False-Alarm-Controllable Detection of Marine Small Targets via Improved Concave Hull Classifier

Abstract

In this paper, a new-brand feature-based detector via an improved concave hull classifier (FB-ICHC) is proposed to detect marine small targets. The dimension of feature space is suggested to be three, making a compromise between high detection accuracy and low computational cost. The main contributions are in the following two aspects. On the one hand, three features are well-designed from time series and Doppler spectrum, called relative phase zero ratio (RPZR), relative variation coefficient (RCV), and whitened peak height ratio (WPHR). RPZR can measure the pseudo-period properties in phase time series, insensitive to SCRs. In the Doppler spectrum, RCV reflects fluctuation variation in high SCR cases and WPHR describes the intensity property after clutter suppression in low SCR cases. On the other hand, in 3D feature space, an improved concave hull classifier is developed to further shrink the decision region, where a fast two-stage parameter search is designed for low computational cost and accurate control of false alarm rate. Finally, experimental results using open-recognized datasets show that the proposed FB-ICHC detector can improve detection performance by over 20% and reduce runtime by over 49%, compared with existing feature-based detectors with three features.

1. Introduction

For marine surveillance radars, small target detection has been an important and challenging task when they perform coast guard, port security, ship monitoring, search and rescue [1,2,3]. These small targets part from its concealment, such as speed boats, lifeboats, unmanned aerial vehicles, and armed frogmen, have weak radar cross sections (RCS) and are often submerged in strong sea clutter, resulting in a low signal-to-clutter ratio (SCR). To improve the low SCR, long-time integration technology is commonly used in radar signal processing, where integration time lasts from tenths of a second to several seconds. In such a long-time scale, high-resolution sea clutter exhibits complex characteristics of temporal nonstationarity, spatial inhomogeneity, and nonGaussianity. Therefore, it is required to develop detection methods to mitigate sea clutter and meet the requirements of low computational cost and high detection accuracy in real-time marine radars.

Currently, several methods have been introduced to find sea-surface small targets and they can be roughly classified into three types. First, the typical one is the well-known fractal-based method, where the sea clutter time series was proved to have fractal characteristics of roughness and irregularity. In [4], the Hurst exponent is extracted from the amplitude time series to form a single-feature-based detector. Similarly, multifractal properties are exhibited in the frequency domain, fractional domain, and spectrum domain [5,6]. In [7], three Hurst exponents extracted from amplitude time series, phase time series, and complex time series are condensed into an all-dimensional Hurst exponent for detection. It is noted that fractal-based detectors have a fast implementation, while the satisfied performance needs the integration time to last over several seconds. Second, the convolution neural network (CNN) method is used to distinguish small targets from sea clutter in 2D images. The main task is to find one image as the input and design the structure of CNN. Different input images can provide different information, such as PPI images, range-Doppler (RD) graphs, and time-frequency (TF) images [8,9]. For good performance, two images are jointly used for dual-channel convolutional neural networks (DCCNN), such as TF graph and amplitude time series [10], connected graph, and recurrence plot [11]. Further, a sequence of RD graphs from multi-frame returns is used [12]. In fact, the CNN-based methods need the amount of two types of balanced training samples and a suitable 2D image to yield a satisfactory performance. Due to the high computational cost of 2D images and commonly limited training samples of targets, it is not suitable for fast detection in real-time radar systems. The third, feature-based method has been widely recognized as an effective way, where differences between sea clutter and returns with the target are condensed into several features, and detection is declared in the feature space. The main difficulties lie in the feature extraction and design of the decision region. In [13], recurrence plots are developed to describe the nonstationary property of amplitude time series, and four measurements are given for detection. In the frequency domain, relative Doppler peak height (RDPH) and relative vector entropy (RVE) are designed to reflect the geometrical characteristics. The above two features plus relative average amplitude (RAA) form the early tri-feature-based detector, embarking the multi-feature framework in small target detection [14]. In [15], three new features from the median normalized Doppler amplitude spectrum are utilized to detect high-velocity small targets. In the TF domain, three TF features are extracted from the normalized TF image in intensity and geometrical properties [16]. In the polarization domain, scattering-intensity features based on the scattering decomposition models are developed [17,18]. Differently, three features are extracted from phase time series [19], offering a new perspective for feature extraction. In fact, the good performance of feature-based detectors lies in feature effectiveness and dimensional gain.

To further improve the detection probability, it naturally tends to exploit multiple complementary features. However, it brings new challenges to obtain the false-alarm-controllable decision region in the multi-dimensional feature space. In low-dimensional (LD) space, the traditional threshold is used for 1D and the decision region becomes space division in the 2D plane or 3D space. In the 3D space, a fast convex hull learning algorithm is used for novelty detection, where samples from sea clutter are surrounded by a convex hull at a given false alarm rate. Based on the convex hull, internal digging, and external filling form a concave hull in detection [20]. While their dimension limitation impedes the cooperation of more than three features. In high-dimensional (HD) space (more than three), there are two approaches. One is to condense the HD feature vector into the LD feature vector by dimensionality reduction methods. In [21], principal component analysis (PCA) is used to reduce the dimension of the feature vector from six to three. Similarly, a feature-compression-based detector utilizes the information from seven existing salient features by the maximum distance compression [22]. Feature compression is not the preferential choice at the cost of performance loss, though it can avoid the difficult design of the HD classifier. The other is to design a classifier in the HD space. In [23], a one-class support vector machine (OCSVM) is suggested to obtain the decision region, where the hyperparameter in SVM is constantly searched to meet the desired false alarm rate. In [24], the isolation forest algorithm with setting parameters selects the average path length as a test statistic for detection. Besides, the decision tree algorithm [25] and K nearest neighbor (KNN) [26] are used for the design of the HD decision region, where two balanced training samples are required and optimal parameters should be searched according to the given false alarm rate. In a word, the HD classifiers are much more complex and invisible. Thus, it is not an advisable way to continually increase the number of features to pursue performance and simultaneously ignore the difficulty in the design of HD classifiers. For example, three features with the smallest mutual information are chosen from eight features for fast detection [27].

This paper aims to propose a new feature-based detector to achieve high detection accuracy and low computational cost. In terms of high detection accuracy, three features are well-designed to describe the phase pseudo-period property, spectrum fluctuation, and maximum target spectrum after clutter suppression. In addition, an improved concave hull classifier replaces the concave hull classifier to further shrink the decision region. In terms of low computational cost, only the time domain and frequency domain are used without the TF domain and the number of features is limited to three for fast implementation. Besides, a two-stage parameter search is designed to reduce computational time, where a small search range and adaptive line search can quickly locate the suitable parameter. Finally, the proposed detector is testified by the real measured data from the IPIX database and CSIR database.

