Small Target Detection Method Based on Low-Rank Sparse Matrix Factorization for Side-Scan Sonar Images

Abstract

Target detection in side-scan sonar images plays a significant role in ocean engineering. However, the target images are usually severely interfered by the complex background and strong environmental noise, which makes it difficult to extract robust features from small targets and makes the target detection task quite challenging. In this paper, a novel small target detection method in sonar images is proposed based on the low-rank sparse matrix factorization. Initially, the side-scan sonar images are preprocessed so as to highlight the individual differences of the target. Then, the problems of target feature extraction and noise removal are characterized as the problem of matrix decomposition. An improved Robust Principal Component Analysis algorithm is used to extract target information, and the fast proximal gradient method is used to optimize the solution. The original sonar image is reconstructed into the low-rank background matrix, the sparse target matrix, and the noise matrix. Eventually, a morphological operation is used to filter out the noise and refine the target edges in the target matrix for improving the accuracy of target detection. Experimental results show that the proposed method not only achieves better detection performance in comparison to the conventional baseline algorithms but also performs robustly in various signal-to-clutter ratio conditions.

1. Introduction

With the development of side-scan sonar (SSS) image systems and image processing technology, underwater target detection has gained increasing importance in oceanographic research [1,2,3,4]. Due to the long detection range and strong penetrability of underwater SSS images, they have been widely used in fields such as marine surveying and mapping, suspicious target detection, and marine resources exploration [5,6,7,8]. However, there are two major challenges in the development of a detection algorithm based on underwater SSS images.

Firstly, the complexity and variability of underwater scenes and the influence of noise have brought great difficulties to the automatic interpretation of SSS images. Due to the transmission and scattering characteristics of acoustic waves, the collected SSS images (low frequency) often have low spatial resolution and image quality problems including serious noise and reverberation [9,10,11]. Small targets are usually buried in complex background clutter with a low signal-to-clutter ratio (SCR), which increases the difficulty of automatic target detection. Furthermore, SSS images weaken the essential characteristics of the target, such as texture and contour. This exacerbates the difficulty of improving the robustness of the algorithms.

Secondly, it is hard to build a universal training dataset for the SSS image. Although the side-scan sonar image data are quite rich, there are currently few public side-scan sonar image datasets since the labeling images require experts with extensive domain knowledge which raises labeling costs. To reduce the difficulty of data annotation, many recent papers address the sample problem through deep learning methods with automatic sample augmentation [12]. However, different seabed environments and observation conditions lead to large differences in the distribution of the target image features. Different seabed backgrounds, different navigation routes, different equipment, viewpoints, and different parameters usually lead to variant shapes of the same target in the image. Detection using data-driven supervised learning algorithms faces the problem of detection robustness in different environments. Figure 1 shows the side-scan sonar images and their 3D visualization results, in which the columnar target is the red rectangular box.

With the development of artificial intelligence, there have been a large number of sonar image processing algorithms based on deep learning [13,14,15], such as real-time underwater maritime object detection algorithm in side-scan sonar images based on transformer-Yolov5 [13], seagrass classification algorithm [16], segmentation algorithm based on pulse coupled neural network [17], and underwater obstacle detection method based on Yolov3 [10]. These kinds of algorithms have high accuracy, especially in harsh environments. However, the detection approaches based on deep learning require a large amount of training data to obtain excellent detection results, meaning a long training time, powerful computational resources, and expensive manual labeling costs. The traditional methods do not need a lot of labeled training data and hardware configuration, which still have high research value in sonar target detection.

Some conventional target detection methods for sonar images can usually achieve good performance in applications by focusing on feature extraction in the pixel domain, such as TOPHAT filtering [18,19], Accumulated Cell Average Constant False Alarm Rate [8] and so on. However, they are sensitive to environmental noise and do not perform robustly when the target size varies over a large range. Similarly, maximizing detection probability in low SCR images without training data remains a problem.

To combat the aforementioned challenges, this paper introduces low-rank sparse matrix factorization in the sonar target detection technology. We proposed an end-to-end sonar small target detection algorithm robust to high background noise, which can directly detect the foreground target without the need to perform image filtering. The innovations of the proposed algorithm are as follows:

(1)

A small target detection algorithm for side-scan sonar based on low-rank sparse matrix factorization is proposed. The proposed framework has good performance in the removal of environmental noise with different variances and the accuracy of the small target detection;

(2)

An improved Robust Principal Component Analysis (RPCA) algorithm is used to extract target information and the fast proximal gradient method is initially employed to optimize the solution in sonar target detection. We explicitly consider the noise information based on the RPCA algorithm, and estimate the low-rank matrix, sparse matrix, and noise matrix at the same time in matrix factorization, and do not need training data;

(3)

Extensive experimental evaluation on real-world SSS image datasets with different noise backgrounds is conducted to show that the algorithm is not only more robust to different target types but also has better detection performance compared with other traditional detection algorithms.

