CN107038456A - A kind of image classification method of the probability linear discriminant analysis based on L1 norms - Google Patents
A kind of image classification method of the probability linear discriminant analysis based on L1 norms Download PDFInfo
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- G06F18/21—Design or setup of recognition systems or techniques; Extraction of features in feature space; Blind source separation
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- G06F18/2415—Classification techniques relating to the classification model, e.g. parametric or non-parametric approaches based on parametric or probabilistic models, e.g. based on likelihood ratio or false acceptance rate versus a false rejection rate
Abstract
The present invention discloses a kind of image classification method of the probability linear discriminant analysis based on L1 norms, can solve the problem that the problem of there is exceptional value in image.From unlike traditional PLDA, the present invention describes noise using laplacian distribution, and Laplce is the probability density function based on L1 norms, it will not fault in enlargement value.By introducing hidden variable, the parameter in variation greatest hope solving model, and dimensionality reduction matrix are used.Matrix after dimensionality reduction is regarded to the feature of sample as, of the invention is that L1 norms describe error in this model, and the dimensionality reduction matrix so solved can lift image classification effect closer to principal direction.
Description
Technical field
The invention belongs to machine learning techniques field, more particularly to a kind of probability linear discriminant analysis based on L1 norms
Image classification method, is particularly suitable for carrying the image classification of exceptional value in image.
Background technology
In to image procossing, image vector is often melted into the data of a higher-dimension.However, high dimensional data is often
It is evenly distributed on a lower dimensional space or popular world.So, find mapping relations of the high dimensional data into lower dimensional space
As a major issue to image classification.Recent decades, the algorithm of Data Dimensionality Reduction has been furtherd investigate.To linear discriminant
It is a kind of dimension reduction method for being widely used in image classification to analyze (LDA).LDA is to be mapped to high dimensional data using projection matrix
Lower dimensional space so that class spacing after mapping with class away from ratio it is maximum.LDA is a kind of method based on algebraically, and algebraically is asked
Solution method only relies on initial data, does not assume any parameter, and the typically degree of belief to model lacks flexibility.
In order to overcome the defect of algebraically, probability linear discriminant analysis (PLDA) method is proposed within 2007.In this model
In, image data table is shown as one-dimensional vector, it is assumed that noise is to obey zero-mean, and covariance is the Gaussian Profile of unit matrix.
PLDA is the theoretical log of applied probability according to dimensionality reduction, solution dimensionality reduction matrix.In the case where noise is the hypothesis of Gaussian Profile, algorithm is obtained
Preferable classification results.
In actual treatment image, the distribution of noise is often complicated, differs and establishes a capital Gaussian distributed, when in image
When there is exceptional value, above method to image when classifying, and accuracy rate will be affected.Because either LDA, goes back
It is PLDA, algorithm is all based on L2- norms, and the method for L2- norms has an obvious defect：L2- norms can be by image
In exceptional value infinitely amplify, when projecting image onto lower dimensional space, the projection matrix gone out with Algorithm for Solving can deviate from very
Positive principal direction.
The content of the invention
The technical problem to be solved in the present invention is to provide a kind of image of the probability linear discriminant analysis based on L1 norms point
Class method, can solve the problem that the problem of there is exceptional value in image.From unlike traditional PLDA, this method is with Laplce point
Cloth describes noise, and Laplce is the probability density function based on L1- norms, it will not fault in enlargement value.It is hidden by introducing
Variable, uses the parameter in variation greatest hope solving model, and dimensionality reduction matrix.Regard the matrix after dimensionality reduction as sample
Feature, of the invention is that L1- norms describe error in this model, the dimensionality reduction matrix so solved closer to principal direction,
Image classification effect can be lifted.