2. Extraction of Three Well-Designed Features and Analysis of Their Complementary

2.1. Description of Radar Detection Problem

To improve the detectability of small targets with low SCRs, long-time observation is required for detection, which can be implemented by the ubiquitous MIMO digital array radars or the marine radars at dwelling mode. In this case, received complex returns time series from N consecutive coherent pulses are jointly used for detection at each range cell. Then, the traditional detection problem of small targets is boiled down to the following binary hypotheses test [1,2,3]

$$\left\{\begin{array}{c}{\mathrm{H}}_0:\left\{\begin{array}{c}\mathbf{z}(n)=\mathbf{c}(n),n=1,2,\dots ,N;\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\\ {\mathbf{z}}_p(n)={\mathbf{c}}_p(n),p=1,2,\dots ,P.\hspace{0.33em}\end{array}\right.\hspace{0.33em}\hspace{0.33em}\hspace{0.17em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\\ {\mathrm{H}}_1:\left\{\begin{array}{c}\mathbf{z}(n)=\mathbf{s}(n)+\mathbf{c}(n),n=1,2,\dots ,N;\\ {\mathbf{z}}_p(n)={\mathbf{c}}_p(n),p=1,2,\dots ,P.\hspace{1em}\hspace{0.33em}\hspace{0.33em}\end{array}\right.\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\end{array}\right.$$

(1)

where z(n) and z_p(n) are received complex time series at a cell under test (CUT) and reference cells (RCs) around CUT, c(n) and c_p(n) are clutter time series at CUT and RCs respectively, s(n) is target returns, and P is the number of RCs. In Equation (1), the RCs at two hypotheses are from the sea clutter time series and it is assumed to have the same statistical probability and Doppler spectrum as the clutter time series at CUT. Thus, RCs can provide clutter information to suppress sea clutter at CUT.

When the observation time is up to the order of seconds, sea clutter and target have totally different characteristics from that at the short observation time of tens of milliseconds. Sea clutter time series exhibits time-varying texture and nonlinear Doppler offsets, deviating from the famous spherical invariant random vector (SIRV) with constant texture [28]. The numerous optimum or asymptotically optimum coherent detection methods that are developed under SIRV are not applicable anymore. In terms of the complicated movements and interaction with rough sea surface, target returns have fluctuated amplitude and nonlinear instant frequency curves. Therefore, it is difficult to establish sea clutter and target returns by simple parametric models. In this case, the traditional detection based on clutter distribution is converted into feature-based detection, where some cognitive differences are extracted from the received time series to distinguish sea clutter and returns with the target. These differences are called features, yielding the feature-based detection method. The features are independent of statistical distributions of clutter and targets, offering an effective approach for small target detection.

2.2. Feature Extraction in Time Domain

In the time domain, the amplitude difference is the most obvious feature. In [14], RAA is defined as the ratio of average amplitude at CUT and that at RCs, suitable for the high SCR cases. SCR is much lower for small targets, and they have heavy amplitude fluctuation in the time scale of the order of seconds, especially for sea-surface small floating targets. Therefore, it is not a wise choice to extract features from amplitude time series. In this way, a new feature is required to be insensitive to SCR.

Inspired by a phase-based detector [19], it is meaningful and feasible to utilize the phase time series from complex time series. The phase time series p(n) is calculated by

$$\mathbf{p}(n)=\left\{\begin{array}{l}\arctan \left\{{\displaystyle \frac{\mathrm{Im}[\mathbf{z}(n)]}{\mathrm{Re}[\mathbf{z}(n)]}}\right\},\mathrm{Re}[\mathbf{z}(n)]>0\\ \pi +\arctan \left\{{\displaystyle \frac{\mathrm{Im}[\mathbf{z}(n)]}{\mathrm{Re}[\mathbf{z}(n)]}}\right\},\mathrm{Re}[\mathbf{z}(n)]<0,\mathrm{Im}[\mathbf{z}(n)]>0\\ \arctan \left\{{\displaystyle \frac{\mathrm{Im}[\mathbf{z}(n)]}{\mathrm{Re}[\mathbf{z}(n)]}}\right\}-\pi ,\mathrm{Re}[\mathbf{z}(n)]<0,\mathrm{Im}[\mathbf{z}(n)]<0\\ -{\displaystyle \frac{\pi }{2}},\mathrm{Re}[\mathbf{z}(n)]=0,\mathrm{Im}[\mathbf{z}(n)]<0\\ {\displaystyle \frac{\pi }{2}},\mathrm{Re}[\mathbf{z}(n)]=0,\mathrm{Im}[\mathbf{z}(n)]>0\end{array}\right.,n=1,2,\dots ,N$$

(2)

where Re[.] and Im[.] represent the real part and imaginary part respectively. The values of p(n) range from −π to π. Figure 1 plots the phase time series during 0.512 s from one IPIX radar dataset [29]. It is found that both sea clutter and returns with target exhibit pseudo-periodic properties in the phase time series. While their frequencies are obviously different. Sea clutter has a high frequency with a fast-varying property, due to its phase angle randomly distributed in the whole phase space. Inversely, returns with targets have a low frequency with a slow-varying property.

To quantitatively measure these geometrical differences, two indexes are proposed to describe the pseudo-periods. One is the number of crossing zeros (NCZ), highly related to the periods. It is calculated by

$$\mathrm{NCZ}=count\left\{p(n)\times p(n+1)<0,n=1,2,\dots ,N-1\right\}$$

(3)

where count(.) is to count the number. The other is called the maximum interval (MI) between consecutive two crossing zeros and it is given by

$$\mathrm{MI}=\max \left\{\left|t_0(i+1)-t_0(i)\right|,i=1,2,\dots ,\mathrm{NCZ}-1\right\}$$

(4)

where t₀ is the time of corresponding crossing zero. In Figure 1, for sea clutter, the NCZ is 161 and MI is 12 ms. For returns with the target, NCZ sharply reduces to 20, but MI increases to 98 ms. Since both NCZ and MI are equipped with the ability to distinguish sea clutter and returns with the target, a fused feature called phase zero ratio (PZR) is defined by

$$\mathrm{P}Z\mathrm{R}=\mathrm{MI}/\mathrm{NZ}$$

(5)

Sea clutter has a small value of PZR because the MI is small and NR is great. While returns with target have the inverse values.

Considering the complex interaction of electrical waves and rough sea surface [28], dynamic sea clutter often exhibits local homogeneity and global heterogeneity. The values of PZR vary with the space and it is meaningless to compare the PZRs from different spaces. In this way, to delete this influence, the relative PZR (RPZR) is computed by

$$\mathrm{RPZR}\left(z\right)={\displaystyle \frac{\mathrm{PZR}\left(z\right)}{{\displaystyle {\sum }_{p=1}^P\mathrm{PZR}\left(z_p\right)}/P}}$$

(6)

Figure 2 plots the histogram distributions of RPZR. It is found that the main values of RPZR for sea clutter focus around one, which is consistent with the fact that sea clutter at CUT and at RCs almost have the same characteristics in the local region. For returns with the target, most values of RPZR are greater than one and its mean value is 2.4. It is much easier to distinguish returns with target from sea clutter when the value of RPZR is far away from one.

