CN105550645A - Least-squares classification method on product Grassmann manifold - Google Patents

Abstract

The invention discloses a least-squares classification method on a product Grassmann manifold. The method has a closed-form solution, and the accuracy rate of identification can be improved. The method comprises the following steps: (1) expressing a video in the form of product Grassmann manifold; (2) establishing a least-square model on the Grassmann manifold and solving the least-square model; and (3) performing least-squares classification, and outputting the classification result.

Description

A kind of product Grassmann flows the least squared classified method on shape

Technical field

The invention belongs to the technical field of pattern-recognition, relate to the least squared classified method on a kind of product Grassmann stream shape particularly.

Background technology

In recent years, linear subspaces method has very important application in computer vision, as target identification, and recognition of face, human body tracking etc.Linear subspaces can reduce calculation cost and can better portray the inherent geometry of data itself.It is a kind of linear subspaces with nonlinear organization that Grassmann flows shape, and the point that video data is shown as on product Grassmann stream shape by high-order SVD breakdown is had significant effect in gesture identification.For identification problem, except the character representation that will find, the sorting technique of robust also plays vital effect to recognition correct rate.

Least square method is as simple effective method most in statistical study, and its research in stream shape space also obtains the concern of a lot of scholar.Lui gives the non-linear least square method on Grassmann stream shape by means of kernel function, and it is measured by geodesic line distance.But use WeightedKarcherMean algorithm in the method Solve problems, it is an iterative algorithm, and what obtain is an approximate solution.

Summary of the invention

Technology of the present invention is dealt with problems and is: overcome the deficiencies in the prior art, provides the least squared classified method on a kind of product Grassmann stream shape, and it has to close separates, and can improve the accuracy of identification.

Technical solution of the present invention is: this product Grassmann flows the least squared classified method on shape, and the method comprises the following steps:

(1) carry out product Grassmann stream shape to video to represent;

(2) on Grassmann stream shape, set up least square model and solve;

(3) least squared classified is carried out, and output category result.

The present invention is equidistantly embedded in Symmetric matrix by the point flowed by Grassmann on shape and carries out error metrics again, separates, can improve the accuracy of identification so have to close.

Accompanying drawing explanation

Fig. 1 shows method frame figure of the present invention.

Embodiment

As shown in Figure 1, this product Grassmann flows the least squared classified method on shape, and the method comprises the following steps:

(1) carry out product Grassmann stream shape to video to represent;

(2) on Grassmann stream shape, set up least square model and solve;

(3) least squared classified is carried out, and output category result.

The present invention is equidistantly embedded in Symmetric matrix by the point flowed by Grassmann on shape and carries out error metrics again, separates, can improve the accuracy of identification so have to close.

Preferably, in described step (1), videometer is shown as the form of tensor wherein I ₁, I ₂, I ₃represent the height of video, wide, length respectively; Change under each pattern can be obtained by high-order SVD, , wherein core tensor, V ⁽¹⁾, V ⁽²⁾, V ⁽³⁾the factor matrix under each pattern respectively, and each V ^(k)being tall and thin orthogonal matrix, is the point that Stiefel flows on shape, so span (V ^(k)) be the point on Grassmann stream shape, (span (V ⁽¹⁾), span (V ⁽²⁾), span (V ⁽³⁾)) be the point on product Grassmann stream shape.

Preferably, least squared classified in described step (2)

$R\left(y\right)\stackrel{\mathrm{\Δ}}{=}{\min }_y\left|\right|{\mathrm{XX}}^T-{\mathrm{\Σ}}_{j=1}^N{D}_j{D_j}^T{y}_j|{|}_F^2---\left(1\right)$

The solution optimizing formula (1) is formula (3)

y ^*＝2(K(D)+K(D) ^T) ^-1K(X，D)(3)

Corresponding error is $\left|\right|{\mathrm{XX}}^T-{\mathrm{\Σ}}_{j=1}^N{D}_j{D_j}^T{y_j}^{*}|{|}_F.$

Preferably, the middle sample (X, Y, Z) of described step (3) is defined as about the residual error of kth class

$\begin{array}{c}{\mathrm{\ϵ}}^k=\left|\right|{\mathrm{XX}}^T-\underset{j=1}{\overset{N_k}{\Σ}}{U}_j^k{U}_j^{k^T}{u}_j^{k*}|{|}_F+||{\mathrm{YY}}^T-\underset{j=1}{\overset{N_k}{\Σ}}{V}_j^k{V}_j^{k^T}{v}_j^{k*}|{|}_F+\\ \left|\right|{\mathrm{ZZ}}^T-{\Σ}_{j=1}^{N_k}{W}_j^k{W}_j^{k^T}{w}_j^{k*}|{|}_F\end{array}$

Wherein be the solution of formula (1) on each submanifold respectively, final classification results is by k ^*=argmin _kε ^kdetermine.

