CN102073875A - Sparse representation-based background clutter quantification method - Google Patents

Abstract

The invention discloses a background clutter quantification method based on sparse representation, which mainly solves the problem that the existing clutter scale cannot well conform to the physical essence of the relativity of the background clutter and cannot fully reflect the visual characteristics of human eyes. The implementation steps are: divide the gray scale background image to be quantified into several small units of equal size, and combine them into a background matrix; extract the main features of the target vector and the background matrix, and obtain the target feature vector and the background feature matrix; The vector and the background feature matrix are normalized; the sparsest representation of the normalized target feature vector in the normalized background feature matrix is calculated; the sum of the absolute values of the sparsest representations is used as the background clutter scale of the entire image. The invention makes full use of the two major features of human eyes when searching, improves the consistency between the predicted target detection probability and the subjective actual target detection probability, and can be used for the prediction and evaluation of the target acquisition performance of the photoelectric imaging system.

Description

Background Clutter Quantization Method Based on Sparse Representation

technical field

The invention belongs to the technical field of image processing, in particular to a background clutter quantification method based on sparse representation, which can be used for the prediction and evaluation of the target acquisition performance of a photoelectric imaging system.

Background technique

The target acquisition performance of electro-optical imaging system is an important concept in military tasks such as electro-optical countermeasures, reconnaissance, early warning and camouflage. In recent years, with the introduction of new materials and the advancement of manufacturing processes, photodetectors have reached or approached the background limit, and the background factor has become a key factor limiting the target acquisition performance of photoelectric imaging systems. The use of a reasonable and accurate background clutter quantization scale in the target acquisition performance characterization model of the optoelectronic imaging system can make its prediction results more accurately reflect the external field performance of the imaging system.

Background clutter is a visual perception effect, which refers to the target-like objects that interfere with target detection. It has two typical characteristics: based on background and target characteristics; relative to the target of interest. Since the 1980s, foreign researchers have done a lot of research on the quantitative representation of background clutter, and proposed a variety of background clutter quantitative description scales, among which the statistical variance scale SV is the most widely used, such as D.E.Schmieder and M.R.Weathersby, "Detection performance in clutter with variable resolution," IEEE Trans.Aerosp.Electron.Syst.AES-19(4), 622-630(1983), and marginal probability scale POE, as in G.Tidhar, G. Reiter, Z. Avital, Y. Hadar, S.R. Rotmam, V. George, and M.L. Kowalczyk, "Modeling human search and target acquisition performance: IV. Detection probability in the cluttered environment," Opt. Eng. 33, 801-808( 1994). However, the statistical variance scale SV is based on the statistical processing of photoelectric images, without considering the visual perception characteristics of the human eye; the edge probability scale POE only uses background information as a reference, which violates the physical essence of background clutter relative to the target. This makes the clutter scale established by the two cannot reasonably reflect the influence of the background on the target acquisition process, and it is difficult to accurately predict and evaluate the field performance of the optoelectronic imaging system.

Contents of the invention

The purpose of the present invention is to overcome the shortcomings of previous background clutter quantification methods, and propose a background clutter scale based on sparse representation, so as to improve the accuracy of prediction and evaluation of the imaging system's external field performance.

In order to achieve such a goal, the present invention transforms the target and background information from the space domain to the feature domain through the dimensionality reduction technique, and quantifies the background clutter by using the sparse representation theory in the feature space. Specific steps are as follows:

(1) Column vectorize the two-dimensional target image to obtain the target vector x;

(2) Divide the background image into N small units of equal size, and the size of each small unit in the horizontal and vertical directions is twice the corresponding size of the target;

(3) Vectorize the columns of each two-dimensional background small unit and combine them into a background matrix Ψ;

和背景特征矩阵Φ；(4) Using principal component analysis PCA to reduce the dimensionality of the target vector x and the background matrix Ψ, and obtain the target eigenvectors respectively

and the background feature matrix Φ;

进行归一化处理，得到归一化目标特征向量

(5) For the target feature vector

Perform normalization processing to obtain the normalized target feature vector

$\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; =</span>\stackrel{<span\; class="notranslate">\; ~</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; /</span>{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\stackrel{<span\; class="notranslate">\; ~</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; 2</span>}}$

Where ||·|| ₂ represents the l ² norm of the vector;

