1. Introduction
With the rise in global warming and technological growth, the Arctic has become a strategically important region for countries around the world and is of great importance in terms of economic and trade flows, military, and tourism [
1,
2,
3]. However, Arctic sea ice, an important factor in the global climate system, is steadily melting [
4,
5], which affects the ecological and climatic environment of the Arctic region [
6,
7,
8]. From other perspectives, the melting of the Arctic sea ice also facilitates the opening of the Arctic corridor and helps to save trade miles and navigation costs, which has important military and trade implications [
9]. In addition, the Arctic is rich in fisheries and mineral resources [
10], and thus the opening of the Arctic passage facilitates countries with a high demand for energy to research and exploit the Arctic resources. Consequently, to better plan the Arctic passage and Arctic resource investigation routes, it is particularly important to monitor the distribution of sea ice efficiently and in real-time.
Human monitoring of the Arctic sea ice has a long history, starting from early sledding expeditions [
11], continuing to aircraft mapping [
12], progressing to the use of submarines, icebreakers [
13], unmanned equipment [
14], and finally to advanced satellite remote sensing technology [
15]. Currently, satellite remote sensing technology is the most efficient way to monitor sea ice, and it has advantages of wide-range detection, real-time, and sustainability differing from traditional methods. It is worth affirming that synthetic aperture radar (SAR) is generally not affected by weather disturbances or cloud cover, which greatly enhances its applicability in acquiring remote sensing imagery [
16,
17]. However, compared to SAR, optical imagery has some unique advantages in polar research. First, optical remote sensing is one of the most direct methods for satellite remote sensing technology and is able to obtain visible light images and panchromatic images with high spatial resolutions that provide clear spatial texture information of the observation target [
18,
19,
20], which is instrumental in the intuitive interpretation of remote sensing images. Second, the acquisition and processing of optical imagery are relatively more straightforward to operate and analyze. Optical images can be directly obtained for free from satellite platforms. It is worth pointing out that acquiring SAR images often involves significant costs. In the processing of SAR images, due to their unique properties, such as speckle noise and polarization characteristics, these images requires higher computational power and specialized expertise to ensure the accuracy and reliability of the image results. Although the use of optical imagery limits the creation of time series, in the Arctic summer, abundant sunlight and less cloud cover make optical imagery an ideal data source for observing sea ice. Furthermore, due to the intuitiveness of optical imagery, it has unique advantages for real-time monitoring and navigation planning [
21]. Based on this, our study aims to propose a rapid and accurate identification method, contributing to the planning of navigation routes and the monitoring of maritime emergencies. Generally speaking, researchers usually adopt three methods for identifying Arctic sea ice: thresholding, texture analysis, and machine learning.
Thresholding initially employed the adaptive global thresholding method that calculates a global threshold by analyzing the grayscale data of the entire image followed by categorizing the pixels into black and white. However, using this method for sea ice images under various lighting conditions is inappropriate, resulting in imprecise segmentation outcomes. To overcome the limitations of global adaptive thresholding, D. Haverkamp et al. [
22] introduced the local adaptive thresholding method. This method leverages the neighborhood information of every pixel to calculate its local threshold and categorizes pixels into black and white accordingly. In contrast to the global thresholding method, the local adaptive thresholding method can precisely identify regions of sea ice. Later, Alexander S. Komarov et al. [
23] achieved commendable experimental results by detecting ice and open water areas in RADARSAT-2 images using adaptive probability thresholding.
The early texture analysis method was based on the gray-level co-occurrence matrix (GLCM) method. By calculating the grayscale value differences between pixels in the image, this method assesses the texture characteristics of diverse regions. Unfortunately, the GLCM-based method is computationally intensive and requires significant data preprocessing, making it impractical for large-scale remote sensing of sea ice. To overcome the drawbacks of the GLCM-based method, ref. [
24] presented an overview of the one-dimensional discrete wavelet transform method leveraging Daubechies wavelet filters represented by finite vectors and matrices. The method takes into account the scale-dependent wavelet variance and covariance and determines their respective confidence intervals. The wavelet transform-based method can handle large-scale remote sensing images of sea ice more quickly than the GLCM-based method.
Numerous machine learning techniques employ hand-crafted feature identification, such as wavelet transform for texture feature identification from images, and using conventional classifiers like support vector machine and decision tree for image segmentation [
25]. These methods necessitate manual feature identification, which is subjective and has certain limitations. Advancements in deep learning technology have facilitated the inclusion of machine learning techniques based on convolutional neural networks (CNN) in remote sensing image processing of sea ice. After using substantial remote sensing images and corresponding labelled data to train CNN models, the model is then deployed to categorize new remote sensing images [
26]. During the development of machine learning techniques, researchers gradually identify the discrepancy problem between remote sensing image data and ground truth label data. Hence, unsupervised learning methods have emerged, such as generative adversarial networks-based techniques [
27,
28]. These techniques can categorize remote sensing images of sea ice, even in the absence of labeled data.
