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Censored harmonic averaging CFAR (CHA-CFAR) detector

The sliding window Constant False Alarm Rate (CFAR) detector compares the Cell Under Test (CUT), denoted as $x_{\textrm{CUT}}$, with an estimate of the background noise level, denoted as $g({x_{1}}, {x_{2}}, \ldots, {x_{N}})$, multiplied by a positive constant $\tau$. The aim is to make a decision on the presence or absence of a target in the CUT as below

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where the secondary data { ${x_1,\ldots,x_N}$ } is assumed to be independent and identically distributed (i.i.d.) random variables containing the background noise. The function $g(\cdot)$ maps the vector $[x_1, \ldots, x_N]$ to a non-negative real number. The hypothesis testing problem in target detection involves making a decision between $H_0$ (indicating that $x_{\textrm{CUT}}$ contains only noise measurements) and ${H_1}$ (indicating that $x_{\textrm{CUT}}$ includes a target component within the noise returns). The threshold multiplier $\tau > 0$ is used to control the required Probability of False Alarm ($P_{\textrm{fa}}$). By adjusting the value of $\tau$, the desired level of false alarms can be achieved. For brevity, we will use $g$ to represent $g({x_{1}}, {x_{2}}, \ldots, {x_{N}})$ throughout this text.

The CHA-CFAR detector was proposed as a new CFAR detector in [1], specifically designed for dense outlier situations. Unlike other methods that eliminate outliers through censoring (as seen in OS-CFAR and TM-CFAR), CHA-CFAR softens the effect of outliers by utilizing the harmonic mean and the Ordered Statistics (OS) principle to estimate the noise level, as shown in equation (2):

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where ${x_{(k)}}$ represents the $k$ th sample of ordered statistics. In CHA-CFAR detector, $k$ represents the number of eliminated secondary data with the smallest amplitudes. It is suggested to choose small values for $k$. In the following, $k$ is set to 8. Indeed, the function $g$ reduces the impact of outliers instead of rejecting them. Consequently, unlike other existing methods, the CHA-CFAR detector does not require prior knowledge of the number of outliers [1].

As mentioned, OS-CFAR and TM-CFAR detectors are based on the Ordered Statistics principle. Also, many CFAR techniques such as the weighted amplitude iteration (WAI)-CFAR detector use iterative algorithms. Some CFAR detectors are summarized in Table 1 [1]. The superiority of CHA-CFAR is illustrated through simulations and real Synthetic Aperture Radar (SAR) images in the following figures.

Table 1. Summary of Various CFAR detectors Unified in (1).

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In the following simulations, we use the results of the CA-CFAR detector in a homogeneous environment as an optimal bound, which called Ideal. Fig. 1 plots $P_{\textrm{d}}$ and $P{\textrm{fa}}$ of different detectors versus SCR (the ratio of target power or outliers power to the noise power). As expected, all detectors have the same performance in a homogeneous environment in Fig. 1 ($P_{\textrm{fa}} = 10^{-3}$ and $N = 32$).

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Fig 1. $P_{\textrm{d}}$ and $P_{\textrm{fa}}$ plot versus SCR in a homogeneous environment.

In the presence of outliers, the CHA-CFAR detector outperforms other methods without knowing the number of high-amplitude outliers and rejecting them. To demonstrate the robustness of the CHA-CFAR detector to a high number of outliers, 15 interfering targets are located in the reference window. As can be seen, the CHA-CFAR detector significantly outperforms others in Fig. 2. Although the performance of the WAI-CFAR detector is better than that of TM- and OS-CFAR detectors, it has a higher processing load. From the computational complexity point of view, the CHA-CFAR detector only needs to censor a few samples with the lowest amplitude, and any additional or iterative processing is not necessary to handle the outliers.

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Fig 2. $P_{\textrm{d}}$ and $P_{\textrm{fa}}$ plot versus SCR for 15 interfering targets.

We compare all detectors for a large vessel as an extended target using real SAR images in Fig. 3. As can be seen, the CHA-CFAR detector exhibits the best detection performance.

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Fig 3. Detection performance comparison on extended target. SLI VV polarization SAR image is applied with $P_{\textrm{fa}} = 10^{−3}$ on TerraSAR-X image for different detectors.

How to Adjust the CFAR Parameters of Detectors Using Monte Carlo

To set the thresholds for CFAR detectors using Monte Carlo, follow these steps:

  1. Open the main_interfering_target.m script and set the number of secondary data contaminated with outliers to zero (Number_Interfering_Target = 0).

  2. Set the values for the false alarm probability ($P_{\textrm{fa}}$) and the number of secondary data $N$. The default values are pfa = 10^-3 and window_size = 32 (Note that N = window_size).

  3. In the CFAR Parameters section of the script, adjust the variables to achieve the desired $P_{\textrm{fa}}$ for all detectors through trial and error.

After adjusting the CFAR parameters to achieve the desired $P_{\textrm{fa}}$, see all detectors have the same $P_{\textrm{fa}}$ when there are no outliers.

To further explore the detectors' behavior:

  • Use the interfering_target.m script to observe outliers by adjusting the Number_Interfering_Target variable and observe its effect on the false alarm probability ($P_{\textrm{fa}}$) and detection probability ($P_{\textrm{d}}$) of different detectors.

  • Employ the edge.m script to observe the impact of a sudden change in the power of background noise on the $P_{\textrm{fa}}$ and $P_{\textrm{d}}$ of different detectors.

  • Utilize the real_SAR_data.m script to examine the $P_{\textrm{d}}$ results of different detectors on real SAR data. Select the area and push submit button to execute the script (Fig. 4).

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Fig 4. The TerraSAR-X images is contains single-look complex images that were acquired from July 2008 to November 2009 over Barcelona, Spain.

Reference

[1] R. G. Zefreh, M. R. Taban, M. M. Naghsh, and S. Gazor, “Robust CFAR detector based on censored harmonic averaging in heterogeneous clutter,” IEEE Transactions on Aerospace and Electronic Systems, vol. 57, no. 3, pp. 1956–1963, Jun. 2021.