CN108389243A - A kind of multiple dimensioned Bézier curve piecewise fitting method of vector line feature - Google Patents
A kind of multiple dimensioned Bézier curve piecewise fitting method of vector line feature Download PDFInfo
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- G—PHYSICS
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- G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
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Abstract
The present invention provides a kind of multiple dimensioned Bézier curve piecewise fitting method of vector line feature, includes carrying out multiple dimensioned segmentation subdivision to the line feature of coordinate points string sampling expression in Distribution GIS, estimates the tangential direction of waypoint respectively；Cubic Bézier curve parameter Estimation is carried out to each point in the coordinate points string of segmentation, carries out least square fitting, and the parameter of curve of iterated revision each point, obtains the best cubic Bézier curve fitting of waypoint string；8 tuple parameters of parameter of cubic Bézier curve and fitting precision are subjected to tissue according to multiple dimensioned segmentation subdivision structure, obtain multiple dimensioned Bézier curve parameter expression model.The present invention devises a kind of new GIS line feature expression technology schemes, is realized and is expressed using parameter curve, and data storage capacity is small, and modelling geometry computational efficiency is high, is conducive to complicated geographical geometry computation model, while showing suitable for multiple dimensioned electrodeless visualization.
Description
Technical field
The invention belongs to technical field of geographic information, are related to a kind of expression of GIS line of vector factor data, especially
A kind of multiple dimensioned Bézier curve piecewise fitting method of vector line feature.
Background technology
It is particularly significant in the expression problem of science and engineering field, quantitative observation data, a frequent suitable function side
Journey can provide great convenience for the solution of problem.As the mapping area of geographic information that science and engineering are combined, mapping
Linear geographical entity river, zone boundary, coastline, contour etc. in geodata, generally use discrete sampling point
Coordinate string (i.e. polygon line) is expressed.The expression-form of polygon line is the excessive sampling expression to Geographic Curve, and being one kind makes
The analog representation that the tracing pattern of geographic element is carried out with the coordinate points of redundancy.The expression-form simple, intuitive of polygon line,
There is certain advantage in data processing simplicity, and visualization drafting convenience, but there is also deficiencies in terms of the two.
On the one hand polygon line is a kind of data storage of redundancy, and mass data processing needs to carry out more redundant computation, computing resource
Consumption is big；On the other hand, polygon line is difficult to meet the electrodeless scalings of map elements and visually requires, the sampling of line feature point string
Precision determines the level of detail that final visualization is drawn, change that cannot be adaptive with the change of facility environment.
Currently, real and trample and praised highly in theoretical research in engineering with geometric function expression geographical entity.With to calculate
Universal, the curve geometric modeling tool of computer-assisted cartography (CAC) technology based on machine Computer Aided Design (CAD) technology
(such as：B é zier, B-spline curves etc.) it is frequently used to expression linear ground object.Have and has researched and proposed the unified geographical entity of several figures
Primitive expression model establishes 14 kinds of strict mathematical relationships between basic-element model and Bézier curve, it is proposed that geographical entity
With the theoretical and related practical application of curve representation.Also there is research using different geometric primitives to point in GIS data expression
The local fit that the GIS line targets that string list reaches are segmented obtains the partitioned representation model of line target, for reducing adopting for data
With frequency, have the function that data compression, is also beneficial to the calculation processing of data.Parameter curve expresses line feature in the fields GIS
Have preliminary application and research, it is contemplated that the parameter curve expression of GIS line feature entirety, and take more rulers of element into account
Characteristic issues are spent, there is also shortcomings with related art scheme for existing research.Thus it can be seen that in the engineering practice of map making
In, the method with the linear geographic element of the parameter curve of segmentation expression is long-standing, but in generalized information system by line feature according to point
There is also deficiencies for the technical method that the mode of section parameter curve is expressed and analyzed.
Invention content
For the above prior art problem, the present invention devises one kind and carrying out a hierarchical subdivision to a string based on BLG tree constructions,
Cubic Bézier curve piecewise fitting is carried out to the waypoint string of different levels, establishes the multiple dimensioned level segmentation B é zier of curve
The method of parameter of curve expression.
