete tangent, defines height by tangent slope
This weighting discrete derivative, the Gauss weight of introducing adjust its tribute to set point discrete derivative according to the distance of adjoint point to set point
It offers, has given full play to the advantage for reducing influence of noise；Second is that the present invention is theoretical using interspace analytic geometry, General Parameters are derived
The geometrical property formula for changing curve weights discrete derivative in conjunction with Gauss, by classical differential geometry discretization, estimates the list of line point cloud
Position tangent vector and curvature, improve the accuracy of estimation.
Detailed description of the invention
Fig. 1 is the method for the present invention flow chart of steps.
Fig. 2 is discrete tangent schematic diagram.
Fig. 3 is tangent vector estimation result figure.
Fig. 4 is Curvature Estimate result figure.
In figure: being (a) vertical line Point cloud curvature estimation result figure, be (b) x wire Point cloud curvature estimation result figure.
Specific embodiment
The present invention will be further explained below with reference to the attached drawings.
As shown in Figure 1, a kind of line point cloud geometrical property evaluation method that discrete derivative is weighted based on Gauss, including following step
It is rapid:
Step 1 obtains line point cloud data, using two dimensional laser scanning rangefinder, scans object scene, obtains object scene
Line point cloud data P={ pi=(xi,yi,zi) | 1≤i≤n }, it is one group and is distributed on the plane of scanning motion and body surface intersection
Orderly discrete point, wherein pi=(xi,yi,zi) it is discrete point on line point cloud P, i is the serial number of discrete point, and n is discrete point
Number；
Step 2 calculates line point cloud parameter, using the method for accumulative Chord Length Parameterization, calculates each discrete point in line point cloud P
piChord length parameter ti, it is described by formula (1),
Parameter ti∈ S and discrete point pi=(xi,yi,zi) ∈ p-shaped is at the relationship α mapped one by oned: S → P, for each
A parameter ti∈ S has a discrete point pi=(xi,yi,zi) ∈ P is corresponding to it, it indicates are as follows: pi=αd(ti)=(xd(ti),yd
(ti),zd(ti)), ti∈ S, wherein S={ ti| 1≤i≤n } it is parameter sets, xi=xd(ti)、yi=yd(ti) and zi=zd
(ti) be parameter discrete function；
Step 3 defines Gauss weighting discrete derivative, leads geometry of numbers meaning using continuous function, defines discrete function
Derivative, calculate discrete function xi=xd(ti)、yi=yd(ti) and zi=zd(ti) derivative, specifically include following sub-step:
Step (a), on geometric meaning, continuous function is equivalent to continuous function in the tangent line of the point in the derivative of set point
Slope therefore define discrete function first in the tangent line of set point, referred to as discrete tangent；Discrete function xi=xd(ti) giving
Pinpoint (ti,xi) discrete tangent be defined as one while meeting the straight line of following three conditions
1) straight lineBy set point (ti,xi)；
2) straight lineWith set point (ti,xi) adjoint point { (tj,xj) | i-m≤j≤i+m } between along x-axis side
The quadratic sum of upward distance is minimum；
3) adjoint point is remoter apart from set point, and the influence to straight line is smaller；
Wherein, m is the radius of neighborhood, and j is the serial number of adjoint point, and a is the slope of discrete tangent, and b is discrete tangent in x-axis
Intercept, as shown in Figure 2；
