Denoising Method for MRI Images Using Modified BM3D Filter with Complex Network and Artificial Neural Networks

Noise is an undesirable and disturbing efect that degrades the quality of an image. Te importance of noise reduction in images and its wide-ranging applications are essential. Most popular image noise flters rely on static parameters that are often challenging to fne-tune. Dynamically adapting these static parameters for image noise flters is a critical area of research. In this study, a combination model between the features of complex networks and artifcial neural networks is proposed to automatically fnd the noise reduction parameter of the block-matching and 3D fltering method. Experimental results on the black and white MRI image set have shown that the model correctly predicted the parameters of the BM3D flter and removed the noise in the images of those MRI images. Te model gave high denoising results with PSNR of 51.94 and SSIM of 0.998.


Introduction
Due to the inherent physical limitations of diferent recording devices, images tend to have random noise during image acquisition.Noise can be understood as the basic signal distortion that hinders the process of observing images and extracting information.With the dramatic increase of digital image generation often taken in low light conditions, image restoration methods have become indispensable tools in the era of image analysis with computer support.Additionally, noise can manifest in the image for various reasons, including fuctuations in probe sensitivity, alterations in the environment, inherent material properties, quantization errors, etc.
In MRI images [1], noise may come from the movement of the patient during the MRI or from the error of the MRI machine.Noise is a huge challenge in the feld of medical imaging research because it reduces important image attributes, making it difcult to diagnose and treat medical examinations.
In the feld of machine learning, specifcally in the process of classifying MRI images or segmenting MRI images using machine learning algorithms, image noise will distort and reduce the quality of machine learning algorithms.Terefore, noise removal is very important in the image preprocessing step for further works.In recent years, noise flters have been widely used in processing MRI images of the human brain [2].Te BM3D flter [3] is a very good performing image noise flter.One drawback is that the flter needs to consider the input parameter.However, the input parameter is very difcult to adjust.It is usually randomly generated, and it is not known what the best value for the input image is.
Complex network [4] was defned as a network that comprises nodes with intricate properties.In order to form a complex network, a large amount of information is needed to fully describe the topology of all network elements.Terefore, complex networks are expected to have outstanding advantages over conventional networks.In real life, complex networks can be used to describe many diferent real-world networks such as social networks, networks of neurons in the brain, biological networks, etc.
By far, the BM3D Filter is one of the most powerful traditional flters.Filter inputs are random parameters, which makes it difcult to fnd input parameters so that the fltered image has the best denoising results.Currently, manually fnding these parameters for the flter takes a lot of time and the efciency is not high.Tis study focuses on developing a method to automate the process of fnding the noise reduction parameter of the BM3D flter [3].In which, the core of the research is the proposal of a new model which is a combination between the features of complex networks and Artifcial Neural Networks (ANN) [5].Te complex network plays the role of extracting features of each brain MRI image.Based on this feature, we can fnd the input parameter of the BM3D flter for the corresponding brain MRI image, so that the PSNR index between the fltered brain MRI image and the original image has the highest result.To evaluate the efectiveness of the proposed method, the two most important parameters in image noise processing, peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM), are used.Te experimental results will also be carefully evaluated and compared with the results of previous studies.
For chronological techniques, the study [6] utilized a self-adjusting parameter GANs network to facilitate the process of extracting smooth edges of noisy digital images, thereby enhancing the actual signal in high-frequency components where the main parts of pixels without noise can be considered as noisy pixels.However, the attempt to remove unwanted noise from the tested images may result in excessive smoothing of the obtained images.Additionally, article [7] proposed a method for rapidly and accurately removing mixed noise by combining pulsecoupled neural network (PCNN) and Perona-Malik equation normalization (P-M equation) to eliminate unwanted noise.
Te article is organized as follows.Section 2 presents previous work on denoising brain MRI images.An overview of brain MRI images and proposed model are introduced in Section 3. Experiment results are described in Section 4. Conclusion and development direction are in Section 5.

