Target Detection in Sea Clutter with Application to Spaceborne SAR Imaging
Shahrokh Hamidi







Target Detection in Sea Clutter with Application to Spaceborne SAR Imaging 



    Shahrokh Hamidi

Department of Electrical and Computer Engineering

University of Waterloo

Waterloo, Ontario, Canada 

shahrokh.hamidi@uwaterloo.ca

    September 9, 2024
=========================================================================================================================================================






§ ABSTRACT 
In this paper, the challenging task of target detection in sea clutter is addressed. We analyze the statistical properties of the signals received from the scene and based on that, we model the amplitude of the signals reflected from the sea clutter according to the Weibull distribution. 
Subsequently, we utilize the aforementioned information to design an adaptive threshold based on the Constant False Alarm Rate (CFAR) algorithm to detect the energy of the targets which have been buried in the sea clutter.

Thorough analysis of the experimental data gathered from the Canadian RADARSAT-1 satellite demonstrates the overall effectiveness of the proposed method. 




Target detection, sea clutter, Weibull distribution, CFAR. 




§ INTRODUCTION

As an all-weather, day-night, and active device, Synthetic Aperture Radar (SAR) is capable of providing crucial information from the surface of the Earth <cit.>. Following the image reconstruction, post processing procedure is performed to obtain more critical information from the scene. Target detection is one of the areas that requires further analysis of the reconstructed image. In this paper, we specifically focus on the targets buried in sea clutter. The goal is to establish a mechanism to extract the energy of the targets from sea clutter. Several techniques have been reported in the literature, such as the wavelet-based analysis <cit.> and fixed threshold-based method <cit.>. The problem with using a fixed threshold to detect targets from clutter is that some of the key features of the desired targets can be ignored while the undesired signals are retained. Moreover, methods such as wavelet-based analysis are not able to capture the statistical properties of the signals. This, in turn, hinders their ability to obtain all the critical information about the desired targets. 

In this paper, we utilize the technique based on the Constant False Alarm Rate (CFAR) method <cit.> which creates an adaptive threshold to extract the energy of the desired target while suppressing the energy of the undesired signals. 
However, in order to be able to implement the CFAR-based adaptive threshold, we need to obtain the information about the statistical properties of the undesired signals which in this case is the sea clutter. To tackle this issue, we study several well-known distributions, namely, Weibull, Log Normal,  Inverse Gaussian, Gamma, and Rayleigh and demonstrate that the Weibull distribution is a more appropriate distribution to model the statistical properties of the sea clutter.

In <cit.>, the authors have studied the statistical properties of the reflected SAR signals. In this paper, however, we specifically focus on sea clutter and also we present more statistical models which distinguishes our work from <cit.>. 
We further describe the SAR imaging procedure in detail and present the experimental results to verify the effectiveness and accuracy of the proposed models which makes the material presented in this paper distinct from <cit.>.
In contrast to <cit.>, we will not consider the K distribution. The reason for this is that, the K distribution is formed by compounding two separate probability distributions, one representing the radar cross-section, and the other representing the speckle noise. However, prior to addressing the target detection in sea clutter, we remove the effect of the speckle noise by applying median filter to the reconstructed SAR image. As a result, the K distribution is not a suitable model anymore. Moreover, based on the experimental data, that we present in this paper, we show that the Weibull distribution  is also capable of modeling the sea clutter, contaminated with speckle noise, highly accurately.   
We present several statistical models to describe the sea clutter and at the end show that the Weibull distribution is the most appropriate distribution to model the statistical properties of the sea clutter. We then utilize the Weibull distribution and establish an adaptive threshold to extract the energy of the desired targets from sea clutter.
The verification of the proposed approach is based on the real data gathered from the Canadian RADARSAT-1 SAR satellite <cit.>. The data has been gathered from English Bay in Vancouver Canada in which several ships have been located inside water which creates a perfect scenario to evaluate the results. 


The organization of the paper is as follows. 
In Section <ref>, we present the system model as well as the Range-Doppler algorithm to be used to reconstruct the SAR images from the raw data. In Section <ref>, we address the statistical analysis of the SAR data. Finally Section <ref> has been dedicated to the experimental results based on the real data gathered from the RADARSAT-1 SAR satellite which is followed by the concluding remarks. 








§ MODEL DESCRIPTION AND IMAGE FORMATION

Fig. <ref> shows the system model in which the satellite is moving along its track and is collecting data from the surface of the Earth. The data collection is based on the strip-map mode. 

