CN103984810B - Part curve and surface small echo method for fairing based on multiresolution analysis - Google Patents
Part curve and surface small echo method for fairing based on multiresolution analysis Download PDFInfo
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Abstract
The invention discloses a kind of small echo method for fairing based on multiresolution analysis, include mainly " reading and processing of data point ", " dyadic wavelet fairing ", " fairing of arbitrary resolution small echo " three steps, it realizes to the small echo fairing with arbitrary control vertex curve, and it can be according to its wavelet scale of curve controlled number of vertex automatic decision, to select most suitable calculating step.The present invention has reached preferable balance in terms of the efficiency of curve smoothing and adaptability.The derivation of the present invention follows multiresolution Analysis Theory with calculating, and there is no the approximation operations such as interpolation and fitting, can realize the Accurate Reconstruction of fair curve, meet the marrow of wavelet analysis.
Description
Technical field
The present invention relates to a kind of part curve and surface small echo method for fairing based on multiresolution analysis belongs to reverse-engineering neck
Domain.
Background technology
It is limited by the accuracy of manufacture and measurement accuracy, the digitized process of part will produce a certain number of noises and mistake
Difference, this will produce certain harmful effect to the reverse precision and its fairness of curve and surface.As must can not in reverse-engineering
Few one ring of key, Curve fairness technology can eliminate these influences as much as possible, to improve the quality of reverse-engineering.
Although traditional Smoothing Algorithm thinking is simply clear, in the case of measurement data is larger, computational efficiency is extremely low.Reverse-engineering
Expect a kind of Smoothing Algorithm that can extract curve and surface intrinsic propesties as far as possible in field.The appearance of multiresolution analysis technology, will
This expectation becomes in order to possible.
Multiresolution analysis can portray the new time frequency analysis mathematical tool of data interdependency as a kind of, while have good
Good time domain locality and frequency domain locality.Start to be applied to computer with increasingly maturation, the multiresolution analysis of the technology
Graphics.1994, Quak etc. applied multiresolution analysis tool, for by 2jA control vertex of+3 (j is binary scale) determines
Curve, be put forward for the first time the curve separating restructing algorithm based on closed interval B-spline small echo.For more resolved lights along this new neck
Domain, domestic and international many scholars expand this research.Ceruti etc. is based on dyadic wavelet Smoothing Algorithm, constructs quickly mostly thin
Ganglionic layer time (Levels ofDetail, LOD) filter group, realizes the small echo fairing of A grades of curves.Pan Yangyu is based on classical more
Differentiate analysis theories, in conjunction with second generation the Upgrade Lattice small echo, by segmentation, prediction, update and etc. the promotion that is formed walk, derive
The Upgrade Lattice Wavelet representation for transient method of quasi- uniform cubic B-spline curve, realizes the fairing of quasi- uniform cubic B-spline curve.
Hussein etc. is based on Lifting Wavelet frame, is generalized to the expression of free form surface, and realize irregular complex free form surface
Filtering and fairing.
Although the studies above realizes the small echo fairing of curve and surface, but in order to reduce the difficulty of Construction of Wavelets, two into
Under the requirement of scale, (2 can only be met certain requirements to control vertex numberj+ (r-1) | r is exponent number) curve and surface carry out light
Suitable, in fact, in reverse-engineering, the quantity of the point actually measured is arbitrary, and corresponding control vertex number is also arbitrary,
Above-mentioned algorithm will be no longer applicable at this time.Above-mentioned algorithm is usually called dyadic wavelet fairing.
In the case of curve and surface control vertex number is arbitrary, Pan Yangyu etc. is asked by the change of knot vector to solve this
Topic, main thought are to increase the quantity of control vertex to meet dyadic wavelet fairing requirement.Wang etc. is according to given weight
Formula is reconstructed orthogonal non-uniform spline wavelets by the removal to knot vector, to realize nurbs curve curved surface
Small echo fairing with simplify.Two scholars such as Pan, Sadeghi use for given knot vector in conjunction with discrete norm concept
Least square fitting algorithm constructs a kind of biorthogonal non-uniform B-spline wavelet, realizes more resolved lights of arbitrary free curve
Suitable, which is greatly improved due to avoiding inner product operation, computational efficiency.But all there is approximate meter in above-mentioned algorithm
It calculates, Accurate Reconstruction can not be carried out to curve and surface.Usually there is no the algorithm of specific requirement to call control vertex quantity above-mentioned
Arbitrary resolution small echo fairing.
Invention content
In view of the above-mentioned problems, applicant on the basis of dyadic wavelet Smoothing Algorithm is studied, proposes that one kind is differentiated more based on
The small echo method for fairing of analysis is realized to the small echo fairing with arbitrary control vertex curve and surface；Simultaneously using C Plus Plus and
C++ numerical analysis class libraries proposes the software implementation method of small echo fairing, according to its small echo ruler of curve controlled number of vertex automatic decision
Degree, to select most suitable computing module, achieves a better balance in terms of the efficiency of curve smoothing and adaptability.
