CN105930648A - Gene expression programming (GEP) bidirectional prediction-based short elliptic arc fitting method - Google Patents

Abstract

The invention discloses a gene expression programming (GEP) bidirectional prediction-based short elliptic arc fitting method, belongs to the technical field of digital measurement, and relates to related knowledge in aspects of GEP prediction modeling and least square fit. The method comprises the step of firstly performing denoising processing on original observation data of a short elliptic arc by adopting a wavelet filtering method, and secondly performing bidirectional finite length prediction on the denoised observation data of the short elliptic arc by utilizing a constructed GEP prediction model, equivalently, extending the length of the short elliptic arc in essence. The invention proposes an adaptive definition method for a mutation rate of a GEP algorithm, which can effectively inhibit the occurrence of an algorithm prematurity phenomenon and increase the algorithm convergence speed. Finally, the predicted observation data of the short elliptic arc is fitted by adopting a weighted least square fit algorithm, so that the fitting precision and stability of the short elliptic arc are remarkably improved.

Description

A kind of short elliptic arc approximating method bi-directional predicted based on gene expression programming (GEP)

Technical field

The present invention " a kind of short elliptic arc approximating method bi-directional predicted based on gene expression programming (GEP) " belongs to digitized and surveys Amount technical field, relates to the theory in terms of GEP prediction modeling, least square fitting and knowledge.

Background technology

Oval as a member important in geometric primitive family, ellipse fitting algorithm is at mechanical engineering, military equipment, image It is used widely in the fields such as process, astronomical surveing, mine surveying, Aero-Space, health care.But, in a lot of situations Under, being limited by measuring condition, Test Cycle, digitized measurement equipment cannot obtain observation station on whole oval circumference, One section of elliptic arc can only be measured.Another, many measurement object (such as: component of machine) inherently one section of elliptic arc bodies. Elliptic arc matching, the fitting algorithm of the shortest elliptic arc is always the challenging difficulty that one, digitized measurement field is generally acknowledged Topic.Tracing it to its cause and be the part that short elliptic arc is whole ellipse, the length of elliptic arc is the shortest, then the sign being dropped Information is the most, and it is the poorest to the representativeness of whole ellipse, thus causes accuracy of measurement poor.Calculate about short elliptic arc matching Method, the most conventional has Direct Least Square method, Hough transform method, smallest subset method, Kalman Filtering method etc..Research finds, When elliptic arc length is less than the 1/4 of whole ellipse, along with the further reduction of elliptic arc length, said method is used to obtain The error of fitting of elliptic arc parameter (centre coordinate, semi-major axis, semi-minor axis, inclination angle) can the most significantly rise.

Summary of the invention

The technical problem to be solved in the present invention is: the error of fitting of existing short elliptic arc fitting algorithm is bigger, above-mentioned for solving Problem, it is provided that a kind of short elliptic arc approximating method bi-directional predicted based on gene expression programming (GEP).

It is an object of the invention to realize in the following manner:

A kind of short elliptic arc approximating method bi-directional predicted based on gene expression programming (GEP), comprises the following steps:

(1) method using wavelet field threshold filter carries out denoising to short elliptic arc observation data；Propose to utilize mixed threshold Function estimates wavelet coefficient, and adjustment factor, threshold value in defining wavelet threshold function are obtained its optimum by genetic algorithm；

(2) build GEP forecast model, utilize the aberration rate self adaptation definition method proposed, short elliptic arc observation data are carried out The two-way limit for length of having predicts；

(3) utilize the thought of " mediant estimation ", determine the weighting in short elliptic arc weighted least-squares fitting algorithm object function Coefficient；

(4) use weighted least-squares fitting algorithm that the observation data after prediction are fitted, obtain the characteristic parameter of elliptic arc.