To achieve the above object, the present invention is adopted the following technical scheme that：
A kind of image classification method of the probability linear discriminant analysis based on L1 norms, comprises the following steps：
Step 1, the image to acquisition set up probability linear discriminant analysis model (PLDA)
OrderIt is one group of independent identically distributed image set, wherein have I classes,
Per class JiIndividual image pattern, and J1+...Ji+...+JI=N, each sample is a column vector and size is RD, D be image to
Line number after quantization, PLDA model is represented by：
xij=μ+Fhi+Gwij+εij (1)
Wherein, vector xijRepresent j-th of image of the i-th class in sample set；μ(μ∈RD) be image set X average；εijFor by mistake
Poor item, F (F ∈ RD×d) and G (G ∈ RD×d) be between class respectively and class in dimensionality reduction matrix, d≤D is the columns after dimensionality reduction；hi(hi
∈Rd) and wij(wij∈Rd) it is xijHidden variable core, i.e. coefficient vector, RdIt is the image feature vector after dimensionality reduction, hiRepresent body
Part variable, for the i-th class, represents the general character of similar image, for wij, represent that class internal variable represents in similar image
Property；
Step 2, set up L1-PLDA models
Under conditions of exceptional value presence, to sample dimensionality reduction, after by center of a sample, PLDA models can be expressed as：
Wherein, by dimensionality reduction matrix merges between class and in class, i.e. B=[F, G] (B ∈ RD ×2d)；The coefficient of dimensionality reduction matrix is closed
And, it is designated asRepresent the hidden variable of j-th of sample of the i-th class；Error term εijObey the drawing of L1- norms
This distribution of pula, it has identical dimension (R with sample dataD)；Assuming that each element in noise vector is independent same
The laplacian distribution of distribution, its probability density function is：
Wherein,The absolute value sum of each element in error vector, σ is error term εijYardstick
Parameter,
For noise item, give parameter σ mono- prior distribution, if it obeys gamma distribution, make the σ of ρ=1/2, then ρ distribution
For：
Wherein, aρIt is form parameter, bρIt is ρ scale parameter；
Step 3, the variation greatest hope of L1-PLDA models are solved
For the variable in model (2), it is estimated using maximum likelihood function, dimensionality reduction matrix { F, G } is solved,
Remember y={ y11,...y1J1；...；yi1,...,yiJi；...；yI1,...,yIJIIt is all hidden variable in model；
Θ={ B, aρ,bρBe model in parameter set,
The form of unlimited Gaussian Profile sum can be expanded to distribution in (3) formula, i.e.,：
(5)
Wherein, β is image detection operator, and η (β) is equal to summation coefficient, is expressed as：
Under the hypothesis of (3)-(5) formula, for given image setBy to hidden variable y and parameter Θ
Edge distribution quadrature, the object function of model can be obtained：
Using the lower bound Ω (Q (y, ρ, β), Θ) of the object function of EM algorithm maximization models, walked in E and update y, ρ and β
Posterior distrbutionp, matrix { B } is updated in M steps, and alternating iteration is walked in E steps and M, stops changing when making object function increase and tend towards stability
In generation, β value is exported, dimensionality reduction matrix is extracted from matrix { B }；
Step 4, image classification
Based on dimensionality reduction matrix F and G, image is classified using Similarity Measures, i.e., two images of calculating is similar
Degree, when this value is less than given threshold value, just provides that two images belong to same class.
Preferably, the model that likelihood function is solved for there is hidden variable generally utilizes greatest hope (EM) algorithm, mould
The lower bound of the object function of type is：
Wherein, Q (y, ρ, β) is hidden variable y, parameter ρ, an APPROXIMATE DISTRIBUTION function of β Posterior distrbutionps；Ω(Q(y,ρ,β),
Θ) be object function lower bound.
Preferably, it is as follows to calculate two image similarities：
If image x1,x2, the probability distribution p (x of two images1) and p (x2) be：
p(x1)=G ' (0, BBT+β)
p(x2)=G ' (0, BBT+β)
Wherein, G ' is gauss of distribution function.