2.3. Feature Extraction in Frequency Domain

Relative to the time domain, the frequency spectrum can offer more effective information in detection. In the frequency domain, the amplitude Doppler spectrum is defined by

$$Z(f_d)={\displaystyle \frac{1}{\sqrt{N}}}\left|{\displaystyle \sum _{n=1}^{N}z(n)\exp \left(-j2\pi n{f}_d/f_r\right)}\right|,f_d\in \left[-{\displaystyle \frac{f_r}{2}}, {\displaystyle \frac{f_r}{2}}\right]$$

(7)

where f_r is the pulse repetition frequency (PRF). Figure 3a shows the Doppler spectrum of sea clutter when N = 512. It has a non-uniform spectrum with a wide main clutter region, ranging from −0.2 Hz to 0.1 Hz in the normalized frequency. Differently, the target has a narrow band around the zero normalized frequency, as shown in Figure 3b. Owing to the high SCR, the peak of the target is much higher than that of sea clutter, which has a great effect on the shape of the spectrum. Then, spectrum fluctuation from returns with the target is much heavier than that of sea clutter. This difference can be condensed into an appropriate feature and the next step is to design an effective feature.

To describe the difference in spectrum fluctuation, coefficient variance (CV) is presented by the ratio of standard deviation to the average.

$$\mathrm{CV}\left(z\right)={\displaystyle \frac{\sqrt{\frac{1}{N-1}{\displaystyle {\sum }_{f_d}{\left(Z\left(f_d\right)-{\displaystyle {\sum }_{f_d}Z\left(f_d\right)/N}\right)}^2}}}{{\displaystyle {\sum }_{f_d}Z\left(f_d\right)/N}}}$$

(8)

Then, relative coefficient variance (RCV) is given by

$$\mathrm{RCV}\left(z\right)={\displaystyle \frac{\mathrm{CV}\left(z\right)}{{\displaystyle {\sum }_{p=1}^P\mathrm{CV}\left(z_p\right)}/P}}$$

(9)

In the H₀ hypothesis, spectrum fluctuation solely results from sea clutter. When a target exists, the Doppler spectrum has great variation from both sea clutter and the target. Certainly, great RCV reflects the presence of targets with a high probability. In Figure 3c,d, most RCV values of sea clutter are around one and those of returns with targets are around 1.5. At the high SCR cases, the values of RCV from returns with target become great, which means good performance in detection.

Currently, RDPH in [14] can find small targets in the Doppler spectrum by calculating the ratio of the peak value to the surrounding local average values, suitable for high SCR cases. For the low SCR cases in Figure 4a, RDPH can only find the peak of sea clutter and result in completely missing the target. To deal with the low SCR cases, whitened spectrum (WS) is defined by the ratio of spectrum at CUT to the average spectrum at RCs

$$\mathrm{WS}(f_d)={\displaystyle \frac{Z(f_d)}{{\displaystyle {\sum }_{p=1}^P{Z}_p(f_d)}/P}}$$

(10)

In Figure 4b, the weak target is totally highlighted in the WS, and sea clutter is heavily mitigated with the main clutter disappeared. In this way, targets and sea clutter have greater differences in intensity than those in the Doppler spectrum. It should be noted that the fundamental premise for successful clutter suppression lies in the homogeneity assumption, where RCs and CUT have the same statistical properties of sea clutter. When sea clutter at RCs exhibits strong spatial non-stationarity, it may violate the homogeneity assumption and reduce its ability to suppress clutter at CUT.

To reflect this property quantitatively, a whitened peak height ratio (WPHR) is proposed by

$$\mathrm{WPHR}\left(z,z_p\right)={\displaystyle \frac{\underset{f_d}{\max }\left\{\mathrm{WS}\left(f_d\right),f_d\in \left[-f_r/2, f_r/2\right]\right\}}{\frac{1}{\#\Theta }{\displaystyle {\sum }_{f_d\in \Omega }\mathrm{WS}\left(f_d\right)}}}$$

(11)

where #Θ is to fix the number of elements in set Θ. The set Θ is given by

$$\begin{array}{l}\Theta =\left[-f_r/2,f_{\max }-\Delta ]\cup [f_{\max }+\Delta ,f_r/2\right]\\ f_{\max }=\arg \underset{f_d}{\max }\left\{\mathrm{WS}\left(f_d\right)\right\}\end{array}$$

(12)

where f_max is the corresponding frequency at the peak spectrum and Δ is the protected region around f_max. Different from RDPH in [14], WPHR can both work in the cases of high SCRs and low SCRs, when the homogeneity assumption holds. Figure 4c,d plots the histograms of WPHR when N = 512. Sea clutter has smaller values of WPHR than returns with the target, where their mean values are 1 and 15 respectively. In a word, all three features have the separating capacity in phase geometric characteristics, fluctuation variation, and intensity ratio after clutter suppression.

2.4. Complementarity Analysis of Three Features

In current feature-based detectors [14,15,16,17,18,19], their performances highly depend on the effectiveness of features and the number of features. The former aims to extract well-designed features from the time domain, frequency domain, time-frequency domain, or other transformed domains. It is required that the designers are equipped with good knowledge of physical scattering schemes and the ability to find the differences between sea clutter and returns with the target. The latter is to increase the number of features since it is commonly recognized that an increase in the number of features can improve the detection probability. Then, many works focus on the extraction of more features from different domains [21,22,23,24,25,26]. However, good performance of dimension improvement lies in the information complementarity from different features. Otherwise, the performance improvement from high dimensions is limited at a high computational cost. In this way, it is necessary to analyze the complementarity of the proposed three features.

First, the correlation between each two features is discussed. A high correlation implies that joint exploitation may offer limited performance improvement. Pearson correlation is a statistic that measures the degree of linear relationship between two variables, with a range between −1 and 1. Assume that the two variables are denoted by x and y. The correlation coefficient is given by

$$\rho (x,y)={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^M\left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)}}{\sqrt{{\displaystyle {\sum }_{i=1}^M{\left(x_i-\overline{x}\right)}^2}{\displaystyle {\sum }_{i=1}^M{\left(y_i-\overline{y}\right)}^2}}}},\overline{x}={\displaystyle \frac{1}{M}}{\displaystyle {\sum }_{i=1}^M{x}_i},\overline{y}={\displaystyle \frac{1}{M}}{\displaystyle {\sum }_{i=1}^M{y}_i}$$

(13)

where M is the total number of variables x and y. To better describe the relationship between the three features, the correlation coefficient matrix is introduced by

$$R_{\rho }=\rho \left({\xi }_i,{\xi }_j\right),i,j=1,2,3$$

(14)

where ${\xi }_1,{\xi }_2,{\xi }_3$ represent variables of the RPZR, RCV, and WPHR respectively. When the absolute correlation coefficient is close to 0, the correlation between two variables is weak. Concretely, it can be classifier into five intervals of [0, 0.2], [0.2, 0.4], [0.4, 0.6], [0.6, 0.8], [0.8, 1] and the corresponding relationships are very weak correlation or no correlation, weak correlation, moderate degree of correlation, strong correlation, and very strong correlation. In Figure 5a, RPZR is independent of the other two features, since feature extraction from the different domains. There is little weak correlation between the RCV and WPHR, due to the same frequency domain.