Observe data for (x for one group that assumes short elliptic arc_i,y_i) i=1,2...N, according to the definition of Generalized Time sequence, it is considered as One time sequences y (n), n=1,2...N utilize Mallat algorithm to realize one-dimensinal discrete small wave transformation to y (n), obtain one group Wavelet coefficient w under each yardstick_j,k, use mixed threshold function to estimate wavelet coefficient, it is defined as follows:

The method of genetic algorithm optimizing obtains optimal adjustment factor a₀, optimum threshold value λ₀, specifically comprise the following steps that

B () carries out real coding to adjustment factor a and threshold value λ；

B () initializes colony；

(c) definition fitness function, and calculate the fitness value of each individuality, definition fitness function is: Here,For root-mean-square error, y (n) is primary signal,For signal after denoising；

(d) set algorithm end condition: object function f ＜ ε₀(such as: ε₀=10^-5), or limit maximum evolutionary generation；If Algorithm meets end condition, then output optimal adjustment coefficient, optimal threshold, obtains estimating little according to the mixed threshold function of definition Wave system number, wavelet reconstruction just obtains filtered data；

E () otherwise, carries out operatings of genetic algorithm to population, including: selecting, intersect, make a variation, selection opertor uses optimal save strategy Strategy, crossover operator uses two-point crossover method, and mutation operator uses Gaussian mutation operator, and sets the crossing-over rate of population, variation Rate；Estimate wavelet coefficientWavelet reconstruction, goes to step (c).

Short elliptic arc based on GEP algorithm observation data double-way has limit for length's prediction steps and method as described below:

Short elliptic arc observation data after (a) input denoising；

(b) defined function collection+,-, * ,/, { x, y}, arrange GEP parameter, including: collection of functions, connects letter for Q}, terminal symbol intersection Number, gene head length, number gene, Population Size, selection strategy, aberration rate, IS insert string rate, IS length of element, RIS Slotting string rate, RIS length of element, gene insert string rate, single-point recombination fraction, recombination fraction, gene recombinaton rates at 2；Produce initial kind Group；Each chromosome is carried out isometric linear string encoding；

C () is decoded operation to each chromosome；

D () definition fitness function, uses the form of Definition of relative error here, calculates the fitness value of each chromosome；

E () judges whether to meet algorithm end condition；

If f () meets algorithm end condition, export optimum individual, the most optimal prediction expression, utilize this prediction expression pair Short elliptic arc observation data carry out the two-way limit for length of having and predict, then terminate program；

If g () is unsatisfactory for algorithm end condition, determine conversation strategy；

H () carries out genetic manipulation, including: select operation, insert string operation, reorganization operation, mutation operation；

It is different from and aberration rate P is set_mG () is a constant, propose a kind of adaptive definition method: P_m(g)=P_m0(1+a)

Variation adjustment factorP_m0For default aberration rate, f (g-1) is the optimal adaptation angle value of previous generation, F (g) is the optimal adaptation angle value of Current generation, N_sFor the algebraically do not evolved continuously, so at the algorithm evolution initial stage, F (g-1) ＜ f (g), N_s=0, then P_m(g)=P_m0；The middle and late stage evolved, f (g-1) ≈ f (g), N_s＞ 0, then P_m(g) ＞ P_m0, and along with non-evolutionary generation N_sIncrease, P_mG () persistently becomes big；

I (), through the operation of various genetic operators, produces a new generation population；Go to step (c).

Relative to prior art, the invention is intended to illustrate a kind of new approaches improving short elliptic arc fitting precision and new method, mainly Contain the content of three aspects.One is to use wavelet filteration method that original short elliptic arc observation data are carried out denoising.Two Being to utilize GEP algorithm that the observation data after short elliptic arc denoising are carried out the two-way limit for length of having to predict, this essentially corresponds to extend The length of short elliptic arc, reduces the error of fitting of short elliptic arc parameter；Three is to use weighted least-squares fitting algorithm to prediction After short elliptic arc observation data be fitted, this weighted least-squares fitting algorithm by definition difference observation data sample at mesh Weight coefficient in scalar functions, can be effectively improved the capacity of resisting disturbance to those observation data that peel off.

Accompanying drawing explanation

Fig. 1 is the flow chart of whole short elliptic arc fitting algorithm.

Fig. 2 is that short elliptic arc is observed data de-noising flow chart by wavelet field threshold filter method.

Fig. 3 is the flow chart having limit for length to predict based on GEP algorithm short elliptic arc observation data double-way.

Fig. 4 is the weighted least-squares fitting algorithm flow chart of elliptic arc.