(6) The result Θ _i obtained by normalizing each vector in the background feature matrix Φ, and subscripted serial numbers in ascending order to form a normalized background feature matrix Θ,

Θ _i = Φ _i /||Φ _i || ₂ , i=1, 2,..., N

Wherein, Φ _i and Θ _i are respectively the ith column vector of the background feature matrix Φ and the normalized background feature matrix Θ, and N is the number of column vectors in the background feature matrix Φ;

在归一化背景特征矩阵Θ中的最稀疏表示，获得相似向量s：即求解满足

的s的最小l⁰范数解：(7) Calculate the normalized target feature vector

In the sparsest representation in the normalized background feature matrix Θ, a similar vector s is obtained: that is, the solution satisfies

The minimum l ⁰ norm solution of s:

min||s|| ₀ satisfies

i＝1，2，...，K，的总和，作为背景杂波量化尺度：

其中K为相似向量s中非零元素的个数。(8) Take the absolute value of the non-zero elements in the similar vector s

The sum of i=1, 2, ..., K, as the background clutter quantization scale:

Where K is the number of non-zero elements in the similarity vector s.

The present invention has the following advantages:

1) The present invention uses the PCA method to simulate the first stage of human vision in the search process—the feature selection stage by using the PCA method to reduce the dimensionality of the target and background information, and by obtaining the sparsest feature vector of the target feature vector in the background feature matrix It is expressed to simulate the second stage of human vision in the search process - the joint search stage, which is in line with the perceptual characteristics of human vision in the process of target acquisition;

2) The present invention not only considers the background feature but also incorporates the relative influence of the target feature in the quantification of the background clutter, which conforms to the physical nature of the background clutter.

Based on the above two points, the background clutter quantification method of the present invention is more consistent with the physical mechanism in which the background information affects the target acquisition process in the field test. The experimental results show that: compared with the clutter quantification method commonly used in the past, the prediction of the target detection probability by the background clutter quantification method of the present invention is more consistent with the target detection probability obtained by the observer's actual test, and the prediction of the target acquisition performance is more accurate .

Description of drawings:

Fig. 1 is the schematic diagram of the realization process of the present invention;

Fig. 2 is the low background clutter image that the present invention uses, target image and similar vector distribution figure;

Fig. 3 is the background clutter image, target image and similar vector distribution diagram used in the present invention;

Fig. 4 is the high background clutter image that the present invention uses, target image and similar vector distribution figure;

Fig. 5 is the fitting curve between the quantization scale of each background clutter and the observer's actual target detection probability, taking the Search_2 image database as the experimental data.

Detailed ways

With reference to Fig. 1, the implementation steps of the background clutter quantization method based on sparse representation in the present invention are as follows:

Step 1, take the pixel values of the target image as the unit, and form the target vector in the order of the initial column number from small to large

x={t _1,1 ,t _2,1 ,t _3,1 ,...,t _C,1 ,t _1,2 ,...,t _C,D } ^T

Among them, t _{i, j} represents the target pixel value at position (i, j), C and D represent the number of rows and columns of the target image respectively, and T represents the transpose operation on the vector.

In step 2, the background image to be quantized is divided into N small units of equal size, and the size of each small unit in the horizontal and vertical directions is twice the corresponding size of the target.

表示取小于或等于x的最大整数。The size of N is determined by the size A×B of the background image to be quantized and the size M=C×D of each small unit, namely Among them, A and B represent the number of rows and columns of the background image, respectively,

Indicates to take the largest integer less than or equal to x.

i＝1，2，...，N，组合背景矩阵 Step 3, the column vector obtained by column vectorizing each background small unit in turn

i=1, 2, ..., N, combined background matrix

Ψ={A ₁ , A ₂ ,..., A _N }

Step 4, use principal component analysis (PCA) to reduce the dimensionality of the target vector x and the background matrix Ψ, the specific steps are as follows:

(4a) Use each element in the background matrix Ψ to subtract the result X _ij obtained by subtracting the mean value of the element in the row, and order the subscript (i, j) to form the background difference matrix X,

${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&Psi;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; -</span>\underset{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; N</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&Psi;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; N</span>}$

Wherein, Ψ _ij and X _ij are the values of the background matrix Ψ and the background difference matrix X at (i, j) respectively, and M and N are the number of rows and columns of the background matrix Ψ respectively;