Illustrate above method below.

1. the product Grassmann stream shape of video represents

Video can be expressed as tensor form as high dimensional data as wherein I ₁, I ₂, I ₃represent the height of video, wide, length respectively.Change under each pattern can be obtained by high-order SVD, namely wherein core tensor, V ⁽¹⁾, V ⁽²⁾, V ⁽³⁾the factor matrix under each pattern respectively, and each V ^(k)be tall and thin orthogonal matrix, therefore can regard the point on Stiefel stream shape as, so span (V ^(k)) be the point on Grassmann stream shape.Therefore (span (V ⁽¹⁾), span (V ⁽²⁾), span (V ⁽³⁾)) be the point on product Grassmann stream shape.

2.Grassmann flows the embedded least square method on shape and solves

Least square technology is a most simple effective method in statistical study.In theorem in Euclid space, parameter can by minimize residual error R (β)=|| y-A β || ²obtain, wherein for training set, for regressand value.Estimated parameter have Explicit solutions shape as now corresponding error is || y-A (A ^ta) ^-1a ^ty|| ².

Order for sample size is the training set of N, wherein represent that Grassmann flows shape, fitting parameter, it is input amendment.Utilize projection mapping

The point flowed by Grassmann on shape is embedded into Symmetric matrix, and wherein Sym (d) represents Symmetric matrix.The distance of upper two point X and Y in such Grassmann space can define by the distance of embedded space, namely $d(X,Y)=\frac{\sqrt{2}}{2}\left|\right|-{\mathrm{XX}}^T-{\mathrm{YY}}^T|{|}_F,$ And the geodesic line distance that this Distance geometry Grassmann flows the upper definition of shape is of equal value.Such distance definition is also convenient to follow-up solving.Being similar to the theorem in Euclid space principle of least square, providing embedded least square for solving following optimization problem,

$R\left(y\right)\stackrel{\mathrm{\Δ}}{=}{\min }_y\left|\right|{\mathrm{XX}}^T-{\mathrm{\Σ}}_{j=1}^N{D}_j{D_j}^T{y}_j|{|}_F^2---\left(1\right)$

Wherein y _jit is a jth element of vectorial y.

Introduce below and how to solve above-mentioned optimization problem.Have

$\begin{array}{c}\left|\right|{\mathrm{XX}}^T-\underset{j=1}{\overset{N}{\Σ}}{D}_j{D_j}^T{y}_j|{|}_F^2=\\ Tr\left(X^T{\mathrm{XX}}^{T}X\right)+\underset{j,i}{\overset{N}{\Σ}}{y}_i{y}_{j}Tr\left({D_i}^T{D}_j{D_j}^T{D}_i\right)-2\underset{j}{\overset{N}{\Σ}}{y}_{j}Tr\left({D_j}^T{\mathrm{XX}}^T{D}_j\right)\end{array}$

Definition

${\[K\left(D\right)\]}_{i,j}=Tr\left({D_i}^T{D}_j{D_j}^T{D}_i\right)=\left|\right|{D_i}^T{D}_j|{|}_F^2,$

${\[K(X,D)\]}_i=Tr\left({D_j}^T{\mathrm{XX}}^T{D}_j\right)=\left|\right|X^T{D}_j|{|}_F^2$

Therefore model (1) becomes

min _y{y ^TK(D)y-2y ^TK(X，D)}(2)

To (2) about y differentiate, and make derivative equal 0, have

(K(D)+K(D) ^T)y-2K(X，D)＝0

So the solution of optimization problem (1) is

y ^*＝2(K(D)+K(D) ^T) ^-1K(X，D)(3)

Therefore corresponding error is $\left|\right|{\mathrm{XX}}^T-{\mathrm{\Σ}}_{j=1}^N{D}_j{D_j}^T{y_j}^{*}\left|\right|F.$

3. based on the principle of classification of embedded least square method

The 3 factorial Grassmann that corresponding video data is discussed below flow shape wherein × represent cartesian product.Suppose that training set has M class, note kth class training set is wherein N _kfor number of samples.Target infers test sample book which kind of belongs to.

