CN107831473A - Based on Gaussian process recurrence apart from instantaneous Doppler image sequence noise-reduction method - Google Patents

Based on Gaussian process recurrence apart from instantaneous Doppler image sequence noise-reduction method Download PDF

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CN107831473A
CN107831473A CN201710954556.8A CN201710954556A CN107831473A CN 107831473 A CN107831473 A CN 107831473A CN 201710954556 A CN201710954556 A CN 201710954556A CN 107831473 A CN107831473 A CN 107831473A
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distance
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CN107831473B (en
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白雪茹
彭鑫
周峰
张毓
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Xidian University
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S7/00—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00
    • G01S7/02—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00
    • G01S7/28—Details of pulse systems
    • G01S7/2813—Means providing a modification of the radiation pattern for cancelling noise, clutter or interfering signals, e.g. side lobe suppression, side lobe blanking, null-steering arrays
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S13/00—Systems using the reflection or reradiation of radio waves, e.g. radar systems; Analogous systems using reflection or reradiation of waves whose nature or wavelength is irrelevant or unspecified
    • G01S13/88—Radar or analogous systems specially adapted for specific applications
    • G01S13/89—Radar or analogous systems specially adapted for specific applications for mapping or imaging
    • G01S13/90—Radar or analogous systems specially adapted for specific applications for mapping or imaging using synthetic aperture techniques, e.g. synthetic aperture radar [SAR] techniques
    • G01S13/904—SAR modes
    • G01S13/9064—Inverse SAR [ISAR]
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S13/00—Systems using the reflection or reradiation of radio waves, e.g. radar systems; Analogous systems using reflection or reradiation of waves whose nature or wavelength is irrelevant or unspecified
    • G01S13/88—Radar or analogous systems specially adapted for specific applications
    • G01S13/89—Radar or analogous systems specially adapted for specific applications for mapping or imaging
    • G01S13/90—Radar or analogous systems specially adapted for specific applications for mapping or imaging using synthetic aperture techniques, e.g. synthetic aperture radar [SAR] techniques
    • G01S13/904—SAR modes

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  • Radar, Positioning & Navigation (AREA)
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Abstract

The invention discloses it is a kind of based on Gaussian process return apart from instantaneous Doppler image sequence noise-reduction method, mainly solve the problems, such as that prior art image quality under low signal-to-noise ratio is low.Its scheme includes：1) ISAR echoes are received to go forward side by side the pulse compression of line-spacing descriscent；2) signal after range pulse is compressed carries out Short Time Fourier Transform along orientation, obtains range Doppler time isometric chart, and carries out value in chronological order and obtain image sequence；3) to the image-region value not comprising target, the standard deviation of noise is calculated；4) value in image sequence carries out maximal possibility estimation and obtains corresponding parameter vector；5) the noise reduction vector of pixel is calculated according to parameter vector, and forms noise-reduced image sequence.The present invention is realized under low signal-to-noise ratio to the noisy noise reduction apart from instantaneous Doppler image sequence, effectively improves picture quality, available for non-stationary moving target detection, imaging and the feature extraction under Low SNR.

Description

Distance-instantaneous Doppler image sequence the noise-reduction method returned based on Gaussian process
Technical field
The invention belongs to Radar Technology field, further relates to distance-instantaneous Doppler image sequence noise-reduction method, can For non-stationary moving target detection, imaging and the feature extraction under Low SNR.
Background technology
Because ISAR ISAR has many advantages, such as to empty day motive target imaging, therefore it is widely used in army The field such as thing, civilian.But the radar return of target suffer from white Gaussian noise interference, so as to have a strong impact on ISAR into As quality, the difficulty that target classification identifies is added.Therefore, ISAR image noise reductions are the committed steps of imaging.Existing radar Image denoising method has two kinds：First, the sparse reconstructing method based on base tracking, this method underuse scattering point in image Change information between sequence, effect is poor under Low SNR；Second, the noise-reduction method based on spatial domain, this method is not sharp With the statistical information of noise, and image detail information retains deficiency.
What Ji-Hoon Bae, Byung-Soo Kang, Kyung-Tae Kim, and Eunjung Yang etc. delivered at it Paper " Performance of Sparse Recovery Algorithms for the Reconstruction of Radar Images From Incomplete RCS Data”(IEEE Geoscience and Remote Sensing Letters, Vol.12, No.4, April 2015) in propose it is a kind of based on base tracking noise reduction method.This method utilizes mesh The sparse characteristic of logo image, imaging problem is converted into l1Norm optimization problem, by solving regularization optimization object function l1It is real Existing sparse ISAR imagings and image noise reduction.Because this method is merely with single image progress noise reduction, not using scattering point in image Variation characteristic in sequence, therefore effect is poor when echo signal to noise ratio is very low.