(4b) Multiply the transposed matrix X ^T by the background difference matrix X to the right to obtain the covariance matrix A:

A=X ^T X;

(4c) Carry out eigenvalue decomposition on the covariance matrix A to obtain its non-zero eigenvalue λ _k and corresponding eigenvector v _k , k=1, 2, ..., t, where t is the non-zero covariance matrix A The total number of eigenvalues, λ ₁ ≥ λ ₂ ≥ Λ ≥ λ _t > 0, the eigenvectors are mutually orthogonal;

(4d) Take 90% of the sum of the non-zero eigenvalues of the covariance matrix A as the threshold, and take the reciprocal of the square root of the first W non-zero eigenvalues to form a diagonal matrix D:

$\mathrm{D.}=\left[\begin{array}{ccc}1/\sqrt{{\mathrm{\&lambda;}}_1}& & \\ & \mathrm{<figure-callout\; id="1"\; label="o"\; filenames="HDA0000042845270000041.png"\; state="\{\{state\}\}">o</figure-callout>}& \\ & & 1/\sqrt{{\mathrm{\&lambda;}}_W}\end{array}\right],$ satisfy $\underset{k=1}{\overset{W}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_k/\underset{k=1}{\overset{t}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_k\&ap;0.9$

At the same time, take the eigenvectors v _k corresponding to the W non-zero eigenvalues, k=1, 2, ..., W, to form a feature matrix: v={v ₁ , v ₂ , ..., v _W };

(4e) Multiply the feature matrix v by the background difference matrix X, and then multiply the diagonal matrix D by the left to obtain the whitening matrix R ^M×W :

R=X*v*D

Among them, the number of rows M of the whitening matrix R is far greater than the number of columns W;

(4f) Multiply the background difference matrix X by the transposition matrix of the whitening matrix R to the left to obtain the background feature matrix:

Φ＝R ^T X;

(4g) Use each element of the target vector x to subtract the result d _i of the mean value of the corresponding row element in the background matrix Ψ, and form the target difference vector d={d ₁ , d ₂ , ..., d _M } ^T ,

${\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; -</span>\underset{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; N</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&Psi;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; ;</span>$

Where x _i is the i-th element of the target vector x, Ψ _ij is the value of the background matrix Ψ at (i, j), M and N are the number of rows and columns of the background matrix Ψ, respectively;

(4h) Multiply the target difference vector d by the transposition matrix of the whitening matrix R to the left to obtain the target feature vector:

$\stackrel{<span\; class="notranslate">\; ~</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; R</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; .</span>$

In this step, the dimensionality reduction method of the target vector and the background matrix can be reduced by the following methods in addition to the PCA described above:

1) Multidimensional scaling method MDS (I.Borg, and P.Groenen, "Modern Multidimensional Scaling: theory and applications," 2nd ed., Springer-Verlag New York, 2005);

2) Independent component analysis (ICA) (A. and E.Oja, "A Fast Fixed-Point Algorithm for Independent Component Analysis," Neural Computation, vol.9, No.7, pp:1, 483-1, 492, Oct.1997.);

3) Non-negative matrix factor method NMF (D.D.Lee and H.S.Seung. "Algorithms for non-negative matrix factorization," In Advances in Neural Information Processing systems, 2001.);

4) LLE (T.R.Sam and K.S.Lawrence, "Nonlinear Dimensionality Reduction by Locally Linear Embedding," SCIENCE, Vol.290No.22, Dec.2000.);

5) Laplacian Eigenmaps LE (M. Belkin and P. Niyogi, "Laplacian Eigenmaps for Dimensionality Reduction and Data Representation," Neural Computation 15, pp: 1373-1396, 2003.).

进行归一化处理得到归一化目标特征向量

Step 5, for the target feature vector

Perform normalization processing to obtain the normalized target feature vector

$\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; =</span>\stackrel{<span\; class="notranslate">\; ~</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; /</span>{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\stackrel{<span\; class="notranslate">\; ~</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; 2</span>}}$

where |||| ₂ denotes the _l2 norm of the vector.

Step 6, normalize the result Θ _i of each vector in the background feature matrix Φ, and form the normalized background feature matrix Θ in the order of subscripted serial numbers from small to large,

Θ _i = Φ _i /||Φ _i || ₂ , i=1, 2,..., N

Among them, _Φi and _Θi are the i-th column vectors of the background feature matrix Φ and the normalized background feature matrix Θ, respectively, and N is the number of column vectors in the background feature matrix Φ.