Related Work
Some methods of denoising MRI images from 2019 up to now are summarized in Table 1.Image noise fltering technologies today are quite diverse, ranging from the use of basic image noise flters such as bilateral flters, median flters, or the combination of basic noise flters with machine learning algorithms to optimize flter parameters.Tere is also a novel approach that involves using pure algorithms for noise fltering.Te common goal of these algorithms is to eliminate various types of noise such as Gaussian, Rician, Shrinking, Free, etc., from MRI images.
Te study of Chang et al. [8] was diferent from the rest of the studies.Tis study focused on the problem of fnding the best input parameter for bilateral flter with MRI image dataset without random selection of parameters like other traditional flters.Results with Gaussian noise at 1% for the highest PSNR index of 39.29, for the highest SSIM index of 0.983.Although the results were not high, this was a new idea, and the system implementation time has been signifcantly reduced.
Te authors in Tripathi et al. [9] proposed a CNN-DMRI model to be trained on a noisy MRI dataset using an automated method to generate training data pairs.Tis model used convolutional layers and ReLU activation layers to learn hidden characteristics of MRI images and produced denoised MRI images, resulting in the highest PSNR index of 43.18, the highest SSIM index of 0.987.
Moreno López et al. [10] used the model using Unsupervised Learning to train noisy and noiseless MRI image datasets.Since then, the features of noisy MRI images have been found to eliminate noise.Te study obtained the highest PSNR index of 38,015, the highest SSIM index of 0.8977.
Te authors in the study by Sreelakshmi et al. [11] proposed a method that combines model of adaptive median flter (ADMF) and convolutional neural networks (CNN) to solve the noise problem in MRI images.Tis method also uses a machine learning system based on Gradient Boosting Machine Learning (GBML) algorithm to classify and extract features from MRI images, thereby helping to improve the quality of MRI images.Te results obtained in that study were very good with the PSNR obtained for Gaussian noise at 50% level was 48.68, with Shrinking noise at 10% was 68.85.
Te study by Wang et al. [12] focused on noise processing methods for magnetic resonance imaging (MRI) images based on two main techniques, namely Nonlocal Structural Similarity and Low-Rank Sparse Representation.Te image data in this study was obtained from the Brainweb 3D T1-weighted dataset.Rician noise was used in the experiment, with a Rician noise ratio set at 4%. Te evaluation results of image quality after noise processing showed a PSNR score of 38.503 and an SSIM of 0.976.Te high PSNR score and SSIM close to 1 indicate that the noise processing method has achieved high performance in noise reduction and preservation of essential information in MRI.
Te study by Mehta et al. [13] focused on noise processing for MRI images using the U-net architecture and image processing techniques.Te image data in this study consists of 253 MRI images.Gaussian noise with a noise level of 25% (Gauss 25%) was used.Te Peak Signal-to-Noise Ratio (PSNR) score was used to evaluate the image quality after noise processing, and the result showed a PSNR of 30.96.
Te most recent study by Kollem et al. [14] introduced a novel method using the difusivity function to process noise for medical MRI images.Te image data in this study are medical MRI images and are assumed to be originally noise-free images but with added Poisson noise.Te evaluation results of image quality after noise processing showed a Peak Signal-to-Noise Ratio (PSNR) score of 42.78 and a Structural Similarity Index (SSIM) of 0.99645.

Materials and Methods
3.1.MRI Image.Magnetic Resonance Imaging-MRI is a technique that uses magnetic felds, radio waves, and computer systems to produce images of parenchymal structures that are clearer and more detailed than conventional diagnostic methods.Based on MRI images, doctors can detect abnormalities in the brain parenchyma in general, as well as vascular tumors, arterial occlusions, invasion of the venous sinuses, as well as the relationship between the tumor and the around structures.MRI has two basic pulse sequences, T1 Weight and T2 Weight.As in Figure 1, a T2weighted image is presented.In addition, there are other pulse sequences such as PD or FLAIR.In general, the efciency of MRI is very high, especially in diagnosing brain tumors.

Gauss Noise.
Tere are many sources of noise in an image and these noises come from many diferent aspects such as image acquisition, transmission and compression.Mathematically, a noisy image v(x) is expressed as the sum of the original, unnoised image, u(x) and the noise function η(x), as described by the following formula: Te goal of noise reduction methods is to reduce noise in natural images while minimizing the loss of original features and enhancing the signal-to-noise ratio (SNR).
Gaussian noise [15] is a popular model for approximating noise in many diferent applications.Te probability distribution density of the noise is a Gaussian function, which is characterized by the mean μ and the variance σ 2 .
where p(z) is the equation for the distribution of Gaussian noise in the MRI image; μ and σ are the mean and standard deviation, respectively.

