Review of Graph Neural Networks for Medical Image Denoising

As deep learning continues to evolve, more and more applications are generating data from non-Euclidean domains and representing them as graphs with complex relationships and interdependencies between objects. This poses a significant challenge to deep learning algorithms. Because, due to the uniqueness of graphs, applying deep learning to ubiquitous graph data is not an easy task. To solve the problem in non-Euclidean domains, graph Neural Networks (GNNs) have emerged. A graph neural network (GNN) is a neural model that captures dependencies between graphs by passing messages between graph nodes. With the continuous development of medical image technology, medical image diagnosis plays a crucial role in clinical practice. However, in practice, medical images are often affected by noise, artifacts, and other interfering factors, which may lead to inaccurate and unstable diagnostic results. Therefore, image-denoising techniques become especially critical in medical image processing. Therefore, researchers have proposed innovative methods based on graph neural networks for effective noise removal, preserving the key features of the image and improving the quality and usability of medical images. This paper reviews the research progress of graph neural networks in the field of medical image denoising. It also summarises the problems and challenges of current research and looks at the future direction of medical image-denoising research.


Introduction
With the rapid development of neural networks in recent years, deep learning has become a "jewel" in the field of artificial intelligence and machine learning [1].Many machine learning tasks [2,3] that used to rely on manual methods to extract feature information (e.g., image recognition, machine translation) have been replaced by a variety of more advanced deep learning methods.Of course, the success of deep learning in image classification [4], video processing [5], speech recognition [6], natural language understanding [7] is no accident.This is due not only to big data [8] and high-performance computing power [9] but also to the effectiveness of deep learning itself in extracting potential representations from Euclidean data [10].For graphs can be regular or irregular.A graph may simultaneously have unordered nodes of different sizes, nodes from the same graph may have different numbers of neighbors as well as each node neighborhood in the graph may be different.This leads to the fact that some operations of deep learning algorithms (e.g., convolution operations [11]) can achieve good results in the Euclidean domain, but are difficult to apply to the graph domain.Graphs are ubiquitous and widely used in the real world to represent objects and their relationships in various domains.
Examples include large-scale social networks [12], traffic networks [13], chemical molecule analysis [14], recommender systems [15], ecosystems, and so on.More and more applications rely on representing data generated in non-Euclidean [16] domains as graphs Jing Wang 2 with complex relationships and interdependencies between objects.The complexity of graph structures poses significant challenges to existing deep-learning algorithms.In recent years, there has been a strong interest in deep learning methods for extending graph data, and graph neural networks (GNNs) [17] have emerged, driven by deep learning algorithms such as convolutional neural networks (CNNs) [18] and recurrent neural networks (RNNs) [19].The emergence of graph neural networks has made it possible to apply deep learning algorithms to solve graph problems in non-Euclidean domains.Graph Neural Networks (GNN) is a deep learning algorithm based on graph structure that learns the representation of nodes and edges in a graph and achieves tasks such as classification, clustering, and prediction of the graph as a whole.Unlike traditional machine learning algorithms that need to transform the graph into vectors or matrices, graph neural networks improve the representation of graph data by performing computations directly on the graph, using the relationships between nodes.For example, in social network analysis, graph neural networks can help us with tasks such as discovering community structure and predicting user interests and behaviors.In chemical molecule analysis, graph neural networks can help us with tasks such as classification, clustering, and prediction of molecules.In recommender systems, graph neural networks can leverage relationships between users to improve recommendations.Early research on Graph Neural Networks (GNN) falls under the category of Recurrent Neural Networks (RecGNN) with high overhead.Ref. [20] introduced neural networks to directed acyclic graphs and promoted the study of GNNs.Gori, et al. [21] were the first to introduce the concept of graph neural networks.[22,23] further elaborated the concept of graph neural networks.In the last few years as the use of non-Euclidean data [24] has become more widespread, more attention has been focused on the study of graph neural networks.Wu, et al. [25] classify graph neural networks into four categories.Zhang, et al. [26] provide a comprehensive review of deep learning methods on different types of graphs.Thomas, et al. [17] categorizes graph neural networks based on their ability to process graph types and attributes.Waikhom and Patgiri [27] outline the learning approaches of graph neural networks.Zhou, et al. [28] propose a generic pipeline design for graph neural network models.There are also many research works on graph neural network learning methods.Cao, et al. [29] extract feature information in hyperspectral classification to avoid the problem of oversmoothing of message passing caused by Yang, et al. [30].As the research work progressed, methods based on contrast learning were also successful.Okuda, et al. [31] proposed unsupervised graph representation learning to discover common objects and a set of specific objects in an image for localization.The node classification and edge detection combine two learning methods, random walk, and language modeling, and the learned representation can be used for downstream tasks.
The rest of the paper is as follows: Section 2 introduces the basic structure of a graph neural network.Section 3 details the graph convolutional neural network, an important part of a graph neural network.Section 4 introduces the graph attention network.Section 5 focuses on the application of graph neural networks in the process of medical image denoising.Section 6 introduces the training and learning methods of graph neural networks.Section 7 explains the importance of denoising medical images.In Section 8, the paper is summarized and the applications and prospects of graph neural networks in medical image denoising are prospected.