Paper " the A speckle reduction algorithm that Jun Z, Xueguang C, Jian L deliver at it by soft-thresholding based on wavelet filters for SAR images”(Fourth International Conference on Signal Processing Proceedings,IEEE,1998:1469- A kind of image denoising method based on wavelet transformation is proposed in 1472vol.2.).This method uses 2-d wavelet basis representation first SAR image, then for SAR image characteristic, select suitable soft-threshold to filter out speckle noise, reconstructed finally by wavelet transformation The SAR image gone out after denoising.This method belongs to traditional airspace filter method, and its performance depends on the selection of window, and to figure As detailed information retains deficiency.
The content of the invention
It is an object of the invention to propose a kind of distance-instantaneous Doppler RID image sequences returned based on Gaussian process Noise-reduction method, to realize effective denoising under Low SNR to movement destination image sequence, focused on well so as to obtain Target image sequence.
The present invention basic ideas be：Based on Gaussian process regression theory, RID image sequence Denoising Problems are converted into number According to regression problem, hyper parameter vector is solved using kernel method, and Accurate Reconstruction is carried out to data, is finally realized to RID image sequences The denoising of row.Meanwhile parametric optimal solution is solved using conjugate gradient method, algorithm complex is low, and accuracy is high.Its implementation bag Include as follows：
Distance-instantaneous Doppler image sequence the noise-reduction method returned based on Gaussian process, including：
(1) linear FM signal is launched to moving target by ISAR, and obtains its echo, take distance to Sampling number is Nr, orientation sampling number is Na, obtain a Nr×NaNoisy radar return matrix Sr；
(2) to noisy radar return matrix SrAlong distance to pulse compression is carried out, distance is obtained to the square after pulse compression Battle array Sd；
(3) adjust the distance to the matrix S after pulse compressiondAlong orientation carry out Short Time Fourier Transform, obtain distance-it is how general Le-time isometric chartWherein r represents distance dimension, and f represents Doppler's dimension, and t represents time dimension, and M is represented Doppler ties up resolution cell number, and N represents time quantum number；
(4) N at different moments of distance-Doppler-in time isometric chart C (r, f, t) is taken from 1 to N sequentially in time Frame pitch is from-instantaneous Doppler image, composition noisy image sequence I；
(5) the pixel point sequence that the i-th row jth in noisy image sequence I arranges is designated as Iij, with sequence IijOrdinate value Form noisy ordinate vector Yij, the square area that no target occurs is chosen in noisy image sequence I, to institute in region There is pixel position value, noisy image sequence I mean noise standard difference σ is calculatedn；
(6) initial value for resetting row subscript i and row subscript j is i=1, j=1, with sequence IijAbscissa value composition Abscissa vector Xij；
(7) according to formulaParameter nuclear matrix is calculated CN, wherein σfFor range parameter, σ is scale parameter, xaRepresent abscissa vector XijIn a-th of element, xbRepresent abscissa to Measure XijIn b-th of element, a and b values are from 1 to N；
(8) function is maximized using conjugate gradient methodJoined Number vector estimateWherein parameter vector θ={ σf, σ },For range parameter estimate, σ*For scale parameter Estimate；
(9) abscissa vector X is takenijIn c-th of elementIf the initial value for circulating subscript c is c=1；
(10) the parameter vector estimate that will be obtained in step (8)Substitute into equation below：
And data core matrix K and core vector K is calculated*, by data core matrix K, core vector K*With noisy ordinate vector YijSubstitute into formulaElement is calculatedCorresponding functional value- 1 represents inversion operation, and T represents to turn Put operation；
(11) circulation subscript c, if c ＜ N, is made into c=c+1, return to step (10) compared with time quantum number N；If c >=N, with these functional valuesForm noise reduction ordinate vectorAnd perform step (12)；
(12) by row subscript i and distance to sampling number NrCompare, row subscript j and Doppler are tieed up into resolution cell number M Compare, if i ＜ NrAnd j ＜ M, then j=j+1 is made, if i ＜ NrAnd j >=M, then make i=i+1, j=1, return to step (7)；If i >= Nr, with these noise reduction ordinates vectorForm noise-reduced image sequence I*。
The invention has the advantages that：
1. the present invention takes full advantage of moving target scattering point contains position distribution and changes in amplitude in image sequence Information, the problems such as image detail caused by avoiding traditional single image noise-reduction method is fuzzy, false target point can not eliminate.