Technical scheme is as follows：
A kind of small echo method for fairing based on multiresolution analysis, includes the following steps：
(1) reading and processing of data point：
Raw data points are read in, and are defined as matrix variables；Data format is often three data of row, indicates a point respectively
X, y, z coordinate；After reading in raw data points, by data point inverse at control vertex, and the quantity s of control vertex is calculated；By
In the present invention, study is cubic B-Spline interpolation, and corresponding exponent number r=4 then calculates corresponding reasonable scale m according to m=s-3；
Corresponding binary scale j is calculated by formula (1)：
J=lg (m)/lg2 (1)
If j is integer, meet the requirement of dyadic wavelet fairing, executes step (2)；If j is not integer, no
Meet dyadic wavelet fairing requirement, executes step (3)；
(2) dyadic wavelet fairing：
According to the binary scale j that formula (1) determines, during restructuring matrix initialization formula (2), (3) calculating fairing
The restructuring matrix P usedjAnd Qj；
By restructuring matrix PjAnd QjRow is to merging into square matrix PjQj=[Pj|Qj]；
By solving decomposition of system of linear equations formula (4) realization to curve；
Wherein, Cj-1For the control vertex of fair curve, Dj-1For the detail section filtered out during small echo fairing；
According to formula (5), restructuring matrix P is utilizedjAnd Qj, Accurate Reconstruction raw data points Cj；
Cj=PjCj-1+QjDj-1 (5)
(3) arbitrary resolution small echo fairing：
It counts according to reading in, determines the reasonable scale m of small echo before fairing；It is required according to fairing, small echo is reasonable after determining fairing
Scale n；
ΦmIndicate the spline space under scale m；ΦnIndicate the spline space under scale n；ΨnIndicate the small echo under scale n
Space；
According to scale m, n and corresponding quasi- uniform cubic B-spline functionIt calculates
Inner product matrix BnBm；BnBmIn each element by formula (6) calculate obtain；
Restructuring matrix Q is calculated by the solution of kernel according to formula (7)mn, it is desirable that QmnFor sparse matrix；
BnBm×Qmn=0 (7)
According to formula (8), arbitrary resolution wavelet function is constructed；
Ψn=ΦmQmn (8)
By wavelet space ΨnIn i-th of Wavelet representation for transient beSpline space ΦmIn i-th of B-spline be expressed as
Spline space ΦnIn i-th of B-spline be expressed asThen BnBn=[<Φn|Φn>], indicate inner product(the n constituted
+ 3) × (n+3) rank matrix；WnWn=[<Ψn|Ψn>], indicate inner product(m-n) × (m-n) the rank matrixes constituted；
WnBm=[<Ψn|Φm>], indicate inner product(m-n) × (m+3) the rank matrixes constituted；Meanwhile according to formula (6), BnBm
It has calculated；
According to formula (9), restructuring matrix P is calculatedmnAnd Qmn；
Calculate split-matrix AmnAnd Bmn, according to the experience of calculating, split-matrix AmnAnd BmnIt is not usually sparse matrix, is
Operation efficiency is improved, during fairing, does not calculate split-matrix directly, but by solving system of linear equations formula (10) come real
It is existing；
Wherein, CnFor the control vertex of fair curve, DnFor the detail section filtered out during small echo fairing；
According to formula (11), restructuring matrix P is utilizedmnAnd Qmn, the original control vertex C of Accurate Reconstructionm；
Cm=PmnCn+QmnDn (11)。
Its further technical solution is：The software implementation method of the step (2) is, by establishment following two into small
The glistening light of waves is realized along function：
(1)double CubicBSpline(double t,int m,int k)；
The function defines a quasi- uniform cubic B-spline basic functionWherein m is flexible scale, right
For dyadic wavelet fairing, m=2j；K is translation scale, and meaning is k-th of batten in B-spline space, k ∈ [1,2j+
3]；
(2)void InitPj(matrix&Pj,int j)；
The function is directed to binary scale j, calculates corresponding restructuring matrix Pj, and result is stored in matrix variables Pj；
(3)void InitQj(matrix&Qj,int j)；
The function is directed to binary scale j, calculates corresponding restructuring matrix Qj, and result is stored in matrix variables Qj；
(4)double DyadicWavelet(double t,int m,int k)；
Uniform cubic B-spline function subject to the functionCorresponding dyadic wavelet basic function ψ (t, m, k) | t ∈
[0,1], wherein m are flexible scale, for dyadic wavelet fairing, m=2j, k is translation scale, and meaning is small echo batten
K-th of spline wavelets in space, k ∈ [1,2j-1]；
(5)void DWFairing(matrix&Cj,int j,matrix&Cjj,matrix&Djj)；
The function is dyadic wavelet fairing function, can be according to the control vertex C of primitive curvejAnd corresponding scale j, meter
The control vertex C of curve after calculation fairingj-1And corresponding detail section control vertex Dj-1, and result is saved in corresponding square
In battle array variable Cjj, Djj；
(6)void DWRecon(matrix&Cj,int j,matrix&Cjj,matrix&Djj)；
The function is dyadic wavelet reconstruction of function, can be according to the control vertex C of curve to be reconstructedj-1And it is corresponding thin
Save part control vertex Dj-1, reconstruct the control vertex C of primitive curvej.It is emphasized that the scale j in this function is to wait for weight
The scale of structure curve.