Basic Structure of GNNs
Graph Neural Network (GNN) is a deep learning model specially designed for processing graph data.Different from traditional neural networks, GNN [32] can capture complex relationships in graph data and thus has a wide range of applications in the fields of social network analysis, recommender systems, bioinformatics, etc.The basic structure of GNN includes components such as graph representation, graph convolutional layer, aggregation function, activation function, graph pooling layer, and graph attention mechanism [33].Among them, the graph convolution layer is the core component, which learns the representation of a node through feature propagation from neighboring nodes.The aggregation function determines how nodes interact with their neighbors and the activation function [34] introduces nonlinearity.The graph pooling layer is used to reduce the size of the graph, while the graph attention mechanism gives different weights to different neighbor nodes to improve the network representation.Multiple graph convolutional layers [35] can be cascaded and stacked to form deep networks that are trained by backpropagation algorithms.GNNs can efficiently learn and express complex relationships in a variety of graphstructured data through the combination of these structures and components.Figure 1 shows the process of updating the information of the central node by aggregating the information of the neighboring nodes based on the message-passing paradigm.that encodes information about that node.These feature vectors are the initial representations of the nodes in the graph.Node embeddings capture attributes of nodes, such as text, images, or numeric data, which form the starting point for computation in a graph neural network.In each layer of a graph neural network, a node collects information about neighboring nodes by aggregating the embeddings of neighboring nodes [36].In different graph neural network architectures, the aggregation may be done in different ways, such as weighted summation, graph convolution, or attention mechanisms.The aggregated information will be used to update the embedding of the nodes.Aggregation functions define how to merge information from neighboring nodes.Common aggregation functions include mean aggregation (averaging), sum aggregation (summing), and more complex operations such as attention-based aggregation, where nodes weigh the importance of neighboring information based on a learned attention score.The choice of aggregation function can greatly affect the performance and expressiveness of the GNN.Common graph neural network models include Graph Convolutional Neural Network (GCN) [37,38], Graph Attention Network (GAT) [39,40], GraphSAGE, Graph Isomorphism Network (GIN), etc.This paper focuses on GCN and GAT.

Graph Convolutional Neural Networks
GCN is a type of convolutional neural network that acts directly on the graph and uses its structural information.The main idea of GCN is that, for each node, we take into account all its neighbors as well as the feature information it contains.Assuming that we use the average() function, then performing the above operations on each node gives us a representation of the average value that can be fed into the neural network.Modern GCNs mimic CNNs by designing convolution and readout functions to learn common local and global structural patterns of graphs.We begin with a discussion of the convolution operation and then move to the readout operation and some other improvements.Convolutional neural networks play a central role in building many other complex GNN models.Graph convolutional neural networks include spatial-based graph convolutional neural networks and spectral-based graph convolutional neural networks.Table 1 explains some of the notations covered in section 3.1.