2. Related Work

The underwater target detection algorithm of a sonar image mainly uses spatial domain information and frequency domain information to capture the difference between the target and the background in several appearances, such as contour, gray distribution state, and local contrast, to separate the target from the background. The mainstream sonar target detection algorithms can be grouped into two categories: the pixel domain and the feature domain. Among them, the algorithms based on pixel domain include the frame difference method, the optical flow method, the morphological filtering, and the constant false alarm rate. Feature-based algorithms can be divided into detection algorithms based on local feature representation, principal component analysis, and deep learning theory.

2.1. Target Detection Based on Pixel Domain

The underwater sonar target detection algorithms based on the pixel domain realize detection through the operation between pixels. In pixel operation, the frame difference method and the optical flow method [20,21] are suitable for target detection. Morphological filtering [22,23] is also a classic target detection algorithm, and Rahnemoonfar et al. proposed a framework to enhance the sonar image based on top-hat transformation and utilized morphological operators to identify seagrass blowouts [16]. The constant False Alarm Rate (CFAR) algorithm [24,25,26] is often used in underwater sonar target detection, which utilizes a set detection threshold to determine the object distances from different pixel cells. In order to improve accuracy, researchers have made some improvements to CFAR, such as the Cell Average Constant False Alarm Rate (CA_CFAR) [27], Ordered Statistical Constant False Alarm Rate (OS_CFAR) [28], and the Accumulated Cell Average Constant False Alarm Rate (ACA_CFAR) [8] algorithm. The underlying principle of these algorithms is relatively simple and does not need to extract the deep features of the target. However, inhomogeneous intensity and complex seabed textures such as sand ripples in sonar imagery cause severe degradation in the performance of existing approaches.

2.2. Target Detection Based on Feature Domain

The underwater sonar target detection algorithms based on feature domain realize detectors by extracting object features. These algorithms extract color, contour, shape, and other information about the targets, and then classify the targets. There are some classical object detection methods such as Markov random field [29], and sonar image segmentation based on level set [30]. In recent years, a large number of deep learning detection algorithms have emerged. The algorithms have excellent performance in large image processing and are widely used in the field of object detection because of their powerful feature extraction abilities. Zhou proposed an image segmentation method based on a simplified version of the pulse coupled neural network [17]. Palomeras designed a convolutional neural network to detect mine-like objects in forward-looking sonar data [31]. Cao applied a one-stage detection network named YOLOv3 to determine obstacle candidate areas in a sonar image [10]. However, most neural network algorithms require training with a large number of sonar images. It should be noted that not only is it difficult to obtain learning samples, but training a neural network requires a long time, which causes the application bottleneck of the algorithm.

2.3. Traditional Detection for Small Targets

Considering the above problems, we investigate traditional small object detection algorithms that do not require training data. Small target detection algorithms based on local feature representation and principal component analysis achieve robust performance. Several algorithms based on local feature representation are proposed including local contrast method (LCM) [32], improved local contrast method (ILCM) [33], and accelerated multiscale weighted local contrast measure (AMWLCM)  [34]. Some algorithms separate sparse components from low-rank components based on principal component analysis and then search for the target in the sparse components. Gao et al. [35] proposed an IPI model for weak and small target detection through image matrix construction and RPCA application to reconstruct the target and background matrices.

Although the IPI method has obtained great success in small target detection, we note that there are still two limitations that prevent it from achieving better performance in sonar image detection. One limitation is that the algorithm does not consider the noise matrix, as it will reduce target signal strength and remain clutter background edges in the detection result. The other limitation is that the local patch image is first constructed from the original image, which will be time-consuming.

Aiming to eliminate the aforementioned limitations, this paper explores the prior of spatial correlation between the target and the background in a SSS image, designs a low-rank sparse matrix factorization model to detect the small target, and estimates the target and noise matrix simultaneously, which can effectively reduce the false alarm rate in SSS images with heavy environment noise and reverberation, thus improving the precision of small target detection and performing robustly in various SCR conditions.

3. The Proposed Method

Considering the sparsity of the small target in the SSS images and the different imaging mechanisms, we design a matrix factorization method for small target detection in the side-scan sonar image. We aim at the problems of low SCR and insufficient target appearance features in SSS images where our proposed matrix factorization method decomposes the origin SSS image into the target matrix, the background matrix, and the noise background.

In the non-target region of the SSS image, there are background obstructions and noise disturbances. While the background obstruction is continuous and generated by the reflection of the sea floor (reverberant noise), the independent Gaussian noise occurs due to the sonar equipment and electronic noise [36]. We uniformly divide the non-Gaussian part into the background. As the small underwater target represents a small proportion of the entire scanned image, we choose the sparse matrix to construct the object model. Besides considering that the SSS image background has a certain correlation between the intensities of neighboring pixels, the image background is modeled as a low-rank matrix. Therefore, the SSS image observation model using the sparse, low-rank, and Gaussian noise is reconstructed. In this paper, the noise parameters are estimated iteratively, so that the low-rank matrix can be recovered accurately from noisy observations.