Detailed description of the invention

As Figure 1-Figure 4, the invention is intended to illustrate a kind of new approaches improving short elliptic arc fitting precision and new method, mainly Contain the content of three aspects.One is to use wavelet filteration method that original short elliptic arc observation data are carried out denoising.Two Being to utilize GEP algorithm that the observation data after short elliptic arc denoising are carried out the two-way limit for length of having to predict, this essentially corresponds to extend The length of short elliptic arc, reduces the error of fitting of short elliptic arc parameter；Three is to use weighted least-squares fitting algorithm to prediction After short elliptic arc observation data be fitted, this weighted least-squares fitting algorithm by definition difference observation data sample at mesh Weight coefficient in scalar functions, can be effectively improved the capacity of resisting disturbance to those observation data that peel off.Specific embodiment and Step is as follows:

(1) short elliptic arc observation data are carried out denoising

As illustrated in fig. 2, it is assumed that the one of short elliptic arc group of observation data is (x_i,y_i) i=1,2...N, according to determining of Generalized Time sequence Justice, is considered as time sequences y (n), n=1,2...N.Famous Mallat algorithm is utilized to realize the one-dimensional discrete to y (n) Wavelet transformation, obtains one group of wavelet coefficient w under each yardstick_j,k.For soft, the deficiency of hard threshold function, propose here Using mixed threshold function to estimate wavelet coefficient, its form of Definition is as follows:

This mixed threshold function avoids the hard threshold function simple zero setting to wavelet coefficient, too increases those more than threshold simultaneously The flexibility of the wavelet coefficient contracted transformation of value, shrinkage degree can be realized by adjustment factor.After determining threshold function table, Another one key issue is how to determine the size of threshold value.To this end, the method that the present invention proposes to use genetic algorithm optimizing is come Obtain optimal adjustment factor, optimum threshold value, and then, utilize the optimum adjustment coefficient obtained and optimal threshold to observation data Realize filtering.Specifically comprise the following steps that

A () carries out real coding to adjustment factor a and threshold value λ.Compared with binary coded form, real coding can effectively be kept away Exempt from the generation of " Hamming steep cliff " phenomenon, and, it is high that real coding has precision, it is simple to the advantages such as large space search.

B () initializes colony.Set population scale, the initialized colony of stochastic generation.

(c) definition fitness function, and calculate the fitness value of each individuality.Fitness function is asked for whole genetic algorithm Solving particularly significant, can choosing of fitness function directly influence the convergence rate of genetic algorithm and find optimal solution.Individual Quality be exactly based on what the fitness function value of its correspondence embodied.Here, definition fitness function is:

Here,For root-mean-square error, y (n) is primary signal,For signal after denoising. With root-mean-square error as object function, root-mean-square error is the least, illustrates that filtered signal is closer to actual signal.

(d) set algorithm end condition: object function f ＜ ε₀(such as: ε₀=10^-5), or limit maximum evolutionary generation.If Algorithm meets end condition, then output optimal adjustment coefficient a₀, optimal threshold λ₀, estimated according to the mixed threshold function of definition Meter wavelet coefficientWavelet reconstruction just obtains filtered data.

E () otherwise, carries out operatings of genetic algorithm to population.Including: selecting, intersect, make a variation, selection opertor uses optimal save strategy Strategy, crossover operator uses two-point crossover method, and mutation operator uses Gaussian mutation operator, and sets the crossing-over rate of population, variation Rate.Estimate wavelet coefficientWavelet reconstruction, goes to step (c).

Meanwhile, according to the research experience of applicant, if it is possible to determine the value of optimizing parameter according to known conditions or priori Interval, so will be obviously improved the convergence rate of genetic algorithm.Such as, in the definition of mixed threshold function, it is known that regulation system Number a ∈ [01], so, during utilizing genetic algorithm for solving, it is possible to limits the span of a as [01].

(2) the short elliptic arc after denoising is observed data and is carried out the two-way limit for length of having and predict by GEP algorithm

The flow chart that short elliptic arc based on GEP algorithm observation data double-way has limit for length to predict is shown in Fig. 3.Detailed step and method are such as Lower described:

Short elliptic arc observation data after (a) input denoising.

