CN102663424A - Total variation and euler elastic rod-based supervised mode identification method - Google Patents

Abstract

Disclosed in the invention is a total variation and euler elastic rod-based supervised mode identification method, comprising: constructing a total variation and Euler elastic rod-based energy function under the framework of least square regularization; utilizing a variational method to convert solution of energy function minimization into solution of a corresponding Euler-Lagrange differential equation; solving the differential equation to further obtain a final classifier; and carrying out mode identification on data by utilizing the classifier. According to the invention, a novel method is provided for solution of a supervised mode identification problem and in general, can be applied to solution of a classification problem like handwriting digit identification; and the provided method enables an effect that is comparable with one caused by an existing popular method to be realized for most of data sets.

Description

Based on total variation and Euler's elastic rod the supervised recognition method arranged

Technical field

The invention belongs to mode identification technology, be specifically related to least square regularization (RLS) framework that is applied to energy minimization in the Flame Image Process based on the model of total variation and Euler's elastic rod is used to have the pattern-recognition of supervision.

Background technology

The supervised learning technology is the big hot topic problem in the area of pattern recognition, has a wide range of applications in a plurality of fields such as computer visions.What its essence research was studied carefully in pattern-recognition is sorting technique, under situation about not producing ambiguity, refers to pattern-recognition with this term of classification in this instructions.Have supervised classification to refer to utilize the training sample set that label is arranged to train a kind of rule (sorter) new sample is predicted, formal description is as the one of which:

Given training set { (x ₁, y ₁) ... (x _n, y _n), x wherein _i∈ R ^d, R ^dFor the d dimension real vector space, for two types of classification problem y _i∈+1 ,-1}.Its essence is and to learn a mapping, the consideration of therefore can getting off the framework that classification problem is grouped into a function learning from vector x to label y.The general regularization framework that supervised classification is arranged can be expressed as:

$\underset{u}{\min }{\mathrm{\Σ}}_{i=1}^{n}L(u\left(x_i\right),y_i)+\mathrm{\λS}\left(u\right)$

Wherein u (x) is the mapping function that will learn, and n is the training sample number, and L represents loss function, and S (u) is that regular terms is used for some character of portrayal mapping u (x), and λ is a parameter, is used for the proportion of regulation loss item and regular terms.When L get predicted value u (x) and actual value y difference square the time, be exactly least square regularization framework, the present invention just is based on the framework under the continuous situation of this least square regularization.

Existing certain methods such as SVM, the hinge loss function of employing, what regular terms was punished is interfacial gap width, but it still is based on Statistical Learning Theory.Though statistical methods such as Bayesian inference commonly used have been obtained great successes; But also expose some shortcoming gradually; Thereby too much cause dimension disaster such as parameter; The classical statistical distribution functions and the actual distribution of True Data are inconsistent, are difficult to the probability density function of effective data estimator etc.Function learning method based on functional energy minimization more and more comes into one's own at present, the least squared classified (RLSC) that R.M.Rifkin proposed based on the Tikhonov regularization in 2003; Kush R in 2010 etc. utilize the method for level set to do the research of supervised classification, use for reference the Level Set Method in the image segmentation, and the surface area of decision boundary (the zero level collection of level set function) as the regularization energy functional, has all been obtained good effect.

The present invention sets up energy functional regularization model from the angle of background mathematics under a continuous framework, and utilizes instruments such as the functional analysis variational method, adopts rational method for solving to realize supervised classification.

In Flame Image Process, total variation image denoising model and Euler's elastic rod image mending model have all been obtained very big success, all are through setting up the functional model, utilizing mathematical measures such as the variational method to find the solution.Can be interpreted as existing label value (pixel value) on known data point to image denoising, image mending; Trying to achieve a function (image of requirement) predicts unknown point or the value on the point of pollution is arranged; The situation of this two dimension is risen to higher-dimension, just be similar to classification problem.The present invention combines the energy minimization framework of regularization to do supervised classification on these two kinds of models, and the regular terms that these two kinds of models provide makes and solves classification problem from the brand-new angle of another kind at the geometric properties of having portrayed the function of asking in varying degrees.

