Extraction of Peak Feature Based on Synthetic Sinc Model in SAR images

Peak is an important feature in Synthetic Aperture Radar(SAR), which represents essence of scattering centre. There are two general approaches in the literature to extract peak. One way is to extract peak after speckle suppression filtering. Using this method, the extracted feature is in accurate, and the algorithm is more complicated. Another is that detecting the amplitude of the peak directly. In order to have a fast and accurate peak extraction, we proposed using the Sinc peak model algorithm in this paper. It directly extracts peak features from the original SAR image without any noise suppression filtering, which is instead of Gauss mask function. The estimation parameters of peaks use QE theorists. Finally, we can get accuracy three parameters to describe peak features. Analytic fluctuation of parameters is compared with Gauss model peaks using truth SAR images. Experimental demonstrate that the new algorithm is more effective than others for extracting peak features in SAR images.


INSTRUCTION
SYNTHETIC aperture radar (SAR) images contain bright spots representing locations where strong back-scatters are present in the scene.These bright spots are pixels having localized peak intensities, called image peaks.Peaks are important features in SAR.They are denoted as local maxima in images, which represent scattering centres.Their relative positions and amplitudes can be shown as the geometric form and the surface structure of the targets.Information obtained about the peaks can be applied to: improve peak matching in the automatic target recognition (ATR) problem; false-alarm rate detection; and identification of imaging artifacts, such as the sidelobes of bright peaks.There are two areas involving peak extraction: electromagnetic field and imagery field.Various numerical methods for calculating electromagnetic field distributions such as the method of moments (MoM), finite-difference-time-domain (FDTD) method and the finite element method (FEM) are considered in the literature.They usually require an accurate discretization of the structure relative to wavelength.At high frequencies, this leads to an immense computational effort for a large structure.Therefore, these methods are usually applied to problems of a smaller scale [1] .Hybrid [2] or asymptotic methods based on geometrical optics (GO) or physical optics (PO) are found more approximate for problem of a large scale.Physical optics may also be extended by physical theory of diffraction (PTD) [1] , the method of equivalent currents (MEC) [3] and impedance boundary condition (IBC) techniques [4] .Canonical point scatters, such as a trihedral, are characterized by the fact that their peak intensities are located at a single location in the image [5] .Peak models have also been developed to predict the intensity of peaks due to canonical point scatters [6] .However, in imagery field, there are few papers about peak extraction, and most papers attributed feature extraction of scattering centers like as [7,8].In the literature, peaks are usually extracted after speckle suppression filtering in order to prevent its influence on the peak in SAR image.There is also a peak extraction method based on the constant false alarm probability (CFAR) principle [9] .However, peaks extracted after speckle suppression filtering are not precise enough and the algorithms are computationally complex.They cannot satisfy the target requirement for target recognition with large samples.Amplitude of peak is instead all of peak features is then proposed to address there issues for target recognition [10] .However, ignores the width of peak feature, and causes an incomplete extraction of peak feature.In addition, its recognition rate is low.In this paper, we propose a physical peak model to have a more complete peaks description instead of width, amplitude, and fluctuation.We then adopt a statistical estimation fluctuation function that provides estimates of feature uncertainty.The remainder of the paper is organized as follows.Section II describes the proposed Synthetic Sinc Model and its comparison with Gauss model.In Section III, two estimation methods are used to determine the parameters of peak feature, and fluctuation algorithm based on the Synthetic Sinc Model is presented.Section IV presents a simple peak parameter estimation and fluctuation algorithm based on Synthetic Sinc Model.Experimental results are obtained from simulated data using isolated point scatter with 10db multiply noises.Comparison with Gauss peak model is also presented.Using the real target chips of SAR image do the experimental results further confirm the benefits of the proposed approach.Section V provides the conclusions.

PEAK MODEL DEVELOPMENT
According to radar imaging theory, an echo can be represented using the distance and direction coordinates.That is, (1) where A is a complex constant given by, And A is the modulus value of A. The window function ω a at the azimuth direction is the Sinc function.The window function ω r at the range direction is the rectangle function.r and x are the range and azimuth direction respectively.s 0 (x, r) is obtained when the ground backscattering coefficient g 0 (x, r) goes through a 2D linear system h(x, r).The system impulse response h(x, r) can be divided into the convolution of the impulse response in azimuth direction h a and in range direction h r as shown below: The impulse responses in azimuth direction h a and in range direction h r are given as follows: Our goal is to obtain an exact value of A. We make convolution between h s (x, r) andh(x, r) to get: The filter that satisfies (7) above can have good performance on solving the convolution.In fact, it can be proved that h(x, r)h s (x, r) in both azimuth and range directions are Sinc function instead of unit impulse response δ(x, r).For Sinc function, there are peak sidelobes, which degrade the quality of imaging.We can make a single point echo.When the point echo is a circle area function Somb, (x 0 , y 0 ) is the center of coordinate , sin The simplified circle area function Somb is approximated by:

Parameter Estimation based on QE peak
Sinc function with Gaussian mask is equal to the Fourier transform of Gaussian function multiplying rectangular window in frequency domain.Here, we transfer the result to frequency domain for solving the problem of parameter estimation.Assume sinc(πx)sinc(πy) is unit 2D Synthetic Sinc Model.The width of Sinc function in time domain is τ x , τ y .The width of Sinc function in frequency domain is τ u , τ v .We can then get the scale variation of τ u ', τ v ' at X and Y direction in frequency domain because of the change of σ x , σ y in time domain.In time and frequency domains, the scaled transfers of the bottom of peak are: , , τ x ', τ y ' denotes the radius of ellipse bottom at x axis and y axis in time domain respectively; τ u ', τ v ' represents radius of rectangle bottom at x axis and y axis in frequency domain respectively.The Fourier Transform of Gaussian Function and Sinc function can be expressed as: Adopting the form of ( 13), ( 14).A simple form of IFT can then be obtained as According to 1D inverse Fourier Transform, the result of 1D IFT is given as, Since ω is independent of x, the integral can be expressed as When σ t ≥1, the Gaussian function Therefore the 2D convolution of f(x, y) can be approximated as the multiple of volume V(G τ f ' (σ x u, σ y v)) and Gaussian function, as be described in Fig. 3 and Fig. 4.
Where, In Figure 3, the original peak model is σ x =1, σ y =1, τ x =2, τ y =2， then τ u '=1, τ v '=1 in frequency domain.We find the value ofσ t ,σ l is the more crossed the σ x , σ y , the result τ u ', τ v ' of frequency is the better than others in Figure 4. And, we can prove that the equations above are correct because the simulation results are close to the Gaussian Function with window.But when σ t ,σ l are over large, the peak will be disappeared.
For extracting peak parameters, we should make sure the position (x, y).The first derivative of a local maximum curve is zero.The peak position can be defined as the point which is both of first derivative zero-crossing points transverse and longitudinal directions.Finding the interested point of two zerogradient level line in x and y directions.So, f x , f y mean firstorder partial derivative in x and y directions and f xx , f yy mean second-order partial derivative.When f x (x 0 , y 0 )=0 f y (x 0 , y 0 )=0 ,and f xx (x 0 , y 0 ) <0, f yy (x 0 , y 0 )<0,we can find the position of peak value (x 0 ,y 0 ).The method same as the Sythtic Gauss Peak model(SGP) in [11].In Figure 5, We can see when mask is 4 pixel, the estimation effect is better than other mask widths with 10dB noise.So we should choose the fit mask width for different Sythtic Sinc Peak model(SSP)σ width for extracting exactly position information.We can then use QE algorithms to detect parameters of peak.
According to f x (x, y)=I(x, y)m x (x, y), f xx (x, y)=I(x, y) m xx (x, y), we have where A means test amplitude, τ u =1, τ v =1.Using ( 23)-( 25), σ x , σ y , H can be solved.In SSP proposed, its size of sample is 41 41  .Assure A=1, width changes from 1 to 5 pixel.We start QE estimation of width and amplitude of peak in Fig. 6 and Fig. 7.The error of width estimation is 10 -2 .The error of amplitude estimation is 10 - 2 .This method is more effective than SGP [11] .

Position Fluctuation Analysis
Because noise causes the fluctuation of the estimation of peak and width estimates, we can use the variance of fluctuation to evaluate different estimation algorithm.
δ is the fluctuation of peak. where In the same way, we can obtain the equation below: The first order and second order derivatives are derived as below: In the simulation, we add the noise which observes Gamma distribution where 2 1 N n   . The first order and second order variance can be calculated as In Figure 8, α refers to the coefficient of fluctuation in one pixel.and α＜1, α=f When x 1 and x 2 is very close, f x1 -f x2 is an approximate of the second order derivative of f.In the Sinc peak, we can obtain the fluctuation of peak is

Fluctuation Analysis of width and amplitude
Using to the same principle, we can obtain the equation below: When are same as results of GAUSS model [11] .Thus, we have In solving E(δfδσ y ), we can obtain the fluctuation variance of width and amplitude.

Distribute of fluctuation experiment and result analyze
The peak simulation results based on fluctuation analysis are shown in Bessel function of the first kind.After performing the Range-Doppler(RD) imaging processing, we obtain the image in Fig. 1.

Figure 1 .
Figure 1.Point echo RD image In Figure 1, the point target after imaging isn't a circle area function Somb while it is a 2D Sinc function which is shown below: interval [π/σ x ,π/σ x ].The integral can be approximated as the area of G 2π/σx (σ x ω): S(G 1/σx (σ x u)) , where ω is angle frequency and f means frequency, and ω=2πf.Then

Figure 5
Figure 5 QE estimation of posion when Sinc peak is at x=0.6, y=0.1 with 10dB noise

Figure 6 Figure 7 .
Figure 6 QE estimation of width when Sinc peak is at x=0.6, y=0.1 with 10dB noise

Figure 11
Figure 11 Peak extract of three kinds targets in SAR image 5. CONCLUSION In order to extract peak feature in SAR image accurately and quickly, this paper proposes to extract peak feature in SAR image based on Synthetic Sinc Model.This extraction method keeps the original peak feature and provides reliable feature vectors because the SAR image is required to have suppression flitting in this case.It is shown to can improve the speed of extraction and realize feature recognition.Further research on point echo imaging shows that Synthetic Sinc Model is suitable for SAR echo imaging because Gauss Model cannot reflect real speckle process.With Gaussian mask, we can perform the fluctuation analysis using derivative equations and least square method to conclude that Synthetic Sinc Model has the best performance on recovering peak in images with the least fluctuation variance.

Table 1 .
The simulation is under Gamma noise with SNR equal to 10dB.

Table 1 .
Fluctuation analysis of three algorithmsIn Table1, we can conclude that Synthetic Sinc Model is more stable and exact.The fluctuation of noise has little influence on Synthetic Sinc Model.The mean of Var(δx), Var(δσ), Var(δH) by Synthetic Sinc Model is 0.0032 which is smaller than that by Gauss Model.It implies that Synthetic Sinc Model can improve the accuracy of reconstructing.4.

Table 2 .
Fluctuation analysis of three algorithms in SAR image peak