Technical solution of the present invention provides a kind of multiple dimensioned Bézier curve piecewise fitting method of vector line feature, including following
Step：
Step 1, the GIS line of vector factor data that coordinate points string list reaches by line abbreviation binary tree structure stratification subdivision,
Multiple dimensioned point sequence curve waypoint is obtained, the tangential direction at waypoint is calculated by the local feature of waypoint, for song
The holding of line geometric continuity between each segmentation Bézier curve when being fitted；
Step 2, for point set that is discrete in each segmentation obtained by step 1, being sequentially connected
It solves one and passes through its head and the tail point P1,PmCubic Bézier curve Q3(t) withIt most preferably approaches, obtains one section of B é zier three times
The interior control point of curve, including first have to estimateMiddle each pair of point answers curve Q3(t) the parameter T={ t putj| j=1,2 ...,
M }, the model of fit of least square is established, Newton iterative method is then carried out and best Bézier curve fitting result is calculated；
Step 3,8 tuple parameters of cubic Bézier curve carry out tissue according to multiple dimensioned segmentation subdivision, obtain more rulers
The Bézier curve parameter expression model of degree.
Moreover, the realization process of step 1 is as follows,
Step 1.1, by Douglas-Peucker algorithms, the line feature that point string list reaches is by recursive subdivision at different levels
Line segment be stored in binary tree, obtain line abbreviation binary tree structure, obtain the multi-level waypoint of curve with this configuration；
Step 1.2, the angular bisector of the warp architecture constituted between continuous three waypoint lines is calculated, angle is calculated
The normal vector of bisector obtains the tangential direction of curve partial structurtes；
Step 1.3, for the tangent line of the head and the tail of curve point, the Trendline by calculating the multiple points of head and the tail local continuous is used as
Tangential direction.
Moreover, the realization process of step 2 is as follows,
Step 2.1, if parameter T={ t of each point for cubic Bézier curve section to be fitted in given waypoint stringj|
J=1,2 ..., m }, the initial value of a parameter is provided with the quotient of the accumulated angular length of data point and total arc length, formula is as follows：
Wherein,For r-th of point PrWith r+1 point Pr+1The distance between,For line string the 1st
Cumulative length between a o'clock to j-th point,For the total length of line string；
Step 2.2, under step 1.2 and step 1.3 gained tangent line constraints, using initial parameter, pass through minimum two
Multiplication carries out the Fitting Calculation of cubic Bézier curve, and the model of fit for obtaining curve initial is as follows so that cost function S is most
It is small,
Wherein, Q3(tj) it is parameter tjPoint on corresponding cubic Bézier curve；
Step 2.3, carry out what Newton iterative calculation was continued to optimize using each point and the error function of matched curve
Model of fit, each point are as follows with the degree of approximation function of curve：
f(tj)=[Q3(tj)-Pj]·[Q3(tj)-Pj]T.
Iterating to calculate formula is：
Wherein,For j-th of point PjThe corresponding parameter value of the l times iteration,For j-th of point PjThe l times iteration is corresponding
Parameter value,It is point PjDegree of approximation function existsWhen derivative value；
Stop iteration when meeting the end condition of iteration.
Moreover, the end condition of iteration is that twice fitting error ε is less than corresponding predetermined threshold value before and after iteration, for each
Section matched curve, usesIn arrive matched curve Q3(t) evaluation of the maximum value of distance as error of fitting, in actual execution
In can control the number l of iteration, error E calculation formula is as follows：
Wherein, E is error of fitting, E2For square of error of fitting；
The difference ε of front and back iteration error twice seeks as follows：
ε=El-El-1
Wherein, ElFor the l times iterative fitting error, El-1For l-1 iterative fitting errors.
Alternatively, the end condition of iteration is to reach scheduled iterations.
Moreover, the realization process of step 3 is as follows,
Step 3.1, the expression of the control point of the cubic Bézier curve for each waypoint string being calculated in step 2 is turned
Change the parameter expression of 8 tuples into；
Step 3.2, multi-level matched curve section is subjected to tissue expression according to line abbreviation binary tree structure, tree node is deposited
The fitting precision value for storing up 8 tuple parameters and curved section obtains different songs by retrieving the curved section of different fitting precisions
Line expression of results.
