CN106291555A - A kind of offshore spilled oil detection method based on C-band polarization SAR image - Google Patents

Abstract

The present invention discloses a kind of offshore spilled oil detection method based on C-band polarization SAR image, belongs to polarimetric radar technical field of remote sensing image processing.For oil spilling scattering signatures in traditional method and the problem such as accuracy of identification relation is indefinite, accuracy of identification is relatively low; the multi-source informations such as district to be detected local wind speed, wave, polarization are introduced in SAR oil spilling detects; the polarization characteristic in oil spilling district with doubtful oil spilling district is combined with marine physics feature; oil spilling detected rule is built on the basis of supervision message; structure result according to rule is treated detection region and is carried out comprehensive descision; improve oil spilling accuracy of detection, to protecting the marine environment and ecological resources are significant.

Description

A kind of offshore spilled oil detection method based on C-band polarization SAR image

Technical field

The invention belongs to polarimetric radar technical field of remote sensing image processing, be a kind of based on C-band polarization SAR The new method that offshore spilled oil is detected by (Polarimetric Synthetic Aperture Radar, POLSAR) image.

Background technology

China has very long coastline base band, and territorial waters area is huge, and economic development is rapid, and sea transport is busy, But the Accidental Oil Spill Frequent Accidents such as thing followed marine oil pipeline ruptures, oil carrier leakage, collision and illegal blowdown, to edge Sea causes extensive damage with Offshore Ecology environment, marine resources with marine economy, has had a strong impact on China's Sustainable Marine and has sent out Exhibition implementation.Statistics show, 1973 to 2010 years, and coastal area of china occurs the big minor accident of ship spill more than 3000 altogether Rising, within the most every four days, occur 1, ship spill total amount reaches more than 40,000 ton.As quick-fried in Talien New Port petroleum pipeline on July 16th, 2010 Fried fire accident, the sea at least having resulted in 50 square kilometres, surrounding waters is polluted；There is leakage of oil in oil field, Bohai Sea Gulf in June, 2011 Accident, causes significant impact to cultivation, tourism, ecology, and accident has resulted in accumulative 5500 square kilometres of seas and polluted. After marine oil spill occurs, can detect that oil spilling is to protecting the marine environment and ecological resources have important meaning in time and accurately Justice.Satellite remote sensing has the ability of quick obtaining large area Earthwatch, has become as most effective in oil spilling detection technique and Important means.Compare optical satellite, owing to oceanographic condition is complicated, round-the-clock, round-the-clock synthetic aperture radar (SAR) satellite Bigger advantage is had at emergency response and long time-series dynamics watcher's mask.At present, utilize SAR satellite to carry out oil spilling on a large scale to visit Surveying and increasingly come into one's own, research institution of various countries has all carried out correlational study, and has built service operation system, such as: Norway The KSAT system of Kongsberg satellite hub exploitation, the OMW system of Satlantic company of Canada exploitation, France The SARTool system etc. of BoostTechnologies company exploitation.China is the most later due to starting, and existing business system is adopted Oil spilling detection method efficiency and precision all Shortcomings, it is difficult in emergency response for decision-making section provide accurate information. Relative to land Oil spills, marine oil spill situation is the most complex.After oil overflows into ocean, the distinctive ring in ocean Under the conditions of border, there are the physics of complexity, chemical and biological change procedure, and changed by these, finally disappear from marine environment Lose.These changes have diffusion, drift about, evaporate, disperse, emulsifying, photochemical oxidation decompose, deposit and biodegradation etc..Therefore Marine oil spill performance on SAR satellite image is comprehensively determined by many factors, and simple dependence microwave remote sensing means tend not to Obtain preferable detection accuracy, oil spilling detection can be improved well in conjunction with marine site, local wind speed, wave, even optical pickocff Precision.

Summary of the invention

The present invention considers the practical situation in oil spilling district, closes with accuracy of identification for oil spilling scattering signatures in traditional method Be indefinite, accuracy of identification is relatively low etc., and problem is studied.SAR oil spilling detect in introduce district to be detected local wind speed, wave, The multi-source informations such as polarization, combine the polarization characteristic in oil spilling district with doubtful oil spilling district with marine physics feature, build high-dimensional Cross complete characteristics and describe collection, under supervision sample guides, carry out linear Laplace mapping graph dimensionality reduction, remove redundancy, structure Oil spilling detected rule, treats detection region according to the structure result of rule and carries out comprehensive descision, improve oil spilling accuracy of detection.