The signal transmitted by the radar is a chirp signal which after being reflected from a point target and upon down-conversion is described as 

    s(t, η) = σ  w_r(t - 2R(η)/c)w_a(η - η_c)
        e^  -j 4 π f_cR(η)/c + j πβ t^2 + j 4 πβ t R(η)/c,

where f_c is the carrier frequency and the parameter β is given as b/T, in which b and T stand for the bandwidth and the chirp time, respectively. In addition, w_r is a rectangular window with length T and t is referred to as fast-time parameter.
In addition, σ is the complex radar cross section for the point reflector, η is referred to as slow-time parameter and R(η) is the instantaneous radial distance between the radar and the target which is given as  R(η) = √(R^2_0 + v^2(η - η_c)^2). Moreover, w_a is a rectangular window with its length equal to the synthetic aperture length divided by v, where v is the speed of the satellite.
The ultimate goal in SAR imaging is to estimate the complex radar  section σ.

The Range-Doppler algorithm is the first algorithm that was developed for spaceborne SAR image reconstruction <cit.>.
The first step in the Range-Doppler algorithm is to perform range compression which results in <cit.>

    s_ rc(t, η) = σ  p_r(t - 2R(η)/c)w_a(η - η_c) 
               e^  -j 4 π f_c R(η)/c,

where p_r(x) = sin(π x)/π x.

Next, we should compensate for the RCM phenomenon which is the result of coupling between the range and azimuth directions that is created because of the non-zero squint angle for the antenna. Consequently, after RCM compensation, we can describe the signal in the Doppler domain as <cit.>

    S_ rcmc(t, f_η)    = σ  p_r(t - 2R_0/c)W_a(f_η - f_η_c) ×
       e^ -j 4 π f_c R_0/c e^  j πf^2_η/K_a,

in which K_a = 2v^2/λ R_0.
The final step for image formation in the Range-Doppler algorithm, is to compress the data in the azimuth direction which upon performing this stage the compressed signal in the range and azimuth directions is given as <cit.>

    s_ rcac(t, η)     = σ  p_r(t - 2R_0/c)p_a(η) e^  -j 4 π f_c R_0/c×
       e^  j 2 π f_η_cη,

where f_η_c is the Doppler centroid frequency <cit.>. 




§ STATISTICAL MODELING

In this section, we attempt to model the amplitude of the reflected signal from sea clutter. The main candidate is the Weibull distribution which its probability density function is described as   <cit.> 

    p_W(x; α, β) =  α/β(x/β)^α-1e^ - (x/β)^α,     α > 0,   β > 0.

Another possible option for consideration to model the sea clutter is the Log-normal distribution which its probability density function is given as  <cit.>

    p_ LN(x; η, γ) =  1/xη√(2 π) e^ - (log x -γ)^2/2 η^2,     γ > 0,   η > 0.

The next distribution that we consider in this paper, is the Inverse Gaussian distribution with the following probability density function  <cit.>

    p_ IG(x; μ, λ) =  √(λ/2 π x^3) e^ - λ(x -μ)^2/2 μ^2 x,     μ > 0,   λ > 0.

The Gamma distribution with 2 degrees of freedom is the next best choice to model the sea clutter. The probability density function for the Gamma distribution is given as    <cit.>

    p_G(x; a, b) = b^a/Γ(a) x^a-1  e^-bx,     a>0,    b > 0.

The last distribution that we study in this paper, is the Rayleigh distribution. The probability density function for the Rayleigh distribution is described as   <cit.>

    p_R(x; σ) = x/σ e^ -x^2/2σ,    σ > 0.

Based on the given probability of false alarm (p_ fa), we can calculate the adaptive threshold T_a as  <cit.> 

    p_ fa = p(X > T_a| H_0) = ∫_ T_a^∞  p(X| H_0)   dX,

where X describes the statistics of the cell under test and T_a is the adaptive threshold. Furthermore, H_0 represents the null hypothesis. 

By using the experimental data that we present in this paper, we will demonstrate that among the proposed distributions, the Weibull distribution is the best distribution to model the sea clutter. Therefore, the calculation of the adaptive threshold is only performed for the Weibull distribution. 
Consequently, based on (<ref>), the adaptive threshold T_ aW for the Weibull distribution, which has been given in (<ref>), is calculated as

    T_ aW = [βlog(1/p_ fa)]^1/α.