Its further technical solution is：The software implementation method of the step (3) is, by working out following arbitrary point
Resolution small echo fairing function is realized：
(1)double CubicBSpline(double t,int m,int k)；
The function defines a quasi- uniform cubic B-spline basic functionWherein m is flexible scale, k
To translate scale, meaning is k-th of batten in B-spline space, k ∈ [1, m+3]；
(2)double MultiBSplineBSpline(double t,vector&p)；
The function calculate two spline functions between product CubicBSpline (t, m, k) × CubicBSpline (t,
N, l), in vector variable p, p [1]=m, p [2]=k, p [3]=n, p [4]=l.
(3)void InnerProductBB(matrix&BnBm,int m,int n)；
The function calculates inner product matrix BnBm=[<Φn|Φm>], and result is saved in matrix variables BnBm.Integral
Section is [0,1]；
(4)void InitQmn(matrix&Qmn,matrix&BnBm,int m,int n)；
The function calculates restructuring matrix Q according to given fairing scale m, nmn, and result is stored in variable Qmn；
(5)double BSplineWavelet(double t,int m,int k,matrix&Qmn)；
The function calculates B-spline wavelet basis function according to formula (8), during returning from scale m to scale n fairing, kth
Small echo corresponds to the functional value of t；
(6)double MultiWaveletWavelet(double t,vector&p,matrix&Qmn)；
Product between the function two wavelet basis functions of calculating, BSplineWavelet (t, m, k, Qmn) ×
In BSplineWavelet (t, m, l, Qmn), vector variable p, p [1]=p [3]=m, p [2]=k, p [4]=l；
(7)double MultiWaveletBSpline(double t,vector&p,matrix&Qmn)；
The function calculates the product between wavelet basis function and B-spline basic function, BSplineWavelet (t, m, l, Qmn)
In × CubicBSpline (t, m, k), vector variable p, p [1]=p [3]=m, p [2]=k, p [4]=l；
(8)void InnerProductWW(matrix&WnWn,int m,int n)；
The function is used for calculating the inner product matrix W of wavelet basis functionnWn=[<Ψn|Ψn>], and result is saved in matrix
In variable WnWn.Integrating range is [0,1]；
(9)void InnerProductWB(matrix&WnBm,int m,int n)；
The function is used for calculating the inner product matrix W between wavelet basis function and B-spline basic functionnBm=[<Ψn|Φm>],
And result is saved in matrix variables WnBm.Integrating range is [0,1]；
(10)void ARWFairing(matrix&Cm,matrix&Cn,matrix&Dn,int m,int n)；
The function is according to given scale m and n, by the control vertex C of primitive curvemResolve into the control top of fair curve
Point CnWith the control vertex D of detail sectionn, and result is saved in corresponding matrix variables Cn and Dn；
(11)void ARWRecon(matrix&Cm,matrix&Cn,matrix&Dn int m,int n)；
The function is according to given scale m and n, by the control vertex C of fair curvenWith the control vertex D of detail sectionn,
Accurate Reconstruction at primitive curve control vertex Cm, and result is saved in matrix variables Cm.
The method have the benefit that：
It is 2 that the small echo Smoothing Algorithm of the present invention, which can handle control vertex number,j+ 3 specific curves, and control top can be handled
Points are arbitrary curve；Can be according to the number of the control vertex of input, in " dyadic wavelet fairing " and " arbitrary resolution is small
The glistening light of waves is suitable " between automatically select most suitable algorithm, therefore reached good balance in terms of computational efficiency and adaptability；This
The derivation of invention follows multiresolution Analysis Theory with calculating, and there is no the approximation operations such as interpolation and fitting, can realize fairing song
The Accurate Reconstruction of line meets the marrow of wavelet analysis.
Description of the drawings
Fig. 1 is that the step of the present invention divides schematic diagram.
Fig. 2 is data point reading and process chart.
Fig. 3 is dyadic wavelet fairing function passes relational graph.
Fig. 4 is arbitrary resolution small echo fairing function passes relational graph.