2. the present invention takes full advantage of noise statisticses information, movement destination image sequence is obtained under Low SNR Good noise reduction.
Technical scheme is described in further detail below in conjunction with the drawings and specific embodiments.
Brief description of the drawings
Fig. 1 is the implementation process figure of the present invention；
Fig. 2 is containing the 22nd width image in -13dB white Gaussian noise image sequences；
Fig. 3 is to the 22nd width image in image sequence after Fig. 2 progress noise reductions using the present invention.
Embodiment
Technical scheme and effect are described in further detail below in conjunction with accompanying drawing.
Reference picture 1 is as follows to the implementation steps of the present invention：
Step 1, the noisy radar return matrix S of ISAR admission moving-targetr。
ISAR launches electromagnetic wave, and the electromagnetic wave runs into the reflection occurred after target, institute in communication process The echo of reflection is received in communication process by noise jamming by radar receiver, obtains noisy radar return matrix Sr, take Distance to sampling number be Nr, orientation sampling number is Na。
Step 2, to noisy radar return matrix SrHandled, obtain noisy image sequence I.
(2a), will be with launching linear FM signal using the distance of ISAR to scene center as reference distance Carrier frequency, frequency modulation rate are identical, and distance for reference distance linear FM signal as reference signal, after taking conjugation to reference signal It is multiplied with the echo-signal of reception, obtains solving the matrix after line frequency modulation reason：
Wherein,For apart from fast time, tmFor orientation slow time, Sref() is reference signal, SrdAfter solution line frequency modulation Matrix, * represent conjugate operation；
(2b) by solve line frequency modulation after matrixFourier transformation is done along distance dimension, obtains distance to pulse compression Matrix Sd；
(2c) adjusts the distance to pulse compression matrix SdAlong orientation carry out Short Time Fourier Transform, obtain distance-Doppler- Time isometric chart；
The long L=31 of window, the Doppler of (2c1) setting Short Time Fourier Transform tie up resolution cell number M=61 and window function Sliding step Sp=20, iterations m initial value are m=1；
(2c2) takes the signal S after pulse pressuredM rows do Short Time Fourier Transform, obtain the when frequency division of a width M × N Butut, and it is stored in distance-Doppler-time isometric chartIn, wherein r represents distance dimension, and f represents more Pu Lewei, t represent time dimension, and M represents Doppler and ties up resolution cell number, and N represents time quantum number, N=(Nr-L/2)/ Sp；
(2c3) is by iterations m and distance to sampling number NrIt is compared, if m≤NrThen make m=m+1, return to step (2c2), if m ＞ NrThen stop iteration；
(2d) takes the N frames of distance-Doppler-in time isometric chart C (r, f, t) at different moments from 1 to N sequentially in time Distance-instantaneous Doppler picture, composition noisy image sequence I.
Step 3, according to noisy image sequence I, calculating average noise standard deviation sigman。
The pixel point sequence that i-th row jth in noisy image sequence I arranges is designated as I by (3a)ijIf row subscript i and row subscript j Initial value be i=1, j=1, with sequence IijOrdinate value form noisy ordinate vector Yij, without target square region Domain length of side D=3；
(3b) is according to formulaNoisy ordinate vector Y is calculatedijNoise Power, wherein Yij(m) noisy ordinate vector Y is representedijIn m-th of element；
(3c) is according to formulaNoisy ordinate vector Y is calculatedijNoise criteria difference σij；
(3d) by row subscript i compared with length of side D, by row subscript j compared with length of side D, if i≤D and j ＜ D, make j=j+ 1, if i≤D and j >=D, make i=i+1, j=1, return to step (3b)；If i ＞ D, step (3e) is performed；
(3e) is according to formulaMean noise standard difference σ is calculatedn。
Step 4, according to noisy image sequence I and noise standard deviation sigmanCalculating parameter nuclear matrix CN。
The initial value that (4a) resets row subscript i and row subscript j is i=1, j=1, with sequence IijValue abscissa value Form abscissa vector Xij；
(4b) is according to formulaParameter nuclear matrix is calculated CN:
Wherein σfFor range parameter, σ is scale parameter, xaRepresent abscissa vector XijIn a-th of element, xbRepresent horizontal stroke Coordinate vector XijIn b-th of element, a and b values are from 1 to N；
(4c) uses parameter magnitudes parameter σf, scale parameter σ composition parameter vectors θ={ σf,σ}。
Step 5, parameter vector θ estimated values theta is solved*。
(5a) gives iteration precision 0≤ε ＜ ＜ 1, cycle-index k=0, initial point θk=(- 1,1), maximum cycle Maxk=50, calculate functionGradientAnd calculate ladder 2 norms of degree | | gk||；
(5b) is by 2 norms of gradient | | gk| | compared with iteration precision ε, if | | gk| |≤ε, then stop calculating, output estimation Value θ*≈θkIf | | gk| | ＞ ε, perform step (5c)；
(5c) calculates the direction of search：
(5d) gives β ∈ (0,1), γ ∈ (0,0.5), makes iterations m=0, maximum iteration maxm=10,
(5e) calculates inequalityβmβ m powers are expressed as, T represents transposition Operation；
Iterations m compared with maximum iteration maxm, is judged whether above-mentioned inequality is set up by (5f), if inequality Establishment or m ＞ maxm, then make step factor λ=βm, perform step (5g)；Otherwise m=m+1, return to step (5e) are made；
(5g) calculates the estimates of parameters θ after renewalk+1=θk+λdk, calculate the gradient after renewalAnd Calculate 2 norms of gradient after updating | | gk+1||；
(5h) is by 2 norms of gradient after renewal | | gk+1| | compared with iteration precision ε, by cycle-index k and largest loop time Number maxk compares, if | | gk+1| |≤ε or k ＞ maxk stop calculating, then output estimation value θ*≈θk, otherwise, make cycle-index k= K+1, return to step (5c).
Step 6, according to estimates of parameters θ*Noise-reduced image sequence I is calculated*。
(6a) takes abscissa vector XijIn c-th of elementIf the initial value for circulating subscript c is c=1；
The estimate that (6b) will be obtained in step 5Substitute into equation below：
And data core matrix K and core vector is calculated K*,
Wherein xaRepresent abscissa vector XijIn a-th of element, xbRepresent abscissa vector XijIn b-th of element, a With b values from 1 to N；
(6c) is by data core matrix K, core vector K*With noisy ordinate vector YijSubstitute into formulaCalculate Obtain elementCorresponding functional valueWherein -1 represents inversion operation, and T represents transposition operation；
(6d) will circulate subscript c compared with time quantum number N, if c ＜ N, make c=c+1, return to step (6b)；If c >=N, with these functional valuesForm noise reduction ordinate vectorAnd perform step (6e)；
(6e) is by row subscript i and distance to sampling number NrCompare, row subscript j and Doppler are tieed up into resolution cell number M Compare, if i ＜ NrAnd j ＜ M, then j=j+1 is made, if i ＜ NrAnd j >=M, then make i=i+1, j=1, return to step (4b)；If i >= Nr, with these noise reduction ordinates vectorForm noise-reduced image sequence I*。
The effect of the present invention can be further illustrated by following emulation：
1. simulation parameter
Using the radar echo signal for being operated in C-band, corresponding carrier frequency is 10GHz, and with a width of 0.6GHz, target includes 3 Individual scattering point, echo signal to noise ratio are -13dB.
2. emulation content
Emulation 1：Enter row distance-instantaneous Doppler imaging to echo-signal, take the 22nd width image in its image sequence, tie Fruit such as Fig. 2.
Emulation 2：Noise reduction is carried out to Fig. 2 imaging results with the inventive method, takes the 22nd width image in its noise reduction sequence, is tied Fruit such as Fig. 3.
It can be obtained by Fig. 2 and Fig. 3 contrasts, the distance-instantaneous Doppler image sequence obtained using the present invention, figure can be reduced As ambient noise so that target point protrudes, while removes false target point, and accurately estimates the position of moving-target.
Simulation result shows that moving target distance-instantaneous Doppler is imaged by the present invention using Gaussian process regression theory Noise reduction problem is converted into the regression problem to pixel time signal, and parametric optimal solution, fully profit are solved using conjugate gradient method Contain position distribution and changes in amplitude information in image sequence with moving target scattering point, obtained under Low SNR Obtained the good moving target distance-instantaneous Doppler image sequence of focusing.