Space-based graph convolutional neural network
The spatial domain-based graph convolutional neural structure mainly consists of three types of operators: neighbor sampling, message computation, and message aggregation.The neighboring nodes represented by a particular node are aggregated using the aggregate operation in GCN as a way to achieve message passing between nodes.Figure 2 represents the passing of node messages in spatial domain-based GCN.The simplest aggregation process is to do the product operation between the node characteristics(x) of the graph and the topological structure information of the graph (adjacency matrix E).The specific process is shown in Figure 3.The spatial approach is to define the convolution directly in the spatial domain, the problem faced is that, because each node's neighbors are not the same size, it is not possible to define a neighborhood of the same size, so the implementation of parameter-sharing faces greater difficulties, but the idea is still, convolution is still a node in the neighboring nodes on the weighted average, so the subsequent many methods aimed at solving the problem of parameter sharing.

Spectral-based graph convolutional neural networks
Convolution based on spectral methods is a special case of convolution based on spatial methods.Spectral domainbased graph convolution via neural network investigates the properties of the graph with the help of eigenvalues and eigenvectors of the Laplace matrix of the graph.Filters are introduced to define convolution from a signal-processing perspective.Firstly, multiplication is done on the signal in the spectral domain using the theorem of convolution.Secondly, the convolution is implemented by transforming the signal into the original space using the Fourier transform [41].This approach avoids the problem of difficulty in defining convolution caused by the fact that the graph data does not satisfy translation invariance.Because the structure of the graph does not satisfy the translation invariance, it is not possible to define the convolution directly in the spatial domain, so the signal is transformed to the frequency domain, the convolution operation is implemented in the frequency domain and converted back to the spatial domain, which is the spectral method.
Graph convolutional neural networks based on spectral methods assume that the graph is undirected.The normalized graph Laplace matrix is the mathematical representation of the undirected graph and is defined as: where  denotes the diagonal matrix of node degrees.  = ∑ ( , )  ,  denotes the adjacency matrix of the graph.Using the property of symmetric positive semidefinite of the graph Laplace matrix, the normalized Laplace matrix can be decomposed as: where  = [  ,   , ⋯ ,  − ] ∈  × is the Feature vector matrix,  is the diagonal matrix of eigenvalues (spectrum),   =   .The feature vectors of the regularised Laplacian matrix form a set of orthogonal bases.In graph signal processing, the signal of a graph  ∈   is a feature vector consisting of the individual nodes of the graph,   representing the  th node.The Fourier transform and Fourier inverse transform of a graph G are defined as: ( ) , where,  � is the result of the Fourier transform.To better understand the Fourier transform of a graph, we can see from its definition that it does project the input graph signal into an orthogonal space whose base is made up of the eigenvectors of the regularised graph Laplacian.The elements of the transformed signal are the coordinates of the graph signal in the new space so that the original input signal can be expressed as: This is the result of the Fourier inverse transform.Next, we can define the graph convolution operation on the input signal .

𝑋𝑋 * 𝐺𝐺 𝑔𝑔
where,  ∈   is the filter we define; ⊙ Indicates the Hadamard product.Suppose we define such a filter: ( ).
T g diag U g Θ = (7) Thus, the graph convolution operation can be represented in a simplified way as: Spectral-based graph convolution networks all follow this pattern, with the key difference between them being the choice of filters.Existing spectral-based graph convolution network models are the following: Spectral CNN, Chebyshev Spectral CNN (ChebNet), and Adaptive Graph Convolution Network (AGCN).

Graphical Attention Network
Graph Attention Network(GAT) [27] consists of many functionally identical blocks (Graph Attention Layer).Its properties include high efficiency, low storage type, inductive learning, and full graph access.The graph attention layer has a feature value of ℎ �⃗ = �ℎ �⃗ 1 , ℎ �⃗ 2 , ⋯ , ℎ �⃗  � , ℎ �⃗  ∈ ℝ  for the node at input.where  represents the number of nodes and  represents the dimensionality of the node features.After a Graph Attention Layer, a new feature vector is output, which can be represented as , assuming that the dimension of the node feature of this feature vector is  ′ .As shown in Figure 5.