的s的最小l⁰范数解：Step 7, find the normalized target feature vector In the sparsest representation in the normalized background feature matrix Θ, a similar vector s is obtained: that is, the solution satisfies

The minimum l ⁰ norm solution of s:

min||s|| ₀ satisfies

The solution steps are as follows:

的s的最小l¹范数解：(7a) to satisfy

The minimum l ¹ norm solution of s:

min||s|| ₁ satisfies

(7b) Relax the formula <1> as:

min||s|| ₁ satisfies

Among them, ε is any constant not less than 0, when ε=0, formula <2> will degenerate into formula <1>;

(7c) Using the LASSO algorithm, the formula <2> is transformed into:

满足||s||₁≤σ                         <3>

Satisfy ||s|| ₁ ≤ σ <3>

Among them, σ is any constant not less than 0;

(7d) Using the Lagrangian algorithm, transform the formula <3> into an unconstrained optimization formula:

${\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{<span\; class="notranslate">\; *</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; arg</span>}\underset{\mathrm{<span\; class="notranslate">\; the\; <figure-callout\; id="1"\; label="s"\; filenames="HDA0000042845270000041.png"\; state="\{\{state\}\}">s</figure-callout></span>}}{\mathrm{<span\; class="notranslate">\; min</span>}}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&Theta;s</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; ></span>$

表示目标函数最小时变量s的值；Among them, α is the Lagrangian multiplier,

Indicates the value of the variable s when the objective function is minimized;

(7e) Using the truncated Newton interior point algorithm, formula <4> is transformed into a quadratic programming formula with inequality constraints:

$\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="min"\; filenames="HDA0000042845270000041.png"\; state="\{\{state\}\}">min</figure-callout></span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&Theta;s</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; N</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; ></span>$

- μ _i ≤ s _i ≤ μ _i , i=1, 2, . . . , N.

Among them, s _i is the i-th element of the similarity vector s, μ _i is the factor that constrains _si , and -μ _i ≤ _{s i} ≤ _{μ i} is the constraint condition;

(7f) Establish a logarithmic barrier function for the constraint condition - μ _i ≤ s _i ≤ μ _i :

Using the logarithmic barrier function, transform Equation <5> into an optimal solution for the central trajectory function F _β (s, μ, α) = 0 defined by the weight factor β:

(7g) Using Newton's iterative method to solve equation <6>, the iterative formula can be obtained:

$\left(\begin{array}{c}{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}\\ {\mathrm{<span\; class="notranslate">\; \&mu;</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}\\ {\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}\end{array}\right)<span\; class="notranslate">\; =</span>\left(\begin{array}{c}{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}\\ {\mathrm{<span\; class="notranslate">\; \&mu;</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}\\ {\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}\end{array}\right)<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; \&dtri;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; \&beta;</span>}}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>{<span\; class="notranslate">\; \&dtri;</span>\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; \&beta;</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; )</span>$

表示求函数的二阶导数，

表示求函数的一阶导数；Among them, s ^(k) , μ ^(k) and α ^(k) represent the results of s, μ and α after the kth iteration respectively, s ^(k+1) , μ ^(k+1) and α ^{(k+ 1)} respectively represent the results of s, μ and α after the k+1th iteration, k is a non-negative integer not greater than 50,

Represents the second order derivative of the function,

Indicates the first derivative of the function;

(7h) Get weight factor β=0.5 and the initial value of solution vector:

$\left(\begin{array}{c}{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span>}\\ {\mathrm{<span\; class="notranslate">\; \&mu;</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span>}\\ {\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span>}\end{array}\right)<span\; class="notranslate">\; =</span>\left(\begin{array}{c}{\mathrm{<span\; class="notranslate">\; \&Theta;</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}\\ \mathrm{<span\; class="notranslate">\; 0.95</span>}<span\; class="notranslate">\; \&CenterDot;</span>\mathrm{<span\; class="notranslate">\; sgn</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&Theta;</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; \&Theta;</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 0.1</span>}\mathrm{<span\; class="notranslate">\; max</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; sgn</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&Theta;</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; \&Theta;</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span>\\ \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right)$

Among them, max means to take the maximum value of the vector element, and sgn means the positive and negative attributes of the vector element:

(7i) Bring the initial value and weight factor into step (7g) for iterative calculation, until the result of two adjacent iterations is brought into formula <5> and the subtraction difference is not greater than 10 ^-3 , the obtained s at this time The value of is the minimum l ¹ norm solution of s in formula <1>, jump to step (7k); if the maximum number of iterations is 50, the optimal solution is still not obtained, go to step (7j);

(7j) Use the final iterative result as the initial value, update the weight factor to 2 times the original value, reset the number of iterations to zero, and return to step (7i);

在归一化背景特征矩阵Θ中的最稀疏表示。(7k) Verify the sparsity of the smallest l ¹ norm solution s for all test images, according to D.Donoho, "For most large underdetermined systems of linear eauations the minimal l ¹ -norm near solution approximates the sparse solution," preprint, 2004 The conclusion proposed in .: when the minimum l ¹ norm solution s has sparse characteristics, it is equivalent to the minimum l ⁰ norm solution. It can be seen that the minimum l ¹ norm solution s obtained by the present invention is the normalization target feature vector

The sparsest representation in the normalized background feature matrix Θ.

To solve the minimum ^l1 norm problem, except the algorithm given in the present invention, the following methods can also be used:

1) Gradient mapping method (M.Figueiredo, R.Nowak, and S.Wright, "Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems," IEEE Journal of Selected Topics in Signal Processing, Vol.1, No. .4, pp: 586-597, 2007.);

2) homotopy method (D.Malioutov, M.Cetin, and A.Willsky, "Homotopy continuation for sparse signal representation," In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005.);

3) Iterative shrinkage threshold method (I.Daubechies, M.Defrise, and C.Mol, "Aniterative thresholding algorithm for linear inverse problems with asparsity constraint," Communications on Pure and Applied Math, Vol.57, pp: 1413-1457, 2004.);

4) Nesterov's method (A.Beck and M.Teboulle, "A fast iterative shrinkage-thresholding algorithm for linear inverse problems," SIAM Journal on Imaging Sciences, Vol.2, No.1, pp: 183-202, 2009.);

5) Cross-direction method (J. Yang and Y. Zhang, "Alternating direction algorithms for l ¹ -problems in compressive sensing," (preprint) arXiv: 0912.1185, 2009.).

其中，K为相似向量s中非零元素的个数。Step 8, take the absolute value of the non-zero elements in the similarity vector s The sum of i=1, 2, ..., K, as the background clutter quantization scale of the whole image:

Among them, K is the number of non-zero elements in the similarity vector s.

Rationality and superiority of the present invention can be further described by following experiments and comparative analysis:

Experimental verification:

1. Experimental conditions

Taking the Search_2 image database provided by TNO Human Factors Research Institute in the Netherlands as an example, the rationality of the image background clutter scale of the present invention and its superiority in target acquisition performance prediction are verified. The Search_2 image database includes 44 high-resolution digital natural scene images with different background complexities, as well as the specific parameters of each scene and the test results of the observer's actual observation experiment. For a detailed description of the database, please refer to the literature A.Toet, P. Bijl, and J.M. Valeton, "Image data set for testing search and detection models," Opt. Eng. 40(9), 1760-1767 (2001); A.Toet, P.Bijl, F.L.Kooi, and J.M.Valeton, " A high-resolution image data set for testing search and detection models,” Report TM-98-A020, TNO Human Factors Research Institute, (1998) and A.Toet, “Errata in Report TNO-TM 1998A020: A high-resolution image data set for testing search and detection models,” (2001).

2. Examples

Fig. 2, Fig. 3 and Fig. 4 have respectively provided the background image, target image and similar vector distribution figure of three kinds of different clutter levels of low, middle and high used in the present invention: Fig. 2 is low background clutter image and target area image , where Figure 2(a) is the low background clutter image, Figure 2(b) is the target area image, that is, the part marked by the white rectangle in Figure 2(a), and Figure 2(c) is the calculated corresponding Similarity vector distribution diagram; Figure 3 is the middle background clutter image and the target area image, where Figure 3(a) is the middle background clutter image, Figure 3(b) is the target area image, that is, the white rectangle in Figure 3(a) The part marked by the box, Fig. 3(c) is the calculated corresponding similarity vector distribution diagram; Fig. 4 is the high background clutter image and target area image, in which Fig. 4(a) is the high background clutter image, Fig. 4 (b) is the image of the target area, that is, the part marked by the white rectangle in Fig. 4(a), and Fig. 4(c) is the corresponding similarity vector distribution map obtained by calculation.