We will introduce in detail the proposed method for detecting small underwater targets in sonar image sequences. Figure 2 shows the complete framework of the method of small target detection proposed in this paper. Initially, the sonar image D is constructed from the original SSS image $D_i$ using image preprocessing. Then the improved convex optimization algorithm is applied to image D to simultaneously estimate the low-rank background image B, the sparse target T, and the noise image N. Following that, a series of morphological operations are carried out on the target image T. With the assistance of the components given above, our algorithm can be divided into the following parts.

3.1. Image Preprocessing

From the perspective of Principal Component Analysis (PCA), the mean is subtracted to regulate the features of the data. For a SSS image, we are not interested in the illuminance of background and noise but focus on salient regions such as the targets. Equation (1) can remove the average brightness value of the image and highlight the significant target, which is more conducive to subsequent detection. Figure 3 shows the result of subtracting the mean from the input image $D_i$, and the image preprocessing is computed as:

$$D=max\left\{D_i-mean\left(D_i\right),0\right\},$$

(1)

where $D_i$ is the input SSS image and $mean$ denotes to calculate the global gray mean of the input image. After subtracting the mean, the negative values were replaced with zero to remove some noise. The first row of Figure 3 shows a representative SSS image and the preprocessed image, and the second row shows the corresponding histograms. We can see from the figures that the preprocessed image can remove a significant part of the noise.

3.2. Low-Rank Matrix and Sparse Matrix

Due to the reason mentioned above, the complex SSS image background satisfies the low-rank characteristics and the small target meets the sparse feature, which is the precondition of our matrix factorization model. First, we take the gray SSS image for a two-dimensional matrix and further analyze the feature of the complex background images. The singular values of the image matrix can present the correlation in different rows, therefore, we select the low-rank matrix to model the background matrix, which contains constant seafloor reflection. As shown in Figure 4, we analyze the feature of the complex background in the preprocessed SSS image, and the first row of Figure 4 shows three representative sonar background images with the same size of 300 × 300 pixels. The second row of Figure 4 shows the singular values of the corresponding preprocessed images; it can be seen that although the original background images of the three images are extremely different, their singular values decrease rapidly to zero. Based on the above discussion, we can consider the background image B as a low-rank matrix given by:

$$rank\left(B\right)\le r,$$

(2)

where r is a constant, $r<<m,n$, the size of a input image is m × n pixels.

Subsequently, according to the few pixels occupation of the small target in the SSS image, we can ignore the appearances of the objects and regard the target matrix as a sparse matrix T. Figure 5 shows the average area proportion of the target in the whole image from the three sequences. The proportion of small targets in SSS images occupies very few pixels compared to the total images, so the sparse target matrix T satisfies the following conditions:

$$\frac{{∥T∥}_1}{{∥D∥}_1}<k,$$

(3)

where ${∥T∥}_1$ represents ${\ell }_1-\mathrm{norm}$ of the target matrix T, ${∥D∥}_1$ represents ${\ell }_1-\mathrm{norm}$ of the target matrix D, and k is a constant much less than 0.01, which means that the matrix T satisfies this sparseness property.

The characteristic of a non-local low-rank exists universally in natural images, which propels many preeminent non-local methods in various fields, such as a non-local low-rank technique for the hyperspectral image (HSI) denoising [37,38,39], compressed HSI reconstruction [40], inpainting [41,42], a non-local low-rank model for infrared small target detection [43], and so on.

It is noteworthy that we can adopt a more general low-rank assumption that all background images come from a mixture of low-rank subspace clusters. The inherent spatial correlation between the pixels of an image indicates that the background is expressed continuously, and the pixels are highly correlated [19]. It can be seen intuitively in Figure 4 that the background has non-local self-correlation property. In such scenarios, the sonar background image can be constructed as one low-rank subspace. Hence we only exploit one low-rank subspace assumption in the matrix decomposition to make it a more simple expression.

3.3. Optimization Method

Based on the property of the side-scan sonar image, the convex optimization problem is to simultaneously separate background, target, and noise images from sonar images. The total optimization problem is translated into three sub-problems: (1) the low-rank background matrix estimation; (2) the sparse target matrix estimation; (3) the noise matrix estimation. To solve the problems efficiently, the soft thresholding operator [44] for target estimation, the singular value thresholding operator [45] for background estimation, and the gradient descent method for noise estimation are utilized in our optimization method. The singular value thresholding operator is the proximity operator associated with the nuclear norm. Details about the proximity operator can be found in Ref. [46].