Summary of the invention

The objective of the invention is to propose a kind ofly new has a supervised recognition method based on the regularization framework.

Technical scheme of the present invention is following:

A kind of have a supervised recognition method based on total variation and Euler's elastic rod; This method is based on continuous least square regularization framework:

wherein x is that the proper vector that is used for a sample of pattern-recognition is represented; The function of u (x) for requiring; Y is the class label value that training data is concentrated sample x, and Ω is the residing field of definition of training sample, and λ is a parameter; S (u) is a regular terms

Said regular terms S (u) adopts the form of total variation and Euler's elastic rod, thereby obtains the new supervised recognition method that has.

Described have a supervised recognition method, comprises the steps (ginseng Fig. 1):

1) at first under said least square regularization framework, constructs energy functional based on total variation and Euler's elastic rod;

2) utilize the variational method that this energy functional minimization problem is converted into then and find the solution corresponding Euler-Lagrangian PDE;

3) said PDE are found the solution, and then obtained final sorter;

4) use this sorter that data are carried out pattern-recognition.

Described have a supervised recognition method, and in the step 3), the time search algorithm that adopts gradient to descend is found the solution said PDE, promptly increases a virtual time variable, adopts the gradient descent method that said PDE are carried out iterative.

Described have a supervised recognition method; In the step 3); Adopt the system of linear equations process of iteration that lags behind that said PDE are found the solution, promptly fix the non-linear partial of optimizing in the cost function, make it become a system of linear equations; Non-linear partial to lagging behind before upgrading again after the Solving Linear iterates and finds the solution.

Described have a supervised recognition method, and the energy functional of total variation described in the step 1) is:

$J_{\mathrm{TV}}\left[u\right]={\∫}_{\mathrm{\Ω}}{(u-y)}^2\mathrm{dx}+\mathrm{\λ}{\∫}_{\mathrm{\Ω}}|\▿u|\mathrm{dx}$

Wherein, x is that the proper vector that is used for a sample of pattern-recognition is represented, the function of u (x) for requiring, y are the class label value that training data is concentrated a sample, J _TV[u] expression is about the energy functional based on total variation of u (x), second

Be the total variation under the continuous situation of u (x), Ω is the field of definition at independent variable place,

Be gradient operator,

Represented the exemplary number of L2 of the gradient of function u (x), λ is can predefined regularization parameter.

Described have a supervised recognition method, and the energy functional of the elastic rod of Euler described in the step 1) is:

$J_{\mathrm{EE}}\left[u\right]={\∫}_{\mathrm{\Ω}}{(u-y)}^2\mathrm{dx}+\mathrm{\λ}{\∫}_{\mathrm{\Ω}}(a+b{\mathrm{\κ}}^2)|\▿u|\mathrm{dx},$ $\mathrm{\κ}=\▿\·\left(\frac{\▿u}{|\▿u|}\right)$

Wherein, x is that the proper vector that is used for a sample of pattern-recognition is represented, the function of u (x) for requiring, y are the class label value that training data is concentrated a sample, J _EE[u] representative is about the energy functional based on Euler's elastic energy of u (x), and κ is the mean curvature of the equivalent hypersurface (level set) of function u, and Ω is the field of definition at independent variable place,

Be the gradient of function u,

Represented the exemplary number of the gradient vector of function u (x),

Be the formal representation of divergence operator, λ is can a predefined regularization parameter, and a and b are two parameters of control total variation and Euler's elastic energy contribution degree.

Beneficial effect of the present invention: the present invention has proposed new method to the supervised recognition problem is arranged, and can be applied to classification problem generally speaking, for example Handwritten Digital Recognition; On most data sets, the method that the present invention proposes can reach the effect that compares favourably with existing popular approach.

Description of drawings

Fig. 1 is a process flow diagram of the present invention.

Fig. 2 is sample synoptic diagram and the classification results signal that the USPS handwriting digital is concentrated.