Claims (8)

1. distance-instantaneous Doppler image sequence the noise-reduction method returned based on Gaussian process, including：
(1) linear FM signal is launched to moving target by ISAR, and obtains its echo, take distance to sampling Count as Nr, orientation sampling number is Na, obtain a Nr×NaNoisy radar return matrix Sr；
(2) to noisy radar return matrix SrAlong distance to pulse compression is carried out, distance is obtained to the matrix S after pulse compressiond；
(3) adjust the distance to the matrix S after pulse compressiondAlong orientation carry out Short Time Fourier Transform, obtain distance-Doppler-when Between isometric chartWherein r represents distance dimension, and f represents Doppler's dimension, and t represents time dimension, and M represents Duo Pu Dimension resolution cell number is strangled, N represents time quantum number；
(4) the N frame pitches at different moments of distance-Doppler-in time isometric chart C (r, f, t) are taken from 1 to N sequentially in time From-instantaneous Doppler image, composition noisy image sequence I；
(5) the pixel point sequence that the i-th row jth in noisy image sequence I arranges is designated as Iij, with sequence IijOrdinate value composition contain Make an uproar ordinate vector Yij, the square area that no target occurs is chosen in noisy image sequence I, to all pixels in region Point position value, noisy image sequence I mean noise standard difference σ is calculatedn；
(6) initial value for resetting row subscript i and row subscript j is i=1, j=1, with sequence IijAbscissa value form horizontal seat Mark vectorial Xij；
(7) according to formulaParameter nuclear matrix C is calculatedN, its Middle σfFor range parameter, σ is scale parameter, xaRepresent abscissa vector XijIn a-th of element, xbRepresent abscissa vector Xij In b-th of element, a and b values are from 1 to N；
(8) function is maximized using conjugate gradient methodObtain parameter to Measure estimateWherein parameter vector θ={ σf, σ },For range parameter estimate, σ*Estimate for scale parameter Value；
(9) abscissa vector X is takenijIn c-th of elementIf the initial value for circulating subscript c is c=1；
(10) the parameter vector estimate that will be obtained in step (8)Substitute into equation below：
<mrow> <msup> <mi>k</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>a</mi> </msub> <mo>,</mo> <msub> <mi>x</mi> <mi>b</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>&amp;sigma;</mi> <mi>f</mi> <mrow> <mo>*</mo> <mn>2</mn> </mrow> </msubsup> <mi>exp</mi> <mo>&amp;lsqb;</mo> <mo>-</mo> <mfrac> <msup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>a</mi> </msub> <mo>-</mo> <msub> <mi>x</mi> <mi>b</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msup> <mi>&amp;sigma;</mi> <mrow> <mo>*</mo> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> <mo>&amp;rsqb;</mo> <mo>+</mo> <msubsup> <mi>&amp;sigma;</mi> <mi>n</mi> <mn>2</mn> </msubsup> <mi>&amp;delta;</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>a</mi> </msub> <mo>,</mo> <msub> <mi>x</mi> <mi>b</mi> </msub> <mo>)</mo> </mrow> <mo>,</mo> </mrow>
And data core matrix K and core vector K is calculated*, by data core matrix K, core vector K*With noisy ordinate vector YijGeneration Enter formulaElement is calculatedCorresponding functional value- 1 represents inversion operation, and T represents transposition behaviour Make；
(11) circulation subscript c, if c ＜ N, is made into c=c+1, return to step (10) compared with time quantum number N；If c >=N, With these functional valuesForm noise reduction ordinate vectorAnd perform step (12)；
(12) by row subscript i and distance to sampling number NrCompare, compared with row subscript j is tieed up into resolution cell number M with Doppler, If i ＜ NrAnd j ＜ M, then j=j+1 is made, if i ＜ NrAnd j >=M, then make i=i+1, j=1, return to step (7)；If i >=Nr, use These noise reduction ordinates vectorForm noise-reduced image sequence I*。
2. according to the method for claim 1, it is characterised in that to noisy radar return matrix S in step (2)rAlong distance to Pulse compression is carried out, is carried out as follows：
(2a) will launch with ISAR and believe using the distance of ISAR to scene center as reference distance Number carrier frequency, frequency modulation rate are identical, distance for reference distance linear FM signal as reference signal Sref；
(2b) is by reference signal SrefTake the noisy radar return matrix S with reception after being conjugatedrIt is multiplied, obtains solving the square after line frequency modulation Battle array Srd；
(2c) is to the matrix S after solution line frequency modulationrdFourier transformation, which is done, along distance dimension obtains distance to the matrix S after pulse compressiond。
3. according to the method for claim 1, it is characterised in that adjusted the distance in step (3) to the matrix S after pulse compressiondEdge Orientation carries out Short Time Fourier Transform, carries out as follows：
The long L=31 of window, the Doppler of (3a) setting Short Time Fourier Transform tie up resolution cell number M=61 and window function slides step Long Sp=20, iterations m initial value are m=1；
(3b) takes the matrix S after pulse pressuredM rows do Short Time Fourier Transform, obtain the time frequency distribution map of a width M × N, And it is stored in distance-Doppler-time isometric chartIn, wherein N=(Nr-L/2)/Sp；
(3c) is by iterations m and distance to sampling number NrCompare, if m≤NrThen make m=m+1, return to step (3b), if m ＞ NrThen stop.