Calculate x1,x2Joint probability distribution p (x1,x2)：
p(x1,x2)=G ' (0, AAT+β')
Wherein,
Dimensionality reduction matrix F and G are calculated by EM algorithms and obtained in β and matrix A, calculate the similarity R of two images：
In the present invention, solution annual reporting law is mainly paid attention to exceptional value sensitive issue in image.On the basis of PLDA,
The present invention replaces Gaussian Profile with laplacian distribution to assume the distribution of noise.Because L1- norms are insensitive to exceptional value, this
The projecting direction that sample is found to image is closer to principal direction.
Brief description of the drawings
Fig. 1 is the flow chart of the image classification method of the probability linear discriminant analysis based on L1 norms；
Fig. 2 is the schematic diagram of rejecting outliers in face database in embodiment；
Fig. 3 is the schematic diagram of Feret storehouses classification results.
Embodiment
As shown in figure 1, the embodiment of the present invention provides a kind of image classification of the probability linear discriminant analysis based on L1 norms
Method, comprises the following steps：
Step 1, the image to acquisition set up probability linear discriminant analysis model (PLDA)
OrderIt is one group of independent identically distributed image set, wherein have I classes,
Per class JiIndividual image pattern, and J1+...Ji+...+JI=N, each sample is a column vector and size is RD, D be image to
Line number after quantization, PLDA model is represented by：
xij=μ+Fhi+Gwij+εij (1)
Wherein, vector xijRepresent j-th of image of the i-th class in sample set；μ(μ∈RD) be image set X average, in reality
In operation, data are done with a pretreatment for subtracting average, data center is allowed；F(F∈RD×d) and G (G ∈ RD×d) be respectively
Dimensionality reduction matrix between class and in class, d≤D is the columns after dimensionality reduction；hi(hi∈Rd) and wij(wij∈Rd) it is xijHidden variable core,
That is coefficient vector, RdIt is the image feature vector after dimensionality reduction；εijFor error term.In PLDA models, make as follows for hidden variable
A priori assumption：
hi:N(hi|0,Id)
wij:N(wij|0,Id)
Wherein, hiRepresentative capacity variable, for the i-th class, it keeps constant, the general character of similar image is represented, so often
Class image only one of which identity variable；For wij, it represents that class internal variable represents the individual character in similar image, in the i-th class
Change with different samples；IdFor d*d unit matrix.
Step 2, set up L1-PLDA models
Present invention understands that PLDA Gaussian Profile is based on L2- norms, quadratic term can exaggerate the effect of exceptional value, for
The principal direction that such data PLDA is found is irrational.Therefore L1-PLDA models proposed by the present invention, first order will not be right
Exceptional value produces influence, so that the principal direction found is more reasonable.
Present invention contemplates that under conditions of exceptional value presence, to sample dimensionality reduction, after by center of a sample, model can
To be expressed as：
Wherein, by dimensionality reduction matrix merges between class and in class, i.e. B=[F, G] (B ∈ RD×2d)；The coefficient of dimensionality reduction matrix also may be used
To merge, it is designated asRepresent the hidden variable of j-th of sample of the i-th class；Error term εijObey L1- norms
Laplacian distribution, it has identical dimension (R with sample dataD)；Assuming that each element in noise vector is only
The laplacian distribution with distribution is found, its probability density function is：
Wherein,The absolute value sum of each element in error vector, σ is error term εijYardstick
Parameter.
For noise item, (whether the noise item is error term εij), give parameter σ mono- prior distribution, if it obeys gamma point
Cloth, calculates make the σ of ρ=1/ for convenience2, then ρ be distributed as：
Here aρIt is form parameter, bρIt is ρ scale parameter.
Step 3, the variation greatest hope of L1-PLDA models are solved
For the variable in model (2), it is estimated using maximum likelihood function, dimensionality reduction matrix { F, G } is solved
Sample can just be classified.
NoteFor hidden variable all in model；Θ={ B, aρ,bρ}
For the set of parameter in model.
