CN103473759B - The twilight image method for extracting remarkable configuration that a kind of WKPCA homogeneity degree correction nCRF suppresses - Google Patents

Abstract

The invention discloses a kind of WKPCA homogeneity degree correction nCRF twilight image method for extracting remarkable configuration suppressed.First propose a kind of WKPCA algorithm, each characteristic vector of high-dimensional feature space is carried out coupling (FAM) weighting of characteristic vector angle, weaken or get rid of characteristic interference ill or abnormal in CRF region, extracting CRF main constituent more accurately；On this basis, a kind of homogeneity degree concept of definition and computational methods, by the projection at center main composition of the nCRF characteristic vector, the homogeneity at computing environment-center；Be finally based on this homogeneity degree each amount of suppression in nCRF is corrected so that the mutual amount of suppression of homogeneous region is big, heterogeneous areas amount of suppression is little or mutually do not suppress, weakened outline element self-inhibiting effect as far as possible simultaneously, thus improving inhibiting accuracy rate.Therefore the model that the present invention proposes can more fully detect the difference at environment-center, reduces noise jamming, suppresses grain details more accurately, improves profile response intensity and integrity.

Description

The twilight image method for extracting remarkable configuration that a kind of WKPCA homogeneity degree correction nCRF suppresses

Technical field

The invention belongs to twilight image method for extracting remarkable configuration under the complex scene of a kind of view-based access control model modeling, the twilight image method for extracting remarkable configuration that particularly a kind of WKPCA homogeneity degree correction nCRF suppresses.

Background technology

Contours extract is understood at night vision image and is played an important role in analysis.The application major part of current night vision target acquisition identification aspect is for natural scene, therefore containing substantial amounts of natural texture (such as tree and grass) in twilight image, the exercising result of traditional edge detection operator retains marginal element's (canny operator) of a large amount of non-profiles.And the noise jamming of twilight image own is strong, how for twilight image feature, remove these non-interest edges, local produced by texture and noise, and to keep the integrity of profile be the main problems faced of night vision image contour detecting.

Contours extract problem for complex scene proposes many solutions, wherein obtains remarkable result based on the contours extract of the non-classical receptive field model of biological vision mechanism in high-quality visible images.CRF is played modulating action by the big periphery (non-classical receptive field nCRF) of visual cortex (V1) neuron receptive field (CRF), and this modulation is mainly inhibition, it is possible to realizes homogeneous region and mutually suppresses, and makes the edge isolated more more notable than colony edge.Based on the bionic model in lateral inhibition district, eliminate the edge that background texture produces preferably.Grigorescu et al. (ContourdetectionbasedonNonclassicalReceptiveFieldinhibit ion;ImprovedContourDetectionbyNon-classicalReceptiveFieldInhibition;Contourandboundarydetectionimprovedbysurroundsuppression oftextureedges) utilize the rejection characteristic of nCRF to carry out contour detecting, utilize environment that the direction at center is suppressed, decrease the impact of environment texture, and propose anisotropy suppression and isotropism inhibition.Giuseppe et al. (ABiologicallyMotivatedMultiresolutionApproachtoContourDe tection) proposes a kind of biological multiresolution contour detecting technology inspired based on Bayes's noise reduction and environment suppression technology.Ursino et al. (Amodelofcontourextractionincludingmultiplescales, flexibleinhibitionandattention) introduces level attention mechanism, it is considered to the contours extract that under different scale, background suppresses.These inhibitions can a degree of suppression background texture, extract remarkable configuration；But center environment is not carried out feature difference analysis, it is impossible to well solve heterogeneous composition mutual inhibition system and outline elements from suppressing problem, it is possible to cause profile response faint and profile is interrupted.

For reducing the effect that conllinear profile suppresses, mulberry planter et al. (Contourdetectionbasedoninhibitionofprimaryvisualcortex) stimulates orientation to stimulate orientation discrepancy with CRF according to nCRF, to inhibitory action weighting, establish the butterfly model based on lateral inhibition district.Zeng et al. (Center-surroundinteractionwithadaptiveinhibition:Acomput ationalmodelforcontourdetection;Contourdetectionbasedonanon-classicalreceptivefieldmodel withbutterfly-shapedinhibitionsubregions) the set direction inhibition of a kind of pair of yardstick contour extraction method and a kind of improvement is proposed, this model adopts butterfly region computing environment to suppress.Giuseppe et al. (Animprovedmodelforsurroundsuppressionbysteerablefiltersa ndmultilevelinhibitionwithapplicationtocontourdetection) utilizes variable filter and multi-layer to suppress to propose a kind of environment inhibition.The inhibition of direction difference weighting decreases profile self-inhibiting effect to a certain extent, but there is the situation that incorgruous heterogeneous areas suppresses mutually.And relative to high-quality visible images, LLL image noise serious interference, profile local orientation feature does not highlight, environment suppresses inaccurate.Strong noise, low contrast cause that twilight image background texture cannot suppress completely on the one hand；Inhibitory action weakened outline intensity on the other hand, profile is subject to the suppression of periphery texture, it is easy to fracture occurs, affects succeeding target identification.

Summary of the invention

It is an object of the invention to provide the WKPCA homogeneity degree correction nCRF twilight image method for extracting remarkable configuration suppressed, the method can solve the problem that the deficiency that single gun parallax weighting suppresses, in conjunction with biological vision mechanism, for twilight image multidimensional characteristic difference, twilight image effectively suppresses noise and texture from complex scene, extracts remarkable configuration.

The technical solution realizing the object of the invention is: first study coupling (FAMFeaturevectorAngleMatching) weighting KPCA (WKPCAWeightingKPCA) method computing environment-center homogeneity degree (HDHomogeneityDegree), feature vectors angle at high-dimensional feature space: the method inhibits the morbid state in CRF pixel or abnormal data, again it is carried out principal component analysis, improve main constituent accuracy rate；Each for nCRF pixel is calculated Qi Yu CRF center homogeneity degree in the projection of center main composition.Secondly based on this environment-center multiple features difference, nCRF is suppressed correction, build a kind of homogeneity degree correction inhibition: this model is while same prime element mutual inhibition, avoid heterogeneous pixel interphase interaction and profile composition from suppressing, realize the noise in twilight image, texture suppresses, and the remarkable configuration background separation of night vision image under complicated natural scene.

The present invention compared with prior art, its remarkable advantage:

(1) the present invention is directed to twilight image multidimensional characteristic analysis, a kind of WKPCA algorithm is proposed first, in high-dimensional feature space feature based vectorial angle matching criterior to CRF Covariance Matrix Weighting, effectively weaken off-note data and noise jamming, improve the accuracy of CRF principal component analysis.