It can be seen from Fig. 2(c), Fig. 3(c) and Fig. 4(c), for the images of three different clutter levels, the similarity vectors of the target image and the background image all have sparse characteristics, thus proving that in the present invention Rationality of the algorithm employed to compute the sparsest representation of the normalized target feature vector among the normalized background feature vectors.

It can be seen from Figure 2(a) and Figure 2(b) that the similarity between the background and the target is low, the background clutter of the entire image is very low, and it is easy to detect the target; it can be seen from Figure 3(a) and Figure 3(b) , the similarity between the background and the target is high, the background clutter of the entire image is high, and it is difficult to detect the target; it can be seen from Figure 4(a) and Figure 4(b), compared with the previous two images, the background and The similarity of the target is the highest, the background clutter of the whole image is the highest, and it is the most difficult to detect the target. Figure 2 (a), Figure 3 (a) and Figure 4 (a) are quantified respectively with the image background clutter quantization scale of the present invention to obtain their background clutter scales to be 2.0195, 1.3060 and 1.0120 respectively, which is different from the above Consistent with the subjective perception of human vision described above, it can be seen that the quantization scale of the present invention can reflect the real situation of background clutter.

3. Experimental results

The 7th, 15th, 23rd, 26th, and 4 images in the Search_2 database were removed when the present invention was verified experimentally. This is because the present invention studies single target detection, and there are double targets in the first four images, which is beyond the scope of the present invention. The scope of the invention, and the target in the last image is too small, which belongs to the weak and small target detection problem, and does not belong to the research field of the present invention. Therefore, in the experiment to verify the superiority of the background clutter quantization scale of the present invention in predicting target acquisition performance, the final valid data are the remaining 39 images.

Fig. 5 is the fitting curve between the results obtained by quantifying these 39 images and the observer's actual target detection probability using POE, SV and the background clutter scale of the present invention, wherein Fig. 5(a), 5(b ) and 5(c) are respectively the fitting curves between POE, SV and the background clutter quantization scale of the present invention and the observer's actual target detection probability. The fitting formula is:

$\mathrm{<span\; class="notranslate">\; PD</span>}<span\; class="notranslate">\; =</span>\frac{{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 50</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; E.</span>}}}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 50</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; E.</span>}}}$

Among them, X represents the background clutter scale; _X50 and E are constants, which can be obtained by fitting with the observer's actual target detection probability; PD is the observer's actual target detection probability obtained from subjective experiments, which can be obtained by the formula:

PD＝N _c /(N _c +N _f +N _m )

Obtained, where N _c , N _f and N _m are the number of people who correctly detected the target, the number of people who misjudged the target and the number of people who did not detect the target corresponding to each image in the Search_2 database.

Table 1 presents the fitting results between each background clutter scale and the target detection probability obtained from the actual subjective experiment in the form of data, including the values of _X50 and E corresponding to each background clutter scale, and the performance measure RMSE , CC and SCC evaluate the consistency between the predicted target detection probability of each background clutter scale and the observer's actual target detection probability. Among them, X ₅₀ and E are curve fitting parameters; RMSE is root mean square error; CC is Pearson correlation coefficient; SCC is Spearman rank correlation coefficient.

Table 1: Performance comparison of the background clutter scale, POE and SV of the present invention

As can be seen from Table 1, the Pearson correlation coefficient and the Spearman rank correlation coefficient of the background clutter quantization scale of the present invention and the observer's actual target detection probability are greater than other background clutter scales, and the root mean square error is smaller than other background clutter scales, thus The superiority of the background clutter quantization scale of the present invention in predicting target acquisition performance is proved.