Embodiment

Embodiment of the present invention is following:

Embodiment one: the USPS Handwritten Digital Recognition

USPS (U.S.Postal Service) data set uses the handwriting digital picture that scans on the United States Post Office envelope, and every pictures is the gray scale picture of 16*16, comprises a numeral, has provided its partial data like Fig. 2 (a) and has showed, from numeral 0 to 9, totally 10 types.Present embodiment has therefrom been randomly drawed 1000 samples and has been done experiment, because the raw data dimension is higher, with principal component analytical method (PCA) it is reduced to 30 dimensions, and normalizes to [0,1] interval.

Step 1: structure energy functional under the regularization framework

(a) total variation energy functional

On the mathematics; The total variation of function f (x) (Total Variation) has form under one-dimensional case and when the function consecutive hours: the total variation of

expression f (x); A and b are interval end points, and f ' is a function derivative (x).Can see that total variation is the tolerance for the summation of all changes of function.

Will be about the total variation of function u (x) regular terms as the functional energy, the energy functional that supervised classification is arranged that obtains is:

$J_{\mathrm{TV}}\left[u\right]={\∫}_{\mathrm{\Ω}}{(u-y)}^2\mathrm{dx}+\mathrm{\λ}{\∫}_{\mathrm{\Ω}}|\▿u|\mathrm{dx}$

J wherein _TV[u] expression is about the energy functional based on total variation (TV is the abbreviation of Total Variation) of u (x), second

Be the total variation under the continuous situation of u (x), Ω is the zone at independent variable place,

Be gradient operator (for example

Represented the exemplary number of L2 of the gradient of function u (x), λ is a parameter.In image denoising, the effect of total variation regular terms is to keep the information at object edge place in the image.General denoising method tends to make the place, image border also by fuzzy, and the total variation model can suppress this fuzzy.Here in the energy functional of supervised classification; Not only restricted function variation summation is less for the total variation item; Make in classification, to obtain a mild function in most of zone, and allow bigger variation, make the border of classification more obvious at the boundary of class and class.Minimize this and can provide final sorter about the function that the energy functional of u obtains, the zero level collection of function is a classification boundaries.

(b) Euler's elastic rod energy functional

The model of Euler's elastic rod is generalized to having in the supervised classification of higher-dimension, and direct unlike the total variation that kind, the model in the image mending is:

$\underset{I}{\min }\mathrm{\λ}{\∫}_{\mathrm{\Ω}}(a+{\mathrm{b\κ}}^2)|\▿I|\mathrm{dxdy}+{\∫}_{\mathrm{\Ω}\backslash D}{(I-I_0)}^2\mathrm{dxdy}$

Wherein I is by being asked image, I ₀Will repair image for original, Ω is the entire image zone, and D is for wanting repairing area, and a, b are parameter, and κ is the medium high curvature of a curve of image, is in form:

$\mathrm{\κ}=\▿\·n,n=\frac{\▿I}{|\▿I|}$

is divergence operator; N is a normal vector outside the isocontour unit; is the gradient of image;

is the gradient-norm of image, is the exemplary number of L2 here.Euler's elastic rod (Euler ' s Elastica) is meant the curve γ that reaches following elastic energy equilibrium state: E ₂[γ]=∫ _γ(a+b κ ²) ds, ds is an arc length, κ is the curvature of γ.In the patch formation model first is the integration to all isocontour Euler's elastic energies in the image.Euler's elastic rod is applied to image mending, it is advantageous that the interpolation ability of elastic rod, this non-linear batten is a strong instrument of filling up the disappearance zone.When being generalized to higher-dimension to this elastic energy; Need to consider that curvature κ is generalized to the situation of higher-dimension, u under two-dimensional case (x, y)=0 (x here; Y is two-dimentional independent variable); Determine a level line y=u (x), its curvature then has form

u under three-dimensional situation (x, y; Z)=0 (x; Y, z are three-dimensional independent variable) what determine is an equivalent curved surface, and should have and the identical form of one-dimensional case lower curve curvature by the equivalence mean curvature of surface.Mean curvature of surface is the external tolerance of curved surface, and it has portrayed the degree of crook that curved surface is embedded into surrounding space.The situation of higher-dimension more is the equivalent hypersurface of function u by the hypersurface of u=0 decision, and Euler's elastic energy is moved to the elastic energy that higher-dimension is exactly these hypersurfaces of portrayal.Table 1 has provided curve or (surpassing) curved surface that determines through implicit function theorem from the expression formula of u, and their form for curvature, and its (on average) curvature has identical form except a constant factor relevant with dimension.