Fig. 5 (a)~Fig. 5 (e) is compressor female rotor fairing of line example.Wherein,
Fig. 5 (a) is constructed part small echo.
Fig. 5 (b) is the comparison of primitive curve and fair curve.
Fig. 5 (c) is the curvature analysis of primitive curve.
Fig. 5 (d) is the curvature analysis of fair curve.
Fig. 5 (e) is the detail curve filtered out.
Fig. 6 (a)~Fig. 6 (e) is turbine blade section fairing of line example.Wherein,
Fig. 6 (a) is constructed part small echo.
Fig. 6 (b) is the comparison of primitive curve and fair curve.
Fig. 6 (c) is the curvature analysis of primitive curve.
Fig. 6 (d) is the curvature analysis of fair curve.
Fig. 6 (e) is the detail curve filtered out.
Specific implementation mode
The following further describes the specific embodiments of the present invention with reference to the drawings.
The invention mainly comprises three steps：" reading and processing of data point ", " dyadic wavelet fairing ", " arbitrary resolution
Rate small echo fairing ".Wherein, " reading and processing of data point " step is responsible for waiting for the reading of fairing point, judges the number of control vertex
Amount and its corresponding scale；" dyadic wavelet fairing " step is responsible for having 2j+ (r-1) (j is binary scale, and r is exponent number) a control
The curve on vertex processed is decomposed, and realizes fairing and the Accurate Reconstruction of curve；Responsible pair of " fairing of arbitrary resolution small echo " step
The fairing of general curve and Accurate Reconstruction.Step divides as shown in Figure 1.
1, the reading and processing of data point
Raw data points are read in first, and are defined as matrix variables matrix InPoints.Data format is often row three
Data indicate that the x, y, z coordinate of a point, data format are as shown in table 1 respectively.
1 Format Data Point of table
Secondly, by data point inverse at control vertex, and the quantity s of control vertex is calculated；Since what the present invention studied is
Cubic B-Spline interpolation, corresponding exponent number r=4 then calculate corresponding reasonable scale m according to m=s-3；By formula (1) calculating pair
The binary scale j answered：
J=lg (m)/lg2 (1)
If j is integer, meet the requirement of dyadic wavelet fairing, can perform " dyadic wavelet fairing "；If j is not whole
Number, then do not meet dyadic wavelet fairing requirement, can only execute at this time " fairing of arbitrary resolution small echo ".Due to " dyadic wavelet light
It is suitable " in restructuring matrix Pj、QjIt is to determine and known, the computational efficiency of " dyadic wavelet fairing " is than " arbitrary resolution small echo
High more of fairing ".The detailed process that scale judges is as shown in Figure 2.
2, dyadic wavelet fairing
2.1, dyadic wavelet Smoothing Algorithm
The realization process of dyadic wavelet fairing is as follows：
(1) the binary scale j determined according to formula (1), passes through restructuring matrix initialization function InitPj (), InitQj ()
Calculate the restructuring matrix P used during fairingjAnd Qj。
Note：Restructuring matrix in above-mentioned dyadic wavelet fairing refers in particular to (3) two kinds of formula (2), formula matrixes：
(2) by restructuring matrix PjAnd QjRow is to merging into square matrix PjQj=[Pj|Qj]。
(3) due to split-matrix AjAnd BjGeneral is not sparse matrix, in order to improve computational efficiency, in dyadic wavelet light
Along during, does not calculate split-matrix directly generally, but the decomposition to curve is realized by solving system of linear equations formula (4).
Wherein, Cj-1For the control vertex of fair curve, Dj-1For the detail section filtered out during small echo fairing.
(4) according to formula (5), restructuring matrix P is utilizedjAnd Qj, Accurate Reconstruction raw data points Cj。
Cj=PjCj-1+QjDj-1 (5)
The major parameter and variable used in calculating process are as shown in table 2.