Denoise Algorithm-BM3D. Block matching and 3D
fltering-BM3D [3], was proposed by K. Dabov, is the most advanced algorithm available for noise reduction.BM3D algorithm uses block-matching technique to detect and search for similar blocks in the image.Ten, these blocks are grouped into groups with similar properties, and 3D fltering is applied to reduce noise for each group.3D fltering technique uses spatial information and frequency information of blocks to efectively reduce noise.Tis helps the BM3D algorithm to achieve good noise reduction performance and low computational complexity, suitable for processing high noise images.BM3D has been widely used in photo and video processing applications, from image noise reduction to video restoration or image compression.Tis algorithm has been improved and developed with new versions to improve accuracy and processing speed.
In the most recent study by Mäkinen et al. [16], it has been shown that one of the most important input parameters of the BM3D flter is the power spectral density standard deviation of the image noise which is denoted by σ psd .Tis parameter σ psd is in the range [0, 1].

Complex Networks.
Complex network has been successfully applied in many felds [17].Tere have been many studies successfully using complex networks by building graphs on image processing problems.In previous research by Lima et al. [18] on image processing using complex networks, the study analyzed basic graph features such as Degree, Centrality, Communities.From these basic features of the graph, the author modeled the image again, thereby giving the basic features of the image.
An image can be described and graphed based on color patterns, textures, and image shapes.An undirected graph G � (V, E) consists of V being the set of non-empty vertices, and E being the set of unordered pairs of diferent elements of V called edges or connections between two pixels i and j.Te features of the graph can be mentioned as follows: (1) Vertex degree: Degree of a given vertex i is the number of vertices connected to it (it's neighboring vertices).
(2) Average degree: Te average degree (∅ μ ) is the sum of the number of edges of the graph divided by the number of vertices of the graph.(3) Average minimum path: Te average minimum path is the average of all the minimum paths of the network.(4) Mean centrality: Te central mean is a measurement that represents the mean of the central peaks (the peaks that matter to the minimum paths of the network).( 5) Number of communities: For a graph G(V, E), a community in this graph is a subgraph G′(V′, E ′ ) in which the vertices are strongly connected.Tere are many ways to measure a subgraph because there are diferent defnitions of community structure.Te most accepted defnition is the one that requires all nodes of the community to be interconnected.Tis leads to the defnition of a cluster.A cluster is the densest subgraph between three or more vertices, meaning that each vertex of the graph needs to be connected to another vertex, in such a way that there are no other adjacent vertices between them.(6) Entropy of subgraphs: Quantify the randomness of the subgraphs G ′ generated in the graph G.
From the above features of the graph, we obtain the features of the image graph, shown in Table 2.