Table 1 is by curve (surpassing) curved surface and the curvature thereof of implicit function theorem decision

The present invention has provided the function learning energy functional of Euler's elastic rod regular terms thus:

$J_{\mathrm{EE}}\left[u\right]={\∫}_{\mathrm{\Ω}}{(u-y)}^2\mathrm{dx}+\mathrm{\λ}{\∫}_{\mathrm{\Ω}}(a+b{\mathrm{\κ}}^2)|\▿u|\mathrm{dx},$ $\mathrm{\κ}=\▿\·\left(\frac{\▿u}{|\▿u|}\right)$

Wherein, J _EE[u] representative is about the energy functional based on Euler's elastic energy (EE is the abbreviation of Euler ' s Elastica) of u (x), and κ is the mean curvature of the equivalent hypersurface of function u.Simplification for form; Still the form that keeps κ in the master pattern has here been saved the coefficient factor

of front and when calculating, this factor has been added on the weight b.Similar in the energy functional of all the other mark implications and TV before.First Squared Error Loss item restricted function be as far as possible near the actual value of training data in the energy functional, and the elastic energy of the equivalent hypersurface of regular terms restricted function u makes little that it tries one's best, and this has minimum elastic energy with regard to meaning the classification boundaries that finally obtains.If make the a=1 in the EE energy functional, b=0, this energy functional just degenerate becomes the TV energy functional that not have energy that consideration brings by the curvature item.

Step 2: utilize the variational method to transfer energy minimization to PDE

Utilize the variational method can the energy functional minimization problem of TV and EE be transformed into and find the solution corresponding Euler-Lagrangian PDE (PDE).The PDE that the total variation energy functional is corresponding is:

$\mathrm{\λ}\▿\·\left(\frac{\▿u}{|\▿u|}\right)-2(u-y)=0---\left(1\right)$

Notice that

is divergence operator here;

is the gradient of function u, and represented the exemplary number of the gradient vector of function u (x).This is a Nonlinear Elliptic Equations.For Euler's elastic rod energy functional, corresponding PDE is:

$\left\{\begin{array}{c}\mathrm{\λ}\▿\·V-2(u-y)=0\\ V=\mathrm{\φ}\left(\mathrm{\κ}\right)\frac{\▿u}{|\▿u|}-\frac{1}{|\▿u|}\▿\left({\mathrm{\φ}}^{\′}\left(\mathrm{\κ}\right)\right|\▿u\left|\right)+\frac{1}{{|\▿u|}^3}\▿u(\▿u\·\▿\left({\mathrm{\φ}}^{\′}\left(\mathrm{\κ}\right)\right|\▿u\left|\right))\end{array}\right.---\left(2\right)$

Wherein adopt mark φ (κ) :=a+b κ ², φ ' (κ) representes derivative about curvature κ.

Step 3: adopt two kinds of methods to find the solution, obtain final sorter

If adopt conventional numerical method (for example finite difference, finite element) can run into dimension disaster, and above-mentioned PDE (1) are very complicated with (2) form, temporarily do not have suitable method on the numerical value under the situation of higher-dimension for PDE.The present invention adopts the thought of approximation to function, and the function functional is minimized the minimization problem that is converted into about linear coefficient, and adopts the time search algorithm (GD) of gradient decline and two kinds of methods of system of linear equations process of iteration (IagLE) of hysteresis to find the solution.

Utilize the thought of approximation to function, function u (x) be expressed as the linear combination of some RBFs:

Wherein m is the number of basis function, w _iBe linear coefficient, Be gaussian radial basis function, c and z _iBe parameter, || x-z _i|| be the L2 norm of vector.Make w=(w ₁, w ₂..., w _m) ^TSo the problem of solved function u (x) becomes finds the solution linear coefficient w.Basis function when implementing

In z _iGet all training sample points.