Step (b), the definition according to discrete tangent are calculated using there is the method for solving of constrained optimization problem given
Point (ti,xi) discrete tangent, there is constrained optimization problem to be described by formula (2),
Wherein,For adjoint point (tj,xj) Gauss weight, be described by formula (3),
Wherein, σ is bandwidth, is using the slope a that method of Lagrange multipliers can solve discrete tangent,
According to the slope a of discrete tangent, discrete function x is definedi=xd(ti) in set point (ti,xi) derivative, by formula
(5) it is described,
Wherein, xi'=x 'd(ti) it is discrete function xi=xd(ti) in set point (ti,xi) derivative；
Step (c), using with step 3 sub-step (a) and (b) identical method, discrete function y can be definedi=yd(ti)
And zi=zd(ti) in set point (ti,yi) and (ti,zi) derivative, be described respectively by formula (6) and (7),
Wherein, (tj,yj) it is set point (ti,yi) adjoint point, (tj,zj) it is set point (ti,zi) adjoint point, σ is bandwidth,
yi'=y 'd(ti) it is discrete function yi=yd(ti) in set point (ti,yi) derivative, z '=z 'd(ti) it is discrete function zi=zd
(ti) in set point (ti,zi) derivative,WithFor Gauss weight, it is described respectively by formula (8) and (9),
Then line point cloud vector discrete function pi=αd(ti)=(xd(ti),yd(ti),zd(ti)) in tiThe Derivative Definition at place
For,
p′i=α 'd(ti)=(x 'd(ti),y′d(ti),z′d(ti)), (10)
Wherein, p 'i=α 'd(ti) it is vector discrete function pi=αd(ti)=(xd(ti),yd(ti),zd(ti)) in tiPlace
Derivative, abbreviation discrete derivative；
Step 4 derives parametric curve computation of geometric property formula, according to interspace analytic geometry theory, is commonly joined
Numberization curve α (t)=(x (t), y (t), z (t)) unit tangent vector v (t) and curvature κ (t) be respectively,
Wherein, α ' (t) is the derivative of α (t), and v ' (t) is the derivative of v (t)；
Step 5, the estimation of line point cloud geometrical property, utilize the geometrical property of discrete derivative and General Parameters curve, estimation
Line point cloud is in tiThe unit tangent vector v at placed(ti) and curvature kd(ti) be respectively,
Wherein, v 'd(ti) it is vector discrete function vd(ti) in tiThe discrete derivative at place utilizes method identical with step 3
It calculates, Fig. 3 gives the tangent vector estimation result figure of a round wire point cloud, Fig. 4 is the face of a true office scenarios
Point cloud data extracts vertical line point cloud according to scanning row number from the point cloud data of face, extracts x wire point cloud according to scanning line number,
Then Curvature Estimate is being carried out to these line point clouds respectively, Curvature Estimate result such as Fig. 4 (a) and Fig. 4 (b) are shown.
The invention has the advantages that: first is that, the method that the present invention utilizes Gauss weighted least-squares calculates discrete tangent, by
Tangent slope defines Gauss and weights discrete derivative, and the Gauss weight of introducing adjusts it to given according to the distance of adjoint point to set point
The contribution of point discrete derivative, has given full play to the advantage for reducing influence of noise；Second is that the present invention is managed using interspace analytic geometry
By, the geometrical property formula of General Parameters curve is derived, weights discrete derivative in conjunction with Gauss, classical differential geometry is discrete
Change, estimates the unit tangent vector and curvature of line point cloud, improve the accuracy of estimation.
