CN106056607A - Monitoring image background modeling method based on robustness principal component analysis - Google Patents

Abstract

The present invention discloses a monitoring image background modeling method based on robustness principal component analysis. The monitoring image background modeling method based on robustness principal component analysis is based on the sparse and low-matrix matrix decomposition theory, takes the robustness principal component analysis (RPCA) as the basis and employs a cut-off nucleus norm to take place of a traditional nucleus norm approximation matrix constraint, and in the frame of the augmentation Lagrange multiplier method, uses an inexact augmented Lagrange multiplier (IALM) with faster convergence to directly separate the prospect object and a background model from the monitoring image. The low-order matrix recovered by the method is a background image matrix, and the sparse large-noise matrix is a prospect object position matrix. The monitoring image background modeling method based on robustness principal component analysis can accurately detect the prospect object in the complex scene such as dynamic texture background, the illumination gradual change and the smog weather, and the recovered background matrix has a lower order to more concisely and effectively solve the real problem of the background modeling.

Description

A kind of monitoring image background modeling method based on Robust Principal Component Analysis

Technical field

The present invention relates to the background modeling method of monitoring image in a kind of safety-protection system.

Background technology

Developing rapidly of adjoint network technology and digital video technology, monitoring technology is day by day towards intelligent, networking side To development, this makes the requirement to monitoring image background extracting technology more and more higher, and good background modeling scheme is also real One of key technology of existing moving object detection and identification.Background modeling is generally used for the scene obtained from a still camera In be partitioned into dynamic object, its method is more, typically has basic background modeling method, statistics background modeling method, blurred background to build Modulus method and background estimating method, and using most is statistics background modeling method.Statistics background modeling method includes based on single Gaussian mode The method of type, method based on mixed Gauss model and modeling method based on Density Estimator etc..The base of these traditional methods This thinking is to first pass through one section of training image sequential extraction procedures of study to go out the background characteristics of this image sequence, thus sets up a number Learn model and describe its background, then by this background model the image sequence needing detection processed and (typically use background Subtractive method), extract pixels different from character in background model in present image, be the dynamic object of image.Due to The scene of video monitoring can vary over (such as illumination, shade and weather condition etc.), and these methods need the most more New background model, thus there is background model and can not adapt to the localized variation problem in scene rapidly and accurately.Simultaneously as Needing learning training sequence structure background model in advance, these all constrain them in video monitoring intellectuality and networking Application.As can be seen here, to not comprising the independent learning training stage and can accurately adapt to background modeling and the motion of scene changes The research tool of object detection method is of great significance.

Rank of matrix is estimated by sparse decomposition with low-rank matrix as one is sparse, it is possible to effectively from those by very noisy Pollute or the observation data of partial loss are focused to find out its low-dimensional eigenspace, and recover original observation signal or data. In monitoring image background modeling, the background parts of image is only by a small amount of controlling factors, thus shows the characteristic of low-rank；And Moving target or prospect can be detected, so monitoring image meets low-rank and adds sparse by the residual error identifying space sparse distribution Structure, can be as Robust Principal Component Analysis problem.And because of rank of matrix function and l in Robust Principal Component Analysis problem₀Norm Nonconvex property, is typically nuclear norm by rank of matrix functional relaxation, l₀Norm relaxes the l into matrix₁Norm.But matrix nuclear norm is same Time reduce all singular values of matrix, have ignored the prior information of singular value, so can not well force in practical situations both Nearly rank of matrix function.

It is desirable to have a kind of monitoring image background modeling method based on Robust Principal Component Analysis to overcome or at least Alleviate the most methodical drawbacks described above.

Summary of the invention

The present invention is directed to the problems referred to above in existing method, on the basis of RPCA, propose employing non-convex blocks nuclear norm Replace traditional convex nuclear norm and approach the constraint of matrix low-rank.Experiment display, the non-convex of this rank of matrix function is lax can be compared The convex lax low-rank feature that symbolizes more accurately, thus recover background more accurately in monitoring image background modeling is applied.