ANN.
An artifcial neural network (ANN) is a type of machine learning model that is inspired by how the human nervous system works.An ANN is a network of nodes (neurons) connected by weights.Te nodes represent inputs and outputs, and the weights represent the importance of each input in computing the output.ANN consists of 3 main components: Input layer, output layer (Tey only include 1 layer) and hidden layer (this layer can have 1 or more layers depending on the specifc problem).ANN works in a way that describes how the nervous system works with its interconnected neurons.
In ANN, except for the input layer, all the nodes of the other layers are fully connected to the nodes of the previous layer.Each node of the hidden layer receives the input matrix from the previous layer and combines it with the weights to get the result.Te purpose of ANN is to learn and automatically fnd complex relationships in the input data by adjusting the weights.ANN networks are capable of learning and self-adjusting weights based on known input-output data pairs, through training.After being trained, ANNs can be used to predict and classify new data.
3.6.Metrics 3.6.1.PSNR.Peak signal-to-noise ratio (PSNR) [19] is a technical term that compares the maximum potential signal power to the power of undesirable noise, afecting its quality.PSNR is determined through mean square error (MSE), for 2 monochromatic MRI images I and K, in which I is the noisy MRI image and K is the original MRI image, MSE is calculated by the following formula: where, m is the number of rows of the pixel matrix; n is the number of columns of the pixel matrix; i is the row index, representing the i th row of the pixel matrix; j is the column index, representing the j th column of the pixel matrix.Terefore, PSNR is defned as follows: where MAX is the maximum pixel value of the MRI image, usually 255 in the baseline MRI.
3.6.2.SSIM.Te SSIM index [19] was used to measure the similarity between two diferent images, specifcally in this study, the initial MRI image and the MRI image after noise fltering.Te SSIM formula is based on three parameters for comparison: luminance, contrast, and structure.Te formula is shown as as follows: where μ x , μ y , σ x , σ y and σ xy are the mean, standard deviation and covariance of x and y images, respectively, and C 1 , C 2 are constants.
3.7.Proposed Model.Te recommendation system consists of two phases: training phase and testing phase, the purpose of the model is to automatically fnd the best σ psd parameter of the BM3D flter for the input image.Te procedure is described as fgures below.
Figure 2 shows the training process to fnd the parameter value σ psd .From the input MRI image, after adding Gaussian noise at 50% level, we get a noisy MRI image.Next, by using a complex network to model the noisy MRI image, 5 features of the image are extracted.Along with this step the best parameter σ psd for the BM3D flter will be estimated.Te PSNR mentioned in Figure 2 shows the correlation between the pixels of the noisy MRI image and the noise-free MRI image.Tis index will be estimated through the parameter σ psd in the range [0, 1].Te σ psd value which gives the highest PSNR, is the best value for that noisy MRI image.In the above diagram, the ANN network is used.Te network is a sequential neural network designed for regression tasks.It begins with a dense layer of 1024 neurons, this is followed by multiple dense layers with decreasing numbers of neurons: 512, 256, 128, 64, and 32, each with "relu" activation, improving the model's ability to learn complex patterns.Te fnal layer has a single neuron, with "he_uniform" initialization, designed to output a continuous value.
Te essence of the problem here is the use of back propagation algorithm [20] and neural networks to increase the accuracy of the training process, thereby obtaining the ANN model.Five features [X 1 , X 2 , X 3 , X 4 , X 5 ] of each noisy MRI image will be extracted by the complex network.Tey are the input to the Back Propagation algorithm and the output is the best parameter σ psd found above, called [Y].In the back propagation algorithm, input values are passed through the neural network to compute the output.Ten the error between Table 2: Some features of images using complex networks.

Features Average degree Mean centrality Number of communities
Average minimum path Entropy of subgraphs the actual output and the predicted output is calculated, the inputs [X 1 , X 2 , X 3 , X 4 , X 5 ] will be respectively with an output of a certain [Y].Te algorithm then propagates the error back from the output layer to the input layer, calculating the partial derivative for each weight in the network.Te weights are then updated using an optimization algorithm such as gradient descent to minimize the error.Terefore, each noisy MRI image has its own characteristics, from which these particular features will correspond to a best noise flter parameter.Te ANN model is used in Figure 3 to test and evaluate the above training process.
Te proposed model in this study is a general model, not only applicable to MRI brain image data but also can be applied to various other image datasets.However, each image dataset has a diferent data distribution, so applying the proposed model needs to be customized to each type of image data to achieve the best results.

Data Collection.
In this study, the dataset was collected from 230 brain tumor patients at Bach Mai Hospital, Hanoi, Vietnam, covering all age groups.Te dataset consists of various magnetic resonance imaging sequences such as T1, T2, T1ce, and T2 Flair.To perform the model described above, this research utilized T2 MRI sequences, which efectively depict the characteristics of white matter and gray matter.Te dataset used includes 500 T2 pulse sequence MRI images of the human brain.Tey are in jpeg format with dimensions of 256 × 256 pixels.Te set of images is divided into 3 sets, the training set includes 300 images, the validation set includes 100 images and the test set includes 100 images.

Results of the Training Process.
To evaluate the efciency of the model training process, in this paper, the Mean Absolute Error (MAE) [21] parameter is used to calculate the loss function for the above proposed ANN model.Mean Absolute Error measures the average magnitude of errors in a set of predictions without considering their direction.It is the mean over the sample of the absolute diference between the prediction and the actual observation.Tis factor is calculated as follows: where n is the number of data points, x i is the actual value (real value of σ psd ), and y i is the predicted value (predicted value of σ psd ) of the model.Te loss function of the proposed model is illustrated in Figure 4. Te model was trained over 300 epochs with an initial learning rate of 0.01. Figure 4 shows that the model converged to its lowest loss value after 200 epochs.