The present invention adopts the following technical scheme that

A kind of offshore spilled oil detection method based on C-band polarization SAR image, comprises the following steps:

Step 1: radar image pretreatment；

Step 2: build high-dimensional polarization characteristic set；

Step 3: build linear Laplace mapping graph dimensionality reduction device dimensionality reduction and carry out k-mean classification；

Step 4: Ocean Wind-field data auxiliary oil spilling detection；

Step 5: precision evaluation.

The radar image preprocessing process of described step 1；

Resolve including header file, choose incident angle SAR image between 23-55 degree and carry out sea and overflow and have detection, many Depending on noise reduction process.

The detailed process that described step 2 builds high-dimensional polarization characteristic set is as follows:

Full-polarization SAR image is carried out polarization decomposing, because offshore spilled oil is extracted by polarization parameter has preferably auxiliary Effect, uses multiple polarization characteristic extracting method to obtain multiple mistakes complete polarization characteristic full-polarization SAR image, and polarization decomposing obtains To multi-Dimensional parameters.

Described step 3 builds linear Laplace mapping graph dimensionality reduction device dimensionality reduction and carries out k-mean classification, its detailed process As follows:

Every class extracts supervision sample (each sample is more than 30 pixels), utilizes manual oversight information to drop feature set Dimension, obtains the new feature of more oil spilling discriminating power, builds LDLE dimensionality reduction device and carries out dimensionality reduction, in order to realize based on higher-dimension scattering spy The information retrieval levied, the present invention is firstly introduced into LDLE and carries out dimensionality reduction, through LDLE dimensionality reduction to 5 dimension, removes redundancy, the most right After dimensionality reduction, feature carries out K-mean (K average) classification, (proposes in the partition process of threshold value to use the value of polarization entropy by threshold value Carry out) obtain sea water-dark target-recognition result (comprising oil spilling, low wind speed district or other sea-surface targets etc. in dark target).

The detailed process of described step 4 Ocean Wind-field data auxiliary oil spilling detection is as follows:

To C-band full-polarization SAR data, CMOD5 mode function is utilized to carry out ocean surface wind speed, Wind-field Retrieval, if not C Wave band data then needs the auxiliary of external source buoy and spaceborne Ocean Wind-field data；Finally remove suspicious low wind according to wind field data Speed shadow district, oil spilling of refining differentiates result.

The precision evaluation process of described step 5 is as follows:

On the basis of the sea true oil spilling data Ground Truth that hook is drawn, the classification results in integrating step 4, utilize Traditional confusion matrix evaluation methodology obtains oil spilling testing result and carries out precision evaluation.

In described step 3, the LDLE method specific implementation of feature set dimensionality reduction is as follows:

Assume the M sample X={x having dimension N₁,x₂,…,x_M}∈R^N×M, Y={y₁,y₂,…,y_M}∈R^d×MAfter dimensionality reduction Data dimension is the lower dimensional space data of d, then the operation of LDLE dimensionality reduction is expressed as matrix product conversion Y=U^TX, wherein U ∈ R^N×d； In LDLE, " after dimensionality reduction, generic sample distance is minimum " is expressed as:

$\arg \underset{Y}{\min }\underset{ij}{\Σ}\left|\right|y_i-y_j|{|}^2{W}_{ij}^{\left(W\right)}$

y_iAnd y_jRepresent 2 different samples in the M sample data after dimensionality reduction respectively, use when weighing sample distance Gaussian kernel function is weighted, weight functionAs follows:

$W_{ij}^{\left(W\right)}=\left\{\begin{array}{cc}\exp (-||x_i-x_j|{|}^2/t),& if{x}_i\∈N_{k_1}^{+}\left(x_j\right)or{x}_j\∈N_{k_1}^{+}\left(x_i\right)\\ 0,& else\end{array}\right.$

Wherein, t is thermonuclear parameter；x_iAnd x_jRepresent 2 different samples in M the sample data chosen respectively,Represent xth_iThe k that individual sample is identical, closest with its classification₁Individual sample x_j；Consider L^(W)=D^(W)-W^(W)With L^(W)Represent the weight function when " generic sample distance is minimum after dimensionality reduction ", sample function The k closed on₁The weight function of individual sample and and close on sample weight function and weigh this difference of function with sample, then formula " similar other style after dimensionality reduction This distance is minimum " can be arranged as:

$\arg \underset{Y}{\min }tr\left({\mathrm{YL}}^{\left(W\right)}{Y}^T\right)$

In LDLE, " after dimensionality reduction, different classes of sample distance is maximum " is represented by:

$\arg \underset{Y}{\min }\underset{ij}{\Σ}\left|\right|y_i-y_j|{|}^2{W}_{ij}^{\left(B\right)}$

Same employing Gaussian kernel function is weighted:

$W_{ij}^{\left(B\right)}=\left\{\begin{array}{cc}\exp (-||x_i-x_j|{|}^2/t),& if{x}_i\∈N_{k_2}^{+}\left(x_j\right)or{x}_j\∈N_{k_2}^{+}\left(x_i\right)\\ 0,& else\end{array}\right.$

Wherein,Represent xth_iIndividual sample and its classification differ, closest k₂Individual sample x_j；Consider L^(B) =D^(B)-W^(B)With L^(W)Represent the power when " different classes of sample distance is maximum after dimensionality reduction " The k that function, sample function close on₁The weight function of individual sample and and close on sample weight function and weigh this difference of function with sample, then formula " fall After dimension, different classes of sample distance is maximum " can be arranged as:

$\arg \underset{Y}{\min }tr\left({\mathrm{YL}}^{\left(B\right)}{Y}^T\right)$

Definition weight beta is as weighing " after dimensionality reduction generic sample distance minimum " and " different classes of sample distance after dimensionality reduction Maximum " percentage contribution, then have after merging formula:

$\arg \underset{Y}{\min }tr\[Y(L^{\left(W\right)}-{\mathrm{\βL}}^{\left(B\right)})Y^T\]$

Substitute into Y=U^TX, the most above-mentioned optimization problem becomes matrix-eigenvector-decomposition problem:

$\begin{array}{ccc}\arg \underset{Y}{\min }tr\left(U^T{\mathrm{XL}}^{*}{X}^{T}U\right)& s.t.& U^T{\mathrm{XX}}^{T}U=I\end{array}$

Wherein L^*=L^(W)-βL^(B)。

Compared with prior art, the innovation of the present invention is: method based on the segmentation of back scattering power threshold is past Toward not distinguishing oil spilling district and the sea water covered without oil spilling very well, passing through to introduce polarization entropy during threshold value divides can to the greatest extent may be used The water body impact on oil spilling in the case of little stormy waves can be got rid of, thus be effectively improved oil identification accuracy；By oil spilling district with doubtful The polarization characteristic in oil spilling district combines with marine physics feature, builds higher-dimension and spends complete characteristics description collection, draws at supervision sample Lead down and carry out linear Laplace mapping graph dimensionality reduction, remove redundancy, build oil spilling detected rule, it is simple to follow-up oil spilling is examined Survey；Utilizing the oil spilling detected rule built to treat the doubtful oil spilling in detection region to differentiate, target dark to sea carries out doubtful oil spilling and beats Divide assessment, according to result of detection, the higher-dimension of input is spent complete characteristics collection and be estimated, filter out and oil spilling is differentiated the most favourable Feature auxiliary follow-up study.Oil spilling detection efficiency and precision are greatly improved.

Accompanying drawing explanation

Fig. 1 is the flow chart of the embodiment of the present invention.

Detailed description of the invention

Technical solution of the present invention is described in detail below in conjunction with drawings and Examples.

Technical solution of the present invention can use computer technology to realize automatically and run, as it is shown in figure 1, the stream of the embodiment of the present invention Journey includes following step:

Choosing and pretreatment of step 1:POLSAR image.

Before oil spilling detects, the present invention needs original POLSAR data are carried out header file parsing, chooses angle of incidence Degree SAR image between 23-55 degree, carries out regarding denoising to it subsequently, and orientation drops to upwards doing regarding of 5*5 with distance more Low phase dry spot noise.

Step 2: build high-dimensional polarization characteristic set.

Full-polarization SAR image is carried out polarization decomposing, because offshore spilled oil is extracted by polarization parameter has preferably auxiliary Effect.Multiple polarization characteristic extracting method is used to obtain a large amount of mistakes complete polarization characteristic polarization image.Polarization decomposing obtains many (multidimensional refers to (a) in polarization characteristic collection table, (b), (d), (h), (i) to dimension parameter, and (k) feature carries out essential, and further feature can High dimensional data is constituted) with any selection；The polarization decomposing parameter extracted in the present embodiment is as shown in the table, totally 49 dimension.

