## 4 Experimental Results and Discussion
### Clinical Liver Ultrasound Images  
The proposed algorithm efficiency was estimated using a set of B-mode greyscale ultrasound liver images. The images were obtained using the ECUBE 12R ultrasound research system from Alpinion medical systems, Seoul, Korea. The components used to generate the ultrasound images include a 128-element linear transducer at a center frequency of 5 MHz, a lateral beam width of 1.5 mm, and a pulse length of 1 mm. In our experiment, sparse coding was performed using two dictionaries with a 64 \(\times\) 256 size, designed to handle patches of 8 \(\times\) 8 size pixels (\(N\) = 64 and \(K\) = 256)--one trained from a noisy image and the other trained from a set of reference images. The training data were constructed from a dataset comprising 3245 reference ultrasound images. The random collection of 16 \(\times\) 16 dictionary atoms (\(K\) = 256) is presented in Figure 5a and the dictionary trained on the noisy image itself by overlapping patches is represented in Figure 5b. Where, every dictionary atom occupies a cell of 8 \(\times\) 8 pixel (\(N\) = 64). We performed the tests on the three ultrasound reference images shown in Figures 6a, 7a and 9a. The KSVD algorithm was initialized with a trained dictionary and executed 180 iterations, as recommended in [29]. The numerical evaluation was performed using PSNR and MSSIM (as discussed in Section 3.1) on the proposed algorithm and compared with the denoising methods [PERSON] filter [10], Lee filter [11], 3 \(\times\) 3 Weiner filter [13], Kuan filter [14], 3 \(\times\) 3 median filter [15], SRAD filter [39], and wavelet filter [40]. Figure 6a, shows a right lobe liver image with size 256 \(\times\) 256 pixels, where the lateral size is given by the \(x\)-axis, and the axial size is given by the \(y\)-axis. In this original image, we included a speckle noise parameter \(\sigma\) = 10 and the PSNR was calculated using Equation (12). It is clear that detailed information of the image is highly distorted, as shown in Figure 6b with a PSNR value of 28.148 dB. Figure 6c,d show the denoising results obtained by the proposed method using Dictionary 1 with a PSNR value of 35.033 dB and Dictionary 2 with a PSNR value of 35.537 dB. It is clear that the SR over learned dictionaries improves both edges and smooth features by eliminating the noise and reconstructs the image as much closer to the original image, as shown in Figure 6a. Figure 7 shows the comparative experimental results obtained on real-time ultrasound images. For this experiment, we obtained a 256 \(\times\) 256-pixel liver image of a healthy person with a PSNR value of 24.6271 dB. The radio frequency (RF) frames were obtained using a linear transducer with a frequency range of 8 MHz. This frequency range was selected because of its suitability for liver imaging, and we considered natural speckle noise for these experiments. The original speckled image was then denoised using the proposed algorithm with both dictionaries and also using conventional algorithms. To assess the speckle reduction, we selected two regions in of the speckled image. The two  
Figure 5: The random collections of 16 \(\times\) 16 atoms (\(K\) = 256) of trained dictionary from (**a**) a reference set of 3245 ultrasound images and (**b**) a noisy image. regions in the case of Figure 7a are displayed as a red square and a green square. The red one indicates the diaphragm of a liver and the green square shows the presence of an excessive noisy region observed from deeper tissue. The differences can be noticed from the filtered images in dashed red and the green square. Figure 7d-f show that detailed information lost by the blurring effect on the results obtained with Frost filter, median filter, and Kuan filter. In particular, the wavelet filter, Weiner filter, and the SRAD filter are not very effective in reducing speckle and perform poorly in retrieving sharp edge information, as can be seen in Figure 7g-i.