For achieving the above object, the present invention adopts the following technical scheme that

In monitoring image background modeling method, the image sequence observed is continuous print n two field picture, each two field picture picture Element value is by row one m dimensional vector of end to end composition, then this observed image sequence table is shown as the matrix D of m × n dimension, has The background parts of great similarity represents by low-rank matrix A to be restored, and the least foreground part of distribution is expressed as dilute Dredge matrix E, and D=A+E.On the basis of RPCA, employing is blocked the nuclear norm traditional nuclear norm of replacement and is approached the constraint of matrix low-rank. Block nuclear norm to define as shown in formula (1):

${||A||}_{T,*}={||A||}_{*}-\underset{{\mathrm{XY}}^T=I,{\mathrm{YY}}^T=I}{max}Tr\left({\mathrm{XAZ}}^T\right)---\left(1\right)$

WhereinRepresent that A's blocks nuclear norm, | | A | |_*Represent the nuclear norm of A, Tr (XAZ^T) it is matrix XAZ^TMark, X, Z are arbitrary matrix, X ∈ R^r×m,Z∈R^r×n, then form the optimization problem shown in formula (2):

$\begin{array}{cccc}\underset{A,E}{\min }& {||A||}_{*}-\underset{{\mathrm{XX}}^T=I,{\mathrm{YY}}^T=I}{\max }Tr\left({\mathrm{XAZ}}^T\right)+\mathrm{\λ}{||E||}_1& s.t.& D=A+E\end{array}---\left(2\right)$

Wherein | | E | |₁The l of representing matrix E₁Norm, λ is for balancing the low-rank degree of A and the sparse degree of E；Ask for solving Topic (2), devises a kind of alternative manner, and the l time iteration is

Step1: calculate A_lSingular value decomposition U Σ V^T, wherein U={u₁,u₂…u_m}∈R^m×m, V={v₁,v₂…v_n}∈R^{n ×n}, With σ={ σ₁,σ₂…σ_n, obtain U_r=(u₁,u₂…u_r)^TAnd V_r=(v₁,v₂…v_r)^T, r Being the order of matrix A, formula (2) can be exchanged into the convex optimization problem as shown in formula (3):

$\begin{array}{cccc}\underset{A,E}{\min }& {||A||}_{*}-Tr\left(U_r^T{\mathrm{AV}}_r\right)+\mathrm{\λ}{||E||}_1& s.t.& D=A+E\end{array}---\left(3\right)$

Step2: use augmented vector approach to solve the convex optimization problem shown in formula (3), the augmentation glug of formula (3) Bright day function L (A, E, Y, μ) is expressed as

$L(A,E,Y,\mathrm{\&mu;})={||A||}_{*}-Tr\left(U_r^T{\mathrm{AV}}_r\right)+\mathrm{\&lambda;}{||E||}_1+<Y,D-A-E>+\frac{\mathrm{\&mu;}}{2}{||D-A-E||}_F^2---\left(4\right)$

WhereinFor matrixMark, Y is linear equality constraints multiplier, and<Y, D-A-E>represents Y and D- The standard inner product of A-E, μ represents the penalty factor being unsatisfactory for linear equality constraints, | | D-A-E | |_FRepresent D-A-E's Frobenius norm；Use non-precision augmented vector approach to solve formula (4), first fix A and Y, ask one to make formula (4) The E minimized, then fixes E and Y, seeks an A making formula (4) minimum, and to kth time iteration, concrete iterative process is as follows:

Step 1: fixing A_kAnd Y_k, then

$E_{k+1}=S_{{\mathrm{\&lambda;\&mu;}}_k^{-1}}(D-A_k+{\mathrm{\&mu;}}_k^{-1}{Y}_k)$

Represent withSoft-threshold operator for threshold value；

Step 2: fixing E_kAnd Y_k, then

Represent withFor the singular value contraction operator of threshold value, according to the threshold value of contraction operatorDetermine r；

Step 3: fixing E_kAnd A_k, then

Y_k+1=Y_k+μ_k(D-A_k+1-E_k+1)；

Step 4: update μ_k

μ_k+1=min{ μ_kρ,μ_max}。

Iteration Step1 and Step2 until formula (3) solve convergence, recover A and E, it is achieved background and the separation of prospect.