4. according to the method for claim 1, it is characterised in that being averaged for noisy image sequence I is calculated in (5) in step Noise criteria difference σn, carry out in accordance with the following steps：
(5a) sets row subscript i and row subscript j initial value as i=1, j=1, the square area length of side D=3 without target；
(5b) is according to formulaNoisy ordinate vector Y is calculatedijNoise work( Rate, wherein Yij(m) noisy ordinate vector Y is representedijIn m-th of element；
(5c) is according to formulaNoisy ordinate vector Y is calculatedijNoise criteria difference σij；
(5d) by row subscript i compared with square area length of side D, by row subscript j compared with length of side D, if i≤D and j ＜ D, make J=j+1, if i≤D and j >=D, make i=i+1, j=1, return to step (5b)；If i ＞ D, step (5e) is performed；
(5e) is according to formulaMean noise standard difference σ is calculatedn。
5. according to the method for claim 1, it is characterised in that parameter nuclear matrix C is calculated in step (7)N, represent such as Under：
Wherein xaRepresent abscissa vector XijIn a-th of element, xbRepresent abscissa vector XijIn b-th of element, a and b Value is from 1 to N.
6. according to the method for claim 1, it is characterised in that function f (θ) is maximized with conjugate gradient method in step (8), Obtain parameter vector θ estimated values theta*, carried out by following iterative step：
(8a) gives iteration precision 0≤ε ＜ ＜ 1, cycle-index k=0, initial point θk=(- 1,1), maximum cycle maxk= 50, calculate the gradient g of functionk=▽ f (θk), and calculate 2 norms of gradient | | gk||；
(8b) is by 2 norms of gradient | | gk| | compared with iteration precision ε, if | | gk| |≤ε, then stop calculating, output estimation value θ* ≈θk；If | | gk| | ＞ ε, perform step (8c)；
(8c) calculates the direction of search
(8d) gives β ∈ (0,1), γ ∈ (0,0.5), makes iterations m=0, maximum iteration maxm=10；
(8e) calculates inequalityβmβ m powers are expressed as, T represents transposition operation；
Iterations m compared with maximum iteration maxm, is judged whether above-mentioned inequality is set up by (8f), if inequality is set up Or m ＞ maxm, then make step factor λ=βm, perform step (8g)；Otherwise m=m+1, return to step (8e) are made；
(8g) calculates the parameter vector estimated values theta after renewalk+1=θk+λdk, calculate the gradient g after renewalk+1=▽ f (θk+1), and Calculate 2 norms of gradient after updating | | gk+1||；
(8h) is by 2 norms of gradient after renewal | | gk+1| | compared with iteration precision ε, by cycle-index k and maximum cycle Maxk compares, if | | gk+1| |≤ε or k ＞ maxk stop calculating, then output estimation value θ*≈θk, otherwise, make cycle-index k=k+ 1, return to step (8c).
7. according to the method for claim 1, it is characterised in that data core matrix K is calculated in step (9), represents such as Under：
Wherein xaRepresent abscissa vector XijIn a-th of element, xbRepresent abscissa vector XijIn b-th of element, a and b Value is from 1 to N.
8. according to the method for claim 1, it is characterised in that core vector K is calculated in step (9)*, represent as follows：
<mrow> <msup> <mi>K</mi> <mo>*</mo> </msup> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msup> <mi>k</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mi>c</mi> <mo>*</mo> </msubsup> <mo>,</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <mrow> <msup> <mi>k</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mi>c</mi> <mo>*</mo> </msubsup> <mo>,</mo> <msub> <mi>x</mi> <mi>b</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <mrow> <msup> <mi>k</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mi>c</mi> <mo>*</mo> </msubsup> <mo>,</mo> <msub> <mi>x</mi> <mi>N</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow>
Wherein, xbRepresent abscissa vector XijIn b-th of element, b values are from 1 to N.
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