The present invention is putting the Multilevel B é zier curve representation structures for factor data of getting lines crossed by establishing, and changes traditional point
The redundant representation method of string sampling, is expressed using parametric function model, on the one hand reduces the volume of data storage, another party
Face is conducive to the modelling that complicated geographical geometry calculates, and computational efficiency is high, while can also realize curve different scale, and difference is set
Adaptive visualization expression under the conditions of standby.
Description of the drawings
Fig. 1 is the flow chart of the embodiment of the present invention.
Fig. 2 is the embodiment of the present invention point string multi-scale division schematic diagram, and wherein Fig. 2 a are a BLG tree constructions for string
Douglas-Peucker algorithms divide schematic diagram, and Fig. 2 b are a string BLG tree construction schematic illustration of tissue.
Fig. 3 is that the tangent line of the embodiment of the present invention curve segmentation point calculates schematic diagram, and wherein Fig. 3 a are contiguous segmentation point offices
Portion's structure tangent line estimates schematic diagram, and Fig. 3 b are waypoint string tangential direction constrained fitting schematic diagrames.
Fig. 4 is the embodiment of the present invention piecewise fitting iteration result schematic diagram, and wherein Fig. 4 a are 1 iterative fitting curve knots
Fruit schematic diagram, Fig. 4 b are 50 iterative fitting Dependence Results schematic diagrames, and Fig. 4 c are 100 iterative fitting Dependence Results schematic diagrames,
Fig. 4 d are 500 iterative fitting Dependence Results schematic diagrames.
Fig. 5 is the embodiment of the present invention Multilevel B é zier Curve Tree structure representation schematic diagrames, and wherein Fig. 5 a are curve point
The schematic diagram of section, the tree structure figure that Fig. 5 b store for it.
Fig. 6 is the multiple dimensioned Bézier curve expression of results figure of the embodiment of the present invention accuracy constraint lower curve, and Fig. 6 a are most
Big drawing error of fitting is less than the segmentation Bézier curve expression of results figure of 50mm, and Fig. 6 b are that maximum drawing error of fitting is less than
The segmentation Bézier curve expression of results figure of 30mm, Fig. 6 c are the segmentation Bézier curves that maximum drawing error of fitting is less than 20mm
Expression of results figure, Fig. 6 d are the segmentation Bézier curve expression of results figures that maximum drawing error of fitting is less than 10mm, and Fig. 6 e are most
Big drawing error of fitting is less than the segmentation Bézier curve expression of results figure of 5mm, and Fig. 6 f are that maximum drawing error of fitting is less than 2mm
Segmentation Bézier curve expression of results figure.
Specific implementation mode
Understand for the ease of those of ordinary skill in the art and implement the present invention, with reference to the accompanying drawings and embodiments to this hair
It is bright to be described in further detail, it should be understood that implementation example described herein is merely to illustrate and explain the present invention, not
For limiting the present invention.
The present invention it is considered that as GIS Linear elements expression an alternative, piecewise fitting parametric function expression
Line data model, which has some, potential advantage.First, it is excessive sampling to curve that point string list, which reaches, has larger data
Redundancy, and Parameter Expression is the modelling fitting to curve, has data back to accumulate small, the high advantage of computational efficiency；Secondly,
The geometry entity of parameter model expression is conducive to the geometrical analysis in GIS in the advantage in terms of numerical computations efficiency and computational accuracy
And calculating；Finally, point string expression way is limited to sampling precision when visualized graphs export, and parameter model expression can basis
The output of visualization condition, scale requirement and graphical display accuracy self-adapting preferably effect of visualization.String list is reached
The expression that GIS line features are converted into parametric function modelling is a curve fit problem, and target is by segmentation few as possible
Function obtains high-precision as possible approach.Meanwhile the Linear element in GIS has Analysis On Multi-scale Features, is establishing line feature segmentation
It is also required to take scale effect into account when curve representation model.