Accompanying drawing illustrates:

Fig. 1 is to use what RPCA method based on nuclear norm and the present invention proposed to obtain based on the RPCA method blocking nuclear norm The monitoring image background modeling contrast effect figure arrived.In Fig. 1: (a) is that monitoring image (is respectively hall monitoring image from the top down 30th frame, the 96th frame and the 132nd frame of sequence)；B background that () obtains for RPCA method based on nuclear norm；C () is based on core The prospect that the RPCA method of norm obtains；D () is the background obtained based on the RPCA method blocking nuclear norm；E () is based on cutting The prospect that the RPCA method of disconnected nuclear norm obtains.

Detailed description of the invention:

Clearer for the purpose making the present invention implement, technical scheme and advantage, below in conjunction with in the embodiment of the present invention Accompanying drawing, the technical scheme in the embodiment of the present invention is further described in more detail.In the accompanying drawings, the most identical or class As label represent same or similar element or there is the element of same or like function.Described embodiment is the present invention A part of embodiment rather than whole embodiments.The embodiment described below with reference to accompanying drawing is exemplary, it is intended to use In explaining the present invention, and it is not considered as limiting the invention.Based on the embodiment in the present invention, ordinary skill people The every other embodiment that member is obtained under not making creative work premise, broadly falls into the scope of protection of the invention.Under Face combines accompanying drawing and is described in detail embodiments of the invention.

Test based on actual monitoring image data herein.The monitoring image sequence data that experiment uses is notable for having The hall monitoring image data set of dynamic object, shade change.Take the front 200 frame gray-scale maps of above-mentioned monitoring image data set respectively As experimental data, every two field picture size is respectively 144 × 176, its structure the observing matrix generated is respectively D (25344 ×200)。

In the background modeling method of monitoring image, the image sequence observed comprises 200 two field pictures, each two field picture picture Element value joins end to end by row and forms 25344 dimensional vectors, then the most available matrix D of this observed image sequence (25344 × 200) representing, the background parts with great similarity represents by low-rank matrix A to be restored, and before distribution is the least Scape is partially shown as sparse matrix E, and D=A+E.On the basis of RPCA use block nuclear norm replace traditional nuclear norm approach Matrix low-rank retrains.Block nuclear norm to define as shown in formula (1):

${||A||}_{T,*}={||A||}_{*}-\underset{{\mathrm{XX}}^T=I,{\mathrm{YY}}^T=I}{\max }Tr\left({\mathrm{XAZ}}^T\right)---\left(1\right)$

WhereinRepresent that A's blocks nuclear norm, | | A | |_*Represent the nuclear norm of A, Tr (XAZ^T) it is matrix XAZ^TMark, X, Z are arbitrary matrix, X ∈ R^r×m,Z∈R^r×n, then form the optimization problem shown in formula (2):

$\begin{array}{cccc}\underset{A,E}{\min }& {||A||}_{*}-\underset{{\mathrm{XX}}^T=I,{\mathrm{YY}}^T=I}{\max }Tr\left({\mathrm{XAZ}}^T\right)+\mathrm{\&lambda;}{||E||}_1& s.t.& D=A+E\end{array}---\left(2\right)$

Wherein | | E | |₁The l of representing matrix E₁Norm, λ is for balancing the low-rank degree of A and the sparse degree of E；Ask for solving Topic (2), devises a kind of alternative manner, and the l time iteration is

Step1: calculate A_lSingular value decomposition U Σ V^T, wherein U={u₁,u₂…u_m}∈R^m×m, V={v₁,v₂…v_n}∈R^{n ×n}, With σ={ σ₁,σ₂…σ_n, obtain U_r=(u₁,u₂…u_r)^TAnd V_r=(v₁,v₂…v_r)^T, r It it is the order of matrix A.Because working as X=U_rAnd Z=V_rTime, Tr (XAZ^T) maximum can be reached.Formula (2) can be exchanged into such as formula (3) institute The convex optimization problem shown:

$\begin{array}{cccc}\underset{A,E}{\min }& {||A||}_{*}-Tr\left(U_r^T{\mathrm{AV}}_r\right)+\mathrm{\&lambda;}{||E||}_1& s.t.& D=A+E\end{array}---\left(3\right)$

Step2: use augmented vector approach to solve the convex optimization problem shown in formula (3), the augmentation glug of formula (3) Bright day function L (A, E, Y, μ) is expressed as

$L(A,E,Y,\mathrm{\&mu;})=\left|\right|A|{|}_{*}-Tr\left(U_r^T{\mathrm{AV}}_r\right)+\mathrm{\&lambda;}|\left|E\right|{|}_1+<Y,D-A-E>+\frac{\mathrm{\&mu;}}{2}\left|\right|D-A-E|{|}_F^2---\left(4\right)$

WhereinFor matrixMark, Y is linear equality constraints multiplier, and<Y, D-A-E>represents Y and D- The standard inner product of A-E, μ represents the penalty factor being unsatisfactory for linear equality constraints, | | D-A-E | |_FRepresent D-A-E's Frobenius norm；Use non-precision augmented vector approach to solve formula (4), first fix A and Y, ask one to make formula (4) The E minimized, then fixes E and Y, seeks an A making formula (4) minimum, and to kth time iteration, concrete iterative process is as follows:

Step 1: fixing A_kAnd Y_k, then

$E_{k+1}=S_{{\mathrm{\&lambda;\&mu;}}_k^{-1}}(D-A_k+{\mathrm{\&mu;}}_k^{-1}{Y}_k)$

Represent withSoft-threshold operator for threshold value；

Step 2: fixing E_kAnd Y_k, then

Represent withFor the singular value contraction operator of threshold value, according to the threshold value of contraction operatorDetermine r；

Step 3: fixing E_kAnd A_k, then

Y_k+1=Y_k+μ_k(D-A_k+1-E_k+1)；

Step 4: update μ_k

μ_k+1=min{ μ_kρ,μ_max}。

Iteration Step1 and Step2 until problem (3) solve convergence, recover A and E, it is achieved background and prospect point From.

The present invention, on the basis of RPCA, proposes to use the nuclear norm traditional convex nuclear norm of replacement that blocks of non-convex to approach square Battle array low-rank constraint, the non-convex of this rank of matrix function is lax convex the relaxing of ratio can symbolize low-rank feature more accurately, thus The application of monitoring image background modeling recovers background more accurately.

For the checking present invention based on blocking the effectiveness of RPCA method and the feasibility of nuclear norm, utilize software MATLAB R2013b is simulated emulation, verifies the effect of monitoring image background modeling.

The 30th frame, the 96th frame and the 132nd frame of hall image sequence is chosen, by based on nuclear norm from results of experimental operation RPCA method and monitoring image background modeling contrast effect figure such as Fig. 1 institute of obtaining based on the RPCA method blocking nuclear norm Show.In processing this practical problem of monitoring image background modeling, based on blocking the RPCA method of nuclear norm compared to based on core The RPCA method of norm, it is possible to adapt to the dynamic change in scene more accurately, its background recovered contain less prospect, Local shades；This phenomenon not only shows its background matrix recovered more low-rank, is Image Data Compression and storage simultaneously Provide a kind of new probability.

Last it is noted that above example is only in order to illustrate technical scheme, it is not intended to limit.To the greatest extent The present invention has been described in detail by pipe with reference to previous embodiment, it will be understood by those within the art that: it is still Technical scheme described in foregoing embodiments can be modified, or wherein portion of techniques feature is carried out equivalent replace Change；And these amendments or replacement, do not make the essence of appropriate technical solution depart from the essence of various embodiments of the present invention technical scheme God and scope.