Claims (1)
1. a kind of line point cloud geometrical property evaluation method for weighting discrete derivative based on Gauss, it is characterised in that including following step
It is rapid:
Step 1 obtains line point cloud data, using two dimensional laser scanning rangefinder, scans object scene, obtains the line of object scene
Point cloud data P={ pi=(xi,yi,zi) | 1≤i≤n }, it is one group and is distributed in the plane of scanning motion and having on body surface intersection
Sequence discrete point, wherein pi=(xi,yi,zi) be line point cloud P on discrete point, i be discrete point serial number, n be discrete point
Number；
Step 2 calculates line point cloud parameter, using the method for accumulative Chord Length Parameterization, calculates each discrete point p in line point cloud Pi's
Chord length parameter ti, it is described by formula (1),
Parameter ti∈ S and discrete point pi=(xi,yi,zi) ∈ p-shaped is at the relationship α mapped one by oned: S → P, for each ginseng
Number ti∈ S has a discrete point pi=(xi,yi,zi) ∈ P is corresponding to it, it indicates are as follows: pi=αd(ti)=(xd(ti),yd(ti),
zd(ti)), ti∈ S, wherein S={ ti| 1≤i≤n } it is parameter sets, xi=xd(ti)、yi=yd(ti) and zi=zd(ti) be
The discrete function of parameter；
Step 3 defines Gauss weighting discrete derivative, leads geometry of numbers meaning using continuous function, defines leading for discrete function
Number calculates discrete function xi=xd(ti)、yi=yd(ti) and zi=zd(ti) derivative, specifically include following sub-step:
(a), on geometric meaning, continuous function is equivalent to continuous function in the tangent slope of the point in the derivative of set point,
Therefore, discrete function is defined first in the tangent line of set point, referred to as discrete tangent；Discrete function xi=xd(ti) in set point
(ti,xi) discrete tangent be defined as one while meeting the straight line x=at+b of following three conditions,
1) straight line x=at+b passes through set point (ti,xi)；
2) straight line x=at+b and set point (ti,xi) adjoint point { (tj,xj) | i-m≤j≤i+m } between along the x-axis direction on away from
From quadratic sum it is minimum；
3) adjoint point is remoter apart from set point, and the influence to straight line is smaller；
Wherein, m is the radius of neighborhood, and j is the serial number of adjoint point, and a is the slope of discrete tangent, and b is discrete tangent cutting in x-axis
Away from；
(b), it according to the definition of discrete tangent, is calculated using there is the method for solving of constrained optimization problem in set point (ti,xi)
Discrete tangent, there is constrained optimization problem to be described by formula (2),
Wherein,For adjoint point (tj,xj) Gauss weight, be described by formula (3),
Wherein, σ is bandwidth, is using the slope a that method of Lagrange multipliers can solve discrete tangent,
According to the slope a of discrete tangent, discrete function x is definedi=xd(ti) in set point (ti,xi) derivative, by formula (5) into
Row description,
Wherein, x 'i=x 'd(ti) it is discrete function xi=xd(ti) in set point (ti,xi) derivative；
(c), using with step 3 sub-step (a) and (b) identical method, discrete function y is definedi=yd(ti) and zi=zd(ti)
In set point (ti,yi) and (ti,zi) derivative, be described respectively by formula (6) and (7),
Wherein, (tj,yj) it is set point (ti,yi) adjoint point, (tj,zj) it is set point (ti,zi) adjoint point, σ is bandwidth, y 'i=
y′d(ti) it is discrete function yi=yd(ti) in set point (ti,yi) derivative, z 'i=z 'd(ti) it is discrete function zi=zd(ti)
In set point (ti,zi) derivative,WithFor Gauss weight, it is described respectively by formula (8) and (9),
Then line point cloud vector discrete function pi=αd(ti)=(xd(ti),yd(ti),zd(ti)) in tiThe Derivative Definition at place is,
p′i=α 'd(ti)=(x 'd(ti),y′d(ti),z′d(ti)), (10)
Wherein, p 'i=α 'd(ti) it is vector discrete function pi=αd(ti)=(xd(ti),yd(ti),zd(ti)) in tiThe derivative at place,
Abbreviation discrete derivative；
Step 4 derives parametric curve computation of geometric property formula, according to interspace analytic geometry theory, obtains General Parameters
Curve α (t)=(x (t), y (t), z (t)) unit tangent vector v (t) and curvature κ (t) be respectively,
Wherein, α ' (t) is the derivative of α (t), and v ' (t) is the derivative of v (t)；
Step 5, the estimation of line point cloud geometrical property estimate line point using the geometrical property of discrete derivative and General Parameters curve
Cloud is in tiThe unit tangent vector v at placed(ti) and curvature κd(ti) be respectively,
Wherein, v 'd(ti) it is vector discrete function vd(ti) in tiThe discrete derivative at place is counted using method identical with step 3
It calculates.
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