Parameter during 2 dyadic wavelet fairing of table realization and variable
2.2, dyadic wavelet fairing function
On the basis of dyadic wavelet Smoothing Algorithm, the present invention utilizes C++Builder tools and C++ numerical analysis class libraries,
Software mode realizes dyadic wavelet fairing.It is worked out specific according to variable-definition, the present invention in dyadic wavelet Smoothing Algorithm and table 2
Power function is as follows：
(1)double CubicBSpline(double t,int m,int k)；
The function defines a quasi- uniform cubic B-spline basic functionWherein m is flexible scale, right
For dyadic wavelet fairing, m=2j；K is translation scale, and meaning is k-th of batten in B-spline space, k ∈ [1,2j+
3]；
(2)void InitPj(matrix&Pj,int j)；
The function is directed to binary scale j, calculates corresponding restructuring matrix Pj, and result is stored in matrix variables Pj；
(3)void InitQj(matrix&Qj,int j)；
The function is directed to binary scale j, calculates corresponding restructuring matrix Qj, and result is stored in matrix variables Qj；
(4)double DyadicWavelet(double t,int m,int k)；
Uniform cubic B-spline function subject to the functionCorresponding dyadic wavelet basic function ψ (t, m, k) | t ∈ [0,
1], wherein m is flexible scale, for dyadic wavelet fairing, m=2j, k is translation scale, and meaning is that small echo batten is empty
Between in k-th of spline wavelets, k ∈ [1,2j-1]；
(5)void DWFairing(matrix&Cj,int j,matrix&Cjj,matrix&Djj)；
The function is dyadic wavelet fairing function, can be according to the control vertex C of primitive curvejAnd corresponding scale j, meter
The control vertex C of curve after calculation fairingj-1And corresponding detail section control vertex Dj-1, and result is saved in corresponding square
In battle array variable Cjj, Djj；
(6)void DWRecon(matrix&Cj,int j,matrix&Cjj,matrix&Djj)；
The function is dyadic wavelet reconstruction of function, can be according to the control vertex C of curve to be reconstructedj-1And it is corresponding thin
Save part control vertex Dj-1, reconstruct the control vertex C of primitive curvej.It is emphasized that the scale j in this function is to wait for weight
The scale of structure curve.
Transitive relation between each function is as shown in Figure 3.
3, arbitrary resolution small echo fairing
3.1, arbitrary resolution small echo Smoothing Algorithm
Arbitrary resolution small echo fairing the specific implementation process is as follows：
(1) according to points are read in, the reasonable scale m of small echo before fairing is determined；It is required according to fairing, there is small echo after determining fairing
Manage scale n.
(2)ΦmIndicate the spline space under scale m；ΦnIndicate the spline space under scale n；ΨnIt indicates under scale n
Wavelet space；
According to scale m, n and corresponding quasi- uniform cubic B-spline functionIt calculates
Inner product matrix BnBm；BnBmIn each element by formula (6) calculate obtain.
(3) in view of the orthogonality of spline space and wavelet space, reconstruct is calculated by the solution of kernel according to formula (7)
Matrix Qmn, since solution is not unique, generally require QmnFor sparse matrix.
BnBm×Qmn=0 (7)
(4) according to formula (8), arbitrary resolution wavelet function is constructed.
Ψn=ΦmQmn (8)
(5) by wavelet space ΨnIn i-th of Wavelet representation for transient beSpline space ΦmIn i-th of B-spline be expressed asSpline space ΦnIn i-th of B-spline be expressed asThen BnBn=[<Φn|Φn>], indicate inner productIt constitutes
(n+3) × (n+3) rank matrixes；WnWn=[<Ψn|Ψn>], indicate inner product(m-n) × (m-n) the rank squares constituted
Battle array；WnBm=[<Ψn|Φm>], indicate inner product(m-n) × (m+3) the rank matrixes constituted；Meanwhile according to formula (6),
BnBmIt has calculated.
(6) according to formula (9), restructuring matrix P is calculatedmnAnd Qmn。
(7) split-matrix A is calculatedmnAnd Bmn, according to the experience of calculating, split-matrix AmnAnd BmnUsually it is not sparse matrix,
To improve operation efficiency, during fairing, do not calculate split-matrix directly, but by solve system of linear equations formula (10) come
It realizes.
Wherein, CnFor the control vertex of fair curve, DnFor the detail section filtered out during small echo fairing.
(8) according to formula (11), restructuring matrix P is utilizedmnAnd Qmn, the original control vertex C of Accurate Reconstructionm。
Cm=PmnCn+QmnDn (11)
The major parameter and variable used in calculating process are as shown in table 3.
Parameter during 3 arbitrary resolution small echo fairing of table realization and variable
3.2, arbitrary resolution small echo fairing function
On the basis of arbitrary resolution wavelet Smoothing Algorithm, the present invention utilizes C++Builder tools and C++ numerical value point
Class libraries is analysed, software mode realizes arbitrary resolution small echo fairing.It is fixed according to variable in arbitrary resolution small echo Smoothing Algorithm and table 3
Justice, the concrete function function that the present invention works out are as follows：
(1)double CubicBSpline(double t,int m,int k)；
The function defines a quasi- uniform cubic B-spline basic functionWherein m is flexible scale, k
To translate scale, meaning is k-th of batten in B-spline space, k ∈ [1, m+3].
(2)double MultiBSplineBSpline(double t,vector&p)；
The function calculate two spline functions between product CubicBSpline (t, m, k) × CubicBSpline (t,
N, l), in vector variable p, p [1]=m, p [2]=k, p [3]=n, p [4]=l.
(3)void InnerProductBB(matrix&BnBm,int m,int n)；
The function calculates inner product matrix BnBm=[<Φn|Φm>], and result is saved in matrix variables BnBm.Integral
Section is [0,1].