Claims (2)

1. a monitoring image background modeling method based on Robust Principal Component Analysis, it is characterised in that: each two field picture Pixel value is by row one m dimensional vector of end to end composition, then continuous print n two field picture is expressed as the matrix D of m × n dimension, has The background parts of great similarity represents by low-rank matrix A to be restored, and the least foreground part of distribution is expressed as dilute Dredging matrix E, and D=A+E, on the basis of Robust Principal Component Analysis, employing is blocked the nuclear norm traditional nuclear norm of replacement and is approached matrix Low-rank retrains, and blocking nuclear norm definition is

$\left|\right|A|{|}_{T,*}=|\left|A\right|{|}_{*}-\underset{{\mathrm{XX}}^T=I,{\mathrm{YY}}^T=I}{max}Tr\left({\mathrm{XAZ}}^T\right)---\left(1\right)$

WhereinRepresent that A's blocks nuclear norm, | | A | |_*Represent the nuclear norm of A, Tr (XAZ^T) it is matrix XAZ^TMark, X, Z are Arbitrary matrix, X ∈ R^r×m,Z∈R^r×n, then form the optimization problem shown in formula (2):

$\begin{array}{cccc}\underset{A,E}{\min }& \left|\right|A|{|}_{*}-\underset{{\mathrm{XX}}^T=I,{\mathrm{YY}}^T=I}{\max }Tr\left({\mathrm{XAZ}}^T\right)+\mathrm{\&lambda;}|\left|E\right|{|}_1& s.t.& D=A+E\end{array}---\left(2\right)$

Wherein | | E | |₁The l of representing matrix E₁Norm, λ is for balancing the low-rank degree of A and the sparse degree of E；For Solve problems (2), devising a kind of alternative manner, the l time iteration is

Step1: calculate A_lSingular value decomposition U ∑ V^T, wherein U={u₁,u₂…u_m}∈R^m×m, V={v₁,v₂…v_n}∈R^n×n,With σ={ σ₁,σ₂…σ_n, obtain U_r=(u₁,u₂…u_r)^TAnd V_r=(v₁,v₂…v_r)^T, r is square The order of battle array A, formula (2) can be exchanged into the convex optimization problem as shown in formula (3):

$\begin{array}{cccc}\underset{A,E}{\min }& \left|\right|A|{|}_{*}-Tr\left(U_r^T{\mathrm{AV}}_r\right)+\mathrm{\&lambda;}|\left|E\right|{|}_1& s.t.& D=A+E\end{array}---\left(3\right)$

Step2: using augmented vector approach to solve the convex optimization problem shown in formula (3), iteration is until converging to this The optimal solution of subproblem, recovers A and E, it is achieved background and the separation of prospect.

Monitoring image background modeling method based on Robust Principal Component Analysis the most according to claim 1, it is characterised in that: Convex optimization problem shown in formula (3), Augmented Lagrangian Functions L (A, E, Y, μ) of formula (3) is expressed as

$L(A,E,Y,\mathrm{\&mu;})=\left|\right|A|{|}_{*}-Tr\left(U_r^T{\mathrm{AV}}_r\right)+\mathrm{\&lambda;}|\left|E\right|{|}_1+<Y,D-A-E>+\frac{\mathrm{\&mu;}}{2}\left|\right|D-A-E|{|}_F^2---\left(4\right)$

WhereinFor matrixMark, Y is linear equality constraints multiplier, and<Y, D-A-E>represents Y and D-A-E Standard inner product, μ represents the penalty factor being unsatisfactory for linear equality constraints, | | D-A-E | |_FRepresent the Frobenius model of D-A-E Number；Use non-precision augmented vector approach to solve formula (4), first fix A and Y, seek an E making formula (4) minimize, so Rear fixing E and Y, seeks an A making formula (4) minimum, and to kth time iteration, concrete iterative process is as follows:

Step 1: fixing A_kAnd Y_k, then

$E_{k+1}=S_{{\mathrm{\&lambda;\&mu;}}_k^{-1}}(D-A_k+{\mathrm{\&mu;}}_k^{-1}{Y}_k)$

Represent withSoft-threshold operator for threshold value；

Step 2: fixing E_kAnd Y_k, then

Represent withFor the singular value contraction operator of threshold value, according to the threshold value of contraction operatorDetermine r；

Step 3: fixing E_kAnd A_k, then

Y_k+1=Y_k+μ_k(D-A_k+1-E_k+1)；

Step 4: update μ_k

μ_k+1=min{ μ_kρ,μ_max}。