In this paper, it is assumed that random noise satisfies the independent and identically distributed condition, D is the SSS preprocessed image, T is the target matrix, B is the background matrix, and N is the noise matrix. Then the problem can be described as an optimization problem:

$$min{∥B∥}_{*}+{\lambda }_1{∥T∥}_1+{\lambda }_2{∥N∥}_F^2,s.t.D=B+T+N,$$

(4)

where ${∥B∥}_{*}$ represents the nuclear norm of the matrix B, which is the sum of its singular values. ${∥T∥}_1$ denotes the of ${\ell }_1-\mathrm{norm}$ of the matrix T, and ${∥N∥}_F$ denotes the Frobenius norm of the matrix N.

The above convex optimization problem is to simultaneously separate background, target, and noise images from SSS images. In SSS images with low SCR, background estimation and noise estimation are significant in many pieces of research, which can be used to evaluate the reliability of the detection method as well as the reverberation estimation in sonar image systems. To solve the background and noise estimation problem, Equation (4) can be further evolved into a dual problem:

$$min{∥B∥}_{*}+{\lambda }_1{∥T∥}_1+{\lambda }_2{∥N∥}_F^2+\frac{1}{2\mu }{∥D-B-T-N∥}_F^2,$$

(5)

where $\mu$ is a given positive parameter, Formula (5) remains an optimization problem and we could solve it based on the accelerated proximal gradient (APG) method because it has a good balance between efficiency and accuracy in solving related RPCA methods [47,48,49,50]. The constraints of Formula (5) are convex relaxed into the objective function, so the function is defined as:

$$\begin{array}{cc}\hfill minF\left(B,T,N,\mu \right)=& \mu ({∥B∥}^{*}+{\lambda }_1{∥T∥}_1+{\lambda }_2{∥N∥}_F^2)+\frac{1}{2}{∥D-B-T-N∥}_F^2.\hfill \end{array}$$

(6)

This is necessary with an appropriate choice of the regularizing parameter ${\lambda }_1>0$ and ${\lambda }_2>0$. Throughout this paper,  unless otherwise stated, we will fix ${\lambda }_1={\lambda }_2=\frac{1}{\sqrt{max(m,n)}}$. The size of an input image is m×n pixels. For simplicity,   we hypothesize $\mathrm{g}\left(B,T,N,\mu \right)=\mu ({∥B∥}^{*}+{\lambda }_1{∥T∥}_1+{\lambda }_2{∥N∥}_F^2)$ and $f\left(B,T,N\right)=\frac{1}{2}{∥D-B-T-N∥}_F^2$, therefore, Equation (6) can be rewritten as:

$$minF\left(B,T,N,\mu \right)=g\left(B,T,N,\mu \right)+f\left(B,T,N\right).$$

(7)

Instead of directly optimizing the formula, the quadratic model is used to approximate Equation (7). $f(Y_{}^B,Y_{}^T,Y_{}^N)$ has a known value and is closer to $f\left(B,T,N\right)$. The partial derivatives of $f\left(B,T,N\right)$ with respect to B, T, and N are solved alternately. It means that only one variable is solved at a time and the other variables are set to constants. Therefore, the optimization problem is transformed into three sub-problems until convergence. For the exact global optimal solution, Ref. [51] proves that the alternatively stepwise result is the exact global optimal solution.

(1)

Estimate the low-rank matrix B.

$$B_{k+1}=\underset{}{arg\underset{B}{min}}{\mu }^k{∥B∥}_{*}+\frac{L_f}{2}{∥B-G_k^B∥}_F^2,$$

(8)

where $G_k^B=Y_k^B+\frac{1}{L_f}(D-Y_k^B-Y_k^T-Y_k^N)$ and $L_f$ is a constant. This is a nuclear norm minimization problem that can be obtained by solving the singular values of the observed data matrix Y using a soft-thresholding operation. Assuming the singular value decomposition of $G_k^B$ where $G_k^B=US{V}^T$, Equation (8) can be solved as:

$$B_{k+1}=U{S}_{\frac{{\mu }_k}{2}}\left[S\right]V^T,$$

(9)

where $V^T$ is the transpose of the matrix V,

$$S_{\frac{{\mu }_k}{2}}\left[x\right]=\left\{\begin{array}{cc}sign\left(x\right)(\left|x\right|-\frac{{\mu }_k}{2})& if\left|x\right|>\frac{{\mu }_k}{2}\\ 0& otherwise\end{array}\right.,$$

(10)

(2)

Estimate the sparse matrix T.

$$T_{k+1}=arg\underset{T}{min}{\lambda }_1{\mu }^k{∥T∥}_1+\frac{L_f}{2}{∥T-G_k^T∥}_F^2,$$

