## 1 Introduction  
Magnetic resonance (MR) imaging is the most widely used technique in the field of medical diagnostics [1, 2]. In particular, MR imaging, using radio-frequency (RF) pulses, is able to non-invasively image the internal structure of the human body and is known to have a very high diagnostic accuracy [3, 4, 5]. In addition to the original general MR imaging techniques such as T1 weighted imaging [6], T2 weighted imaging [7], and proton density (PD) weighted imaging [8], more modern specialized techniques such as diffusion weighted imaging (DWI) [9, 10, 11], diffusion tensor imaging (DTI) [12, 13, 14], magnetic resonance angiography (MRA) [15, 16], functional MR imaging (fMRI) [17, 18], and MR spectroscopy (MRS) [19] are actively used to achieve a relatively high disease detection accuracy. Recently, an ultra-high field MR imaging system was developed and applied in numerous studies to improve the signal to noise ratio (SNR) [20, 21, 22]. In particular, [PERSON] et al. and [PERSON] et al. proposed newly designed RF coils to improve the SNR and to achieve a highly homogeneous B1-field [20, 21]. However, MR images include various sources of noise, including limitations in acquisition time, patient movement, susceptibility differences between different tissues, external signals, and the mechanical defects of the device [23; 24]. In particular, because the intensity of the MR signal is low, the MR image contains a significant amount of noise, which degrades the image quality. Generally, the noise in the MR image forms a Gaussian and Rician distribution, and the basic formula can be written as follows [25; 26]:  
\[\mathrm{P}(M|S,\sigma)=\frac{M}{\sigma^{2}}e^{\frac{-(M^{2}+S^{2})}{2\sigma^{2 }}}\gamma_{0}\Big{(}\frac{MS}{\sigma^{2}}\Big{)}\,M>0, \tag{1}\]  
where \(|M|\) is the Rician distribution, \(S\) is the original MR image, \(\sigma\) is the standard deviation or equivalently the noise variance for the Gaussian distribution, \(\gamma_{0}\) is the modified Bessel function, and \(M\) is the observed MR image. Additionally, the data from clinical MR imaging is corrupted by noise from the acquisition process and this noise degrades the SNR, which is a key factor in MR image quality evaluation [23]. Therefore, noise reduction is a very important part of MR imaging and the methods of removing such noise can be roughly divided into two types: (1) the improvement of hardware and (2) the use of software (image processing algorithm). While using an ultra-high field magnet or developing a high-performance coil can reduce noise, there is an associated problematic increase in cost. To mitigate the high price of hardware development, noise reduction algorithms are actively being studied by many researchers [2; 27; 28; 29]. Various filtering methods have been proposed to remove different components of noise. Filters can be divided into two types according to their regional characteristics: (1) local filters and (2) non-local filters. Local filters use the average or median value based on the relationship between noisy pixels and neighboring pixels. On the contrary, non-local filters compare the area containing the noisy pixel and the neighboring area to obtain a weight, and subsequently filters using a Gaussian kernel and an average value. Among the noise reduction algorithms using the non-local filter method, the total variation-based method and non-local means (NLM) method were introduced and they are widely used in the field of MR imaging [30; 31; 32]. In particular, the NLM noise reduction algorithm was proposed to minimize the loss of the inherent information in the image and to selectively remove only the noise. This algorithm measures the similarity of the intensity and edge information in an image after setting areas with the same sized mask located around the region of interest (ROI). Additionally, the higher the degree of similarity, the higher the assigned weight used during the image processing.