Claims (3)

1. A background clutter quantification method based on sparse representation, comprising the following process: (1) Column vectorize the two-dimensional target image to obtain the target vector x; (2) Divide the background image into N small units of equal size, and the size of each small unit in the horizontal and vertical directions is twice the corresponding size of the target; (3) Vectorize the columns of each two-dimensional background small unit and combine them into a background matrix Ψ; (4) Using principal component analysis PCA to reduce the dimensionality of the target vector x and the background matrix Ψ, and obtain the target eigenvectors respectively

and the background feature matrix Φ; (5) For the target feature vector

Perform normalization processing to obtain the normalized target feature vector $\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; =</span>\stackrel{<span\; class="notranslate">\; ~</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; /</span>{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\stackrel{<span\; class="notranslate">\; ~</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; 2</span>}}$ Among them, ||·|| ₂ represents the l ² norm of the vector; (6) The result Θ _i obtained by normalizing each vector in the background feature matrix Φ, and subscripted serial numbers in ascending order to form a normalized background feature matrix Θ, Θ _i = Φ _i /||Φ _i || ₂ , i=1, 2,..., N Wherein, Φ _i and Θ _i are respectively the ith column vector of the background feature matrix Φ and the normalized background feature matrix Θ, and N is the number of column vectors in the background feature matrix Φ; (7) Calculate the normalized target feature vector

In the sparsest representation in the normalized background feature matrix Θ, a similar vector s is obtained: that is, the solution satisfies

The minimum l ⁰ norm solution of s: min||s||0 satisfies

(8) Take the absolute value of the non-zero elements in the similar vector s

The sum of i=1, 2, ..., K, as the background clutter quantization scale:

Where K is the number of non-zero elements in the similarity vector s. 2. background clutter quantification method according to claim 1, wherein step (4) described by means of principal component analysis PCA to target vector x and background matrix Ψ dimensionality reduction, carry out as follows: (4a) Use each element in the background matrix Ψ to subtract the result X _ij obtained by subtracting the mean value of the element in the row, and order the subscript (i, j) to form the background difference matrix X, ${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&Psi;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; -</span>\underset{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; N</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&Psi;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; N</span>}$ Wherein, Ψ _ij and X _ij are the values of the background matrix Ψ and the background difference matrix X at (i, j) respectively, and M and N are the number of rows and columns of the background matrix Ψ respectively; (4b) Multiply the transposed matrix X ^T by the background difference matrix X to the right to obtain the covariance matrix A: A=X ^T X (4c) Carry out eigenvalue decomposition on the covariance matrix A to obtain its non-zero eigenvalue λ _k and corresponding eigenvector v _k , k=1, 2, ..., t, where t is the non-zero covariance matrix A The total number of eigenvalues, λ ₁ ≥ λ ₂ ≥ Λ ≥ λ _t > 0, the eigenvectors are mutually orthogonal; (4d) Take 90% of the sum of the non-zero eigenvalues of the covariance matrix A as the threshold, and take the reciprocal of the square root of the first W non-zero eigenvalues to form a diagonal matrix D: $\mathrm{D.}=\left[\begin{array}{ccc}1/\sqrt{{\mathrm{\&lambda;}}_1}& & \\ & o& \\ & & 1/\sqrt{{\mathrm{\&lambda;}}_W}\end{array}\right],$ satisfy $\underset{k=1}{\overset{W}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_k/\underset{k=1}{\overset{t}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_k\&ap;0.9$ At the same time, take the eigenvectors v _k corresponding to the W non-zero eigenvalues, k=1, 2, ..., W, to form a feature matrix: v={v ₁ , v ₂ , ..., v _W }; (4e) Multiply the feature matrix v by the background difference matrix X, and then multiply the diagonal matrix D by the left to obtain the whitening matrix R ^M×W : R=X*v*D Among them, the number of rows M of the whitening matrix R is far greater than the number of columns W; (4f) Multiply the background difference matrix X by the transposition matrix of the whitening matrix R to the left to obtain the background feature matrix: Φ＝R ^T X; (4g) Use each element of the target vector x to subtract the result d _i of the mean value of the corresponding row element in the background matrix Ψ, and form the target difference vector d={d ₁ , d ₂ , ..., d _M } ^T , ${\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; -</span>\underset{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; N</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&Psi;</span>}}_{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; ;</span>$ Among them, _xi is the i-th element of the target vector x, Ψ _ij is the value of the background matrix Ψ at (i, j), and M and N are the number of rows and columns of the background matrix Ψ, respectively; (4h) Multiply the target difference vector d by the transposition matrix of the whitening matrix R to the left to obtain the target feature vector: $\stackrel{<span\; class="notranslate">\; ~</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; R</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; .</span>$ 3. background clutter quantification method according to claim 1, wherein the calculation normalization target feature vector described in step (7)