As shown in Figure 1, a kind of multiple dimensioned Bézier curve piecewise fitting side of vector line feature that the embodiment of the present invention proposes
Method includes the following steps：
Step 1：Multiple dimensioned point is carried out to the line feature of coordinate points string sampling expression in GIS-Geographic Information System (GIS) first
Section subdivision, estimates the tangential direction of waypoint, includes passing through BLG to the GIS line of vector factor datas that coordinate points string list reaches respectively
(Binary Line Generalization-Tree, line feature abbreviation binary tree) layer of structure subdivision is set, is obtained multiple dimensioned
Curve segmentation point, the tangential direction at waypoint is calculated by the local feature of waypoint, is used for when curve matching each point
The holding of geometric continuity between section Bézier curve；
The step 1 of embodiment is realized using following sub-step：
Step 1.1：By Douglas-Peucker algorithms, the line feature that point string list reaches is by recursive subdivision at different levels
Line segment be stored in binary tree, obtain line abbreviation binary tree structure, obtain the multi-level waypoint of curve with this configuration：Such as figure
2, by Douglas-Peucker algorithms, the line feature that point string list reaches is stored in two by recursive subdivision at the line segment of different levels
It pitches in tree, i.e. BLG tree constructions, obtains the multi-level waypoint of curve with this configuration；
Step 1.2：The angular bisector of the warp architecture constituted between continuous three waypoint lines is calculated, angle is calculated
The normal vector of bisector obtains the tangential direction of curve partial structurtes：Waypoint tangent line computation model, S as shown in the left sides Fig. 31,S2,
S3It is continuous three waypoints respectively, they control a geographical bending features under some scale, can use ∠
S1S2S3Angular bisector as curve along waypoint S2Normal directionThus it is assured that waypoint S2Tangential directionBase
In this, the right sides Fig. 3 are parameter curve restricted model, wherein C0,C3For waypoint, that is, it is fitted the end control point of Bézier curve,
C1,C2For control point in Bézier curve to be solved,It is waypoint C respectively0,C3The tangent line rector at place, Prescribed Properties
Step 1.3：For the tangential then Trendline by originating with continuous 3 points in end of the waypoint of curve head and the tail
(fitting a straight line) is estimated.
Step 2：Cubic Bézier curve parameter Estimation is carried out to each point in the coordinate points string of segmentation, carries out least square fitting,
And using the parameter of curve of the continuous iterated revision each point of Newton method, the best cubic Bézier curve fitting of waypoint string is obtained, including
For including m point P that are discrete, being sequentially connected in each segmentation in step 1j, constitute point set
J is serial number a little, solves one and passes through its head and the tail point P1,PmCubic Bézier curve Q3(t) withMost preferably approach, that is,
Solve the interior control point of one section of cubic Bézier curve.It first has to estimateIn each point with curve Q3(t) apart from closest approach on
Parameter T={ tj| j=1,2 ..., m }, the model of fit formula of least square is established, Newton iteration method iterative calculation is then carried out
Obtain best Bézier curve fitting result；Wherein, t is the parameter of Bézier curve function, tjIndicate that correspond at j-th point
The parameter value of Bézier curve.
The step 2 of embodiment is realized using following sub-step：
Step 2.1：Parameter T={ t of each point for cubic Bézier curve section to be fitted in given waypoint stringj|j
=1,2 ..., m }, the initial value of a parameter is provided with the quotient of the cumulative length of data point and total lengthCalculation formula is
Wherein,For r-th of point PrWith r+1 point Pr+1The distance between,For line string the 1st
Cumulative length between a o'clock to j-th point,For the total length of line string.
Step 2.2：In step 1.2 and step 1.3 under gained tangent line constraints, using initial parameter, pass through minimum
Square law carries out the Fitting Calculation of cubic Bézier curve, obtains the initial model of curve, computation model is：
Wherein, Q3(tj) it is parameter tjPoint on corresponding cubic Bézier curve.
The two ends control point P of i.e. known curve1,Pm, model contains there are two parameter to be solved, that is, asks and controlled in two
Point C1,C2(such as Fig. 3) so that cost function S is minimum, that is, needs to meet equation or less to C1,C2Partial derivative be 0
Equation solution obtains：
WhereinFor Bernstein basic functions three times：
For parameter t=tjWhen basic function value.