(2) the definition one homogeneity degree concept of novelty of the present invention and computational methods thereof, adopt WKPCA algorithm to carry out CRF and nCRF region homogeneous assay, at each pixel eigenvector projection of high-dimensional feature space nCRF to CRF center main composition, both calculating homogeneity degree.The method can the difference at indicator environment-center accurately more comprehensively.

(3) present invention proposes a kind of WKPCA homogeneity degree correction nCRF inhibition, according to each pixel of nCRF and CRF center homogeneity degree, its amount of suppression is corrected.While the more effective eliminating noise jamming of this model, weakening background detail, improve the suppression degree of homogeneity composition more accurately, reduce the probability that the mutual suppression between heterogeneous pixel, outline elements, raising profile Whole Response and minimizing profile are interrupted.

Accompanying drawing explanation

Fig. 1 is nCRF range-attenuation function.

Fig. 2 is that WKPCA homogeneity degree correction nCRF suppresses twilight image method for extracting remarkable configuration flow chart.

Fig. 3 is multidimensional feature space non-classical receptive field structure.

Fig. 4 is the nCRF schematic enlarged-scale view centered by real goal profile pixel, and each pixel characteristic vector and CRF central mean vector FAM weights graphics in nCRF.

Fig. 5 is that PCA, KPCA, WKPCA algorithm classifying quality to Gaussian data compares.

Fig. 6 is PCA, KPCA, WKPCA algorithm homogeneous assay results contrast.

Fig. 7 is the WKPCA homogeneity degree correction inhibition analysis at background, target, profile center.Row is to being artwork, the output of WKPCA homogeneity angle value, the output of Gabor energy extreme value from top to bottom respectively and effectively suppressing output (homogeneity angle value and Gabor energy extreme value product).

Fig. 8 is that each model is to twilight image contours extract effectiveness comparison under complicated natural scene.

Detailed description of the invention

Below in conjunction with accompanying drawing, the application is described further.

1, nCRF environment suppresses:

Two-dimensional Gabor function can describe the receptive field section of visual cortex simple cell effectively, by the odd even reaction mould (Gabor energy) to simple receptive field wave filter, can simulate the fundamental characteristics of typical complex cell well.These complex cells can regard local azimuthal energy operator as, and graph edge and line can be accurately positioned by the maximum movable with complex cell, and therefore the present invention carrys out the response of Simulation of Complex cell by Gabor energy.Two-dimensional Gabor filter is expressed as follows.

Whereinθ is the preference orientation of CRF；It it is difference；ByWithRepresent odd even Gabor filter respectively；Length-width ratio γ determines the eccentricity that Gauss is oval；λ is wavelength；Gauss standard difference σ determines CRF active area；σ/λ representation space frequency bandwidth.

The response of simple cellConvolution for Gabor function Yu input picture I.The neuron response E stimulated by orthogonal simple cell response definition CRF_σ(x,y;θ_i)。

$E_{\mathrm{\σ}}^2(x,y;{\mathrm{\θ}}_i)=r_{\mathrm{\λ\σ}0}^2(x,y;{\mathrm{\θ}}_2)+r_{\mathrm{\λ\σ\π}/2}^2(x,y;{\mathrm{\θ}}_i)---\left(3\right)$

Wherein Gabor energy is divided into N_θIndividual response orientation: θ_i=iπ/N_θ,i=0,1,…,N_θ-1.Parameter is set according to [9]: N_θ=12, γ=0.5, σ/λ=0.56, select σ as independent variable.The response of CRF is output as each orientation Gabor energy extreme value.

E_σ(x,y)=max{E_σ(x,y;θ_i)|i=0,1,…,N_θ-1}(4)

Biological study shows that the nCRF of V1 district cell is mainly inhibition, and this environment inhibitory action is intended to separate homogeneous region, highlights profile.NCRF model adopts range-attenuation function ω_σ(x y) describes the space structure that environment suppresses.ω_σ(x y) is defined as the coaxial Gaussian difference function DOG of half-wave correction and L1 specification_σ,ρσ(x,y)。

${\mathrm{DOG}}_{\mathrm{\σ},\mathrm{\ρ\σ}}(x,y)=\frac{1}{\sqrt{2\mathrm{\π}{\left(\mathrm{\ρ\σ}\right)}^2}}\exp (-\frac{x^2+y^2}{2{\left(\mathrm{\ρ\σ}\right)}^2})-\frac{1}{\sqrt{2\mathrm{\π}{\mathrm{\σ}}^2}}\exp (-\frac{x^2+y^2}{2{\mathrm{\σ}}^2})---\left(5\right)$

ω_σ(x,y)=|DOG_σ,ρσ(x,y)|⁺/‖|DOG_σ,ρσ(x,y)|⁺‖₁

${\left|z\right|}^{+}=\left\{\begin{array}{cc}0& z<0\\ z& z\&GreaterEqual;0\end{array}\right.---\left(6\right)$

Suppression form t_σ(x y) is defined as the distance decay Local Orientation Energy of each pixel in annular background.

t_σ(x,y)=ω_σ(x,y)*E_σ(x,y)(7)

As shown in Figure 1, | DOG |⁺Guarantee ω_σ(x y) acts only on environmental area.σ and ρ σ determines CRF area S respectively_cWith nCRF area S_nc.Biotic experiment shows that nCRF diameter can reach 2-5 times of CRF diameter, therefore takes ρ=4.Pass through DOG_σ,kσ(x, y)=0, it is possible to obtain CRF radius r_cWith nCRF radius ρ r_c。

$r_c=\sqrt{2\ln \mathrm{\&rho;}/{\mathrm{\&rho;}}^2-1}\&CenterDot;\mathrm{\&rho;\&sigma;}---\left(8\right)$

NCRF environment suppresses output to be:

R_σ(x,y)=|aE_σ(x,y)-bt_σ(x,y)|⁺

(9)

2, the correction of homogeneity degree suppresses

As shown in Figure 2, for improving the accuracy that environment-center suppresses, make to provide bigger amount of suppression, heterogeneous region to misalign the heart with the region of center homogeneity to suppress, and contour area is from suppressing minimizing, each environment pixel and center, on nCRF model basis, are carried out homogeneous assay by the present invention, it is proposed to a kind of homogeneity degree computational methods, environment is suppressed composition correction, it is achieved the lower twilight image background texture of very noisy interference suppresses and remarkable configuration extracts.