Next, mutual information [27] is introduced to measure the dependence between two features. Relative to the Pearson correlation, it can capture broader relationships of linear and nonlinear relationships. The expression is given by

$$I(\mathrm{x},y)=H(x)+H(y)-H(x,y)={\displaystyle \sum _{x,y}p(x,y){\log }_2\left(\frac{p(x,y)}{p(x)p(y)}\right)}$$

(15)

where H(x) and H(y) are the edge entropy, H(x,y) is the combination entropy, p(x), p(y), p(x,y) are the probability density functions. The lower bound of mutual information is 0. While its upper bound is not 1, with a smaller value of the entropy of the two variables. In this way, normalized mutual information (NMI) is calculated by

$$\mathrm{NMI}(\mathrm{x},y)=={\displaystyle \frac{2\times I(x,y)}{H(x)+H(y)}}$$

(16)

When the two features are completely independent, the value of NMI is 0. In Figure 5b, the values of NMI between any two features are almost zeros. Therefore, it can be summarized that any two features from the proposed three features can offer complementary information without mutual information.

Finally, we test the detection performance of each feature in different clutter environments. In Figure 5c, each feature serves as one detector called an RPZR-based detector, RCV-based detector, and WPHR-based detector. There are forty dataset labels of ten datasets with four polarizations and the detailed information about datasets refers to Section IV. RPZR-based detector achieves the best overall performance, followed by WPHR-based detector and RCV-based detector. However, it is found that no detector maintains the highest detection probability on all datasets. Though the three detectors all have low or high detection probabilities at some datasets, such as the 4th and 10th labels, they indeed have different moving trajectories during the whole observation time. It implies that three features are indeed complemented and can be jointly used in 3D space for high detection accuracy.

3. Novelty Detection Using Improved Concave Hull Classifier in 3D Feature Space

3.1. Novelty Detection in 3D Feature Space

After feature extraction, the received complex time series at CUT and RCs are condensed into three features. Considering their complementarity, a three-dimensional (3D) vector is constructed as the test statistic.

$$\xi ={\left[\mathrm{RPZR}\left(z,z_p\right),\mathrm{RCV}\left(z,z_p\right),\mathrm{WPHR}\left(z,z_p\right)\right]}^T,\xi \in {\mathrm{R}}^3$$

(17)

where ${\mathrm{R}}^3$ represents 3D feature space. In this way, the next step is to determine which hypothesis this 3D vector belongs to. According to Equation (1), a traditional detection problem is transformed into a classification problem in 3D feature space. It is given by

$$\left\{\begin{array}{c}{\mathrm{H}}_0:\xi ={[{\xi }_1\left(c,c_p\right),{\xi }_2\left(c,c_p\right),{\xi }_3\left(c,c_p\right)]}^T\hspace{0.17em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\\ {\mathrm{H}}_1:\xi ={[{\xi }_1\left(s+c,c_p\right),{\xi }_2\left(s+c,c_p\right),{\xi }_3\left(s+c,c_p\right)]}^T\hspace{0.33em}\end{array}\right.$$

(18)

It belongs to a single classification or binary classification. In binary classification, both samples from sea clutter and returns with the target are required for training binary classifiers, where a 3D vector is one sample. However, due to the spatial sparsity of targets and the diversity of target types, it is generally difficult to collect numerous samples from returns with targets. Differently, large amounts of sea clutter are variable from the current environment once marine radars start to work. In this way, the serious imbalance between two kinds of sample sizes can result in great performance loss in training optimal parameters of binary classifiers, though it can bring more information in classification than single classification. Therefore, a single classification is more suitable for radar detection. This is also called novelty detection or abnormal detection [14,25].

In novelty detection, only one kind of sample is positive samples and the others are negative samples. Due to the easy acquisition of sea clutter, samples from sea clutter are often regarded as positive samples. In 3D feature space, positive samples can provide a decision region at a given false alarm rate. The target is declared if the sample falls outside the decision region. According to the Neyman Pearson criterion, it is required to maximize the detection probability at a given false alarm rate P_fa,

$$\max \left\{1-{\displaystyle {\iiint }_{\Omega }p\left(\xi \left|{\mathrm{H}}_1\right.\right)}d\xi \right\}, \mathrm{s}.\mathrm{t}. {\displaystyle {\iiint }_{\Omega }p\left(\xi \left|{\mathrm{H}}_0\right.\right)}d\xi =1-P_{fa}$$

(19)

where $\Omega \in {\mathrm{R}}^3$ is the decision region, $p\left(\xi \left|{\mathrm{H}}_0\right.\right),p\left(\xi \left|{\mathrm{H}}_1\right.\right)$ are the probability density functions (PDF) of vectors under the H₀ hypothesis and H₁ hypothesis respectively. When their PDFs are known, the detection probability is determined by the decision region. The smaller the decision region is, the higher the detection probability is. However, PDFs are unknown and hardly available. Thus, the main aim is to minimize the decision region at unknown PDFs. Assume that the positive samples are uniformly distributed in the whole 3D space. The minimality of the decision region is equivalent to minimizing the volume of the decision region. It satisfies that

$$\underset{\Omega }{\min }\left|\Omega \right|, \mathrm{s}.\mathrm{t}. {\displaystyle {\iiint }_{\Omega }p\left(\xi \left|{\mathrm{H}}_0\right.\right)}d\xi =1-P_{fa}$$

(20)

where |Ω| is to have a volume of Ω. Due to the complex extraction of features and the three dimensions, it is difficult to obtain an analytical expression for Equation (20). In addition, the decision region is highly related to the designed algorithm. Therefore, we must use the Monte Carlo method to achieve the asymptotically optimal solutions or optimal solutions.

Figure 6 gives the flow diagram of the feature-based detector using an improved concave hull classifier (FB-ICHC). It contains an online detection branch and an offline training branch. In the online detection branch, both observation vectors at CUT and RCs are first utilized to extract three features in the time domain and in the frequency domain. Next, three features form a 3D feature space, and it functions as the test statistic. Finally, when the test statistic falls outside the decision region, it declares that CUT belongs to the H₁ hypothesis. In the offline training branch, large amounts of sea clutter are collected to extract features. The positive samples of 3D vectors are trained for the ICHC with a controllable false alarm rate, which is provided for the detection branch.

3.2. Improved Concave Hull Classifier with Controllable False Alarm Rate

With the help of the well-known Monte-Carlo method, the asymptotically optimal solutions to Equation (20) can be given using some single classifiers. Assume that numerous samples are collected from sea clutter and the total number of samples is M. All samples of 3D vectors form a training set, denoted by $A=\left\{{\xi }_1,{\xi }_2,\dots ,{\xi }_M\right\}$. At a given P_fa, the number of false alarms is $[M\times P_{fa}]$, where [.] is the integer operator. Then, the main task is to have a decision region Ω with the minimized volume

$$\underset{\Omega }{\min }\left|\Omega \right|, \mathrm{s}.\mathrm{t}{. \mathrm{A}}_0=\left\{{\xi }_i,i=1,2,\dots ,M-[M\times P_{fa}]\right\}\in \Omega$$

(21)

Only $[M\times P_{fa}]$ samples in A fall outside the decision region Ω, which should hold in the designed single classifier.