In order to simplify results expression, variable is enabled：
So interior control point result is：
Step 2.3：Iterative process as shown in Figure 4, wherein Fig. 4 a, Fig. 4 b, Fig. 4 c, Fig. 4 d are 1 time respectively, 50 times, 100
Secondary, 500 iterative fitting Dependence Results schematic diagrames, corrected parameter T needs to calculate each point PjIt is with parameter on homologous thread
tjPoint Q3(tj) degree of approximation, evaluation function f (tj) be：
f(tj)=[Q3(tj)-Pj]·[Q3(tj)-Pj]T
Iterative formula is following formula, and l is iterations：
Wherein,For j-th of point PjThe corresponding parameter value of the l times iteration,For j-th of point PjThe l times iteration is corresponding
Parameter value,It is point PjDegree of approximation function existsWhen derivative value.
Successive ignition undated parameter value can calculate new interior control point, until fitting precision reach certain requirement or
Reach scheduled iterations, such as control the number of iteration in actual execution, such as l≤200.In embodiment, repeatedly
The terminating point in generation is that the difference ε of twice fitting error before and after iteration is less than corresponding predetermined threshold value.For each section of matched curve,
Use point setMiddle each point is to matched curve Q3(t) evaluation of the maximum value of distance as error of fitting.For one section of matched curve
The l times iterative fitting error E can be defined as：
Wherein, E is error of fitting, E2For square of error of fitting.
So each time during iterative fitting before and after iteration the difference ε of twice fitting error seek it is as follows：
ε=El-El-1
Wherein, ElFor the l times iterative fitting error, El-1For l-1 iterative fitting errors.
Step 3：8 tuple parameters of cubic Bézier curve are subjected to tissue according to multiple dimensioned segmentation subdivision, are obtained more
The Bézier curve parameter expression model of scale.
The step 3 of embodiment is realized using following sub-step：
Step 3.1：The control point expression of the cubic Bézier curve for each waypoint string being calculated in step 2 is turned
Multi-level matched curve section is carried out tissue expression, tree node storage by the parameter expression for changing 8 tuples into according to BLG tree constructions
8 tuple parameters values (the i.e. cubic polynomial letter of curve cross, ordinate of the parameter expression form of one cubic Bézier curve section
Respective four parameter values of number) and its fitting precision value, schematic construction such as Fig. 5, Fig. 5 a are the schematic diagram of curve segmentation, are schemed
5b for its storage tree structure figure, for example, it is initial it is rough be fitted to AL curved sections, AL can also be further segmented into AD
And DL, the middle section of fitting are further divided into AB such as AC sections, BC sections, successively recursive subdivision set until curve is expressed completely
What each node stored is 8 tuple parameters and its fitting precision value of cubic Bézier curve in structure.
Step 3.2：By retrieving the curved section of different fitting precisions, different curve representations is obtained as a result, such as Fig. 6,
Fig. 6 a, Fig. 6 b, Fig. 6 c, Fig. 6 d, Fig. 6 e, Fig. 6 f are maximum drawing error of fitting small 50mm, 30mm, 20mm, 10mm, 5mm respectively,
The segmentation Bézier curve expression of results figure of 2mm, i.e., be the fitting result after becoming larger with engineer's scale respectively.
When it is implemented, the automatic running that computer software technology realizes the above flow can be used.
It should be understood that the part that this specification is not described in detail belongs to the prior art.
It should be understood that the description of above-described embodiment is more detailed, therefore can not be considered to protect patent of invention
The limitation of range is protected, those skilled in the art is not departing from what the claims in the present invention were protected under the inspiration of the present invention
Under ambit, replacement or deformation can also be made, is each fallen within protection scope of the present invention, the protection model that the present invention asks
Enclosing should be determined by the appended claims.
Claims (6)
1. a kind of multiple dimensioned Bézier curve piecewise fitting method of vector line feature, which is characterized in that include the following steps：
Step 1, the GIS line of vector factor data that coordinate points string list reaches is obtained by line abbreviation binary tree structure stratification subdivision
Multiple dimensioned point sequence curve waypoint calculates the tangential direction at waypoint by the local feature of waypoint, quasi- for curve
When conjunction it is each segmentation Bézier curve between geometric continuity holding；
Step 2, for point set that is discrete in each segmentation obtained by step 1, being sequentially connected
It solves one and passes through its head and the tail point P1,PmCubic Bézier curve Q3(t) withIt most preferably approaches, obtains one section of B é zier three times
The interior control point of curve, including first have to estimateMiddle each pair of point answers curve Q3(t) the parameter T={ t putj| j=1,2 ...,
M }, the model of fit of least square is established, Newton iterative method is then carried out and best Bézier curve fitting result is calculated；
Step 3,8 tuple parameters of cubic Bézier curve carry out tissue according to multiple dimensioned segmentation subdivision, obtain multiple dimensioned B
é zier parameter of curve expression models.