Although these three methods have achieved constructive results in the identification of Arctic sea ice, there are still some shortcomings. For example, threshold-based methods are not good at distinguishing thin ice from seawater and are easily affected by light interference. In addition, texture analysis-based methods require classification of different types of sea ice, which requires professional knowledge reserves and experience in acquiring remote sensing images, limiting the technical application scope of such methods. Furthermore, although machine learning methods have achieved good results in image training, they require a large amount of training data and computational resources, as well as the design of professional models, which inevitably consumes a lot of time.
In conclusion, the efficient and accurate identification of Arctic sea ice from remote sensing images is a pressing issue. To approach this problem, a constrained energy minimization (CEM) method is considered [
29,
30,
31], which solely requires the target spectrum without the need for image background information. By employing predetermined constraints, a finite impulse response filter is developed to enable the desired target signal to pass through, while effectively suppressing output energy introduced by secondary signals. CEM is highly versatile and efficient, making it an ideal approach for numerous remote sensing image processing tasks. It is frequently used for identifying various target objects such as land-cover [
32], farmland [
33], coastal change [
34], and so on [
35,
36,
37]. These applications have all yielded impressive results. Despite the CEM method being effective at target identification, it is still susceptible to ambient noise, including unavoidable noises generated from working devices. Consequently, identifying targets using CEM is still restricted by the working environment. To address this shortcoming, this paper proposes utilizing a model that possesses noise immunity to enhance the effectiveness of the CEM method for Arctic sea ice identification.
The CEM method can be transformed into a linear constrained optimization problem. Neural dynamics (ND) shows satisfactory performance in solving optimization problems, such as fast computational speed and good convergence, and has been studied by many scholars [
38,
39,
40,
41]. For example, it can be applied to the motion generation and control of redundant manipulators [
42], the tracking control of chaotic systems [
43], and even to biological population control [
44]. In practical applications, continuous models need to be discretized before being applied to hardware devices. This is different from most of the existing ND models that apply Euler difference formulas to discrete continuous models [
45,
46,
47].
This paper applies a Taylor numerical difference formula [
48] for discretization, which enables its steady-state error to reach
smaller than that of Euler difference formulas with
being the sampling period. Additionally, the absence of noise is always a prerequisite for the implementation of some models, but the presence of noise is unavoidable in practical working situations. To this end, a neural dynamics model with an integration term using the Taylor numerical difference formula for discretization, named error-accumulation enhanced neural dynamics (EAEND) model, is proposed to improve the stability and accuracy of the CEM method for Arctic sea ice identification. Thereafter, relevant convergence analysis and stability proofs are provided, which theoretically illustrate the convergence and noise immunity of the CEM method with the proposed EAEND model in noisy environments. Moreover, to assess the effectiveness of the CEM method integrated with the EAEND model for identifying Arctic sea ice, a series of comparative experiments are conducted on various sea ice observation images abiding by different noise environments. The outcomes highlight the superior noise immunity and benefits of incorporating the EAEND model.
Figure 1 illustrates the flowchart representing the CEM method’s process for Arctic sea ice identification assisted by the EAEND model. Specifically, this paper transforms the CEM scheme into an optimization problem. Subsequently, the EAEND model proposed in this paper is used to solve the optimization problem. To minimize the error and obtain an optimal filter coefficient for Arctic sea ice identification, an error function is constructed and the ND formula with an integral term is introduced. The error is then minimized using error-accumulative enhanced neural dynamics. Thus, the Arctic sea ice can be successfully identified.
In the subsequent sections of this paper, the organization is as follows.
Section 2 provides the model construction of the EAEND model;
Section 3 gives the model simplification as well as theoretical proofs for the convergence and robustness of the EAEND model;
Section 4 and
Section 5 present comparative identification experiments of different Arctic sea ice remote sensing images, while the proposed EAEND model is compared with other models; in
Section 6, the experimental results are discussed and analyzed.
Section 7 summarizes the whole paper. The contribution points of this paper are provided below.
The proposed EAEND model, using a Taylor numerical difference formula to enable steady-state error to reach , is capable of improving the Arctic sea ice identification accuracy of the CEM method.
The proposed EAEND model eliminates the computing error by using an error integration term, and therefore effectively suppresses noises and improves the solution accuracy, which is able to assist the CEM method in identifying Arctic sea ice in noisy environments.
Theoretical analyses and comparative experiments confirm that the CEM method with the proposed EAEND model has high accuracy and strong stability.