While, the traditional single classifiers, such as the convex hull learning algorithm [14], OCSVM [23], and isolation forest (iForest) [24], do not consider the false alarm rate in their structures. There are mainly two approaches to modify the existing single classifiers with controllable false alarm rate. One is a direct approach. In the convex hull learning algorithm, desired false alarm samples are continually excluded by shrinking the decision region with the minimum volume. The other is an indirect approach. Because there is an implicit functional relationship between one parameter and the false alarm rate, a specified parameter of the single classifier is selected to control the false alarm rate. In OCSVM, a proper hyper-parameter is globally searched to reach a given false alarm. On the whole, the direct approach has an accurate control of the false alarm rate, while it is often difficult to eliminate samples one by one in a visual way. Differently, the indirect approach is simple in theory, while a search of specified parameters may result in a small deviation in the false alarm rate. Therefore, how to accurately control the false alarm rate becomes the focus and difficulty in the design of a single classifier.

3.2.1. Traditional α-Shape Concave Hull Algorithm

In theory, a concave hull is bound to have a smaller volume than a convex hull. To further minimize the decision region, the α-shape concave hull algorithm is introduced in 3D space [30]. The final concave hull is determined by the input training set and a unique sphere of radius parameter. Its fundamental idea is to roll a small sphere of radius α over a set of samples. The triangular surfaces formed by the three points intersecting the sphere are the final concave hull, which is also called the rolling sphere method. The specific steps of the traditional α-shape concave hull algorithm are as follows.

Step 1: Input data

Assume that the input data is training set $A=\left\{{\xi }_1,{\xi }_2,\dots ,{\xi }_M\right\}$ and a sphere of the radius is α.

Step 2: Determine the local region

For any sample ξ_i, find the local region B_i when the samples have a distance less than 2α to ξ_i,

$$B_i=\left\{\left|{\xi }_i-{\xi }_j\right|<2\alpha ,j=1,2,\dots ,M,j\ne i\right\}$$

(22)

where |.| is to calculate the modulus of a vector, equivalent to finding the Euclidean distance between two vectors.

Step 3: Search the boundary samples

In B_i, choose a pair of samples (ξ_ij, ξ_ik). By passing the three samples ξ_i, ξ_ij, ξ_ik, two spheres are created with a radius of α and their centers are denoted by c₁ and c₂. When the distances of the remaining samples in B_i to any one center are greater than the radius α, the three samples are recognized as boundary samples, and the search is terminated. Otherwise, repeat searching for another pair of samples until all the samples in B_i are involved.

Step 4: Output the concave hull

Repeat steps 2 to 3 until all samples in training set A are covered. Finally, the boundary samples can be joined in pairs to generate triangular surfaces. All the triangular surfaces are connected to form a concave hull.

Figure 7 shows the generation of a concave hull when the sphere of radius α has different values. The green area represents the decision region generated from sea clutter samples, and the blue dots indicate false alarm points. As the sphere of radius α becomes larger, the volume of the concave hull is greater. If α tends to infinite, the concave hull degrades into a convex hull. That is, a convex hull is a special case of a concave hull with infinite radius. In the 3D feature space, the sample falling outside the concave hull is declared as a negative sample, which is a false alarm sample. The number of false alarm samples is 16, 8, and 2 respectively when α = 0.8, 1, and 3. Besides, there is a monotone decreasing relationship between parameter α and P_fa, though it is often difficult to their analytical expression. Therefore, the false alarm rate can be controlled by finding a suitable parameter α.

3.2.2. Improved Concave Hull Classifier Using Two-Stage Parameter Search

When the training set from sea clutter is available, the main task is to have a concave hull decision region at a given false alarm rate using the traditional α-shape algorithm. The main difficulty is how to find the suitable parameter of sphere radius α according to the false alarm rate P_fa. Since the theoretical function is difficult to derive, the false alarm rate is calculated by

$$P_{fa}={\displaystyle \frac{N_{pf}}{M}}={\displaystyle \frac{\mathrm{count}\left\{{\xi }_i\notin Concave\left(A,\alpha \right)\right\}}{M}}$$

(23)

where N_pf is the number of false alarm samples, Concave(A, α) is to generate a concave hull at input training set A and a given sphere radius α.

A direct method is to globally search the suitable sphere radius α in radar target detection. First, a range of parameter α is determined. The left boundary is set to 0. The right boundary of a large value ensures all the samples in the concave hull, which is calculated by finding the farthest distance of the samples in training set A to its center. Then, the method of bisection is used to fast search for the suitable parameter α. The global search will encounter two problems high computing costs and rough control of false alarms. The former results from a large range of searches, though the method of bisection is utilized to accelerate the parameter search. The latter comes from the discreteness of the false alarm sample number and the continuity of the parameter.

To solve the above two problems, an improved concave hull classifier (ICHC) is proposed and its main innovations focus on the control of false alarm rate from the refined control and low time cost. In terms of low time cost, a large range of searches is replaced by a small range of parameters to save search time, and the method of bisection is substituted by the method of adaptive line search to fast locate the suitable parameter. In terms of refined control of false alarms, a two-stage search is specially designed. The first stage is a rough search and extra false alarm samples are added to prevent missing the accurate value. The second stage is a refined search to remove the extra false alarm samples. The second stage is to make up for the first stage when the first stage cannot reach the given number of false alarm samples.

Below, the detailed steps of ICHC are given as follows.

Step 1: Input data

Assume that the input data is training set $A=\left\{{\xi }_1,{\xi }_2,\dots ,{\xi }_M\right\}$, false alarm rate P_fa, and extra false alarm samples s = 1 or 2.

Step 2: Initialize a small range of α

Because the shape of the concave hull is determined by the boundary samples, the first stage is to determine the boundary samples located in the training set A. The convex hull learning algorithm is used to obtain the boundary samples to form a boundary set, denoted by $BS=\left\{{\xi }_i=ch\left(A\right),i=1,2,\dots ,Q\right\}$.

In the α-shape concave hull algorithm, the value of the sphere radius is highly dependent on the distance between boundary samples. In this way, the mean K-nearest neighbors of each boundary sample are calculated and the minimum and maximum values serve as the small search range

$$\alpha \in [{\alpha }_{\mathrm{L}}=\min ({\overline{d}}_i),{\alpha }_{\mathrm{R}}=\max ({\overline{d}}_i)],{\overline{d}}_i=\left\{{\displaystyle \frac{1}{K}}{\displaystyle \sum _{k=1}^K{d}_k},i=1,2,\dots ,Q\right\}$$

(24)

where α_L and α_R are the left boundary and right boundary, $d_j=\left|{\xi }_i-{\xi }_j\right|,j=1,2,\dots Q,i\ne j$ and it is arranged in an ascending order $d_1\le d_2\le \dots \le d_{Q-1}$.

Step 3: Count the number of false alarms

When the training set A and sphere radium α are given, a concave hull is generated by the α-shape algorithm. Then, establish two concave hulls of A with α_L, α_R respectively. Using Equation (23), the corresponding number of false alarm samples are M_L and M_R.

Step 4: Search in the first stage

It is found that there is a monotone decreasing relationship between parameter α and P_fa. Because it is hard to derive a specific functional relationship between α and P_fa, a simple linear function is assumed as prior information to implement adaptive search in a fast way. So, an adaptive parameter α is selected by

$${\alpha }_{adaptive}={\alpha }_R-{\displaystyle \frac{[M\times P_{fa}]-M_R}{M_L-M_R}}\left({\alpha }_R-{\alpha }_L\right)$$

(25)

Then, a new concave hull is established at a given α and training set A. Using (23), the number of false alarm samples is $N_{pf}=count\left\{{\xi }_i\notin Concave\left(A,{\alpha }_{adaptive}\right)\right\}$.