(4)void InitQmn(matrix&Qmn,matrix&BnBm,int m,int n)；
The function calculates restructuring matrix Q according to given fairing scale m, nmn, and result is stored in variable Qmn.
(5)double BSplineWavelet(double t,int m,int k,matrix&Qmn)；
The function calculates B-spline wavelet basis function according to formula (8), during returning from scale m to scale n fairing, kth
Small echo corresponds to the functional value of t.
(6)double MultiWaveletWavelet(double t,vector&p,matrix&Qmn)；
Product between the function two wavelet basis functions of calculating, BSplineWavelet (t, m, k, Qmn) ×
In BSplineWavelet (t, m, l, Qmn), vector variable p, p [1]=p [3]=m, p [2]=k, p [4]=l.
(7)double MultiWaveletBSpline(double t,vector&p,matrix&Qmn)；
The function calculates the product between wavelet basis function and B-spline basic function, BSplineWavelet (t, m, l, Qmn)
In × CubicBSpline (t, m, k), vector variable p, p [1]=p [3]=m, p [2]=k, p [4]=l.
(8)void InnerProductWW(matrix&WnWn,int m,int n)；
The function is used for calculating the inner product matrix W of wavelet basis functionnWn=[<Ψn|Ψn>], and result is saved in matrix
In variable WnWn.Integrating range is [0,1].
(9)void InnerProductWB(matrix&WnBm,int m,int n)；
The function is used for calculating the inner product matrix W between wavelet basis function and B-spline basic functionnBm=[<Ψn|Φm>],
And result is saved in matrix variables WnBm.Integrating range is [0,1].
(10)void ARWFairing(matrix&Cm,matrix&Cn,matrix&Dn,int m,int n)；
The function is according to given scale m and n, by the control vertex C of primitive curvemResolve into the control top of fair curve
Point CnWith the control vertex D of detail sectionn, and result is saved in corresponding matrix variables Cn and Dn.
(11)void ARWRecon(matrix&Cm,matrix&Cn,matrix&Dn int m,int n)；
The function is according to given scale m and n, by the control vertex C of fair curvenWith the control vertex D of detail sectionn,
Accurate Reconstruction at primitive curve control vertex Cm, and result is saved in matrix variables Cm.
Transitive relation between each function is as shown in Figure 4.
4, more resolved lights are along example
The example of two complex curve fairing is now provided, and corresponding curve is drawn according to the above method, to verify we
The correctness and stability of method.
Example 1：Compressor female rotor fairing of line example
Fig. 5 (a)~Fig. 5 (e) is compressor female rotor fairing of line example.Preliminary offset point quantity is 196, therefore corresponding
Control vertex number be 196+2=198, corresponding original scale be m=198-3=195；It is scale n=128 by curve smoothing
Curve, at this time corresponding control vertex number be 128+3=131, can inverse at 131-2=129 data point；It can construct
M-n=195-129=66 small echo, shown in the part small echo such as Fig. 5 (a) constructed.Primitive curve and fair curve are drawn, and
It is compared, as shown in Fig. 5 (b), partial enlarged view shows primitive curve there are larger fluctuation, and fair curve fluctuates smaller, light
Along with obvious effects；Further to analyze fairing effect, curvature analysis, analysis result are carried out to primitive curve and fair curve respectively
As shown in Fig. 5 (c) and Fig. 5 (d), before fairing, the maximum curvature of primitive curve is 8.80807mm-1, after fairing, fair curve is most
Deep camber is reduced to 4.51303mm-1, 48.76% is reduced, preferable fairing effect is achieved；Fig. 5 (e) be primitive curve to
During fair curve fairing, the detail section that wavelet filter group is filtered out, for the ease of showing and analyzing, detail portion
Divide and is exaggerated 200 times；By fair curve and detail curve, this algorithm being capable of Accurate Reconstruction primitive curve.
Example 2：Turbine blade section fairing of line example
Fig. 6 (a)~Fig. 6 (e) is turbine blade section fairing of line example.Preliminary offset point quantity is 300, therefore right
The control vertex number answered is 300+2=302, and corresponding original scale is m=302-3=299；It is scale n=by curve smoothing
197 curve, at this time corresponding control vertex number be 197+3=200, can inverse at 200-2=198 data point；It can
M-n=299-197=102 small echo is constructed, shown in the part small echo such as Fig. 6 (a) constructed.It is bent with fairing to draw primitive curve
Line, and being compared, as shown in Fig. 6 (b), partial enlarged view shows primitive curve there are larger fluctuation, fair curve fluctuation compared with
Small, fairing is with obvious effects；Further to analyze fairing effect, curvature analysis is carried out to primitive curve and fair curve respectively, point
It analyses shown in result such as Fig. 6 (c) and Fig. 6 (d), before fairing, the maximum curvature of primitive curve is 0.840148mm-1, after fairing, fairing
Curve maximum curvature is reduced to 0.623921mm-1, 25.74% is reduced, preferable fairing effect is achieved；Fig. 6 (e) is original
Curve is to during fair curve fairing, the detail section that wavelet filter group is filtered out, for the ease of showing and analyzing,
Detail section is exaggerated 100 times；By fair curve and detail curve, this algorithm being capable of Accurate Reconstruction primitive curve.