For each pixel of computing environment and center homogeneity degree δ_w(x, y), at multidimensional feature space, intends adopting KPCA method that CRF center pel carries out principal component analysis, and at CRF principal component projection, pixel each in nCRF is calculated both diversityes.But morbid state or abnormal data (background data etc. being mingled with in the target data that is mingled with in noise data, background, target) is there is due to CRF center, cause that Principle component extraction is inaccurate, therefore the present invention proposes feature vectors angle coupling (FAM) weighting KPCA method (WKPCA), suppress morbid state or abnormal data, improve principal component analysis accuracy rate.

2.1 multidimensional characteristic analyses

Above-mentioned annular nCRF structure suppresses on an equal basis in each orientation, can cause mutually suppressing between heterogeneous pixel and outline elements, it is easy to weakened outline response and destruction integrality of outline.Butterfly inhibition adopts a kind of environment-center hold difference method of weighting exactly, the interaction between conllinear outline elements is reduced while suppressing grain details, high-quality visible images contours extract performance is better than isotropism, anisotropic model by it, but still suffers from the incorgruous mutual inhibitory action of heterogeneous pixel.And compared to high-quality visible images, in twilight image, edge is inaccurate even without obvious directivity by noise jamming Gabor oriented energy, and the butterfly therefore only with direction difference weighting suppresses cannot extract twilight image profile very well.

Biotic experiment shows that inhibitory action is the strongest when the graphic feature in CRF and nCRF is consistent, and when both there are differences, inhibitory action weakens or disappears, and this correlated characteristic includes orientation, spatial frequency and contrast etc..Therefore for making up the deficiency that single direction difference weighting suppresses, the present invention adopts multidimensional characteristic analysis, calculates nCRF and CRF difference more comprehensively accurately at feature space.For effective indicator twilight image feature, reduce computing redundancy simultaneously, choosing the statistical nature of 4 invariable rotaries in each pixel 5 × 5 neighborhood and the direction feature set as pixel, wherein 4 statistical natures include spatial frequency, contrast, gray scale symbiosis statistical entropy and gray scale symbiosis statistic correlation.Direction indicator gradient information；Spatial frequency and contrast reflection local gray level statistical distribution；When being set to 1 or 2 when image space-between, gray scale symbiosis statistical property can describe local high-frequency information, and be different from Gabor filter and adopt Gaussian function localized region weighting, gray scale symbiosis statistical property is assumed in neighborhood window for being uniformly distributed, thus to local insensitive for noise.Therefore merge various features and form the feature set of more indicator.

Definition fx=[O_c,SF_c,CON_c,cooE_c,cooCOR_c]^T, fy=[O_nc,SF_nc,CON_nc,cooE_nc,cooCOR_nc]^TRepresent each pixel characteristic vector in CRF and nCRF respectively.Here O_c、O_nc、SF_c、SF_nc、CON_c、CON_nc、cooE_c、cooE_nc、cooCOR_cAnd cooCOR_ncRepresent the normalized preference direction of each pixel, spatial frequency, contrast and gray scale symbiosis statistical entropy and dependency in CRF and nCRF respectively.

2.2 homogeneous assays

For calculating the difference at each environment pixel of nCRF and CRF center in feature space, it is possible to adopt principal component analysis (PCA) algorithm, namely calculate the main constituent of center pel feature set, and environment pixel characteristic vector is calculated both homogeneity degree at principal component projection.Linear PCA algorithm requires data Gaussian distributed, and real scene is difficult to meet by this condition；KPCA algorithm utilizes the dependency between each dimensional feature, is mapped in high-dimensional feature space by linear space characteristic signal and carries out Difference test, it is possible to excavate the nonlinear transformations between each feature, better distinguishes profile and background.

KPCA

KPCA algorithm, centered by test point, calculates its CRF and nCRF data high-dimensional feature space difference.If subspace dimension is P, define P × N_cMatrixRepresent CRF multi-dimensional feature data, sample fx_iFor P dimensional feature vector fx=[O_c,SF_c,CON_c,cooE_c,cooCOR_c]^T, N_cFor pixel number in CRF；P×N_ncMatrixRepresent nCRF multi-dimensional feature data, sample fy_jFor P dimensional feature vector fy=[O_nc,SF_nc,CON_nc,cooE_nc,cooCOR_nc]^T, N_ncFor pixel number in nCRF.After utilizing nonlinear mapping function φ that FX, FY are mapped to high-dimensional feature space, C_φIt is defined as CRF high-dimensional feature space covariance:

$C_{\mathrm{\&phi;}}=\frac{1}{N_c}\underset{i=1}{\overset{N_c}{\mathrm{\&Sigma;}}}(\mathrm{\&phi;}\left({\mathrm{fx}}_i\right)-{\mathrm{\&mu;}}_{\mathrm{\&phi;}}){(\mathrm{\&phi;}\left({\mathrm{fx}}_i\right)-{\mathrm{\&mu;}}_{\mathrm{\&phi;}})}^T---\left(10\right)$

WhereinV_φFor C_φNonzero eigenvalue Λ_φCharacteristic of correspondence vector.

The each pixel of nCRF and CRF center homogeneity degree δ (r) are defined as

${\mathrm{\&delta;}}^{-1}\left(r\right)={(\mathrm{\&phi;}\left(r\right)-{\mathrm{\&mu;}}_{\mathrm{\&phi;}})}^T{V}_{\mathrm{\&phi;}}{V}_{\mathrm{\&phi;}}^T(\mathrm{\&phi;}\left(r\right)-{\mathrm{\&mu;}}_{\mathrm{\&phi;}})---\left(11\right)$

Wherein r ∈ FY is each pixel characteristic vector of nCRF, namely in the high-dimensional feature space computing environment pixel feature projection property at center main composition.

WKPCA

KPCA can more effective extraction and utilize the nonlinear characteristic of data, but KPCA method there is also certain defect, when even practical center data are that in morbid state distribution (noise jamming), background composition, in impact point number or subject component, background dot number is more, high-dimensional feature space covariance matrix can not be fully described target or background data.CRF correlation matrix C is can be seen that by formula 10_φIn the weight of each pixel equal, if but centre data be morbid state distribution, central. set becomes background data impact point and mixes or target data background dot mixes, C_φCRF data distribution cannot be fully described, and principal component analysis is inaccurate.So the CRF covariance matrix C of high-dimensional feature space test point neighborhood_φIn each pixel be both needed to introduce weight factor, it is suppressed that or reduce the background dot in target data or the impact point in abnormity point, background data or abnormity point to C_φImpact, C_φTarget, background data distribution character just can be described more accurately.