When $N_{pf}=\left[M\times P_{fa}\right]$, go to step 6. If $N_{pf}<\left[M\times P_{fa}\right]$, set ${\alpha }_R={\alpha }_{daptive}$. If $N_{pf}>\left[M\times P_{fa}\right]+s$, set ${\alpha }_L={\alpha }_{daptive}$. Return to the start of step 4 until $\left[M\times P_{fa}\right]<N_{pf}\le \left[M\times P_{fa}\right]+s$. Finally, the corresponding ${\alpha }_{adaptive}$ is the output of a rough search.

Step 5: Search in the second stage

For the ${\alpha }_{adaptive}$, the training set is split into two parts normal samples and false alarm samples. By selecting one from false alarm samples, the smallest concave hull containing the sample and normal samples is established. Then, the smallest volume of concave hulls is found by going through all the false alarm samples and the corresponding false alarm sample is regarded as a normal sample. When the current number of false alarm samples is equal to $[M\times P_{fa}]$, this step stops.

Step 6: Output the decision region

Finally, the decision region without false alarms is determined, denoted by Ω.

Figure 8 plots an example of two decision regions. The green area represents the decision region generated from sea clutter samples, the blue points indicate the locations of false alarm points, and the red points represent the samples from returns with the target. The total number of training samples is 10,210 and the false alarm rate is 0.001. Thus, 10 samples from sea clutter outside the decision region, are regarded as false alarm points. It is clearly found that the ICHC can offer a smaller decision region than the convex hull. Their volumes are 20 and 53 respectively. In the decision judgment, a smaller volume of decision region can improve the detection probability.

3.3. Uniqueness of Concave Hull Decision Region

In theory, a convex hull is unique from different algorithms. However, different algorithms will generate different concave hull decision regions at the same training set. In other words, a concave hull is not unique in theory, which has a great influence on detection probability. At the same training set, the shape of decision regions may be different for different concave hull algorithms. However, whether the shape of the decision region is unique for the same concave hull algorithm. In [31], it is mentioned that the α-shape concave hull is uniquely determined by the input set and sphere radium parameter α. For the proposed ICHC, it is meaningful and necessary to research the uniqueness of decision regions and false alarm samples.

For a convex hull classifier, it is determined by the boundary samples and the triangular surfaces. In this way, the main index to measure the similarity of two decision regions is the sum of the distance between the nearest two boundary samples (SDBS). It is defined by

$$\mathrm{SDBS}\left(A\right)={\displaystyle {\sum }_{i=1}^Q\left|x_i-y_i\right|}$$

(26)

where x_i and y_i are the corresponding boundary samples in the two concave hull decision regions. When SDBS is zero, two concave hull decision regions have the same boundary samples.

The second index is to measure the consistency of false alarm samples outside two concave hull decision regions. The sum of the distance between the nearest two false alarm samples (SDFA) is calculated by

$$\mathrm{SDFA}\left(A\right)={\displaystyle {\sum }_{i=1}^{[M\times P_{fa}]}\left|p_i-q_i\right|}$$

(27)

where p_i and q_i are the corresponding samples to trigger false alarms in the two concave hull decision regions respectively. When SDFA is zero, two concave hull decision regions have the same false alarm samples.

Figure 9 shows the demonstration of concave hull decision regions at different runs. When the number of runs is 0, a decision region using the proposed ICHC is first generated and the false alarm samples are distinguished, which serves as the benchmark for verification. Then, the decision region is constantly reproduced using the same training sample as the number of runs increases. Two indexes can be calculated by comparing the new decision region and the benchmark. It is found that two indexes keep zeros and volumes of decision regions are the same as the number of runs increases from 1 to 10. Therefore, it can be concluded that the proposed ICHC can offer a unique concave hull decision region at the same training samples.

4. Experimental Results and Evaluation

4.1. IPIX Database

It has been well-known that IPIX datasets are open datasets for the verification of small target detection [29]. Ten datasets are collected by IPIX experimental radar deployed on the east coast of Canada in 1993, where the range resolution is 30 m, the PRF is 1000 Hz and four polarizations of HH, HV, VH, and VV are available. The wind speeds and significant wave heights are all given in [16,27]. The test target is a ball wrapped in wire, with a diameter of 1 m. Being anchored in the sea, the target rises and falls with the waves.

To analyze the measured data properties, average SCR in the time domain and a Doppler distance factor (DDF) are discussed. The former describes the intensity difference of sea clutter and returns with the target. The latter evaluates the degree of separation between the clutter spectrum and the target spectrum. It is defined by

$$\mathrm{DDF}=1-{\displaystyle \frac{{\displaystyle {\int }_{-f_r/2}^{f_r/2}{P}_{\mathrm{s}}(f_d)P_c(f_d)}d{f}_d}{\sqrt{{\displaystyle {\int }_{-f_r/2}^{f_r/2}{P}_{\mathrm{s}}^2(f_d)}d{f}_d{\displaystyle {\int }_{-f_r/2}^{f_r/2}{P}_c^2(f_d)}d{f}_d}}}$$

(28)

where P_c and P_s are average Doppler power spectrum (ADPS) from clutter and targets. A great value of DDF implies that two ADPS are separate, where Doppler shifts of targets fall outside the main clutter region.

Figure 10 lists the average SCRs and DDFs in the ten IPIX datasets at four polarizations. The average SCRs have a large range from −2 dB to 18 dB, where cross-polarizations of HV and VH have higher SCRs than like-polarizations of HH and VV. The values of average SCR are smallest at VV polarization, and it may result in the worst performance in detection. The latter five datasets have greater values of DDF than the former five datasets. Among the former five datasets, the Doppler shifts of test targets fall inside the clutter regions in a great probability, especially the #30 dataset. In this way, it can be roughly classified into four groups. The first group is #17, #54, #311, and #320, with high SCRs and great DDFs. The second group is #280, #310 with low SCRs and great DDFs. The third group is #26, #30, and #31 with low SCRs and small DDFs. The fourth group is #40 with high SCRs and small DDFs. Besides, the ten IPIX datasets cover the sea state from 2 to 4. Therefore, it can provide a variety of clutter environments for a reliable performance evaluation.

4.2. Influence of Decision Regions and Features

In fact, decision regions and features have a great influence on detection performance. A smaller volume of decision regions and better features yield a greater detection probability. In Figure 11, the convex hull classifier and FB-ICHC detector use the same three features for fairness. Their average detection probabilities are 0.83, 0.86, 0.86, 0.78, and 0.76, 0.80, 0.80, and 0.71 at HH, HV, VH, and VV polarizations. Thus, relative to the convex hull classifier, ICHC can improve overall average performance by 8% and individual performance by up to 22%. In terms of features, TF features [16], phase features [19] and proposed features are compared using the same ICHC for decision region. From the 10 datasets at four polarizations, the proposed features consistently outperform the others, with an overall average improvement of 11% and 13%. The superiority of performance results from the complementarity among the proposed three features. RPZR offers phase geometrical information and ensures it is insensitive to SCR. RCV is designed for high SCR cases and WPHR is for low SCR cases when RCs have the same statistical properties of clutter as CUT.