The form of unlimited Gaussian Profile sum can be expanded to distribution in (3) formula, i.e.,：
Wherein, β is image detection operator, and η (β) is equal to summation coefficient, is expressed as：
Under the hypothesis of (3)-(5) formula, for given image setBy to hidden variable y and parameter Θ
Edge distribution quadrature, the object function of model can be obtained：
Greatest hope (EM) algorithm is generally utilized for this model that there is hidden variable solution likelihood function, EM algorithms lead to
The lower bound that Jensen inequality finds likelihood function is crossed, the actual value that lower limit function approaches likelihood function is then maximized,
The lower bound of object function：
Wherein, the second row make use of Jensen inequality, and Q (y, ρ, β) is hidden variable y, parameter ρ, one of β Posterior distrbutionps
APPROXIMATE DISTRIBUTION function；Ω (Q (y, ρ, β), Θ) is the lower bound of object function.
Using EM algorithms maximization function Ω (Q (y, ρ, β), Θ), the Posterior distrbutionp for updating y, ρ and β is walked in E, in M steps more
New matrix { B }.Alternating iteration is walked in E steps and M, stops iteration when object function is increased and is tended towards stability, β value is exported, from square
Battle array { B } extraction dimensionality reduction matrix F, G, wherein, by β, when there is exceptional value in data, the rejecting outliers of image are come out.
Step 4, image classification
Image is classified using Similarity Measures, that is, calculates the similarity of two images, is given when this value is less than
During fixed threshold value, just provide that two images belong to same class；Wherein, two image similarities are calculated as follows：
If image x1,x2, the probability distribution p (x of two images1) and p (x2) be：
p(x1)=G ' (0, BBT+β)
p(x2)=G ' (0, BBT+β)
Wherein, G ' is gauss of distribution function (G being changed into G ', if accurate).
Calculate x1,x2Joint probability distribution p (x1,x2)：
p(x1,x2)=G ' (0, AAT+β')
Wherein,
Dimensionality reduction matrix F and G are calculated by EM algorithms and obtained in β and matrix A, calculate the similarity R of two images：
Whether the description of above yellow flag is accurate
Embodiment 1
The image classification method experimental result of the present invention is as follows：
1st, L1-PLDA rejecting outliers result
For face database, by taking Yale storehouses as an example, illustrate β detection effect；It can be seen that from the experimental result in Fig. 2
When being blocked in face in the presence of one piece, display β image can accurately find the position for blocking block.
2nd, L1-PLDA classification results
The present invention, which has been applied to L1-PLDA algorithms in face database, to be classified, and achieves obvious effect.With
Exemplified by Yale storehouses and Feret storehouses, the effect of classification is illustrated.
1) Yale storehouses have 15 classes, and everyone has 11 pictures, and the present invention is from 8 training in an experiment, and 3 are tested,
Every class in training set blocks block with the presence of 2 pictures；The gray-scale map that it is 64 × 64 that image is unified.By the method for the present invention
(L1-PLDA) it is compared with other three kinds of methods, is PLDA, Fisher face (LDA) and based on L1 norms respectively
Linear discriminant analysis (LDA-L1), method of the invention has obvious advantage in classification, as shown in table 1.
The Yale storehouses classification results of table 1
2) 50 classes are randomly selected from Feret storehouses, everyone there are 7 pictures, the present invention is from 5 training, 2 in an experiment
Test is opened, every class in training set blocks block with the presence of 1 pictures；The gray-scale map that it is 50 × 50 that image is unified.By the present invention
Method (L1-PLDA) and PLDA, LDA, LDA-L1, be compared, method of the invention has obvious advantage in classification, such as
Shown in Fig. 3.