Therefore the present invention proposes a kind of WKPCA algorithm, the method at high-dimensional feature space with C_φIn each pixel characteristic vector φ (fx_i) and data center's vector mean μ_φCharacteristic angle coupling (FAM) be criterion, to C_φIn each pixel characteristic vector introduce corresponding weight factor.Even pixel characteristic vector is less with the angle of data center's vector average, represents that both characteristic informations are similar, then this pixel obtains bigger weights, and vice versa, so removes or suppresses C_φIn abnormal data.Weighting CRF covariance matrix C in high-dimensional feature space_φwIt is expressed as:

$C_{\mathrm{\&phi;w}}=\frac{1}{N_c}\underset{i=1}{\overset{N_c}{\mathrm{\&Sigma;}}}{w}_{{\mathrm{fx}}_i}(\mathrm{\&phi;}\left({\mathrm{fx}}_i\right)-{\mathrm{\&mu;}}_{\mathrm{\&phi;}}){(\mathrm{\&phi;}\left({\mathrm{fx}}_i\right)-{\mathrm{\&mu;}}_{\mathrm{\&phi;}})}^T/\underset{i=1}{\overset{N_c}{\mathrm{\&Sigma;}}}{w}_{{\mathrm{fx}}_i}---\left(12\right)$

In formulaFAM weight factor for each pixel:

$w_{{\mathrm{fx}}_i}=\cos (\mathrm{\&phi;}\left({\mathrm{fx}}_i\right),{\mathrm{\&mu;}}_{\mathrm{\&phi;}})=\frac{{\mathrm{\&phi;}}^T\left({\mathrm{fx}}_i\right){\mathrm{\&mu;}}_{\mathrm{\&phi;}}}{\left|\right|{\mathrm{\&phi;}}^T\left({\mathrm{fx}}_i\right)\left|\right|\left|\right|{\mathrm{\&mu;}}_{\mathrm{\&phi;}}\left|\right|}=\frac{\frac{1}{N_c}\underset{m=1}{\overset{N_c}{\mathrm{\&Sigma;}}}k({\mathrm{fx}}_i,{\mathrm{fx}}_m)}{\sqrt{k({\mathrm{fx}}_i,{\mathrm{fx}}_i)}\sqrt{\frac{1}{N_c^2}\underset{m=1}{\overset{N_c}{\mathrm{\&Sigma;}}}\underset{n=1}{\overset{N_c}{\mathrm{\&Sigma;}}}k({\mathrm{fx}}_m,{\mathrm{fx}}_n)}}---\left(13\right)$

I=1 ..., N_c, wherein make use of kernel function quality: k (fx, fy)=<φ (fx), φ (fy)>.

Namely WKPCA algorithm calculates C in high-dimensional feature space_φwCharacteristic vector V_φwAnd in nCRF each pixel characteristic vector φ (r) at V_φwOn projection < V_φw,φ(r)>。

DefinitionHigh-dimensional feature space weighted data ${\{{\mathrm{\&phi;}}_w\left({\mathrm{fx}}_i\right)=\sqrt{w_{{\mathrm{fx}}_i}/w}(\mathrm{\&phi;}\left({\mathrm{fx}}_i\right)-{\mathrm{\&mu;}}_{\mathrm{\&phi;}})\}}_{i=1}^{N_c},$ Can be released by formula 12:

$C_{\mathrm{\&phi;w}}=\frac{1}{N_c}\underset{i=1}{\overset{N_c}{\mathrm{\&Sigma;}}}{\mathrm{\&phi;}}_w\left({\mathrm{fx}}_i\right){\mathrm{\&phi;}}_w{\left({\mathrm{fx}}_i\right)}^T---\left(14\right)$

For C_φwThe diagonal matrix that nonzero eigenvalue is constituted,For each eigenvalue characteristic of correspondence vector.By formula 14 it can be seen that each characteristic vectorAll at φ_w(FX) in metric space, thereforeCan be expressed asLinear combination:

$V_{\mathrm{\&phi;w}}^l=\underset{i=1}{\overset{N_c}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_i^l{\mathrm{\&phi;}}_w\left({\mathrm{fx}}_i\right)={\mathrm{\&phi;}}_w\left(\mathrm{FX}\right){\mathrm{\&alpha;}}^l---\left(15\right)$

Wherein ${\mathrm{\&alpha;}}^l={[{\mathrm{\&alpha;}}_1^l,{\mathrm{\&alpha;}}_2^l,...,{\mathrm{\&alpha;}}_{N_c}^l]}^T.$

14 formulas and 15 formulas are substituted intoCan obtain:

$N_c{\mathrm{\&lambda;}}_w^l{\mathrm{\&phi;}}_w\left(\mathrm{FX}\right){\mathrm{\&alpha;}}^l={\mathrm{\&phi;}}_w\left(\mathrm{FX}\right){\mathrm{\&phi;}}_w^T\left(\mathrm{FX}\right){\mathrm{\&phi;}}_w\left(\mathrm{FX}\right){\mathrm{\&alpha;}}^l---\left(16\right)$

By 16 formula both sides premultiplicationsAndCan obtainThen nuclear matrix K_wEigenvalue be covariance matrix C_φwN_cTimes, characteristic of correspondence vectorUtilize kernel function quality by K_wIt is expressed as:

$\begin{array}{cc}{K}_w=D_w{N}_c{\mathrm{\&Lambda;}}_{\mathrm{\&phi;w}}{D}_w^T={\left[{\mathrm{\&phi;}}_w^T\left({\mathrm{fx}}_i\right){\mathrm{\&phi;}}_w\left({\mathrm{fx}}_j\right)\right]}_{N_c\&times;N_c}& i,j=1,...,N_c\end{array}$

$=\begin{array}{cc}{\left[\frac{\sqrt{w_{{\mathrm{fx}}_i}{w}_{{\mathrm{fx}}_j}}}{w}{(\mathrm{\&phi;}\left({\mathrm{fx}}_i\right)-{\mathrm{\&mu;}}_{\mathrm{\&phi;}})}^T(\mathrm{\&phi;}\left({\mathrm{fx}}_j\right)-{\mathrm{\&mu;}}_{\mathrm{\&phi;}})\right]}_{N_c\&times;N_c}& i,j=1,...,N_c---\left(17\right)\end{array}$

$=\begin{array}{cc}{\left[\frac{\sqrt{w_{{\mathrm{fx}}_i}{w}_{{\mathrm{fx}}_j}}}{w}(k({\mathrm{fx}}_i,{\mathrm{fx}}_j)-\frac{1}{N_c}\underset{m=1}{\overset{N_c}{\mathrm{\&Sigma;}}}k({\mathrm{fx}}_i,{\mathrm{fx}}_m)-\frac{1}{N_c}\underset{m=1}{\overset{N_c}{\mathrm{\&Sigma;}}}k({\mathrm{fx}}_m,{\mathrm{fx}}_j)+\frac{1}{N_c^2}\underset{m=1}{\overset{N_c}{\mathrm{\&Sigma;}}}\underset{n=1}{\overset{N_c}{\mathrm{\&Sigma;}}}k({\mathrm{fx}}_m,{\mathrm{fx}}_n))\right]}_{N_c\&times;N_c}& i,j=1,...,N_c\end{array}$