(11)

$$S_{\frac{\lambda {\mu }_k}{2}}\left[x\right]=\left\{\begin{array}{cc}sign\left(x\right)(\left|x\right|-\frac{\lambda {\mu }_k}{2})& if\left|x\right|>\frac{\lambda {\mu }_k}{2}\\ 0& otherwise\end{array}\right.,$$

(12)

(3)

Estimate the noise matrix N.

$$N_{k+1}=arg\underset{N}{min}{\lambda }_2{\mu }^k{∥N∥}_F^2+\frac{L_f}{2}{∥N-G_k^N∥}_F^2,$$

(13)

where $G_k^N=Y_k^N+\frac{1}{L_f}(D-Y_k^B-Y_k^T-Y_k^N).$ The above equation can be solved using the gradient descent method, taking the derivative of Equation (14) by N and making the result equal to 0; the final representation is then obtained:

$$L_f(N-G_N^k)+2\lambda {\mu }_{k}N=0\Rightarrow N_{k+1}=\frac{L_f}{2\lambda {\mu }_k+L_f}{G}_k^N.$$

(14)

The pseudocode of low-rank and sparse and noise matrix decomposition is shown in Algorithm 1. It can be seen that this algorithm takes the sonar image matrix $D_i$ for input, initializes iteration variables, and searches for the optimal solution according to the enhanced RPCA algorithm. The output of this algorithm is the background matrix B, the target matrix T, and the noise matrix N.

Algorithm 1 Low-Rank and Sparsity and Noise Matrix Decomposition

Input:  Sonar image matrix $D_i\in R^{m\times n}$ Output:  $B=B_k,T=T_k,N=N_k$ 1: $D=max\left\{D_i-mean\left(D_i\right),0\right\}$ 2: $B_0=B_{-1}=0,T_0=T_{-1}=0,t_0=t_{-1}=1$ 3: while not converged do 4:    $Y_k^B=B_k+\frac{t_{k-1}-1}{t_k}(B_k-B_{k-1})$, $Y_k^T=T_k+\frac{t_{k-1}-1}{t_k}(T_k-T_{k-1})$, $Y_k^N=N_k+\frac{t_{k-1}-1}{t_k}(N_k-N_{k-1})$ 5:    $G_k^B=Y_k^B+\frac{1}{L_f}(D-Y_k^B-Y_k^T-Y_k^N)$ 6:    $(U,S,V)=\mathrm{svd}\left(G_k^B\right),B_{k+1}=U{S}_{\frac{{\mu }_k}{2}}\left[S\right]V^T$ 7:    $G_k^T=Y_k^T+\frac{1}{L_f}(D-Y_k^B-Y_k^T-Y_k^N)$ 8:    $G_k^N=Y_k^N+\frac{1}{L_f}(D-Y_k^B-Y_k^T-Y_k^N)$ 9:    Estimate the low-rank matrix B: $B_{k+1}=U{S}_{\frac{{\mu }_k}{2}}\left[S\right]V^T$ 10:    Estimate the sparse matrix T: $T_{k+1}=S_{\frac{\lambda {\mu }_k}{2}}\left[G_k^T\right]$ 11:    Estimate the noise matrix N: $N_{k+1}=\frac{L_f}{2\lambda {\mu }_k+L_f}{G}_k^N$ 12:    $t_{k+1}=\frac{1+\sqrt{4{t_k}^2+1}}{2}$, ${\mu }_{k+1}=max(\eta {\mu }_{k+1},\overline{\mu })$ 13:    $k=k+1$ 14: end while 15: return $B_k,T_k,N_k$

3.4. Morphological Operation

In estimating the optimal solution of the target matrix and the noise matrix at the same time, few losses incur in the object outline features of estimating the target matrix. Therefore, we use the greatest similarity matching to restore the target after determining its approximate position from the optimal target matrix. By visualizing the partial sparse matrix (the target image), it is discovered that the target contains some isolated noise or holes that necessitate mathematical morphology operations such as cleaning and filling. We employ morphological approaches to solve it to increase the precision of target identification since mathematical morphology has high filtering qualities and can sharpen the edge of the target.

After getting the target matrix T, the image erosion operation is conducted on the target matrix to generate the eroded image $T_1$, which is defined as follows:

$$T_1=T\ominus o_1=\left\{z\right|{\left(o_1\right)}_z\subseteq T\},$$

(15)

where $o_1$ is a structuring element created as a flat disk with a radius of 1 in this paper.

The image morphological dilation technique is conducted on $T_1$ to obtain the dilated target image $T_2$. Dilation is the dual operation of erosion while the operation “dilation” is defined as follows:

$$T_2=T_1\oplus o_2=\left\{z\right|{\left({\widehat{o}}_2\right)}_z\cap T_1]\ne \varnothing \},$$

(16)

where $o_2$ is a structuring element created as a rectangular matrix with size 3 × 3 in experiments. $T_2$ is the final target result. The experiment found that the morphological operation is necessary in order to get a clear and complete target.