(b) defined function collection+,-, * ,/, { x, y}, arrange GEP parameter, including: collection of functions, connects letter for Q}, terminal symbol intersection Number, gene head length, number gene, Population Size, selection strategy, aberration rate, IS insert string rate, IS length of element, RIS Slotting string rate, RIS length of element, gene insert string rate, single-point recombination fraction, recombination fraction, gene recombinaton rates at 2.Produce initial kind Group.Each chromosome is carried out isometric linear string encoding.

C () is decoded operation to each chromosome.

D () definition fitness function, uses the form of Definition of relative error here, calculates the fitness value of each chromosome.

E () judges whether to meet algorithm end condition.Can be by arranging what maximum evolutionary generation or algorithm solved after the most how many generations Fitness terminates without algorithm during significant change.

If f () meets algorithm end condition, export optimum individual, the most optimal prediction expression, utilize this prediction expression pair Short elliptic arc observation data carry out the two-way limit for length of having and predict.Then program is terminated.

If g () is unsatisfactory for algorithm end condition, determine conversation strategy.As: can optimum maintaining strategy

H () carries out genetic manipulation.Including: select operation, insert string operation, reorganization operation, mutation operation.

It is different from and aberration rate P is set_mG () is a constant, the present invention proposes a kind of adaptive P_m(g) definition method:

P_m(g)=P_m0(1+a)

Variation adjustment factorP_m0For default aberration rate, f (g-1) is the optimal adaptation angle value of previous generation, F (g) is the optimal adaptation angle value of Current generation, N_sFor the algebraically do not evolved continuously.So at the algorithm evolution initial stage, F (g-1) ＜ f (g), N_s=0, then P_m(g)=P_m0；The middle and late stage evolved, f (g-1) ≈ f (g), N_s＞ 0, then P_m(g) ＞ P_m0, and along with non-evolutionary generation N_sIncrease, P_mG () persistently becomes big.

I (), through the operation of various genetic operators, produces a new generation population.Go to step (c).

(3) realization of elliptic arc weighted least square algorithm

Assume that the pretreated short elliptic arc observation data obtained according to above-mentioned two step are (x_i y_i) i=1,2 ... K, altogether K group is had to observe data.

A () uses " 5 fixed ellipses " method to obtainGroup elliptic arc parameter.

B () calculates the center often organizing elliptic arc to the distance of zero:

$r_{oi}=\sqrt{x_{oi}^2+y_{oi}^2},i=1,2,...C_N^5$

Here, (x_oi,y_oi) represent the centre coordinate of i-th group of elliptic arc.Use mediant estimation rule, select r_med=med (r_oi) institute Corresponding elliptic arc parameter is as initial elliptic arc parameter.

C () assumes r_oNormal Distribution, estimates to obtain its normalized probability density function.

$f\left(r_o\right)=e^{-\frac{{(r_o-u)}^2}{2{\mathrm{\σ}}^2}}$

Here, u is average, and σ is standard deviation.

D (), according to 3 σ principles, is distributed in the data sample outside [u-3 σ u+3 σ] for those, it is believed that they are thick Error, putting corresponding weight coefficient is zero, is rejected.Data sample within [u-3 σ u+3 σ] is distributed in for those, According to the probability density function obtained, give corresponding weight coefficient.

The weighted least-squares object function of (e) definition elliptic arc.

$f(A,B,C,D,E,F)=\underset{i=1}{\overset{K}{\Σ}}{k}_i{\[{\mathrm{Ax}}_i^2+{\mathrm{Bx}}_i{y}_i+{\mathrm{Cy}}_i^2+{\mathrm{Dx}}_i+{\mathrm{Ey}}_i+F\]}^2$

F () uses classical Levenberg-Marquardt (LM) iterative algorithm, solve object function, obtain Whole short elliptic arc parameter.Select the elliptic arc initial estimation parameter obtained by mediant estimation algorithm initial as the iteration of LM algorithm Value, so can effectively accelerate LM convergence of algorithm speed.

Claims (4)

1. one kind based on the bi-directional predicted short elliptic arc approximating method of gene expression programming (GEP), it is characterised in that: include following step Rapid: (1) uses the method for wavelet field threshold filter that short elliptic arc observation data are carried out denoising；Propose to utilize mixed threshold Function estimates wavelet coefficient, and adjustment factor, threshold value in defining wavelet threshold function are obtained its optimum by genetic algorithm；

(2) build GEP forecast model, utilize the aberration rate self adaptation definition method proposed, short elliptic arc observation data are carried out two-way Limit for length is had to predict；

(3) utilize the thought of " mediant estimation ", determine the weight coefficient in short elliptic arc weighted least-squares fitting algorithm object function；

(4) use weighted least-squares fitting algorithm that the observation data after prediction are fitted, obtain the characteristic parameter of elliptic arc.

