Fluctuation Analysis of width and amplitude Distribute of fluctuation experiment and result analyze

1 2 3 4 5 6 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 aEstimation Peak Amplitude for Centeredx=0.6,y=0.1SSP By QE Width of SSP with addnoise and multiplnoise E st im a te d P e a k A m p li tu d e A Mask width=1p Mask width=2p Mask width=4p Best Maskw line 1 2 3 4 5 6 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 bstd. of Estimation Peak Amplitude for Centeredx=0.6,y=0.1SSP By QE Width of SSP with addnoise and multiplnoise st d . o f E st im a te d P e a k A m p li tu d e A -A Mask width=1p Mask width=2p Mask width=4p Figure 7. QE estimation of amplitude when Sinc peak is at x=0.6, y=0.1 with 10dB noise 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] . 3. SYNTHETIC SINC MODEL FLUCTUATION ANALYSIS 3.1 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.     , , h x y I x y I    26 δ is the fluctuation of peak.         , , , , f x y h x y m x y I m I m f x y f            , 37 where   , f I m x y     27 In the same way, we can obtain the equation below:       , , , x x f x y I x y m x y   28       , , , xx xx f x y I x y m x y   29 The first order and second order derivatives are derived as below:   , x x f I m x y     30   , xx xx f I m x y     31 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   2 1 4 N l t Var f      32   2 5 3 16 x N l t Var f      33   2 7 15 32 xx N l t Var f      34 In Figure 8, α refers to the coefficient of fluctuation in one pixel. and α＜1, α=f x1 f x1 ﹣f x2 ， x t    35 t means one pixel, x    ,     2 1 1 2 2 1 2 1 x x x x x x x f f f f f f        36       2 1 2 1 x x x Var x Var f f f     37 1- αt αt X 1 x f 2 x f x f Figure 8 Peak position f Fluctuation 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   6 2 2 2 2 2 5 4 3 16 t l N u v l t Var x H          38

3.2 Fluctuation Analysis of width and amplitude

Using to the same principle, we can obtain the equation below:       2 4 2 , 1 , xx x t t f x y x x f x y       39 When xx f A f  , we have: 2 1 xx xx f A f f f f       40          2 4 3 2 2 1 xx xx xx xx f f Var A Var f E f f Var f f f f             41 Due to f  come form noise, so Varδf、Eδfδf xx 、 Var δf xx are same as results of GAUSS model [11] . Thus, we have   2 4 2 1 x t t x x A       42    4 2 x x t x x A x x         43   4 2 t x x x A x x x x         44     4 4 2 2 x t x t x x A x x x x                 45         4 2 2 4 3 5 2 7 6 2 2 4 2 2 2 5 2 3 4 2 1 1 15 4 16 32 2 4 3 16 t x x t xx xx l t l t t l x t l N t u v l t Var x x x f f f f f x x H                                   46     2 2 2 2 2 2 2 y x t l x y x x t l u v H f e             47 2015 International Workshop on Image and Data Fusion, 21 – 23 July 2015, Kona, Hawaii, USA This contribution has been peer-reviewed. doi:10.5194isprsarchives-XL-7-W4-45-2015 48             2 2 2 2 2 x X y y H H Var H Var f Var f H H H Var Var x Var y x y                                                            2 x y x y H H H H E f E f f f                                                  59 where，       4 4 2 2 x t x x t x x E f E A f x f x x x                             48 Due to     E f x E f y       49     x y E x E y       50     4 2 2 1 2 t xx x xx x f E f E f f f x x x f f                                4 2 1 2 t xx xx x f Var f E f f x x x f f                51 In solving E δfδσ y , we can obtain the fluctuation variance of width and amplitude.

3.3 Distribute of fluctuation experiment and result analyze

The peak simulation results based on fluctuation analysis are shown in Table 1. The simulation is under Gamma noise with SNR equal to 10dB. Algorithms Var δx Varδσ VarδH Ave GAUSS by QE 0.0407 0.0062 0.0184 0.0218 GAUSS by WLS 0.0407 0.0046 0.0230 0.0228 SINC by QE 0.0028 0.0017 0.0051 0.0032 Table 1. Fluctuation analysis of three algorithms In Table 1, 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, Varδσ, VarδH 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. EXPERIMENT AND RESULT ANALYZE OF SAR IMAGE PEAK EXTRACT