Figure 5: Attention layer in GAT
The purpose of using Self-attention is to improve the expressiveness of ℎ ′ ��⃗ .In the Graph Attention Layer, a weight matrix  ∈ ℝ  ′ × is first applied to each node using a weight matrix, and then self-attention is used for each node to calculate an attention coefficient, the shared self-attention mechanism used here, denoted as: represents the importance of node  for node .In theory, we can calculate the weight of any node in the graph to the central node.In GAT, to simplify the calculation, the nodes are restricted to the one-hop neighbors of the central node, and in addition, the nodes take themselves into account as neighboring nodes.In the existing studies, there are many ways to choose.For example, choosing a singlelayer feedforward network with parameter  ⃗ ∈ ℝ 2 ′ and then using LeakyReLU to do a non-linearisation gives.
To improve the generalization of the attention mechanism, GAT chose to use a multi-headed attention layer, i.e. a single-headed attention layer from a set of K mutually independent graph attention layers, and then stitch their results together.At this point, ℎ  ′ is: where ∥ represents the splice operation,   () represents the weight factor calculated from the  th group of attention mechanisms, and   is the weight factor of the  th module.To reduce the dimensionality of the feature vector, we can also use the averaging operation instead of the splicing operation, as shown in the following equation.

Application of graph neural networks in medical image denoising
Graph neural networks have a wide range of potential applications in medical image denoising.Medical images [42] usually have complex structures and are highly noisy, e.g., X-rays [43,44], CT scans [45], MRIs [46], etc., which must be of high quality in diagnosis and analysis.The main application directions of graph neural networks in medical image denoising [47,48] include image noise reduction, image enhancement, motion artifact removal, data recovery, super-resolution reconstruction, and sequence image denoising.These applications will be specifically described below.Figure 6 is the comparison of medical images before and after denoising.
After denoising image Before denoising image (v) Super-resolution reconstruction.Graph neural networks can be used for super-resolution reconstruction of medical images, i.e., recovering high-resolution images from low-resolution images to improve the clarity and detail of the images.(vi) Sequence image denoising.In medical images, some data exists in the form of sequences, such as timesequenced images in magnetic resonance imaging (MRI).Graph neural networks can be applied to these sequential data for denoising in the temporal and spatial domains to improve the quality of the images.
In conclusion, graph neural networks can better cope with the complex structure and highly noisy features of medical images by learning the relationship between pixel points or pixel blocks in medical images, providing clearer and more accurate images for medical imaging diagnosis.

Training and Learning Methods of GNNs
The training and learning of graph neural network (GNN) is similar to that of traditional neural networks, but since graph neural networks are trained on graph-structured data, the following special factors also need to be considered.
Forward propagation and backpropagation.During forward propagation, the GNN passes information through the graph.In each GNN layer, nodes aggregate information from their neighbors based on aggregation functions (e.g., mean, sum, maximum).This aggregated information is used to update the node embeddings [49][50][51].The GNN uses gradient-based optimization techniques such as backpropagation to learn the model parameters, compute the loss function (a measure of the error between the predicted value and the actual value), and update the model's weights by backpropagating the gradients through the graph neural network layers.
Initialization and parameter learning.GNNs usually start with random or pre-trained node embeddings, which are used as initial node representations.Meanwhile, GNNs learn parameters through training to improve the model's effectiveness.Commonly used parameter optimization algorithms include stochastic gradient descent (SGD) algorithm [52] or its variants, etc. Figure 7 illustrates the stochastic gradient descent.
Semi-supervised learning.GNN [53] is very effective in semi-supervised learning environments where only a small fraction of the nodes in the graph are labeled.During training, the GNN makes predictions using both labeled and unlabelled nodes.Unlabelled nodes benefit from the information propagated through the graph, which helps to improve generalization.The loss function typically consists of a supervised loss term for the labeled nodes and a regularisation term, which encourages smoothness in the embedding of neighboring nodes.The regularisation term ensures that nodes with similar network neighborhoods have similar embeddings.Overfitting and regularisation.Like other neural networks, GNNs can suffer from overfitting problems, especially when the graph is small or the model is too complex.To prevent overfitting, regularisation techniques such as dropout and L2 regularisation can be used.In addition, aggregation functions (e.g.attention-based mechanisms) in the GNN architecture can weigh neighboring nodes so that they learn to focus on important neighboring nodes, which can be used as a form of implicit regularisation.
Graph Convolution and Aggregation Functions.The choice of aggregation function is crucial in GNN training.Different functions lead to different results in information propagation.Researchers often experiment with various aggregation functions to find the best fit for a particular task.Architectures such as Graph Convolutional Neural Networks (GCN), Graph Attention Networks (GAT), and GraphSAGE use specific aggregation functions that are effective in different scenarios.Learning the parameters of these functions is an important part of GNN training.