The maximum likelihood estimates for the parameters of the Weibull distribution, given in (<ref>), are expressed as <cit.>

    β̂ = (1/N∑_i = 1^N x^α̂_i)^1/α̂, 
    α̂ = N/1/β̂∑_i = 1^N x^α̂_i log x_i - ∑_i = 1^Nlog x_i,

in which x_i is the i^ th realization of the random variable x.  

In this paper, we implement the 2D CFAR <cit.> algorithm.  
The implementation of the 2D CFAR method is based on the structure which has been shown in Fig. <ref>. 

Consequently, the decision making process for the cell under test is expressed as <cit.>

    X_ CUT≶_H_1^H_0μ_c + σ_c  Q,

where X_ CUT is  the amplitude of the cell under test. Furthermore, μ_c and σ_c are the sample mean and standard deviation computed from the clutter data of the local background. 
Moreover, Q = T_ aW/μ̂ is the detector design parameter which defines the p_ fa and is set empirically in which T_ aW is the adaptive threshold which has been described in (<ref>) and μ̂ denotes the mean value estimated from the underlying clutter model. In addition, the H_0 and H_1 are the null and alternative hypothesis, respectively <cit.>.    

The null hypothesis, H_0, represents the case in which the cell under test has been occupied by clutter only and the alternative hypothesis, H_1, on the other hand, is for the case in which the target is present and the cell under test contains the energy of the target.   

In the next section, we present the experimental results. 


§ EXPERIMENTAL RESULTS

In this section, we present the result of SAR image reconstruction based on experimental data gathered from the Canadian RADARSAT-1 satellite. The data is from English Bay in Vancouver Canada. 
The specifications for the RADARSAT-1 satellite have been given in Table.<ref>.

We select the data related to English Bay. The reason is that, there are several ships in this scene that play the role of isolated strong reflectors which will assist to observe the effect of the RCM clearly and will also be utilized to perform Doppler centroid estimation. 
Furthermore, the ships in the sea water are the main subject of the paper. In other words, we attempt to detect the ships in sea clutter which is the main goal of the paper.
 
In order to reconstruct the image, we apply the Range-Doppler algorithm to the raw data. Fig. <ref>-(a) shows the range compressed data based on (<ref>).
The range compressed energy of the ships can be seen as a few skewed vertical lines. The skew demonstrates the effect of the RCM.
In order to perform the RCM compensation we need to estimate the unambiguous Doppler centroid frequency.

As we mentioned before, if the Doppler centroid frequency is larger than the PRF of the RADAR, there will be an ambiguity in the Doppler centroid frequency estimation.
One way to estimate the unambiguous value for the Doppler centroid frequency is by analyzing the trajectory of the strong isolated targets.
Fig. <ref> shows the range compressed image shown of several ships in sea water.
The skew in the trajectory of these targets is due to the non-zero Doppler centroid frequency. The energy of each one of them is spread over several different range cells.
The slope can easily be calculated and is equal to 0.034 range samples per azimuth samples. To estimate the Doppler centroid frequency we should first multiply the slope by c/2Fr to convert the slope from range samples to range distance and then multiply it by PRF to convert it from azimuth samples to azimuth time.
Hence, we have dR(η)/dη = 198.23   m/s. As a result, from f_ dc = -2/λdR(η)/dη <cit.>, the Doppler centroid frequency can be calculated as f_ dc = -7009   Hz.

Next, we estimate the fractional part of the Doppler centroid frequency which is f^'_ dc <cit.>. The fractional part is essential in focusing the energy of the targets in the azimuth direction. 
Fig. <ref> illustrates the power spectrum of the data in azimuth direction versus slow time frequency f_η. In order to reduce the effect of the noise, we have added the power spectrum corresponding to 2048 range cells. In order to be able to calculate the frequency component at which the signal reaches its maximum value, we have performed curve fitting.
From Fig. <ref>, we can estimate the fractional part of the Doppler centroid frequency as f^'_ dc = 531  Hz.
Finally, at the last stage we perform the azimuth localization. Consequently, the reconstructed image based on the Range-Doppler algorithm is obtained which has been presented in Fig. <ref>.

