## 3 Sparse Representation Framework for Speckle Reduction  
As discussed above, we define our proposed scheme for ultrasound speckle reduction by considering the multiplicative noise model [37] obtained by an ultrasound transducer. Equation (1) can thus rewritten as  
\[y_{\partial}(i,j)=x_{\Re}(i,j)n_{\sigma}(i,j), \tag{2}\]  
where \(y_{\partial}(i,j)\) is the degraded B-mode image signal [46], \(x_{\Re}(i,j)\) represents the ideal image that must be recovered, and \(n_{\sigma}(i,j)\) represents the speckle noise, generally modelled as a Rayleigh probability density function with random variables [11; 47]. Each term includes coordinates \((i,j)\) defined according to the acquisition geometry. In general, a homomorphic filter [48] is a well-proven technique for converting multiplicative noise. In this study, we modified it by taking the log of the multiplicative noisy signal and filtering the image using a Butterworth high-pass (BW-HP) filter to attenuate low frequencies in the transmitted signal while preserving the high frequencies in the reflected component. The equation of the BW-HP filter is  
\[H_{B}(u,v)=\frac{1}{1+\left[D_{0}/\sqrt{u^{2}+v^{2}}\right]^{2}}, \tag{3}\]  
where, \(D_{0}\) is the cut-off frequency and \(f\) is the order of the filter. We varied the frequency values \(u\) and \(v\) of the \(i\) and \(j\) spatial coordinates. We used the BW-HP filter because it generates fewer ringing artifacts on the image signal. We also used a Gaussian low pass (GLP) filter to smooth the low-frequency signal component in the log domain. The equation of the GLP filter is  
\[H_{G}(u,v)=e^{-D^{2}(u,v)/2D_{0}}^{2}, \tag{4}\]  
where \(D(u,v)\) is the distance from the origin in the frequency plane. Finally, the additive noise signals were estimated by applying inverse transform. Figure 1 shows the steps used to convert an original noisy image into an image with additive noise using the enhanced homomorphic transform. This technique consists of five steps. We first take the log on both sides of Equation (2) and use a two-dimensional fast Fourier transform (FFT)  
to represent the image in the frequency domain. Then, the Fourier image is filtered with two filter functions, those are the BW-HP and GLP filters [12]. The BW-HP filter increases the contrast of the image signal corresponding to the high-frequency component. The GLP filter smooths the noise signal without eliminating the entire low-frequency component. Both filtered signals are applied to the two-dimensional inverse fast Fourier transform (IFFT). Finally, taking the exponent of the image, we obtain the transformed image. This process is discussed in detail below. Step 1: Take the log on both sides of the \(x_{\Re}(i,j)\) and the \(n_{\sigma}(i,j)\) signal; now the multiplicative noise can written as  
\[\log(y_{\partial}(i,j))=\log(x_{\Re}(i,j))+\log(n_{\sigma}(i,j)), \tag{5}\]  
After being transformed logarithmically, the signal now contains Gaussian additive noise [49]. We remove \(\log(x_{\Re}(i,j))\) from the speckled ultrasound image by applying an additive noise suppression algorithm. Thus, the problem is now to estimate \(\log(x_{\Re}(i,j))\) from noisy data. Step 2: Apply FFT to convert the image into the frequency domain. Equation (5), thus becomes,  
\[y_{\partial}(u,v)=F_{x_{\Re}}(u,v)+F_{n_{\sigma}}(u,v), \tag{6}\]  
where, \(F_{x_{\Re}}(u,v)\) and \(F_{n_{\sigma}}(u,v)\) are the FFT of \(\log(x_{\Re}(i,j))\) and \(\log(n_{\sigma}(i,j))\), respectively. Step 3: Apply BW-HP and GLP to the \(y_{\partial}(u,v)\) by means of two filter function \(H_{B}(u,v)\) and \(H_{G}(u,v)\) from Equations (3) and (4) respectively in the frequency domain.