Can being obtained by 15 formulas, nCRF pixel characteristic vector φ (r) is at high-dimensional feature space weighted feature vector V_φwOn be projected as:

$<V_{\mathrm{\&phi;w}},\mathrm{\&phi;}\left(r\right)>=D_w^T{\mathrm{\&phi;}}_w^T\left(\mathrm{FX}\right)\mathrm{\&phi;}\left(r\right)=D_w^T{k}_w(Z,r)---\left(18\right)$

In formula $\begin{array}{cc}{k}_w(Z,r)={\mathrm{\&phi;}}_w^T\left(\mathrm{FX}\right)\mathrm{\&phi;}\left(r\right)={\left[\sqrt{w_{{\mathrm{fx}}_i}/w}(k({\mathrm{fx}}_i,r)-\frac{1}{N_c}\underset{m=1}{\overset{N_c}{\mathrm{\&Sigma;}}}k({\mathrm{fx}}_m,r))\right]}_{N_c\&times;1}& i=1,...,N_c\end{array}$

The each pixel of high-dimensional feature space nCRF and center CRF homogeneity degree δ_wR () is expressed as:

${\mathrm{\&delta;}}_w^{-1}\left(r\right)={(\mathrm{\&phi;}\left(r\right)-{\mathrm{\&mu;}}_{\mathrm{\&phi;}})}^T{V}_{\mathrm{\&phi;w}}{V}_{\mathrm{\&phi;w}}^T(\mathrm{\&phi;}\left(r\right)-{\mathrm{\&mu;}}_{\mathrm{\&phi;}})$

$={(\mathrm{\&phi;}\left(r\right)-{\mathrm{\&mu;}}_{\mathrm{\&phi;}})}^T{\mathrm{\&phi;}}_w\left(\mathrm{FX}\right)D_w{D}_w^T{\mathrm{\&phi;}}_w^T\left(\mathrm{FX}\right)(\mathrm{\&phi;}\left(r\right)-{\mathrm{\&mu;}}_{\mathrm{\&phi;}})---\left(19\right)$

$={(k_w(Z,r)-\frac{1}{N_c}\underset{i=1}{\overset{N_c}{\mathrm{\&Sigma;}}}{k}_w(Z,{\mathrm{fx}}_i))}^T{D}_w{D}_w^T(k_w(Z,r)-\frac{1}{N_c}\underset{i=1}{\overset{N_c}{\mathrm{\&Sigma;}}}{k}_w(Z,{\mathrm{fx}}_i))$

Therefore suitable kernel function k is selected to construct the nuclear matrix K of positive definite_w, calculate its characteristic vector D_w, the homogeneity degree δ of each environment pixel and center can be drawn_w(x,y).Radial basis kernel function is selected by many experimentsIt can capacity volume variance between characteristic feature vector；Simultaneously adopt FAM function can characteristic feature to the shape difference of discharge curve, such WKPCA algorithm had both considered characteristic energy difference between pixel, it is also considered that characteristic curve shape difference between pixel, more fully distinguished multi-dimensional feature data.

2.3 homogeneity degree corrections suppress

Still consider that amount of suppression is with range attenuation characteristic, in conjunction with homogeneity degree δ_w(x, y) with range-attenuation function ω_σ(x, y) corrects each pixel amount of suppression in nCRF, and environment in nCRF model is suppressed T_σ(x, y) revision is as follows:

T_σ(x,y)=(δ_w(x,y)×ω_σ(x,y))*E_σ(x,y)

(20)

More accurately calculate CRF center main composition due to WKPCA, target and background can effectively be distinguished.It is background interior pel situation for test point, it is limited mainly by the suppression with its homogeneous region, nCRF effectively suppress the number of pixel and homogeneity angle value big, owing to natural texture Gabor oriented energy extreme value is big and distribution is mixed and disorderly, even if being modulated by range-attenuation function, CRF Spot detection point still obtains bigger amount of suppression and stimuli responsive is seriously impaired；Same target internal test point is limited mainly by the suppression with its homogeneous region, the number of pixel is effectively suppressed to depend on homogeneous region size (i.e. target sizes) in nCRF, target area homogeneity angle value is big, its Gabor oriented energy extreme value is distributed mainly on target internal and edge, if target is relatively big, in nCRF, homogeneity district is big, CRF focus target pixel obtains bigger amount of suppression and responds weak, if target is less, in nCRF, homogeneity district is little, CRF focus target pixel is not inhibited, and retains overall Small object；And for well-marked target contour detecting point, extremely data are disturbed owing to WKPCA eliminates, CRF main constituent is high with target or background characteristics degree of agreement, in nCRF, the false edge of target internal or background texture homogeneity angle value are high, owing to the multidimensional characteristic of profile pixel is different from target and background, imaging space resolution constraint causes that object edge produces to obscure pixel, its spectral signature produces deviation with center main composition, profile pixel homogeneity angle value own is relatively low, and its Gabor oriented energy extreme value focuses primarily upon edge, therefore profile pixel in center is limited mainly by being derived from the pixel suppression of same profile, effective suppression pixel number is little and homogeneity angle value is less, in addition range attenuation modulation, profile pixel is little affected by suppressing and responding strong.Described on end, the correction suppression of WKPCA homogeneity degree can effectively suppress environment texture, it is thus achieved that large wheel exterior feature responds, and at utmost reduces profile from suppressing, and retains integrity profile.

Below in conjunction with embodiment, the present invention will be further described.

With reference to flow chart 2, the present invention to implement step as follows:

Step 1, inputs twilight image.

Step 2, arranges non-classical receptive field model parameter: nCRF and CRF zone radius is than ρ, Gauss standard difference σ, kernel function yardstick σ_k, index of modulation a ∈ [0,1], b ∈ [0,1].CRF and nCRF zoned circular radius r is calculated according to formula 1 and parameter_cWith ρ r_c。

$r_c=\sqrt{2\ln \mathrm{\&rho;}/{\mathrm{\&rho;}}^2-1}\&CenterDot;\mathrm{\&rho;\&sigma;}---\left(21\right)$

Step 3, progressively scans each pixel, calculates each pixel Gabor oriented energy extreme value E_σ(x,y)；The distance weighted function ω in nCRF region is calculated by formula 2_σ(x,y)。

ω_σ(x,y)=|DOG_σ,ρσ(x,y)|⁺/‖|DOG_σ,ρσ(x,y)|⁺‖₁(22)

${\mathrm{DOG}}_{\mathrm{\&sigma;},\mathrm{\&rho;\&sigma;}}(x,y)=\frac{1}{\sqrt{2\mathrm{\&pi;}{\left(\mathrm{\&rho;\&sigma;}\right)}^2}}\exp (-\frac{x^2+y^2}{2{\left(\mathrm{\&rho;\&sigma;}\right)}^2})-\frac{1}{\sqrt{2\mathrm{\&pi;}{\mathrm{\&sigma;}}^2}}\exp (-\frac{x^2+y^2}{2{\mathrm{\&sigma;}}^2})---\left(23\right)$

Step 4, in conjunction with twilight image characteristic, calculates each pixel characteristic vector of image, including direction, spatial frequency, contrast, gray scale symbiosis statistical entropy and gray scale symbiosis statistic correlation.