4. Experimental Results and Discussion

Real data experiments were undertaken to demonstrate the effectiveness of the proposed method for sonar object detection. We selected seven different detection methods for comparison, i.e., IPI [35], LCM [32], AMWLCM [34], ILCM [33], TOPHAT [18,19], MAXMEAN [52,53], and ACA_CFAR [8]. For all the experiments, the parameter selection was consistent with the description from the original papers.

4.1. Real Sonar Datasets Description

We conducted real-world experiments to obtain actual underwater SSS images for the validation of the performance of our proposed method. In the real-world datasets, three sequences of sonar images were adopted. Side-scan sonar is mounted on an underwater unmanned platform, and the experimental scheme with only one side of the side-scan sonar transmit beam is shown in Figure 6.

The beam emission direction is perpendicular to the carrier navigation direction, and the experiment collects SSS images under various conditions according to the experimental conditions. The working underwater depth of the Autonomous Underwater Vehicle (AUV) was changed for the sonar scanner system. The tilt angle of scanning was adjusted constantly and the target images were collected under different beam angles. At the same depth, the heading angles are 0°, 30°, 60°, and 90°, respectively. In addition, we set an autonomous mode for the AUV’s running mode, and the initial direction of navigation is different, hence the motion direction of the side-scan sonar was changed to scan the target from multiple angles. Figure 7 shows the appearance of our AUV platform. The collected targets were randomly distributed in real field lakes with a maximum depth of more than 100 m, and the maximum range of our sonar system was set to 80 m (up to 100 m) for a 0.04 m range resolution.

Table 1. Specifications of the side-scan sonar.

Parameter	Value
Operating frequency	260 kHz/800 kHz
Transducer beam width	260 kHz: 2.2° × 75°/800 kHz: 0.7° × 30°
Range resolution	0.2% of range
Maximum imaging range	100 m
Image size	2000 × 500 pixels
Frame rate	6–20 fps
Depth rating	300 m

lists the specifications of the sonar. Next, we captured images by changing the tilt angle and altitude of the SSS. We conducted 11 experiments scanning the underwater objects and took 300 frames of SSS images. Three groups of images with different signal-to-clutter ratios were collected, 100 images were collected in each group, and the targets contain cylinders, spheres, and a model of the human body. The targets are arranged in different areas. Some obtained images are shown in Figure 8, which shows examples of images obtained in different conditions.

4.2. Evaluation

To get an accurate evaluation of the proposed method, the criteria of the receiver operating characteristic (ROC) curve are adopted, and the probability of detection $P_d$ and false alarm rate $F_a$ are defined as:

$$P_d=\frac{N}{M},F_a=\frac{F}{S},$$

(17)

in the $P_d$ calculation, N is the total number of true targets detected, and M is the number of actual targets. In the $F_a$ calculation, F is the total number of false detection, and S is the total number of images. When the detection box overlaps with the ground truth box and the center point distance of the two boxes is lower than the set threshold with 4 pixels [53], we regard that the detection object is correctly detected. For quantitative analysis, both experiments have fixed false-alarm rates $F_a$ by changing the binarization thresholds for each group.

4.3. Comparison with Other Methods

In this section, we first introduce the evaluation metrics and the data for comparison. Then, we use real sonar image sequences to demonstrate the effectiveness and practicality of the proposed method. The experiments were conducted on a computer with 16-GB memory, Intel Core CPU i7-11370H, 3.3 GHz processor, and the code was implemented in MATLAB.

In order to visually show the effectiveness of the algorithm in improving image SCR and suppressing the background, simulation tests are carried out under three typical sequences with different SCR of sequence 1, sequence 2, and sequence 3 with 100 images in each sequence. The specific image properties and introduction are shown in

Table 2. Details of the three sequences.

Sequence	Number	The Target Size in the Whole Image	The Mean of SCR	The Variances of SCR
Seq1 (High SCR)	100	0.011	6.36	0.16
Seq2 (Medium SCR)	100	0.012	4.91	0.53
Seq3 (Low SCR)	100	0.009	1.08	0.37

.

We conduct testing experiments and quantitative analysis on actual datasets, appropriately comparing them with classical and state-of-the-art methods: IPI, LCM, AMWLCM, ILCM, TOPHAT, MAXMEAN, and ACA_CFAR. It should be highlighted that IPI designs a sliding window to traverse the original image, constructs a block image matrix after vectorization of the block image, and applies RPCA to reconstruct the component matrix of the target and background from this matrix. LCM is based on local feature representation and introduces multi-scale theory. AMWLCM and ILCM are improved algorithms based on LCM. MAXMEAN is designed to filter horizontally and vertically, calculating the mean and median values in each direction. Top-hat calculates the difference between the original image and its morphology to eliminate the background and extract the regional edge and target information. ACA_CFAR has better detection performance in a uniform environment by adaptively selecting reference units. The results of the proposed and comparison algorithms are demonstrated in Figure 9, which clarifies that the proposed algorithm exhibits better performance visually for almost all sequences, especially when the noise interference is serious. The experimental parameters $L_f=2$, $\lambda =1/\sqrt{max(m,n)}$, $\eta =0.9$, and the radius of the morphological structure is set to 1. The results of the evaluation metrics obtained by the proposed method and the comparison algorithms are listed in

Table 3. Probability of detection for different algorithms.