What has been described above is only a preferred embodiment of the present invention, and present invention is not limited to the above embodiments.It is appreciated that this
The other improvements and change that field technology personnel directly export or associate without departing from the spirit and concept in the present invention
Change, is considered as being included within protection scope of the present invention.
Claims (3)
1. a kind of part curve and surface small echo method for fairing based on multiresolution analysis, it is characterised in that include the following steps：
(1) reading and processing of data point：
Raw data points are read in, and are defined as matrix variables；The raw data points come from compressor female rotor molded line or steamer
Machine blade profile molded line；Data format is often three data of row, indicates the x, y, z coordinate of a point respectively；Read in initial data
After point, by data point inverse at control vertex, and the quantity s of control vertex is calculated；It is corresponding due to being cubic B-Spline interpolation
Exponent number r=4 then calculates corresponding reasonable scale m according to m=s-3；Corresponding binary scale j is calculated by formula (1)：
J=lg (m)/lg2 (1)
If j is integer, meet the requirement of dyadic wavelet fairing, executes step (2)；If j is not integer, do not meet
Dyadic wavelet fairing requirement, executes step (3)；
(2) dyadic wavelet fairing：
According to the binary scale j that formula (1) determines, formula (2) is initialized by restructuring matrix, (3) calculate fairing and use in the process
Restructuring matrix PjAnd Qj；
By restructuring matrix PjAnd QjRow is to merging into square matrix PjQj=[Pj|Qj]；
By solving decomposition of system of linear equations formula (4) realization to curve；
Wherein, Cj-1For the control vertex of fair curve, Dj-1For the detail section filtered out during small echo fairing；
According to formula (5), restructuring matrix P is utilizedjAnd Qj, Accurate Reconstruction raw data points Cj；
Cj=PjCj-1+QjDj-1 (5)
(3) arbitrary resolution small echo fairing：
It counts according to reading in, determines the reasonable scale m of small echo before fairing；It is required according to fairing, determines the reasonable scale of small echo after fairing
n；
ΦmIndicate the spline space under scale m；ΦnIndicate the spline space under scale n；ΨnIndicate that the small echo under scale n is empty
Between；
According to scale m, n and corresponding quasi- uniform cubic B-spline function It calculates
Inner product matrix BnBm；BnBmIn each element by formula (6) calculate obtain；
Restructuring matrix Q is calculated by the solution of kernel according to formula (7)mn, it is desirable that QmnFor sparse matrix；
BnBm×Qmn=0 (7)
According to formula (8), arbitrary resolution wavelet function is constructed；
Ψn=ΦmQmn (8)
By wavelet space ΨnIn a-th of Wavelet representation for transient beSpline space ΦmIn a-th of B-spline be expressed asBatten
Space ΦnIn a-th of B-spline be expressed asThen BnBn=[<Φn|Φn>], indicate inner product(n+3) constituted
× (n+3) rank matrixes；WnWn=[<Ψn|Ψn>], indicate inner product(m-n) × (m-n) the rank matrixes constituted；WnBm
=[<Ψn|Φm>], indicate inner product(m-n) × (m+3) the rank matrixes constituted；Meanwhile according to formula (6), BnBm
Through calculating；
According to formula (9), restructuring matrix P is calculatedmn；
Calculate split-matrix AmnAnd Bmn, according to the experience of calculating, split-matrix AmnAnd BmnIt is not sparse matrix, to improve operation
Efficiency does not calculate split-matrix directly, but is realized by solving system of linear equations formula (10) during fairing；
Wherein, CnFor the control vertex of fair curve, DnFor the detail section filtered out during small echo fairing；
According to formula (11), restructuring matrix P is utilizedmnAnd Qmn, the original control vertex C of Accurate Reconstructionm；
Cm=PmnCn+QmnDn (11)
Wherein, the types of variables for the major parameter used in step (3) includes matrix and vector.