Build multi-dimensional feature data collection in each pixel CRF regionWherein fx=[O_c,SF_c,CON_c,cooE_c,cooCOR_c]^T, N_cFor pixel number in CRF, O_c、SF_c、CON_c、cooE_cAnd cooCOR_cFor direction normalized in CRF region, spatial frequency, contrast, gray scale symbiosis statistical entropy and gray scale symbiosis statistic correlation；Multi-dimensional feature data collection in nCRF regionWherein fy=[O_nc,SF_nc,CON_nc,cooE_nc,cooCOR_nc]^T, N_ncFor pixel number in nCRF, O_nc、SF_nc、CON_nc、cooE_ncAnd cooCOR_ncFor direction normalized in nCRF region, spatial frequency, contrast, gray scale symbiosis statistical entropy and gray scale symbiosis statistic correlation.

Step 5, based on WKPCA algorithm, adopts Radial basis kernel function, formula 4 calculate the characteristic vector angle coupling FAM value of each pixel of CRFI=1 ..., N_c；According to FAM value to each pixel characteristic vector weighting of high-dimensional feature space CRF, CRF multi-dimensional feature data collection main constituent after calculating weighting, and each pixel characteristic vector projection in CRF main constituent in nCRF, each pixel and CRF center homogeneity degree δ in the nCRF region of each pixel of image is namely calculated by formula 6_w(r).Wherein r ∈ FY, D_wFor nuclear matrix K_wCharacteristic vector.

$w_{{\mathrm{fx}}_i}=\frac{\frac{1}{N_c}\underset{m=1}{\overset{N_c}{\mathrm{\&Sigma;}}}k({\mathrm{fx}}_i,{\mathrm{fx}}_m)}{\sqrt{k({\mathrm{fx}}_i,{\mathrm{fx}}_i)}\sqrt{\frac{1}{N_c^2}\underset{m=1}{\overset{N_c}{\mathrm{\&Sigma;}}}\underset{n=1}{\overset{N_c}{\mathrm{\&Sigma;}}}k({\mathrm{fx}}_m,{\mathrm{fx}}_n)}}---\left(24\right)$

$k(\mathrm{fx},\mathrm{fy})=\exp [(-{\left|\right|\mathrm{fx}-\mathrm{fy}\left|\right|}^2)/{2\mathrm{\&sigma;}}_k^2]---\left(25\right)$

${\mathrm{\&delta;}}_w^{-1}\left(r\right)={(k_w(Z,r)-\frac{1}{N_c}\underset{i=1}{\overset{N_c}{\mathrm{\&Sigma;}}}{k}_w(Z,{\mathrm{fx}}_i))}^T{D}_w{D}_w^T(k_w(Z,r)-\frac{1}{N_c}\underset{i=1}{\overset{N_c}{\mathrm{\&Sigma;}}}{k}_w(Z,{\mathrm{fx}}_i))---\left(26\right)$

$K_w=\begin{array}{cc}{\left[\frac{\sqrt{w_{{\mathrm{fx}}_i}{w}_{{\mathrm{fx}}_j}}}{w}(k({\mathrm{fx}}_i,{\mathrm{fx}}_j)-\frac{1}{N_c}\underset{m=1}{\overset{N_c}{\mathrm{\&Sigma;}}}k({\mathrm{fx}}_i,{\mathrm{fx}}_m)-\frac{1}{N_c}\underset{m=1}{\overset{N_c}{\mathrm{\&Sigma;}}}k({\mathrm{fx}}_m,{\mathrm{fx}}_j)+\frac{1}{N_c^2}\underset{m=1}{\overset{N_c}{\mathrm{\&Sigma;}}}\underset{n=1}{\overset{N_c}{\mathrm{\&Sigma;}}}k({\mathrm{fx}}_m,{\mathrm{fx}}_n))\right]}_{N_c\&times;N_c}& i,j=1,...,N_c---\left(27\right)\end{array}$

$\begin{array}{cc}{k}_w(Z,r)={\mathrm{\&phi;}}_w^T\left(\mathrm{FX}\right)\mathrm{\&phi;}\left(r\right)={\left[\sqrt{w_{{\mathrm{fx}}_i}/w}(k({\mathrm{fx}}_i,r)-\frac{1}{N_c}\underset{m=1}{\overset{N_c}{\mathrm{\&Sigma;}}}k({\mathrm{fx}}_m,r))\right]}_{N_c\&times;1}& i=1,...,N_c---\left(28\right)\end{array}$

Step 6, in conjunction with homogeneity degree δ_w(x, y) with range-attenuation function ω_σ(x, y) corrects each pixel amount of suppression in nCRF, formula 9 and formula 10 calculate the environment amount of suppression T of each pixel of image_σ(x, y) and suppress output R_σ(x,y).Wherein δ_w(x y) is δ_w(r)。

T_σ(x,y)=(δ_w(x,y)×ω_σ(x,y))*E_σ(x,y)(29)

R_σ(x,y)=|aE_σ(x,y)-bT_σ(x,y)|⁺(30)

Step 7, to output R_σ(x, y) carries out non-maxima suppression, arranges gray threshold and profile length threshold value, non-maxima suppression result is carried out binaryzation, obtains final profile output.

The effect of the present invention can be further illustrated by following simulation result:

1 from fig. 4, it can be seen that be mainly goal pels in CRF window in structure chart, its characteristics of mean vector approximation target property.Therefore the FAM value of CRF window and nCRF ring-like window internal object pixel relatively big (white) in FAM graphics, the FAM value of backdrop pels is less (black).Therefore FAM weight factor can effectively weaken the abnormal data in CRF covariance matrix.Experiment arranges σ=2.0, CRF windows radius 5, nCRF windows radius 20.