Polarization scattering feature set

Step 3: dimensionality reduction device dimensionality reduction is also to build LDLE (Linear Discriminative Laplacian Eigenmaps) Carry out k-mean classification.

Manually choose sea classification, from from hooking every apoplexy due to endogenous wind extraction supervision 30 pixels of sample taken, utilize manual oversight to believe Breath carries out dimensionality reduction to feature set, thus the new feature structure LDLE dimensionality reduction device obtaining more oil spilling discriminating power carries out dimensionality reduction, for Realizing information retrieval based on higher-dimension scattering signatures, the present invention is firstly introduced into linear Laplace mapping graph dimensionality reduction device (Linear Discriminative LaplacianEigenmaps, LDLE) carries out dimensionality reduction.

For the sake of ease of implementation, it is provided that (linear Laplace mapping graph drops the LDLE of the feature set dimensionality reduction of embodiment Dimension device (Linear Discriminative LaplacianEigenmaps, LDLE) method specific implementation is as follows:

Assume the M sample X={x having dimension N₁,x₂,…,x_M}∈R^N×M, Y={y₁,y₂,…,y_M}∈R^d×MAfter dimensionality reduction Data dimension is the lower dimensional space data of d.Then the operation of LDLE dimensionality reduction can be expressed as matrix product conversion Y=U^TX, wherein U ∈ R^{N ×d}.The core concept of supervision LDLE is: for each sample data and the most some neighborhood samples, find a throwing Shadow transformation matrix U, under this projector space, belong to that same category of sample distance is the least, the sample that belongs to a different category away from From big as far as possible, dimensionality reduction data are made to have good separability.In LDLE, " after dimensionality reduction, generic sample distance is minimum " can represent For:

$\arg \underset{Y}{\min }\underset{ij}{\Σ}\left|\right|y_i-y_j|{|}^2{W}_{ij}^{\left(W\right)}$

y_iAnd y_jRepresent 2 different samples in the M sample data after dimensionality reduction respectively, use when weighing sample distance Gaussian kernel function is weighted, weight functionAs follows:

$W_{ij}^{\left(W\right)}=\left\{\begin{array}{cc}\exp (-||x_i-x_j|{|}^2/t),& if{x}_i\∈N_{k_1}^{+}\left(x_j\right)or{x}_j\∈N_{k_1}^{+}\left(x_i\right)\\ 0,& else\end{array}\right.$

Wherein t is thermonuclear parameter；x_iAnd x_jRepresent 2 different samples in M the sample data chosen respectively,Represent x_iIndividual belong to and x_jThe k that classification is identical, closest₁Individual sample；Represent x_jIndividual belong to and x_iClass Not identical, closest k₁Individual sample.Consider L^(W)=D^(W)-W^(W)WithThen formula " after dimensionality reduction generic sample away from From minimum " can be arranged as:

$\arg \underset{Y}{\min }tr\left({\mathrm{YL}}^{\left(W\right)}{Y}^T\right)$

Wherein,L^(W)Represent the weight function when " generic sample distance is minimum after dimensionality reduction ", sample function The k closed on₁The weight function of individual sample and and close on sample weight function and weigh this difference of function with sample；

In LDLE, " after dimensionality reduction, different classes of sample distance is maximum " is represented by:

$\arg \underset{Y}{\min }\underset{ij}{\Σ}\left|\right|y_i-y_j|{|}^2{W}_{ij}^{\left(B\right)}$

Same employing Gaussian kernel function is weighted:

$W_{ij}^{\left(B\right)}=\left\{\begin{array}{cc}\exp (-||x_i-x_j|{|}^2/t),& if{x}_i\∈N_{k_2}^{+}\left(x_j\right)or{x}_j\∈N_{k_2}^{+}\left(x_i\right)\\ 0,& else\end{array}\right.$

WhereinRepresent x_iBelong to and x_jClassification differs, closest k₂Individual sample；WhereinTable Show x_jBelong to and x_iClassification differs, closest k₂Individual sample.Consider L^(B)=D^(B)-W^(B)WithThen formula " fall After dimension, different classes of sample distance is maximum " can be arranged as:

$\arg \underset{Y}{\min }tr\left({\mathrm{YL}}^{\left(B\right)}{Y}^T\right)$

Wherein,L^(B)Represent the weight function when " different classes of sample distance is maximum after dimensionality reduction ", sample letter The k that number closes on₁The weight function of individual sample and and close on sample weight function and weigh this difference of function with sample；Definition weight beta is as measurement " after dimensionality reduction, generic sample distance is minimum " and the percentage contribution of " after dimensionality reduction, different classes of sample distance is maximum ", after merging formula Then have:

$\arg \underset{Y}{\min }tr\[Y(L^{\left(W\right)}-{\mathrm{\βL}}^{\left(B\right)})Y^T\]$

Substitute into Y=U^TX, the most above-mentioned optimization problem becomes matrix-eigenvector-decomposition problem:

$\begin{array}{ccc}\arg \underset{U}{\min }tr\left(U^T{\mathrm{XL}}^{*}{X}^{T}U\right)& s.t.& U^T{\mathrm{XX}}^{T}U=I\end{array}$

Wherein L^*=L^(W)-βL^(B)。

Through LDLE dimensionality reduction to 5 dimension, eliminate redundancy, then feature after dimensionality reduction is carried out K-mean classification (MATLAB software), obtains sea water-dark target-recognition result by threshold value (by introducing polarization entropy in the partition process of threshold value) (dark target comprises oil spilling, low wind speed district or other sea-surface targets etc.).

Step 4: Ocean Wind-field data auxiliary oil spilling detection.

If SAR data is C-band, then it is carried out CMOD5 model ocean surface wind speed, Wind-field Retrieval (uses existing NEST Software), the auxiliary of external source buoy and spaceborne Ocean Wind-field data is then needed if not C-band data；Final according to wind field number According to removing suspicious low wind speed district shadow, oil spilling of refining differentiates result.

Step 5: precision evaluation.

On the basis of the Ground Truth (truthful data) that hook is drawn, the classification results in integrating step 4, utilize traditional Confusion matrix evaluation methodology obtains oil spilling testing result and carries out precision evaluation.

Claims (7)

1. an offshore spilled oil detection method based on C-band polarization SAR image, it is characterised in that comprise the following steps:

Step 1: radar image pretreatment；

Step 2: build high-dimensional polarization characteristic set；

Step 3: build linear Laplace mapping graph dimensionality reduction device dimensionality reduction and carry out k-mean classification；

Step 4: Ocean Wind-field data auxiliary oil spilling detection；

Step 5: precision evaluation.

A kind of offshore spilled oil detection method based on C-band polarization SAR image the most according to claim 1, its feature exists In: the radar image preprocessing process of described step 1；

Resolve including header file, choose incident angle SAR image between 23-55 degree and carry out sea and overflow and have detection, regard fall more Make an uproar process.

A kind of offshore spilled oil detection method based on C-band polarization SAR image the most according to claim 2, its feature exists In: the detailed process that described step 2 builds high-dimensional polarization characteristic set is as follows:

Full-polarization SAR image is carried out polarization decomposing, uses multiple polarization characteristic extracting method to obtain full-polarization SAR image many The complete polarization characteristic of individual mistake, polarization decomposing obtains multi-Dimensional parameters.

A kind of offshore spilled oil detection method based on C-band polarization SAR image the most according to claim 3, its feature exists In: described step 3 builds linear Laplace mapping graph dimensionality reduction device dimensionality reduction and carries out k-mean classification, and its detailed process is as follows:

Manually choose sea classification, from from hooking the every apoplexy due to endogenous wind extraction supervision sample taken, utilize manual oversight information that feature set is entered Row dimensionality reduction, obtains the new feature of more oil spilling discriminating power, builds LDLE dimensionality reduction device and carries out dimensionality reduction, through LDLE dimensionality reduction to 5 dimension, Remove redundancy, then feature after dimensionality reduction is carried out K-mean classification, obtain sea water-dark target-recognition result by threshold value.

A kind of offshore spilled oil detection method based on C-band polarization SAR image the most according to claim 4, its feature exists In: the detailed process of described step 4 Ocean Wind-field data auxiliary oil spilling detection is as follows:

To C-band full-polarization SAR data, CMOD5 mode function is utilized to carry out ocean surface wind speed, Wind-field Retrieval, if not C-band Data then need the auxiliary of external source buoy and spaceborne Ocean Wind-field data；Finally remove suspicious low wind speed according to wind field data dark Shadow zone, oil spilling of refining differentiates result.