The image shown in Fig. <ref> suffers from speckle noise <cit.>. In order to remove the effect of the speckle noise, we introduce a 2D m × n filter and slide it over the reconstructed image while solving the following optimization problem,

    min_a          ∑_i=1^n∑_j=1^m| a_ ij-a|,

where a_ ij is the value for the ( ij)^ th pixel and a is the value chosen by the optimization problem for the (⌊n-1/2⌋ + 1  ⌊m-1/2⌋ + 1)^ th pixel. In fact, by solving the optimization problem in (<ref>) we are filtering the image by the median filter. In other words, we replace the value of each pixel with the median of the neighboring pixels. This proves to be a powerful method to decrease the effect of the speckle noise which, as a result, the fine structures of the image can be revealed. 
It should be noted that, the common practice to alleviate the effect of the speckle noise in SAR images is multi-look processing <cit.>. However, multi-look processing sacrifices the azimuth resolution in order to remove the speckle noise. In contrast, the median filtering approach, which we have proposed, can efficiently remove the speckle noise while it leaves the azimuth resolution intact. Furthermore, the median filtering does not smear the edges in the image which is considered to be a significant property. 

Fig. <ref> shows the result of speckle noise reduction for the reconstructed image depicted in Fig. <ref>.  To remove the effect of the speckle noise we have applied the filter given in (<ref>) with m=n=6.

In order to analyze the effect of sea clutter on target detection, we focus on the specific part of the  reconstructed image shown in Fig. <ref> which contains several ships in sea water. The related part of the data-set has been highlighted as the area inside the red rectangle in  Fig. <ref>. The area inside the green rectangle contains the energy from sea water and is utilized to analyze the statistical properties of the sea clutter. Fig. <ref> shows the histogram for part of the image related to the area which has been located inside the green rectangle. The estimated probability density functions for the Weibull,  Log-normal, Inverse Gaussian, Gamma, and Rayleigh distributions are based on (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>), respectively.

From Fig. <ref>, it is evident that the Weibull distribution models the sea clutter much better than the other distributions. 
Fig. <ref> illustrates the result of applying the 2D CFAR algorithm to the part of the data-set shown as red rectangle in Fig. <ref>. The result presented in Fig. <ref>-(b) is remarkable in a sense that, by modeling the sea clutter with Weibull distribution and applying the 2D CFAR technique we have been able to detect the ships in sea clutter.  

To achieve the result presented in Fig. <ref>-(b), we have set the number of guard cells for the upper, lower, left, and right part of the cell under test to 10. We have also selected the number of training cells for the upper, lower, left, and right part of the cell under test to be equal to 10.

We consider another case in which one of the ships, as has been highlighted as the area inside the yellow rectangle  in Fig. <ref>, is being buried in the sea clutter. Moreover, it should be noted that, the image presented in Fig. <ref> is contaminated with speckle noise. Our goal is to consider the effect of the speckle noise in modeling the histogram of the reconstructed image. 

In Fig. <ref>, we have presented the reconstructed image from a single ship which has been highlighted as the area inside the yellow box in Fig. <ref>. 
 
The part of the image, shown in Fig. <ref>-(b), which has been highlighted as the area inside the red rectangle is utilized to estimate the histogram of the sea clutter and the result has been illustrated in Fig. <ref>. 
 
As it is clear from Fig. <ref>, the Weibull distribution can model the histogram with high accuracy. 
 
Next, we apply the 2D CFAR technique to the reconstructed image presented in Fig. <ref>. We have presented this remarkable result in Fig. <ref>. To achieve the result presented in  Fig. <ref>, we have set the number of guard cells for the upper and lower part of the cell under test to 100 and for the left and right part to 20. We have also selected the number of training cells  for the upper, lower, left, and right part of the cell under test to be equal to 20.





§ CONCLUSION

The SAR image reconstruction was discussed in detail. The statistical properties of the sea clutter was investigated and the correct model based on the Weibull distribution was applied to the sea clutter. The validity of the proposed model was verified based on the real data and the results were presented.

We firmly believe that, the material presented in this paper can have a significant impact on further research in the field of target detection in sea clutter and will provide the researchers in this field with valuable information.    



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[
    < g r a p h i c s >
]Shahrokh Hamidi was born in 1983, in Iran. He received his B.Sc., M.Sc., and Ph.D. degrees all in Electrical and Computer Engineering. He is with the faculty of Electrical and Computer Engineering at the University of Waterloo, Waterloo, Ontario, Canada. His current research areas include statistical signal processing, mmWave imaging, Terahertz imaging, image processing, system design,  multi-target tracking, wireless communication, machine learning, optimization, and array processing.