Two kinds of methods are found the solution coefficient w:

(a) the time search algorithm (GD) of gradient decline

Because the total variation method is a special case of Euler's elastic energy method, from the find the solution of Euler's elastic energy method, finding the solution of total variation is similar.Introduce time variable t, from the PDE (2) that the variational method obtains, the gradient that obtains function u is:

$\frac{\∂u(x,t)}{\∂t}=\mathrm{\λ}\▿\·V(x,t)-2(u(x,t)-y)$

U is expressed as the form that matrix multiply by vector: u=Φ w; Wherein Φ is a matrix; Its element is fixed Φ for

; Convert the gradient of u the gradient of coefficient vector w into, so the iterative formula of gradient descent algorithm becomes:

$W^{(k+1)}=w^{\left(k\right)}-\mathrm{\τ}\frac{{\∂W}^{\left(k\right)}}{\∂t}=W^{\left(k\right)}-\mathrm{\τ}{\mathrm{\Φ}}^{-1}\left[\begin{array}{c}\frac{{\∂u}^{\left(k\right)}}{\∂t}|x=x_1\\ \·\\ \·\\ \·\\ \frac{{\∂u}^{\left(k\right)}}{\∂t}|x=x_n\end{array}\right]---\left(4\right)$

Wherein τ is a time step, in the formula

first

by following form calculate (through a series of derivations and omit u three rank and with upper derivate):

$\▿\·V=\mathrm{\κ}-\frac{4\mathrm{b\κ\Δu}}{{|\▿u|}^3}(\▿\mathrm{uH}\left(u\right))\·\frac{\▿u}{|\▿u|}{+\mathrm{b\κ}}^3+2b(\frac{2\mathrm{\Δu}}{{|\▿u|}^5}+\frac{\mathrm{\κ}}{{|\▿4|}^4})(\▿\mathrm{uH}\left(u\right))\·(\▿\mathrm{uH}\left(u\right))$

$+2b\{-\frac{3}{{|\▿u|}^5}(\▿\mathrm{uH}\left(u\right))\·\▿u+\frac{\▿u}{{|\▿u|}^3}\}(-\frac{2\mathrm{\Δu}}{{|\▿u|}^2}+\frac{\mathrm{\κ}}{|\▿u|})\▿u\·(\▿\mathrm{uH}\left(u\right)).$

Wherein

Δ is a Laplace operator, and H (u) is the Hessian battle array of u.When b gets zero, just can obtain the decline formula of total variation.According to above-mentioned iterative formula (4), initial value is elected w as ⁽⁰⁾=(Φ ^TΦ+η I) ^-1Φ ^TY, η are parameter, y=(y ₁, y ₂..., y _n) ^T, per step is upgraded w ^(k)Until its convergence.

(b) the system of linear equations process of iteration (IagLE) that lags behind

Because the Euler of TV Lagrange PDE (1) have better simply relatively form, at first derive from finding the solution of TV.The divergence operator of the nonlinear terms in the PDE formula (1)

is launched to obtain:

$-\frac{\mathrm{\λ}}{|\▿u|}(\mathrm{\Δu}-\frac{\▿u^{T}H\left(u\right)\▿u}{\▿u^T\▿u})+2(u-y)=0$

With form formula (3) the substitution following formula of the gaussian radial basis function of u combination, and with parameter z _iWith all training sample point x _iReplace, the number m of basis function just equals the number n of training sample at this moment, through obtaining following system of equations after a series of derivations:

$\left\{\begin{array}{c}\underset{i-1}{\overset{n}{\mathrm{\Σ}}}{w}_i{\mathrm{\φ}}_i\{d-1-c{|x-x_i|}^2+c\frac{g^T(x-x_i){(x-x_i)}^{T}g}{g^{T}g}+\frac{\left|g\right|}{\mathrm{\λ}}\}=\left|g\right|y\\ g=\underset{i-1}{\overset{n}{\mathrm{\Σ}}}{w}_i{\mathrm{\φ}}_i(x-x_i)\end{array}\right.---\left(5\right)$

Wherein d is the dimension of data, || be the L2 norm of vector, fix g can obtain system of linear equations about w.So utilize the thought that lags behind, the invention provides other a kind of iterative solution method, the system of linear equations process of iteration (IagLE) that is referred to as to lag behind:

1) gives initial value w at random of w ⁽⁰⁾Calculate g then ⁽⁰⁾

2) utilize least square method to find the solution system of linear equations (5), provide a new w about w ^(k)And calculating g ^(k)

3) iteration carries out the 2nd) go on foot and restrain until w.

