## 1 Introduction
Moreover, even with a high-performance coil capable of generating a strong magnetic field and signal, appropriate detectors and partial coils that can accommodate them are also necessary. Noise can be removed by applying algorithms to acquired images and restoring damaged signals based on the principles of signal recovery due to the drawbacks of hardware development mentioned above [13]. Software application is straightforward and requires less time compared to hardware development. Additionally, using the same parameters allows for reproducibility control. However, this method involves mathematical computations to calculate already acquired signals, and applying incorrect parameters can cause image distortion. Hence, the appropriate implementation is necessary to efficiently reduce noise and restore image quality. Among the noise reduction algorithms, linear filtering, which was initially developed, is a basic image-processing method. Linear filters remove noise through convolution operations that have a linear function with respect to the input signal. Although linear filters are computationally efficient and easy to implement, they have the disadvantage of not detecting complex noise patterns or edge information in images effectively. The Gaussian and median filters used in this study belong to the category of linear filters. On the other hand, non-linear filters reduce noise using non-linear operations and thresholding. The Wiener filter and BM3D algorithm used in this study are examples of non-linear filters [14; 15]. Nonlinear filters possess characteristics opposite to those of linear filters. Therefore, selecting an appropriate filter that suits the specific task in essential in their application. With the development of algorithms for noise reduction, various modified methods have been devised and used to obtain improved results. Among them, the block-matching and 3D filtering (BM3D) algorithm is a noise reduction method in the frequency domain that uses sparse representation [16]. The sparse representation is a method of expressing a signal or data as a linear combination of basis vectors where the key idea is to estimate the original signal using only a subset of the basis vectors. As a result, most of the weights become zeros, and only a few basis vectors play a significant role. Consequently, sparse representation simplifies the representation compared to operating on the entire pixel values, reducing the required computational values and speeding up the processing. From the perspective of noise reduction, sparse representation restores images in a way that minimizes the impact of noise by making the weights of basis vectors affected by the noise close to zero. The concept of the BM3D algorithm was first introduced as a wide-range scanning approach for motion estimation in video compression. Attempts to apply the BM3D algorithm to the field of medical image processing are gradually increasing, and it is being used in various fields, such as nuclear medicine and CT [17; 18]. However, optimization of the \(\sigma\)-value, which determines the degree of smoothing for noise reduction, has not been performed in the field of MR image processing. Therefore, in this study, Rician noise was arbitrarily added to a brain T2-weighted image (T2 WI) acquired using a simulation to derive the \(\sigma\)-value representing the optimal image characteristics. In addition, the results of the application were compared with those of conventional noise reduction algorithms using open-access data for clinical MR images to confirm the effectiveness of the BM3D algorithm with optimized \(\sigma\)-values in clinical images.