## 2 Background
### Related Work on Multiplicative Noise Reduction  
Several algorithms have been proposed to deal with more complex multiplicative and additive speckle noise models [38]. For instance, the [PERSON], [PERSON], [PERSON] filters, and speckle reducing anisotropic diffusion (SRAD) filter [39] are effective on the multiplicative noise model. Other filters, specifically the median, Wiener, and wavelet filters [40], are designed for the additive noise model [4]. However, each filter has certain advantages and limitations [38]. In a few filter models, the quality of the processed image is affected by the window size: large window sizes cause image blurring, degrading the fine details of an image. Conversely, small window sizes do not denoise the image sufficiently. Other widely used multiplicative noise reduction algorithms are based on the TV regularization term [23; 41], nonlocal methods [42; 43], and wavelet-based approaches [16]. Total variation-based methods effectively remove flat-region-based noise and preserve the edges of images. However, fine details are lost because of over-smoothed textures. Nonlocal algorithms depend on similarities of image patches. Their performance is limited by dissimilar image patches. However, wavelet-based approaches preserve texture information better than TV-based methods. This approach assumes that images in the SRs are based on a fixed dictionary [29; 36]. However, certain characteristics of the processed image might not be captured because the dictionary does not contain any similar image content. To overcome the above disadvantages, over the past few years, researchers have sought to develop an algorithm based on SR in the field of image and signal processing [32]. This is because the pattern similarities of image signals such as textures and flat regions, mean that the signal can be efficiently approximated as a linear combination using a dictionary of only a few functions called atoms [29; 34; 36]. [PERSON] and [PERSON] [36] proposed an image denoising algorithm using an adaptive dictionary called KSVD that is based on sparse and redundant representations. It includes sparse coding and dictionary atoms that are updated to better fit the data. The advantage of KSVD compared to fixed dictionaries is that it is effective at removing additive Gaussian noise using the linear combinations of a few atoms, by learning a dictionary from noisy image patches and then reconstructing each patch. A dictionary \(A\in\mathbb{R}^{N_{r}\times N_{c}}\), composed of \(N_{c}\) columns of \(N_{r}\) elements, is called a sparse-land model [36]. K-singular value decomposition seeks the best signal representation of image signal \(y\) from the sparsest representation \(\alpha\):  
\[\widehat{\alpha}=\text{argmin}\|\alpha\|_{0}\text{ subject to }\left\|y-A \alpha\right\|_{2}\leq\varepsilon,\]  
where the vectorization of \(y(i,j)\) is denoted by vector \(y\in\mathbb{R}^{M\times 1}\) and \(\varepsilon\) is the few number of non-zero entries in \(\alpha\). K-singular value decomposition replaces the dictionary update and sparse coding stages with a simple singular value decomposition. The orthogonal matching pursuit (OMP) method [44] is an effective method to find the sparse approximation. In the OMP, if the noise level is below the approximation, the image patches are rejected. The singular value decomposition constructs better atoms by combining patches to reduce noise for ultrasound speckle reduction. K-singular value decomposition has also proved to effectively reduce the speckle produced by additive white Gaussian noise on corrupted images [29; 36]. The filtering algorithm comprises two steps. First, the dictionary is trained from a set of image data patches or from noisy image patches based on KSVD. The next step uses \(\widehat{\alpha}\) to compute SR using dictionary \(A\) and denoises the image [29]. The method proposed in [45] also uses a dictionary learning approach for denoising ultrasound images.