In real radar detection, the time for detection is often limited and it is required for fast implementation. In this way, we testify to the runtime of the proposed ICHC. The laptop computer is used with the Intel i9-13900HX CPU, 32G RAM, NVIDIA GeForce RTX 4070 laptop GPU, and MATLAB 2022b software in Windows 11. Figure 12 shows the training time for the concave hull with a controllable false alarm rate. The total number of training samples is 10,240. The first algorithm is the method of bisection for the suitable parameter in a wide range, according to the given false alarm rate. Based on the bisection search, the second algorithm has a small range using the prior knowledge of boundary samples. The third algorithm proposed in this paper further uses the adaptive search to quickly find the suitable parameter. Narrowing the boundary range of the α parameter cannot significantly reduce the training time when the other settings are the same. However, the adaptive search method can lead to a significant reduction in training time. The average training time for the bisection method across all data sets is 0.83 s, while the average runtime for the adaptive search method is 0.227 s, improving efficiency by over 70%. It demonstrates that the proposed ICHC can indeed establish the concave hull decision region for controlling false alarms more efficiently.

4.3. Performance Comparison Using Real Datasets

Due to the limitation of samples, observation vectors are generated by a sliding window of the length 512 and a sliding interval of 128 at each range cell, where the observation time corresponds to 0.512 s. The false alarm rate P_fa is set 10⁻³ and the number of reference cells P is 9.

Figure 13 gives the detection results at ten IPIX datasets. Among the six detectors, the all-dimensional Hurst exponent (ADHE) detector [7] has the worst performance with average probabilities of 0.516, 0.577, 0.586, and 0.474 at HH, HV, VH, and VV polarization. The main reason lies in the poor ability of a single feature. At the same time, the fractal feature only utilizes the time series to discriminate sea clutter and returns with target and the observation time is far away from several seconds. Differently, the tri-feature-based detector [14] improves average probabilities of 0.525, 0.597, 0.597, and 0.515 respectively at each polarization, owing to the combination of three features. However, relative to the ADHE detector [7], the tri-feature-based detector [14] has comparable performance, and improvement from dimension is limited. This is because the three Hurst exponents extracted from complex time series, amplitude time series, and phase time series are fused into a single feature. In addition, the good performance of the tri-feature-based detector [14] heavily depends on the high SCRs, since the first group datasets with high SCRs obtain great improvement. Though Doppler shifts of targets fall outside the main clutter region in the second group datasets of #280, and #310, the tri-feature-based detector [14] still can hardly find the small target at the low SCR cases. The phase-feature-based detector [19] has the corresponding average probabilities of 0.630, 0.693, 0.683, and 0.615, with a total performance improvement of 21% and 17% compared to the previous two detectors, respectively. It is due to the three-phase features extracted from phase time series and little relationship with SCRs, where datasets with low SCRs still have high detection probabilities, especially in #280, and #310. It implies that the phase feature also plays an important role in detection when the clutter environments are complicated and varied.

For the TF-feature-based detector [16], it attains the average detection probabilities of 0.702, 0.707, 0.700, and 0.615 at HH, HV, VH, and VV polarization. Relative to the phase-feature-based detector [19], it still has a further performance improvement of 11% at HH polarization. In the first and second group datasets, good performance of TF features mainly depends on great DDF. When Doppler shifts of target fall outside the main clutter region, TF image can fully exploit the 2D spectrum properties in both intensity and geometry. However, the TF-feature-based detector [16] suffers heavy performance loss in the third and fourth-group datasets with small DDFs. Indeed, the loss results from the normalized TF image, where both sea clutter and targets have great suppression in the clutter region when Doppler shifts of targets fall inside the main clutter region. To avoid great performance loss, eight features are jointly used in the KNN-based detector [25], containing a tri-feature-based detector and a TF-feature-based detector. The KNN-based detector [25] achieves the average detection probabilities of 0.785, 0.814, 0.811, 0.737, with a performance improvement of 16% relative to the phase-feature-based detector [19]. Thanks to the eight features, the KNN-based detector [25] inherits the good performance of a tri-feature-based detector [14] and a TF-feature-based detector [16]. Thus, it is meaningful to exploit more complemental features without consideration of computational cost. However, real-time detection, fast implementation, and good performance are both required in radar systems.

On the whole, the proposed FB-ICHC detector attains the average detection probabilities of 0.827, 0.859, 0.860, and 0.784 at four polarizations, with a total performance improvement of 6% relative to the KNN-based detector [25] using eight features. In the proposed FB-ICHC detector, RPZR extracted from the phase time series reflects the geometric characteristics and robustness of the different SCR and DDF cases. In the Doppler spectrum, RCV is designed for high SCR cases. Inversely, WPHR is suitable for the low SCR and large DDF cases, where Doppler shifts of targets fall outside the main sea clutter region and the homogeneity assumption is satisfied. Therefore, the designed three features are effective and complementary, which can guarantee robustness to various clutter environments. For example, the detection probability of the #30 dataset is the worst among the ten datasets. This is due to the low SCRs of −0.3 dB, 3.6 dB, 3.5 dB, and 2.0 dB and the small DDFs of 0.68, 0.53, 0.52, and 0.28 at four polarizations, as shown in Figure 10. At low SCR cases, amplitude in the time domain and the two Doppler features [14] are invalid. In small DDF cases, the TF-feature-based detector suffers great performance loss and fails to work. In this way, the phase feature of RPZR plays a main role in the FB-ICHC detector, thanks to its insensitivity to SCR and DDF. Besides, the improved concave hull classifier in 3D space can also make contributions to detection improvement.

Figure 14 plots the influence of ASCR and DDF on the detection performance of the proposed FB-ICHC detector, where N = 512 and P_fa =10⁻³. In Figure 14a, as the values of ASCR increase, the detection probabilities are close to one. While, it has a little dependence on the ASCR, especially in the low ASCR cases. This means that there exist other factors playing the main role in detection. The DDF is discussed in Figure 14b. Indeed, the proposed FB-ICHC detector has a close relationship with DDF and detection probabilities. The reason is obvious. The phase feature of RPZR reduces the dependence on ASCR since ASCR is only related to the amplitude. The remaining two features of RCV and WPHR are determined by the Doppler spectrum and normalized Doppler spectrum. Especially, the greater the values of DDF are, the higher the detection probabilities of WPHR are.

Besides, the influences of observation time on average detection probability are listed in

Table 1. Average detection probability of ten datasets at different observation times.