Have complicated Eular-Lagrange equations (2) for the EE method, the item that will contain curvature is fixing, is designated as K=a+b κ ²So, can obtain similar system of equations:

$\left\{\begin{array}{c}\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{w}_i{\mathrm{\φ}}_i\{d-1-c{|x-x_i|}^2+c\frac{g^T(x-x_i){(x-x_i)}^{T}g}{g^{T}g}+\frac{\left|g\right|}{\mathrm{\λK}}\}=\left|g\right|y\\ g=\mathrm{\Σ}{w}_i{\mathrm{\φ}}_i(x-x_i)\\ K=a+b{(\frac{1}{\left|g\right|}\mathrm{\Σ}{w}_i{\mathrm{\φ}}_i(1-d+c{\left|\right|x-x_i\left|\right|}^2-c\frac{g^T(x-x_i){(x-x_i)}^{T}g}{g^{T}g}))}^2\end{array}\right.$

G and K are fixed the system of linear equations that can obtain about w.The same initial value of w of giving is used the alternative manner identical with TV, and per step is upgraded g and K restrains until w.

When finding the solution system of linear equations, suppose that this system of linear equations is Ψ w=y in form, Ψ is the matrix of in form simplifying.Because the mould of w makes system of linear equations become ill-condition sometimes very greatly, a reasonable solution strategies is the least square problem of finding the solution following regularization:

$\underset{w}{\min }\left|\right|\mathrm{\Ψw}-y{\left|\right|}_2^2+\mathrm{\η}{\left|\right|w\left|\right|}_2^2$

Wherein η is a regularization parameter, and it separates (the Ψ into w= ^TΨ+η I) ^-1Ψ ^TY.So three parameters are arranged in the system of linear equations process of iteration that lags behind: the parameter c in the gaussian radial basis function, regularization factor lambda in the energy functional and the parameter η that finds the solution system of linear equations.

With the linear combination expression formula (3) of the w substitution basis function that obtains, can obtain the approximate analysis expression formula of function u.The zero level collection of this approximate analysis expression formula is two types of classification boundaries, and what obtain at first is two types of sorters, when prediction, in sample substitution u, if functional value is 1 greater than zero prediction label, is-1 smaller or equal to zero label promptly.And totally 10 types of handwriting digitals among this embodiment are multicategory classification problems, adopt a kind of strategy of one-to-many commonly used, promptly train 10 two types of sorters, again these 10 two types of classifiers combination are obtained final sorter.What Fig. 2 (b) represented is the recognition result for Fig. 2 (a).

In solution procedure for the fixing b=0.01 of EE method; So all have only two parameter c and λ for the GD solution of TV and EE; The lagLE solution has three parameter c; λ and η adopt here data set five folding cross validations are selected optimized parameter, and carry out the contrast of classifying quality with SVM and BPNN (reverse transmittance nerve network algorithm).What table 2 provided is the accuracy rate contrast of the whole bag of tricks, and the accuracy rate here is the corresponding optimum accuracy rate of optimized parameter in each method five folding cross validation.Can find out that from table 2 the two kinds of methods effect on the USPS data set based on total variation and Euler's elastic energy that the present invention proposes has surpassed SVM and BPNN.