Importance of Medical Image Denoising
Medical image denoising plays a vital role in all aspects of healthcare and medical research, Table 2 describes several common methods of denoising medical images.The importance of denoising medical images includes several aspects.
Improved diagnosis and treatment.Medical images such as X-rays, CT scans, MRI scans, and ultrasound images are essential tools for medical professionals to diagnose diseases and develop treatment plans.Noise in these images can distort important features, making it difficult to detect subtle anomalies.By reducing noise in medical images through denoising technology, medical practitioners can obtain clearer and more accurate anatomical structures, lesions, and other pathological features.This will lead to more confident diagnoses, better treatment planning, and ultimately improv ed patient outcomes.

Total Variation Denoising
Nonlinear denoising method for smoothing images, preserving edge information.

K-SVD Denoising
Based on dictionary learning, the image is decomposed into linear combinations of bases, and noise is removed.
Reduce radiation exposure.In the case of X-rays and CT scans, denoising medical images helps to reduce the required radiation dose.Reducing the amount of radiation is essential for patient safety, especially for patients who require frequent imaging, such as cancer patients undergoing radiation therapy.Denoising algorithms can improve the quality of low-dose images, making them diagnostic while minimizing the potential risks associated with radiation exposure.This ensures that patients are not unnecessarily harmed while undergoing the necessary Telemedicine and tele-consultation.With the rise of telemedicine and remote consultations, especially in underserved or remote areas, denoising technology can help improve the quality of transmitted medical images.Telemedicine relies heavily on the accurate interpretation of images by remote medical specialists.Removing noise from images before transmission ensures that telemedicine providers make accurate diagnoses and treatment recommendations.This is especially important in emergency cases and in situations where specialized medical services are limited.

Conclusions
This paper describes graph neural networks and the application and impact of graph neural networks in medical image denoising.With the continuous deepening of research, graph neural network technology is also developing, and new models and algorithms are emerging.Therefore, we can foresee that the future research and application of graph neural networks in medical image denoising will be more extensive and in-depth.However, the current application of graph neural networks in medical image denoising also faces many challenges.For example, (1) insufficient data volume and labeling difficulties.Medical image [54] data is usually relatively limited, and labeling medical image data is a very expensive and timeconsuming task.Therefore, the lack of large-scale and high-quality labeled data is a challenge, especially when a large amount of labeled data is required in deep-learning models.(2) Complex noise types and distribution.There are various types of noise in medical images, including random noise, artifacts, motion artifacts, etc., and the distribution may be very complex.Different types of noise require different processing methods, and it is a challenge to design graph neural networks that can handle various types of noise.(3) Computational complexity.Graph neural networks usually require a large amount of computational resources, especially for complex graph structures and large-scale medical image datasets.In practical applications, it is necessary to consider how to efficiently train and deploy these models in environments with limited computational resources.In the future, research on the application of graph neural networks in medical image denoising should be considered in the context of these issues.

Figure 3 :
Figure 3: Information aggregation process of GCN based on spatial domains

Figure 4 :
Figure 4: Comparison of Recurrent-based and Composition-based method in spatial-based GCNs

Figure 6
Figure 6 Comparison of images before and after denoising

Figure 7 :
Figure 7: Stochastic Gradient Descent (SGD) medical imaging.Contribute to research and training.Denoising medical images is critical to medical research because clean and high-quality images are needed to study diseases, develop EAI Endorsed Transactions on e-Learning | Volume 9 | 2023 | new treatments, and evaluate the effectiveness of medical interventions.Medical professionals and researchers also use denoised images for educational and training purposes.Clear images are essential for the teaching of medical students and for the training of medical practitioners in image interpretation and surgical planning.

Table 2 :
Medical image denoising methods