Detectors	OT	HH	HV	VH	VV
ADHE detector [7]	0.128 s	0.340	0.402	0.394	0.252
	0.256 s	0.429	0.507	0.504	0.373
	0.512 s	0.516	0.577	0.586	0.474
	1.024 s	0.606	0.682	0.684	0.549
Tri-feature-based detector [14]	0.128 s	0.429	0.501	0.502	0.403
	0.256 s	0.501	0.585	0.577	0.505
	0.512 s	0.525	0.597	0.597	0.515
	1.024 s	0.583	0.670	0.654	0.553
TF-feature-based detector [16]	0.128 s	0.640	0.638	0.640	0.537
	0.256 s	0.673	0.661	0.663	0.566
	0.512 s	0.702	0.707	0.700	0.615
	1.024 s	0.757	0.771	0.755	0.702
Phase-feature-based detector [19]	0.128 s	0.476	0.552	0.544	0.457
	0.256 s	0.562	0.633	0.623	0.538
	0.512 s	0.630	0.693	0.683	0.615
	1.024 s	0.705	0.758	0.761	0.698
KNN-based detector [25]	0.128 s	0.653	0.687	0.704	0.593
	0.256 s	0.729	0.748	0.770	0.679
	0.512 s	0.785	0.814	0.811	0.737
	1.024 s	0.840	0.861	0.860	0.796
Proposed FB-ICHC detector	0.128 s	0.740	0.766	0.764	0.648
	0.256 s	0.793	0.823	0.828	0.725
	0.512 s	0.827	0.859	0.860	0.784
	1.024 s	0.892	0.903	0.896	0.837

OT is short for observation time.

, where observation time is set to 0.128 s, 0.256 s, 0.512 s, and 1.024 s. Three conclusions are drawn as follows. Firstly, six detectors work well as the observation time increases. It indeed proves the capacity of long-time technology to improve the detection performance. Secondly, the performance gap between the existing five detectors and the proposed FB-ICHC detector gets small in N = 1024. It is because the rough Doppler spectrum in a short time is replaced by a precise Doppler spectrum to provide more elaborative features when observation time is doubled from 0.512 s to 1.024 s. Thirdly, the proposed FB-ICHC detector has overall good and robust performance in the different clutter environments. Additionally, it only needs to compute the 1D Doppler spectrum, which can be well implemented by Fourier transform in a real radar system. However, the 2D TF images significantly increase computational cost and structural complexity in TF-feature-based detectors [10] and KNN-based detectors [25]. Therefore, the proposed FB-ICHC detector is a better choice under tested conditions in comprehensive consideration of both detection performance and computation cost, which is the potential for marine radars to detect sea-surface small targets.

Finally, the performances of six detectors are testified by using another open CSIR dataset [32]. The C-band Fynmeet radar is deployed on the west coast of South Africa, with a pulse repetition frequency of 2500 Hz, range resolution of 15 m, beam wide of 1.8 degrees, and VV polarization at the tracking mode. The dataset was collected in 2006, called TFA10-007. There are 64 range cells and the observation time is about 60 s. The wind speed is 2.4 m/s with a direction of 135 degrees and the significant wave height is 1.84 m. In Figure 15a, the power distributions of clutter and targets are given. The test target is a small boat located in the 16th to 18th range cells. The target is out of tracking during the 25 s to 30 s. The average SCR is calculated as 16.07 dB. In frequency-time distribution (TFD), the main sea clutter region ranges from 0 to 100 Hz in Figure 15b. The instant frequency curve of the target is clear around the zero and time-varying, covering a wide range of DDFs.

Figure 16 shows the detection results of the six detectors using the CSIR dataset. The ADHE detector [7] has a detection probability of 0.568. The moving trajectory of the test target covers a range from the 16th range cell to the 18th range cell. It is hard to find the test targets during the observation time ranging from 18 s to 40 s, where the instantaneous Doppler shifts of the target fall inside the main clutter region and the test target suddenly disappears as shown in Figure 15b. The existing three detectors of the tri-feature-based detector [14], TF-feature-based detector [16], and phase-based detector [19] have almost comparable detection probabilities of 0.741, 0.740, and 0.744 respectively. The proposed FB-ICHC detector and KNN-based detector [25] attain the detection probabilities of 0.898 and 0.867 respectively. The former only uses three features, exploiting the information from phase time series, the Doppler spectrum, and the whitened Doppler spectrum. The latter utilizes eight features from the amplitude time series, Doppler spectrum, and TFD. It is proved that the well-designed features can make up the performance from dimension increase. Simply increasing the number of features does not necessarily lead to better performance. In a certain dimension, the improvement of performance largely depends on the superiority of features themselves and the complementarity among features. In terms of fast implementation and good performance, the number of dimensions is suggested for three in real radar detection environments.

4.4. Performance Comparison with Single-Class Detectors

Below, a full performance evaluation is given to the mainstream single-class detectors, including an OCSVM-based detector [23], an iForest-based detector [24], and an existing concave hull-based (ECH-based) detector [20,27].

In Figure 17, both the OCSVM-based detector [23] and iForest-based detector [24] have eight features extracted from the time domain, frequency domain, and TF domain. At HH, HV, VH, and VV polarizations, the average detection probabilities of the OCSVM-based detector [23] are 0.696, 0.717, 0.717, 0.625 respectively, and that of the iForest-based detector [24] are 0.723, 0.750, 0.751, 0.673. While the eight features are partly correlated there must exist redundant information. In this way, the ECH-based detector [27] selects the optimal three features from eight features, achieving the average probabilities of 0.798, 0.810, 0.799, and 0.745. Relative to the ECH-based detector [27], the proposed FB-ICHC detector improves the detection probability by up to 21% on individual datasets, with an overall improvement of 5.5%. It verifies the performance superiority from well-designed three features and ICHC decision region.

Among the four single-class classifiers, the proposed ICHC and ECH are visualized in the 2D and 3D space. ICHC is obtained by rolling a small sphere of radius α over a set of samples, which has clear geometric meanings. The ECH is constructed based on a convex hull by digging internal points and filling external points. In this way, ECH is more complicated than the convex hull, burdening the computational cost.

Table 2. Training time for different single-class classifiers.

Single-Class Classifier	False Alarm Rata Pfa
	10−3	10−2	10−1
Convex hull [14]	1.21 s	12.56 s	206.29 s
iForest [24]	3.31 s	3.24 s	3.30 s
Proposed ICHC	0.61 s	3.82 s	22.05 s

shows the training time under different fixed false alarm rates, where the total number of training samples is 10,210. The training time of the convex hull classifier dramatically increases when the P_fa ranges from 10⁻³ to 10⁻¹. The iForest keeps the same training time because it can condense the eight features into one value called average path length. Relative to the iForest and convex hull, the proposed ICHC with a small search range and adaptive line search can achieve a reduction of 81.8% and 49.1% when P_fa is set as 10⁻³. In terms of fault tolerance, OCSVM is the worst, since a parameter search can only roughly control the false alarm rate due to the discrete nature of false-alarm points. In summary, the proposed ICHC is suitable for 2D or 3D space, especially when P_fa is lower than 10⁻³.

5. Conclusions

In this paper, an FB-ICHC detector is proposed to find marine small targets at a low computational cost. Three features and the false-alarm-controllable concave hull classifier are well designed to improve the detection performance. The former can fully exploit the difference between sea clutter and returns with target when the number of features is limited to three. The latter can quickly realize accurate control of the false alarm rate. Experimental results using IPIX datasets confirm that the proposed FB-ICHC detector attains robust performance with an implementable structure, suitable for shore-based pulse Doppler (PD) radars. Considering the abundant information on synthetic aperture radar (SAR) images, a combination of PD radars and SAR may provide a potential way in future work.