The comparing result (%) of table 2 the whole bag of tricks accuracy rate on the USPS data set

Embodiment two: be applied to some other common classification data

Present embodiment comprises two types and multi-class data based on the data set of 8 classification in Iibsvm website and the UCI machine learning storehouse.All normalize to [0,1] interval for all data sets, identical among operation steps and the embodiment one.All still choose optimized parameter with five folding cross validations for all methods, the optimum accuracy rate contrast of each method is as shown in table 3:

The classification accuracy contrast (%) of each method of table 3

The IagLE solution that from table 3, can find out TV that the present invention proposes and EE mode identification method is good than GD on the whole, and the effect of EE is slightly better than the effect of TV, and all the accuracy rate than BPNN is high on all data sets for they.The EE method is surpassing SVM under the IagLE solution on six data sets; Effect and SVM on all the other two data also are more or less the same, and can find out that the mode identification method based on total variation and Euler's elastic energy that the present invention proposes can compare favourably with very ripe now SVM algorithm.

Claims (6)

1. one kind has a supervised recognition method based on total variation and Euler's elastic rod; This method is based on continuous least square regularization framework:

wherein x is that the proper vector that is used for a sample of pattern-recognition is represented; The function of u (x) for requiring; Y is the class label value that training data is concentrated sample x, and Ω is the residing field of definition of training sample, and λ is a parameter; S (u) is a regular terms

It is characterized in that,

Said regular terms S (u) adopts the form of total variation and Euler's elastic rod, thereby obtains the new supervised recognition method that has.

2. as claimed in claim 1 have a supervised recognition method, it is characterized in that, comprises the steps:

1) at first under said least square regularization framework, constructs energy functional based on total variation and Euler's elastic rod;

2) utilize the variational method that this energy functional minimization problem is converted into then and find the solution corresponding Euler-Lagrangian PDE;

3) said PDE are found the solution, and then obtained final sorter;

4) use this sorter that data are carried out pattern-recognition.

3. as claimed in claim 2 have a supervised recognition method; It is characterized in that in the step 3), the time search algorithm that adopts gradient to descend is found the solution said PDE; Promptly increase a virtual time variable, adopt the gradient descent method that said PDE are carried out iterative.

4. as claimed in claim 2 have a supervised recognition method; It is characterized in that, in the step 3), adopt the system of linear equations process of iteration that lags behind that said PDE are found the solution; Promptly fix the non-linear partial of optimizing in the cost function; Make it become a system of linear equations, the non-linear partial to lagging behind before upgrading again after the Solving Linear iterates and finds the solution.

5. as claimed in claim 2 have a supervised recognition method, it is characterized in that the energy functional of total variation described in the step 1) is:

$J_{\mathrm{TV}}\left[u\right]={\∫}_{\mathrm{\Ω}}{(u-y)}^2\mathrm{dx}+\mathrm{\λ}{\∫}_{\mathrm{\Ω}}|\▿u|\mathrm{dx}$

Wherein, x is that the proper vector that is used for a sample of pattern-recognition is represented, the function of u (x) for requiring, y are the class label value that training data is concentrated a sample, J _TV[u] expression is about the energy functional based on total variation of u (x), second

Be the total variation under the continuous situation of u (x), Ω is the field of definition at independent variable place,

Be gradient operator,

Represented the exemplary number of the gradient vector of function u (x), λ is can a predefined regularization parameter.

6. as claimed in claim 2 have a supervised recognition method, it is characterized in that the energy functional of the elastic rod of Euler described in the step 1) is:

$J_{\mathrm{EE}}\left[u\right]={\∫}_{\mathrm{\Ω}}{(u-y)}^2\mathrm{dx}+\mathrm{\λ}{\∫}_{\mathrm{\Ω}}(a+b{\mathrm{\κ}}^2)|\▿u|\mathrm{dx},$ $\mathrm{\κ}=\▿\·\left(\frac{\▿u}{|\▿u|}\right)$

Wherein, x is that the proper vector that is used for a sample of pattern-recognition is represented, the function of u (x) for requiring, y are the class label value that training data is concentrated a sample, J _EE[u] representative is about the energy functional based on Euler's elastic energy of u (x), and κ is the mean curvature of the equivalent hypersurface of function u, and Ω is the field of definition at independent variable place,

Be the gradient of function u,

Represented the exemplary number of the gradient vector of function u (x),

Be the formal representation of divergence operator, λ is can a predefined regularization parameter, and a and b are two parameters of control total variation and Euler's elastic energy contribution degree.

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