2. the multiple dimensioned Bézier curve piecewise fitting method of vector line feature according to claim 1, it is characterised in that：Step
1 realization process is as follows,
Step 1.1, by Douglas-Peucker algorithms, line of the line feature that point string list reaches by recursive subdivision at different levels
Section is stored in binary tree, is obtained line abbreviation binary tree structure, is obtained the multi-level waypoint of curve with this configuration；
Step 1.2, the angular bisector of the warp architecture constituted between continuous three waypoint lines is calculated, angle bisection is calculated
The normal vector of line obtains the tangential direction of curve partial structurtes；
Step 1.3, for the tangent line of the head and the tail of curve point, the Trendline by calculating the multiple points of head and the tail local continuous is used as tangent line
Direction.
3. the multiple dimensioned Bézier curve piecewise fitting method of vector line feature according to claim 1, it is characterised in that：Step
2 realization process is as follows,
Step 2.1, if parameter T={ t of each point for cubic Bézier curve section to be fitted in given waypoint stringj| j=1,
2 ..., m }, the initial value of a parameter is provided with the quotient of the accumulated angular length of data point and total arc length, formula is as follows：
Wherein,For r-th of point PrWith r+1 point Pr+1The distance between,For the 1st point of line string
Cumulative length between j-th point,For the total length of line string；
Step 2.2, under step 1.2 and step 1.3 gained tangent line constraints, using initial parameter, pass through least square method
The Fitting Calculation of cubic Bézier curve being carried out, the model of fit for obtaining curve initial is as follows so that cost function S is minimum,
Wherein, Q3(tj) it is parameter tjPoint on corresponding cubic Bézier curve；
Step 2.3, the fitting that Newton iterative calculation is continued to optimize is carried out using the error function of each point and matched curve
Model, each point are as follows with the degree of approximation function of curve：
f(tj)=[Q3(tj)-Pj]·[Q3(tj)-Pj]T.
Iterating to calculate formula is：
Wherein,For j-th of point PjThe corresponding parameter value of the l times iteration,For j-th of point PjThe corresponding parameter of the l times iteration
Value,It is point PjDegree of approximation function existsWhen derivative value；
Stop iteration when meeting the end condition of iteration.
4. the multiple dimensioned Bézier curve piecewise fitting method of vector line feature according to claim 3, it is characterised in that：Iteration
End condition be that twice fitting error ε is less than corresponding predetermined threshold value before and after iteration, for each section of matched curve, use
In arrive matched curve Q3(t) evaluation of the maximum value of distance as error of fitting can control time of iteration in actual execution
Number l, error E calculation formula are as follows：
Wherein, E is error of fitting, E2For square of error of fitting；
The difference ε of front and back iteration error twice seeks as follows：
ε=El-El-1
Wherein, ElFor the l times iterative fitting error, El-1For l-1 iterative fitting errors.
5. the multiple dimensioned Bézier curve piecewise fitting method of vector line feature according to claim 3, it is characterised in that：Iteration
End condition be to reach scheduled iterations.
6. according to claims 1 or 2 or the 3 or 4 or 5 multiple dimensioned Bézier curve piecewise fitting methods of vector line feature,
It being characterized in that, the realization process of step 3 is as follows,
Step 3.1, the expression of the control point of the cubic Bézier curve for each waypoint string being calculated in step 2 is converted into
The parameter expression of 8 tuples；
Step 3.2, multi-level matched curve section is subjected to tissue expression, tree node storage 8 according to line abbreviation binary tree structure
The fitting precision value of tuple parameters and curved section obtains different curves by retrieving the curved section of different fitting precisions
Expression of results.
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