${\mathit{F}}_{\mathit{a}}$	Method	${\mathit{P}}_{\mathit{d}}$ (Seq1)	${\mathit{P}}_{\mathit{d}}$ (Seq2)	${\mathit{P}}_{\mathit{d}}$ (Seq3)
0.5	IPI	0.850	0.852	0.711
	LCM	0.825	0.855	0.666
	AMWLCM	0.599	0.778	0.140
	ILCM	0.640	0.853	0.583
	TOPHAT	0.598	0.245	0.350
	MAXMEAN	0.622	0.511	0.347
	ACA_CFAR	0.661	0.781	0.525
	OURS	0.895	0.900	0.732
1.0	IPI	0.877	0.875	0.743
	LCM	0.861	0.887	0.754
	AMWLCM	0.625	0.823	0.210
	ILCM	0.733	0.877	0.665
	TOPHAT	0.702	0.315	0.379
	MAXMEAN	0.721	0.547	0.465
	ACA_CFAR	0.750	0.799	0.665
	OURS	0.920	0.902	0.783

.

From Table 3, we can see that the proposed method has better performance with a low false alarm rate and is more robust to different SCR conditions compared with other baseline methods. Table 3 also illustrates that all detection methods have superior detection capability for high SCR sonar images than low SCR ones. Despite this, when the background noise is low in sequence 1, some baseline methods such as IPI and LCM also achieve a high probability of detection, but the proposed method gets the highest detection rate of 0.920 with the lowest false alarm rate of 1.0. When the detector is accurate and the false alarm rate is near 0, our method can still maintain a detection rate of 0.8. Besides, Table 3 also illustrates the performance comparison of baseline methods in a quite challenging environment (sequence 3) where the background noise is loud and includes seafloor reverberation, and the detection rates of baseline method TOPHAT, MAXMEAN, and AMWLCM are all less than 0.5 with the lowest false alarm rate of 0.5, which can be deemed as detection failure. Additionally, our algorithm’s detection rate is 0.732 with the same false alarm rate, which is higher than the other comparative methods. The ROC curves for our method in Figure 9 have a bigger region contained by the horizontal axis and a better capacity to detect. The suggested algorithm achieves a high detection rate with a low false alarm rate.

The 3D visualization result of our algorithm is shown in Figure 10, where Line 1 is the original sonar image, and the red box in the image is the ground truth of the target. Line 2 is the 3D visualization result of the original image, and Line 3 is the detection result of the proposed algorithm.

In order to evaluate the visual performance of the proposed method, we conduct qualitative analysis. Figure 11 shows the sonar images of three cylindrical targets and the resultant visualization of the eight detection methods. Among them, the first line is the original SSS image samples of the three sequences and the red box in the image is the target ground truth. The detection result comparison in lines 2–9 is OURS, IPI, LCM, AMWLCM, ILCM, TOPHAT, MAXMEAN, and ACA_CFAR.

It can be seen in the visualization results of all algorithms (Figure 11) that most baseline algorithms can detect the object in severe reverberation sonar images, but there are some problems in the high false detection rate of the background noises and poor integrity detection of the target area. Due to weak detection performance, some baseline methods (TOPHAT, MAXMEAN, and AMWLCM) have a high false alarm rate and misclassify a large number of noisy backgrounds as targets. While the ILCM algorithm based on local contrast can enhance the real target, it has a weak ability to suppress the background and cannot effectively distinguish the real target and bright spot noise in the background. In addition, the ACA_CFAR method can detect the objects well in weak noise, with the enhancement of background noise in sequence 3. The algorithm is affected by speckles, which leads to the degradation of detection performance. Besides, as a general rule for the appearance of the detected target, the more accurate model cannot accurately segment foreground and background in low SCR scenes. The baseline methods (LCM, AMWLCM) can roughly detect the location of the target, but the completeness of the target is poor. Compared with other baseline algorithms, the proposed algorithm eliminates a large number of clutter and false alarms, and is robust to small object detection in sonar images with different SCR.

4.4. Ablation Experiments

In order to better understand the performance of the algorithm, we analyze the effect of each of the steps. First, to evaluate the effectiveness of the decomposition for the noise matrix (improved RPCA) in matrix factorization, we compared the algorithm to detect the foreground without considering the decomposition for the noise matrix (RPCA) and considering noise matrix (OURS), and tested it on images with different SCR. Second, the performance of the algorithm with image preprocessing and improved RPCA was analyzed. Finally, the performance of the algorithm containing image preprocessing, improved RPCA, and morphological operation was analyzed. The resultant ROC curve obtained can be seen in Figure 12.