Te Results of Image Denoising by the Proposed Model.
With the proposed model combined with a training dataset of 300 MRI images, this proposal can denoise any other brain MRI images with a very good PSNR result.Test results with 10 MRI images with Gaussian noise at 50% are shown in Table 3. Experimental results show that the PSNR results of 10 images are quite high, the average is 51.83.From the results shown in Table 3, it can be seen that the best parameter values of σ psd are in the range [0.3 -0.5].
Figure 5 displays the outcomes of denoising an MRI image corrupted by 50% Gaussian noise.Figure 5(a) represents the original MRI image, Figure 5(b) depicts the noisy MRI image, and Figure 5(c) showcases the image after denoising with our proposed method.Clearly, from these images, it is evident that our method excels in restoring noisy images.Distinguishing between Figures 5(a) and 5(c) with the naked eye is quite challenging.

Calculation Time. Using our proposed model in
Gaussian denoising for MRI images has also reduced the processing time relative to some other methods.Finding the best σ psd value is the process of denoising the MRI image for the highest PSNR value To fnd the best σ psd parameter for the BM3D flter manually, the required time is 156s, which when using the proposed model, the time to fnd the parameter σ psd only takes 10s.Tis time is 15 times faster than manually searching for parameters.Obviously, with the same PSNR value obtained, the proposed method took signifcantly less time than the manual method.proposed method is more efcient in restoring original images from 50% Gaussian noise.Furthermore, our study has achieved an impressive PSNR value not only higher than previous works but also when compared to the same type of noise (Gaussian 50%), such as Sreelakshmi et al. [11] (PSNR 48.68).Another signifcant diference is that we have performed denoising of brain MRI images with a relatively high noise level (Gaussian 50%), while some previous studies focused on weaker noise.Tis demonstrates the broad applicability of our method in real-world scenarios when brain MRI images exhibit strong noise.In this section, the SSIM index will be the comparison index.Tis is an index that measures the similarity of an image.Te results of previous studies and this study are listed in Table 5.Based on the information from the research works mentioned above, our study has several important strengths.First, compared to previous studies such as Chang et al. [8], Tripathi et al. [9], Wang et al. [12], and Kollem et al. [14], our method employs BM3D in combination with a proposed model designed to handle high-level Gaussian noise up to 50%.Tis increases the accuracy of the denoising process and achieves a relatively high SSIM value of 0.998.Compared to Moreno López et al. [10], who used unsupervised machine learning to denoise with a standard deviation σ � 50, our method also achieves a higher SSIM value.Comparing with the study by Kollem et al. [14], it is evident that our method ofers better accuracy and is a promising choice for denoising in images, even though the specifc noise type is not explicitly defned in their study [15].

Conclusion
Noise fltering for images is a long-standing research interest with many highly efective techniques.In this study, an automatic method for fnding the power spectral density standard deviation of the image noise for the BM3D flter based on complex networks and ANNs is proposed.Tis method is not only applicable to fnd flter parameters for BM3D image noise flter, but also to other traditional noise flters whose input is random parameters.Experimental results have shown that MRI images with Gaussian noise have been restored quite efectively with signifcantly shortened execution time compared to traditional methods where the selection of parameters is a random process.Te results of the study are very promising and are expected to be implemented practically in healthcare facilities based on embedded devices.In the future, the combination of complex network models and ANNs could be used to develop other biomedical data processing methods such as lung CT images, MRI joint scans, ECG.

Figure 3 :Figure 4 :
Figure 3: Te process of re-testing the proposed model after training.

Table 1 :
Summary of studies on denoise MRI image.
Te high PSNR score and SSIM close to 1 indicate that the noise 2 Journal of Electrical and Computer Engineering Table 4 provides a comparison based on the PSNR index between recent research works and our current study.Our present research in brain MRI denoising has achieved several notable advantages compared to previous studies.First, we employ a combination of the BM3D method and

Table 3 :
Experimental results with 10 MRI images with 50% gaussian noise.