The sparsest representation in the normalized background feature matrix Θ is calculated as follows: (7a) to satisfy

The minimum l ¹ norm solution of s: min||s|| ₁ satisfies (7b) Relax the formula <1> as: min||s|| ₁ satisfies

Among them, ε is any constant not less than 0, when ε=0, formula <2> will degenerate into formula <1>; (7c) Using the LASSO algorithm, the formula <2> is transformed into:

Satisfy ||s|| ₁ ≤ σ <3> Among them, σ is any constant not less than 0; (7d) Using the Lagrangian algorithm, transform the formula <3> into an unconstrained optimization formula: ${\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{<span\; class="notranslate">\; *</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; arg</span>}\underset{\mathrm{<span\; class="notranslate">\; the\; s</span>}}{\mathrm{<span\; class="notranslate">\; min</span>}}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&Theta;s</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; ></span>$ Among them, α is the Lagrangian multiplier,

Indicates the value of the variable s when the objective function is minimized; (7e) Using the truncated Newton interior point algorithm, formula <4> is transformed into a quadratic programming formula with inequality constraints: $\mathrm{<span\; class="notranslate">\; min</span>}\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&Theta;s</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; N</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; ></span>$ - μ _i ≤ s _i ≤ μ _i , i=1, 2, . . . , N. Among them, s _i is the i-th element of the similarity vector s, μ _i is the factor that constrains _si , and -μ _i ≤ _{s i} ≤ _{μ i} is the constraint condition; (7f) Establish a logarithmic barrier function for the constraint condition - μ _i ≤ s _i ≤ μ _i :

Using the logarithmic barrier function, transform Equation <5> into an optimal solution for the central trajectory function F _β (s, μ, α) = 0 defined by the weight factor β:

(7g) Using Newton's iterative method to solve equation <6>, the iterative formula can be obtained: $\left(\begin{array}{c}{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}\\ {\mathrm{<span\; class="notranslate">\; \&mu;</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}\\ {\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}\end{array}\right)<span\; class="notranslate">\; =</span>\left(\begin{array}{c}{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}\\ {\mathrm{<span\; class="notranslate">\; \&mu;</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}\\ {\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}\end{array}\right)<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; \&dtri;</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; \&beta;</span>}}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>{<span\; class="notranslate">\; \&dtri;</span>\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; \&beta;</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; )</span>$ Among them, s ^(k) , μ ^(k) and α ^(k) represent the results of s, μ and α after the kth iteration respectively, s ^(k+1) , μ ^(k+1) and α ^{(k+ 1)} respectively represent the results of s, μ and α after the k+1th iteration, k is a non-negative integer not greater than 50,

Represents the second order derivative of the function, Indicates the first derivative of the function; (7h) Get the initial value of weight factor β=0.5 and solution vector: $\left(\begin{array}{c}{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span>}\\ {\mathrm{<span\; class="notranslate">\; \&mu;</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span>}\\ {\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span>}\end{array}\right)<span\; class="notranslate">\; =</span>\left(\begin{array}{c}{\mathrm{<span\; class="notranslate">\; \&Theta;</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}\\ \mathrm{<span\; class="notranslate">\; 0.95</span>}<span\; class="notranslate">\; \&Center\; Dot;</span>\mathrm{<span\; class="notranslate">\; sgn</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&Theta;</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; \&Theta;</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 0.1</span>}\mathrm{<span\; class="notranslate">\; max</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; sgn</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&Theta;</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; \&Theta;</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span>\\ \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right)$ Among them, max means to take the maximum value of the vector element, and sgn means the positive and negative attributes of the vector element:

(7i) Bring the initial value and weight factor into step (7g) for iterative operation, until the result of two adjacent iterations is brought into formula <5> and the subtraction difference is not greater than 10 ^-3 , the obtained s at this time The value of is the minimum ^l1 norm solution of s in formula <1>, jump to step (7k); if the maximum number of iterations is 50, the optimal solution is still not obtained, go to step (7j); (7j) Use the final iterative result as the initial value, update the weight factor to 2 times the original value, reset the number of iterations to zero, and return to step (7i); (7k) Verify the sparsity of the minimum l ¹ norm solution s for all test images, and use s as the normalized target feature vector

The sparsest representation in the normalized background feature matrix Θ.

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