2. the part curve and surface small echo method for fairing based on multiresolution analysis according to claim 1, it is characterised in that：Institute
The software implementation method for stating step (2) is to be realized by working out following dyadic wavelet fairing function：
(1)double CubicBSpline(double t,int m,int k)；
The function defines a quasi- uniform cubic B-spline basic functionWherein m is reasonable scale, for two
Into for small echo fairing, m=2j；K is translation scale, and meaning is k-th of batten in B-spline space, k ∈ [1,2j+3]；
(2)void InitPj(matrix&Pj,int j)；
The function is directed to binary scale j, calculates corresponding restructuring matrix Pj, and result is stored in matrix variables Pj；
(3)void InitQj(matrix&Qj,int j)；
The function is directed to binary scale j, calculates corresponding restructuring matrix Qj, and result is stored in matrix variables Qj；
(4)double DyadicWavelet(double t,int m,int k)；
Uniform cubic B-spline function subject to the functionCorresponding dyadic wavelet basic function ψ (t, m, k) | t ∈ [0,1],
Wherein m is reasonable scale, for dyadic wavelet fairing, m=2j, k is translation scale, and meaning is in small echo spline space
K-th of spline wavelets, k ∈ [1,2j-1]；
(5)void DWFairing(matrix&Cj,int j,matrix&Cjj,matrix&Djj)；
The function is dyadic wavelet fairing function, can be according to the control vertex C of primitive curvejAnd corresponding scale j, calculate light
Along the control vertex C of rear curvej-1And corresponding detail section control vertex Dj-1, and result is saved in corresponding matrix and is become
It measures in Cjj, Djj；
(6)void DWRecon(matrix&Cj,int j,matrix&Cjj,matrix&Djj)；
The function is dyadic wavelet reconstruction of function, can be according to the control vertex C of curve to be reconstructedj-1And corresponding detail portion
Divide control vertex Dj-1, reconstruct the control vertex C of primitive curvej；It is emphasized that the scale j in this function is to wait for that reconstruct is bent
The binary scale of line.
3. the part curve and surface small echo method for fairing based on multiresolution analysis according to claim 1, it is characterised in that：Institute
The software implementation method for stating step (3) is to be realized by working out following arbitrary resolution small echo fairing function：
(1)double CubicBSpline(double t,int m,int k)；
The function defines a quasi- uniform cubic B-spline basic functionWherein m is reasonable scale, and k is flat
Scale is moved, meaning is k-th of batten in B-spline space, k ∈ [1, m+3]；
(2)double MultiBSplineBSpline(double t,vector&p)；
The function calculates product CubicBSpline (t, m, k) × CubicBSpline (t, n, l) between two spline functions,
In vector variable p, p [1]=m, p [2]=k, p [3]=n, p [4]=l；
(3)void InnerProductBB(matrix&BnBm,int m,int n)；
The function calculates inner product matrix BnBm=[<Φn|Φm>], and result is saved in matrix variables BnBm, integrating range is
[0,1]；
(4)void InitQmn(matrix&Qmn,matrix&BnBm,int m,int n)；
The function calculates restructuring matrix Q according to given reasonable scale m, nmn, and result is stored in variable Qmn；
(5)double BSplineWavelet(double t,int m,int k,matrix&Qmn)；
The function calculates B-spline wavelet basis function according to formula (8), and during returning from scale m to scale n fairing, kth item is small
Wave corresponds to the functional value of t；
(6)double MultiWaveletWavelet(double t,vector&p,matrix&Qmn)；
Product between the function two wavelet basis functions of calculating, BSplineWavelet (t, m, k, Qmn) ×
In BSplineWavelet (t, m, l, Qmn), vector variable p, p [1]=p [3]=m, p [2]=k, p [4]=l；
(7)double MultiWaveletBSpline(double t,vector&p,matrix&Qmn)；
Product between function calculating wavelet basis function and B-spline basic function, BSplineWavelet (t, m, l, Qmn) ×
In CubicBSpline (t, m, k), vector variable p, p [1]=p [3]=m, p [2]=k, p [4]=l；
(8)void InnerProductWW(matrix&WnWn,int m,int n)；
The function is used for calculating the inner product matrix W of wavelet basis functionnWn=[<Ψn|Ψn>], and result is saved in matrix variables
In WnWn, integrating range is [0,1]；
(9)void InnerProductWB(matrix&WnBm,int m,int n)；
The function is used for calculating the inner product matrix W between wavelet basis function and B-spline basic functionnBm=[<Ψn|Φm>], and will knot
Fruit is saved in matrix variables WnBm, and integrating range is [0,1]；
(10)void ARWFairing(matrix&Cm,matrix&Cn,matrix&Dn,int m,int n)；
The function is according to given scale m and n, by the control vertex C of primitive curvemResolve into the control vertex C of fair curven
With the control vertex D of detail sectionn, and result is saved in corresponding matrix variables Cn and Dn；
(11)void ARWRecon(matrix&Cm,matrix&Cn,matrix&Dn int m,int n)；
The function is according to given scale m and n, by the control vertex C of fair curvenWith the control vertex D of detail sectionn, accurately
It is reconstructed into the control vertex C of primitive curvem, and result is saved in matrix variables Cm.
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