Claims (3)
1. a kind of image classification method of the probability linear discriminant analysis based on L1 norms, it is characterised in that comprise the following steps：
Step 1, the image to acquisition set up probability linear discriminant analysis model (PLDA)
OrderIt is one group of independent identically distributed image set, wherein having I classes, per class Ji
Individual image pattern, and J1+...Ji+...+JI=N, each sample is a column vector and size is RD, D is after image vector
Line number, PLDA model is represented by：
xij=μ+Fhi+Gwij+εij (1)
Wherein, vector xijRepresent j-th of image of the i-th class in sample set；μ(μ∈RD) be image set X average；εijFor error term,
F(F∈RD×d) and G (G ∈ RD×d) be between class respectively and class in dimensionality reduction matrix, d≤D is the columns after dimensionality reduction；hi(hi∈Rd)
And wij(wij∈Rd) it is xijHidden variable core, i.e. coefficient vector, RdIt is the image feature vector after dimensionality reduction, hiRepresentative capacity becomes
Amount, for the i-th class, represents the general character of similar image, for wij, represent that class internal variable represents the individual character in similar image；
Step 2, set up L1-PLDA models
Under conditions of exceptional value presence, to sample dimensionality reduction, after by center of a sample, PLDA models can be expressed as：
Wherein, by dimensionality reduction matrix merges between class and in class, i.e. B=[F, G] (B ∈ RD×2d)；The coefficient of dimensionality reduction matrix merges, note
ForRepresent the hidden variable of j-th of sample of the i-th class；Error term εijObey the Laplce of L1- norms
Distribution, it has identical dimension (R with sample dataD)；Assuming that each element in noise vector is independent identically distributed
Laplacian distribution, its probability density function is：
Wherein,The absolute value sum of each element in error vector, σ is error term εijScale parameter,
For noise item, give parameter σ mono- prior distribution, if it obeys gamma distribution, make the σ of ρ=1/2, then ρ be distributed as：
Wherein, aρIt is form parameter, bρIt is ρ scale parameter；
Step 3, the variation greatest hope of L1-PLDA models are solved
For the variable in model (2), it is estimated using maximum likelihood function, dimensionality reduction matrix { F, G } is solved,
Remember y={ y11,...y1J1；...；yi1,...,yiJi；...；yI1,...,yIJIIt is all hidden variable in model；Θ=
{B,aρ,bρBe model in parameter set,
The form of unlimited Gaussian Profile sum can be expanded to distribution in (3) formula, i.e.,：
Wherein, β is image detection operator, and η (β) is equal to summation coefficient, is expressed as：
1
Under the hypothesis of (3)-(5) formula, for given image setPass through the side to hidden variable y and parameter Θ
Fate cloth is quadratured, and can obtain the object function of model：
Using the lower bound Ω (Q (y, ρ, β), Θ) of the object function of EM algorithm maximization models, the posteriority for updating y, ρ and β is walked in E
Distribution, matrix { B } is updated in M steps, walks alternating iteration in E steps and M, iteration is stopped when object function is increased and is tended towards stability,
β value is exported, dimensionality reduction matrix is extracted from matrix { B }；
Step 4, image classification
Based on dimensionality reduction matrix F and G, image is classified using Similarity Measures, that is, calculates the similarity of two images, when
When this value is less than given threshold value, just provide that two images belong to same class.
2. the image classification method of the probability linear discriminant analysis as claimed in claim 1 based on L1 norms, it is characterised in that
The model that likelihood function is solved for there is hidden variable generally utilizes greatest hope (EM) algorithm, the lower bound of the object function of model
For：
Wherein, Q (y, ρ, β) is hidden variable y, parameter ρ, an APPROXIMATE DISTRIBUTION function of β Posterior distrbutionps；Ω(Q(y,ρ,β),Θ)
It is the lower bound of object function.
3. the image classification method of the probability linear discriminant analysis as claimed in claim 1 based on L1 norms, it is characterised in that
Calculate two image similarities as follows：
If image x1,x2, the probability distribution p (x of two images1) and p (x2) be：
p(x1)=G ' (0, BBT+β)
p(x2)=G ' (0, BBT+β)
Wherein, G ' is gauss of distribution function.
Calculate x1,x2Joint probability distribution p (x1,x2)：
p(x1,x2)=G ' (0, AAT+β')
Wherein,
Dimensionality reduction matrix F and G are calculated by EM algorithms and obtained in β and matrix A, calculate the similarity R of two images：
2
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