Sample (X, Y, Z) is defined as about the residual error of kth class

$\begin{array}{c}{\mathrm{\ϵ}}^k=\left|\right|{\mathrm{XX}}^T-\underset{j=1}{\overset{N_k}{\Σ}}{U}_j^k{U}_j^{k^T}{u}_j^{k*}|{|}_F+||{\mathrm{YY}}^T-\underset{j=1}{\overset{N_k}{\Σ}}{V}_j^k{V}_j^{k^T}{v}_j^{k*}|{|}_F+\\ \left|\right|{\mathrm{ZZ}}^T-{\Σ}_{j=1}^{N_k}{W}_j^k{W}_j^{k^T}{w}_j^{k*}|{|}_F\end{array}$

Wherein the solution of each submanifold upper returning problem (1) respectively.Final classification results is determined by following formula

k ^*＝argmin _kε ^k

Experimental verification is carried out to above-mentioned model, and has achieved obvious effect.In an experiment, select the gesture database of Cambridge University, it comprises 900 videos, totally 9 different gestures, and it is divided into 5 set according to different illumination conditions.Set 5 is general as training set, and 1-4 is as test set in set.In experiment, original series is converted into gray level image and adjusts size to 14 × 32 × 23.Current high-caliber 2 kinds of recognition methodss and the correct recognition rata of the inventive method respectively on four test sets is listed in table 1.The inventive method correct recognition rata, apparently higher than the discrimination of the PM method of Lui, is on close level with the best result kgLLC of people's methods such as Harandi.This description of test the inventive method is simply effective.

Table 1

The above; it is only preferred embodiment of the present invention; not any pro forma restriction is done to the present invention, every above embodiment is done according to technical spirit of the present invention any simple modification, equivalent variations and modification, all still belong to the protection domain of technical solution of the present invention.

Claims (4)

1. product Grassmann flows the least squared classified method on shape, it is characterized in that: the method comprises the following steps:

(1) carry out product Grassmann stream shape to video to represent;

(2) on Grassmann stream shape, set up least square model and solve;

(3) least squared classified is carried out, and output category result.

2. product Grassmann according to claim 1 flows the least squared classified method on shape, it is characterized in that: in described step (1), videometer is shown as the form of tensor wherein I ₁, I ₂, I ₃represent the height of video, wide, length respectively; Change under each pattern can be obtained by high-order SVD decomposition, wherein core tensor, V ⁽¹⁾, V ⁽²⁾, V ⁽³⁾the factor matrix under each pattern respectively, and each V ^(k)being tall and thin orthogonal matrix, is the point that Stiefel flows on shape, so span (V ^(k)) be the point on Grassmann stream shape, (span (V ⁽¹⁾), span (V ⁽²⁾), span (V ⁽³⁾)) be the point on product Grassmann stream shape.

3. product Grassmann according to claim 2 flows the least squared classified method on shape, it is characterized in that: least squared classified in described step (2)

$R\left(y\right)\stackrel{\mathrm{\Δ}}{=}{\min }_y\left|\right|{\mathrm{XX}}^T-{\mathrm{\Σ}}_{j=1}^N{D}_j{D_j}^T{y}_j|{|}_F^2---\left(1\right)$

Wherein input amendment, represent that Grassmann flows shape, for sample size is the training set of N, fitting parameter, || || _frepresent Frobenius norm.

The solution optimizing formula (1) is formula (3)

y ^*＝2(K(D)+K(D) ^T) ^-1K(X，D)(3)

Corresponding error is $\left|\right|{\mathrm{XX}}^T-{\mathrm{\Σ}}_{j=1}^N{D}_j{D_j}^T{y_j}^{*}|{|}_F.$

Wherein for optimum solution, and

${\[K\left(D\right)\]}_{i,j}=Tr\left({D_i}^T{D}_j{D_j}^T{D}_i\right)=\left|\right|{D_i}^T{D}_j|{|}_F^2,$

${\[K(X,D)\]}_i=Tr\left({D_j}^T{\mathrm{XX}}^T{D}_j\right)=\left|\right|X^T{D}_j|{|}_F^2$

4. product Grassmann according to claim 3 flows the least squared classified method on shape, it is characterized in that: sample in described step (3) residual error about kth class is defined as

$\begin{array}{c}{\mathrm{\ϵ}}^k=\left|\right|{\mathrm{XX}}^T-\underset{j=1}{\overset{N_k}{\Σ}}{U}_j^k{U}_j^{k^T}{u}_j^{k*}|{|}_F+||{\mathrm{YY}}^T-\underset{j=1}{\overset{N_k}{\Σ}}{V}_j^k{V}_j^{k^T}{v}_j^{k*}|{|}_F+\\ \left|\right|{\mathrm{ZZ}}^T-{\Σ}_{j=1}^{N_k}{W}_j^k{W}_j^{k^T}{w}_j^{k*}|{|}_F\end{array}$

Wherein be that 3 factorial Grassmann flow shape on kth class training set, N _kfor number of samples be the solution of formula (1) on each submanifold respectively, final classification results is by k ^*=argmin _kε ^kdetermine.