In Figure 12, when the background noise is small (sequence 1), a higher detection rate can be achieved with RPCA. However, when the background noise is severe and has bottom reverberation (sequence 2), and the algorithm only does preprocess operation and uses the original RPCA, the result will be affected by noise. In this circumstance, the proposed algorithm has the lowest misinformation rate. When the false alarm rate is 0.5 or tends to 0, the proposed algorithm carries out image preprocessing and improved RPCA at the same time, and the detection rate is above 0.8. In sequence 3, the SCR is low, so when the noise matrix is not considered, the detector fails because the detection rate is below 0.5. The detection rate of the proposed algorithm reaches 0.732 when the false alarm rate is 0.5, which is better than the RPCA method.

For a more intuitive comparison of ablation experiments, the detection rate of each sequence image is shown in Figure 13 where the false alarm rate $F_a$ is 0.5, 1, and 1.8, respectively. The results of evaluation metrics obtained by the proposed and comparative algorithms are tabulated in

Table 4. Probability of detection for the ablation experiment.

${\mathit{F}}_{\mathit{a}}$	Image Preprocessing	RPCA	Improved RPCA	Morphological Operation	${\mathit{P}}_{\mathit{d}}$ (Seq1)	${\mathit{P}}_{\mathit{d}}$ (Seq2)	${\mathit{P}}_{\mathit{d}}$ (Seq3)
	✘	✔	✘	✘	0.474	0.368	0.301
0.5	✘	✘	✔	✘	0.526	0.672	0.579
	✔	✘	✔	✘	0.862	0.840	0.706
	✔	✘	✔	✔	0.895	0.900	0.732
	✘	✔	✘	✘	0.737	0.605	0.403
1.0	✘	✘	✔	✘	0.837	0.805	0.753
	✔	✘	✔	✘	0.899	0.881	0.737
	✔	✘	✔	✔	0.920	0.902	0.783
	✘	✔	✘	✘	0.868	0.642	0.516
1.8	✘	✘	✔	✘	0.842	0.811	0.758
	✔	✘	✔	✘	0.921	0.887	0.780
	✔	✘	✔	✔	0.947	0.903	0.815

.

As shown in Figure 13, among the three detection sequences with different SCR, for the same false-alarm rate, the method with image preprocessing and improved RPCA performs the best. To compare the effects of the algorithms more intuitively, the following figure shows the visual results of the detection algorithm. Among them, the first line is the original side-scan sonar image, the other lines are the detection results of the ablation experiment, and the red box in the images is the ground truth result.

From the visualization results of the ablation experiment shown in Figure 14, the performance of the algorithm is poor when using the RPCA method, and the detection result is good when using the improved RPCA method without image preprocessing, however, some noisy background may not be completely filtered out. When the image preprocessing, improved RPCA method, and morphological operations are carried out at the same time, the algorithm eliminates a large number of clutter and false alarms, and is robust to small object detection in sonar images with different SCR.

Finally, we compare the average computational cost of the eight detection methods (see

Table 5. Computational cost comparison among the proposed algorithm and other algorithms. (unit: s).

Sequence	IPI	LCM	AMWLCM	ILCM	TOPHAT	MAXMEAN	ACA_CFAR	OURS
1	1.832	0.309	4.902	0.270	0.260	0.316	33.499	1.853
2	2.799	0.371	6.360	0.319	0.275	0.304	46.8	2.943
3	1.486	0.303	3.972	0.280	0.257	0.288	28.754	1.495

). From the experimental results, we can get the conclusion that TOPHAT method and MAXMEAN algorithm can achieve a fast detection speed, ACA_CFAR requires more sorting operations, and our algorithm can achieve a faster detection speed than ACA_CFAR and AMWLCM.

5. Conclusions

In this paper, an effective sonar target detection algorithm employing matrix decomposition was presented. The key idea of the proposed method is to use the improved Robust Principal Component Analysis algorithm to extract target information, and use the fast proximal gradient method to optimize the solution. The problems of sonar image feature extraction and noise removal are characterized as matrix decomposition. The experimental results showed that our method improves the probability of detection and significantly outperforms the conventional methods including TOPHAT, MAXMEAN, and ACA_CFAR. The proposed method does not need to rely on big data training and performs robustly in strong noise. Consequently, we can conclude that our method is suitable for SSS small target detection.

However, there are some limitations to the proposed method, the detected target must satisfy the sparse property, and some strong clutter signals or abnormal signals may have sparse characteristics similar to the target and cause false alarm in some cases. In the future, our research will focus on the robustness of detection in the presence of noise.