2, as seen from Figure 5, analog data is emulated, test and comparison PCA, KPCA and WKPCA algorithm performance.Compared to linear PCA, a degree of differentiation target and background of non-linear KPCA, but cannot further to different distributions target classification；And WKPCA can not only separate targets and background, and can effectively distinguish the target of different distributions.Experiment arranges σ_k=0.5.

3, as shown in Figure 6, WKPCA algorithm performance advantage shows in true twilight image data analysis further.CRF center is based on target, and KPCA is than PCA principal component analysis accurately (a).But by abnormal data (background data etc. in noise, target) interference, KPCA main constituent produce deviation, cause target, background interior pel homogeneity degree all relatively low；And objective contour pixel multidimensional statistics feature comprises target and background information, its characteristic vector is easily similar to interference main constituent, causes the homogeneity degree higher (b) of objective contour.WKPCA is owing to eliminating the interference of abnormal data, and center main composition calculates accurately, thus target internal homogeneity degree is higher；The gray-scale statistical characteristics of objective contour comprises target and background information, characteristic vector is inconsistent with center main composition and homogeneity degree is slightly lower；Background homogeneity degree relatively low (c).Experiment arranges σ=2.0, σ_k=0.5.

Employing formula 31 calculates PCA homogeneity degree, and in formula, μ is CRF center pel characteristic mean vector, and v is the main constituent adopting PCA algorithm selected；Employing formula 32 calculates KPCA homogeneity degree, and w is the main constituent that KPCA algorithm is selected；Employing formula 19 calculates WKPCA homogeneity degree.

δ^-1(r)=(r-μ)^Tvv^T(r-μ)(31)

${\mathrm{\&delta;}}^{-1}\left(r\right)={(k(Z,r)-\frac{1}{N_c}\underset{i=1}{\overset{N_c}{\mathrm{\&Sigma;}}}k(Z,{\mathrm{fx}}_i))}^T{\mathrm{ww}}^T(k(Z,r)-\frac{1}{N_c}\underset{i=1}{\overset{N_c}{\mathrm{\&Sigma;}}}k(Z,{\mathrm{fx}}_i))$

(32)

Wherein $k(Z,r)={\mathrm{\&phi;}}^T\left(\mathrm{FX}\right)\mathrm{\&phi;}\left(r\right)=\begin{array}{cc}{[k({\mathrm{fx}}_i,r)-\frac{1}{N_c}\underset{m=1}{\overset{N_c}{\mathrm{\&Sigma;}}}k({\mathrm{fx}}_m,r)]}_{N_c\&times;1}& i=1,...,N_c.\end{array}$

4 as it is shown in fig. 7, correct, for twilight image checking WKPCA homogeneity degree in natural scene, the reasonable effectiveness suppressed.For background texture (row 1-2) nCRF and CRF homogeneity degree high (row 2), Gabor oriented energy extreme value mixed and disorderly (row 3), cause that center is suppressed by nCRF major part pixel, effectively suppressing pixel number pixel inhibiting value big, each big (row 4), therefore the response of background center is weak；For same target part and CRF homogeneity degree high (row 2) in target internal (row 3-4) nCRF, Gabor oriented energy extreme value concentrates on edge and target internal (row 3), cause that in nCRF, center is all suppressed by goal pels with higher value, effectively suppressing pixel number to depend on target sizes, inhibiting value big (row 4), therefore target's center's response is weak；Outline portion (row 5-6) is acted on by WKPCA, target part homogeneity degree consistent with CRF main constituent in nCRF high (row 2), outline portion homogeneity degree is slightly lower, Gabor oriented energy extreme value focuses primarily upon edge (row 3), the false edge (row 5 row 3) of target internal, the Gabor energy of background texture (row 6 row 3) is relatively low, therefore contour detecting pixel is limited mainly by being derived from the pixel suppression of same profile, effectively suppress pixel number little, and homogeneity angle value less (row 4), in addition range correction, each pixel inhibiting value is little, therefore the response of profile center is strong.Experiment arranges σ=2.0, σ_k=0.5.

5, as shown in Figure 8, the application method is utilized to extract remarkable configuration in twilight image.Pretreatment adopts low pass filter noise reduction；Post processing carries out non-maxima suppression and binaryzation gets rid of false edge, to select the edge of higher modulation response and greater depth to export as profile.Still suffering from noise jamming even across low-pass filtering, butterfly suppresses to have remained a large amount of texture and false edge；The correction of linear PCA homogeneity degree suppresses owing to considering multiple features, reduces noise jamming；The correction of KPCA homogeneity degree suppresses to adopt nonlinear mapping, improves center main composition accuracy, more effective differentiation target and background, weakens texure and false edge response further；The correction of WKPCA homogeneity degree suppresses to remove the abnormal data in central sample, farthest suppresses homogeneous region response, retains profile information.Experiment arranges σ=2.0, a=1.0, b=0.8, σ_k=0.5.

Claims (3)

1. the twilight image method for extracting remarkable configuration that a WKPCA homogeneity degree correction nCRF suppresses, it is characterised in that step is as follows:

Step 1, inputs twilight image；

Step 2, arranges non-classical receptive field model parameter: nCRF and CRF zone radius is than ρ, Gauss standard difference σ, kernel function yardstick σ_k, index of modulation a ∈ [0,1], b ∈ [0,1], calculate CRF and nCRF zoned circular radius r according to formula 1 and parameter_cWith ρ r_c；

$r_c=\sqrt{2ln\mathrm{\&rho;}/{\mathrm{\&rho;}}^2-1}\&CenterDot;\mathrm{\&rho;}\mathrm{\&sigma;}---\left(1\right)$

Step 3, progressively scans each pixel, calculates each pixel Gabor oriented energy extreme value E_σ(x,y)；The distance weighted function in nCRF region is calculated by formula 2(x, y) for pixel position in the picture；

${\mathrm{\&omega;}}_{{\mathrm{\&sigma;}}_1}(x,y)=|{\mathrm{DOG}}_{\mathrm{\&sigma;},\mathrm{\&rho;}\mathrm{\&sigma;}}(x,y){|}^{+}/|\left|\right|{\mathrm{DOG}}_{\mathrm{\&sigma;},\mathrm{\&rho;}\mathrm{\&sigma;}}(x,y){|}^{+}|{|}_1---\left(2\right)$

${\mathrm{DOG}}_{\mathrm{\&sigma;},\mathrm{\&rho;}\mathrm{\&sigma;}}\left(x,y\right)=\frac{1}{\sqrt{2\mathrm{\&pi;}{\left(\mathrm{\&rho;}\mathrm{\&sigma;}\right)}^2}}\exp \left(-\frac{x^2+y^2}{2{\left(\mathrm{\&rho;}\mathrm{\&sigma;}\right)}^2}\right)-\frac{1}{\sqrt{2{\mathrm{\&pi;\&sigma;}}^2}}\exp \left(-\frac{x^2+y^2}{2{\mathrm{\&sigma;}}^2}\right)---\left(3\right)$