One the most according to claim 1 based on the bi-directional predicted short elliptic arc approximating method of gene expression programming (GEP), its It is characterised by: assume that one group of short elliptic arc is observed data for (x_i,y_i) i=1,2...N, according to the definition of Generalized Time sequence, can Regarding it as time sequences y (n), n=1,2...N utilize Mallat algorithm to realize the one-dimensinal discrete small wave transformation to y (n), obtain One group of wavelet coefficient w under each yardstick_j,k, use mixed threshold function to estimate wavelet coefficient, it is defined as follows:

One the most according to claim 2 based on the bi-directional predicted short elliptic arc approximating method of gene expression programming (GEP), its It is characterised by: the method for genetic algorithm optimizing is to obtain optimal adjustment factor a₀, optimum threshold value λ₀, specifically comprise the following steps that

A () carries out real coding to adjustment factor a and threshold value λ；

B () initializes colony；

(c) definition fitness function, and calculate the fitness value of each individuality, definition fitness function is:This In,For root-mean-square error, y (n) is primary signal,For signal after denoising；

(d) set algorithm end condition: object function f ＜ ε₀(such as: ε₀=10^-5), or limit maximum evolutionary generation；If algorithm Meet end condition, then output optimal adjustment coefficient, optimal threshold, obtain estimating wavelet systems according to the mixed threshold function of definition Number, wavelet reconstruction just obtains filtered data；

E () otherwise, carries out operatings of genetic algorithm to population, including: selecting, intersect, make a variation, selection opertor uses optimum maintaining strategy, Crossover operator uses two-point crossover method, and mutation operator uses Gaussian mutation operator, and sets the crossing-over rate of population, aberration rate；Estimate Meter wavelet coefficientWavelet reconstruction, goes to step (c).

One the most according to claim 1 based on the bi-directional predicted short elliptic arc approximating method of gene expression programming (GEP), its It is characterised by: short elliptic arc based on GEP algorithm observation data double-way has limit for length's prediction steps and method as described below:

Short elliptic arc observation data after (a) input denoising；

(b) defined function collection+,-, * ,/, Q}, terminal symbol intersection x, y}, arrange GEP parameter, including: collection of functions, connectivity function, Gene head length, number gene, Population Size, selection strategy, aberration rate, IS insert string rate, IS length of element, RIS insert String rate, RIS length of element, gene insert string rate, single-point recombination fraction, recombination fraction, gene recombinaton rates at 2；Produce initial population； Each chromosome is carried out isometric linear string encoding；

C () is decoded operation to each chromosome；

D () definition fitness function, uses the form of Definition of relative error here, calculates the fitness value of each chromosome；

E () judges whether to meet algorithm end condition；

If f () meets algorithm end condition, export optimum individual, the most optimal prediction expression, utilize this prediction expression to short ellipse Circular arc observation data carry out the two-way limit for length of having and predict, then terminate program；

If g () is unsatisfactory for algorithm end condition, determine conversation strategy；

H () carries out genetic manipulation, including: select operation, insert string operation, reorganization operation, mutation operation；

It is different from and aberration rate P is set_mG () is a constant, propose a kind of adaptive definition method: P_m(g)=P_m0(1+a)

Variation adjustment factorP_m0For default aberration rate, f (g-1) is the optimal adaptation angle value of previous generation, F (g) is the optimal adaptation angle value of Current generation, N_sFor the algebraically do not evolved continuously, so at the algorithm evolution initial stage, F (g-1) ＜ f (g), N_s=0, then P_m(g)=P_m0；The middle and late stage evolved, f (g-1) ≈ f (g), N_s＞ 0, then P_m(g) ＞ P_m0, and along with non-evolutionary generation N_sIncrease, P_mG () persistently becomes big；

I (), through the operation of various genetic operators, produces a new generation population；Go to step (c).