A kind of offshore spilled oil detection method based on C-band polarization SAR image the most according to claim 5, its feature exists In: the precision evaluation process of described step 5 is as follows:

On the basis of the sea true oil spilling data that hook is drawn, the classification results in integrating step 4, utilize traditional confusion matrix to comment Valency method obtains oil spilling testing result and carries out precision evaluation.

A kind of offshore spilled oil detection method based on C-band polarization SAR image the most according to claim 4, its feature exists In: in described step 3, the LDLE method specific implementation of feature set dimensionality reduction is as follows:

Assume the M sample X={x having dimension N₁,x₂,…,x_M}∈R^N×M, Y={y₁,y₂,…,y_M}∈R^d×MFor data dimension after dimensionality reduction Degree is the lower dimensional space data of d, then the operation of LDLE dimensionality reduction is expressed as matrix product conversion Y=U^TX, wherein U ∈ R^N×d；In LDLE " after dimensionality reduction, generic sample distance is minimum " is expressed as:

$\arg \underset{Y}{\min }\underset{ij}{\Σ}\left|\right|y_i-y_j|{|}^2{W}_{ij}^{\left(W\right)}$

y_iAnd y_jRepresent 2 different samples in the M sample data after dimensionality reduction respectively, use when weighing sample distance Gaussian kernel function is weighted, weight functionAs follows:

$W_{ij}^{\left(W\right)}=\left\{\begin{array}{cc}\exp \left(-\left|\right|x_i-x_j|{|}^2/t\right),& if{x}_i\∈N_{k_1}^{+}\left(x_j\right)or{x}_j\∈N_{k_1}^{+}\left(x_i\right)\\ 0,& else\end{array}\right.$

Wherein, t is thermonuclear parameter；x_iAnd x_jRepresent 2 different samples in M the sample data chosen respectively, Represent xth_iThe k that individual sample is identical, closest with its classification₁Individual sample x_j,Represent x_jIndividual belong to and x_iClassification phase Same, closest k₁Individual sample；Consider L^(W)=D^(W)-W^(W)WithThen " after dimensionality reduction, generic sample distance is for formula Little " can be arranged as:

$\arg \underset{Y}{\min }tr\left({\mathrm{YL}}^{\left(W\right)}{Y}^T\right)$

Wherein,L^(W)Represent that the weight function when " generic sample distance is minimum after dimensionality reduction ", sample function close on K₁The weight function of individual sample and and close on sample weight function and weigh this difference of function with sample；

In LDLE, " after dimensionality reduction, different classes of sample distance is maximum " is represented by:

$\arg \underset{Y}{max}\underset{ij}{\Σ}\left|\right|y_i-y_j|{|}^2{W}_{ij}^{\left(B\right)}$

Same employing Gaussian kernel function is weighted:

$W_{ij}^{\left(B\right)}=\left\{\begin{array}{cc}\exp \left(-\left|\right|x_i-x_j|{|}^2/t\right),& if{x}_i\∈N_{k_2}^{-}\left(x_j\right)or{x}_j\∈N_{k_2}^{-}\left(x_i\right)\\ 0,& else\end{array}\right.$

Wherein,Represent xth_iIndividual sample and its classification differ, closest k₂Individual sample x_j,Table Show x_jBelong to and x_iClassification differs, closest k₂Individual sample；Consider L^(B)=D^(B)-W^(B)WithThen formula " fall After dimension, different classes of sample distance is maximum " can be arranged as:

$\arg \underset{Y}{max}tr\left({\mathrm{YL}}^{\left(B\right)}{Y}^T\right)$

Wherein,L^(B)Represent that the weight function when " different classes of sample distance is maximum after dimensionality reduction ", sample function face Near k₁The weight function of individual sample and and close on sample weight function and weigh this difference of function with sample；Definition weight beta is as weighing " dimensionality reduction Rear generic sample distance is minimum " and the percentage contribution of " after dimensionality reduction, different classes of sample distance is maximum ", then have after merging formula:

$\arg \underset{Y}{\min }tr\[Y(L^{\left(W\right)}-{\mathrm{\βL}}^{\left(B\right)})Y^T\]$

Substitute into Y=U^TX, the most above-mentioned optimization problem becomes matrix-eigenvector-decomposition problem:

$\begin{array}{ccc}\arg \underset{U}{\min }tr\left(U^T{\mathrm{XL}}^{*}{X}^{T}U\right)& s.t.& U^T{\mathrm{XX}}^{T}U=I\end{array}$

Wherein L^*=L^(W)-βL^(B)。