Step 4, calculates each pixel characteristic vector of twilight image, builds multi-dimensional feature data collection in each pixel CRF regionMulti-dimensional feature data collection in nCRF regionCharacteristic vector fx in CRF region_iElement is Gabor direction, spatial frequency, contrast, gray scale symbiosis statistical entropy and gray scale symbiosis statistic correlation, fx_i=[O_c,SF_c,CON_c,cooE_c,cooCOR_c]^T, i=1 ..., N_c, N_cFor pixel number in CRF, O_c、SF_c、CON_c、cooE_cAnd cooCOR_cFor Gabor direction normalized in CRF region, spatial frequency, contrast, gray scale symbiosis statistical entropy and gray scale symbiosis statistic correlation；Characteristic vector fy in nCRF region_j, j=1 ..., N_nc, fy_j=[O_nc,SF_nc,CON_nc,cooE_nc,cooCOR_nc]^T, N_ncFor pixel number in nCRF, O_nc、SF_nc、CON_nc、cooE_ncAnd cooCOR_ncFor Gabor direction normalized in nCRF region, spatial frequency, contrast, gray scale symbiosis statistical entropy and gray scale symbiosis statistic correlation；

Step 5, adopts WKPCA algorithm, calculates each pixel and CRF center homogeneity degree δ in the nCRF region of each pixel of image_w(r)；

Step 6, in conjunction with homogeneity degree δ_w(x, y) and range-attenuation functionEach pixel amount of suppression in nCRF is corrected, formula 9 and formula 10 calculates the environment amount of suppression T of each pixel of image_σ(x, y) and suppress output R_σ(x,y)；Wherein δ_w(x y) is δ_w(r)；

$T_{\mathrm{\&sigma;}}(x,y)=\left({\mathrm{\&delta;}}_w\right(x,y)\&times;{\mathrm{\&omega;}}_{{\mathrm{\&sigma;}}_2}(x,y\left)\right)*E_{\mathrm{\&sigma;}}(x,y)---\left(9\right)$

R_σ(x, y)=| aE_σ(x,y)-bT_σ(x,y)|⁺(10)

Step 7, to output R_σ(x, y) carries out non-maxima suppression, arranges gray threshold and profile length threshold value, non-maxima suppression result is carried out binaryzation, obtains final profile output.

2. the twilight image method for extracting remarkable configuration that WKPCA homogeneity degree according to claim 1 correction nCRF suppresses, it is characterized in that: the WKPCA algorithm in described step 5 is based on KPCA algorithm, in high-dimensional feature space CRF covariance matrix, introduced feature vectorial angle coupling FAM value is to each pixel characteristic vector weighting of CRF, and FAM value is the vector angle of each pixel characteristic vector of CRF and CRF characteristic data set vector average in high-dimensional feature space.

3. the twilight image method for extracting remarkable configuration that WKPCA homogeneity degree according to claim 1 and 2 correction nCRF suppresses, it is characterised in that: the step that the WKPCA algorithm in described step 5 calculates CRF characteristic data set main constituent is as follows:

First the characteristic vector angle coupling FAM value of each pixel of CRF is calculated by formula 4I=1 ..., N_c, and according to FAM value to each pixel characteristic vector weighting of high-dimensional feature space CRF；Calculate CRF multi-dimensional feature data collection main constituent after weighting again, and the projection in CRF main constituent of each pixel characteristic vector and δ in nCRF_w(r), r ∈ FY, D in formula 6_wFor nuclear matrix K_wCharacteristic vector, k () is Radial basis kernel function；

$w_{{\mathrm{fx}}_i}=\frac{\frac{1}{N_c}\underset{m=1}{\overset{N_c}{\&Sigma;}}k\left({\mathrm{fx}}_i,{\mathrm{fx}}_m\right)}{\sqrt{k\left({\mathrm{fx}}_i,{\mathrm{fx}}_i\right)}\sqrt{\frac{1}{N_c^2}\underset{m=1}{\overset{N_c}{\&Sigma;}}\underset{n=1}{\overset{N_c}{\&Sigma;}}k\left({\mathrm{fx}}_m,{\mathrm{fx}}_n\right)}}---\left(4\right)$

$k(fx,fy)=\exp \&lsqb;(-||fx-fy|{|}^2)/2{\mathrm{\&sigma;}}_k^2\&rsqb;---\left(5\right)$

${\mathrm{\&delta;}}_w^{-1}\left(r\right)={\left(k_w\right(Z,r)-\frac{1}{N_c}\underset{i=1}{\overset{N_c}{\&Sigma;}}{k}_w(Z,{\mathrm{fx}}_i\left)\right)}^T{D}_w{D}_w^T\left(k_w\right(Z,r)-\frac{1}{N_c}\underset{i=1}{\overset{N_c}{\&Sigma;}}{k}_w(Z,{\mathrm{fx}}_i\left)\right)---\left(6\right)$

$K_w={\&lsqb;\frac{\sqrt{w_{{\mathrm{fx}}_i}{w}_{{\mathrm{fx}}_j}}}{w}\left(k\left({\mathrm{fx}}_i,{\mathrm{fx}}_j\right)-\frac{1}{N_c}\underset{m=1}{\overset{N_c}{\&Sigma;}}k\left({\mathrm{fx}}_i,{\mathrm{fx}}_m\right)-\frac{1}{N_c}\underset{m=1}{\overset{N_c}{\&Sigma;}}k\left({\mathrm{fx}}_m,{\mathrm{fx}}_j\right)+\frac{1}{N_c^2}\underset{m=1}{\overset{N_c}{\&Sigma;}}\underset{n=1}{\overset{N_c}{\&Sigma;}}k\left({\mathrm{fx}}_m,{\mathrm{fx}}_n\right)\right)\&rsqb;}_{\begin{array}{cc}{N}_c\&times;N_c& i,j=1,...,N_c\end{array}}---\left(7\right)$

$k_w\left(Z,r\right)={\mathrm{\&phi;}}_w^T\left(FX\right)\mathrm{\&phi;}\left(r\right)={\&lsqb;\sqrt{w_{{\mathrm{fx}}_i}/w}\left(k\left({\mathrm{fx}}_i,r\right)-\frac{1}{N_c}\underset{m=1}{\overset{N_c}{\&Sigma;}}k\left({\mathrm{fx}}_m,r\right)\right)\&rsqb;}_{\begin{array}{cc}{N}_c\&times;1& i=1,...,N_c\end{array}}---\left(8\right).$