## 3 Sparse Representation Framework for Speckle Reduction
The filtered version of \(S(u,v)\) is written as  
\[S(u,v)=H_{B}(u,v)y_{\partial}(u,v)+H_{G}(u,v)y_{\partial}(u,v). \tag{7}\]  
Step 4: Take the inverse Fourier transform of Equation (7) to get the converted signal in the spatial domain  
\[\overline{S}(i,j)=\mathcal{F}^{-1}\{S(u,v)\}.\]  
Step 5: Finally, we obtain the transformed image \(t(i,j)\) by taking the exponent of the image using the following equation  
\[t(i,j)=\exp\big{\{}\overline{S}(i,j)\big{\}}.\]  
In this paper, we model the transformed image as additive noise degradation \(W(i,j)\) of the original image \(x_{\Re}(i,j)\), i.e.,  
\[t(i,j)x_{\Re}(i,j)+W(i,j). \tag{8}\]  
Figure 1: Flow diagram of the enhanced homomorphic filter. FFT: fast Fourier transform; IFFT inverse fast Fourier transform. This completes how we have used the homomorphic filter to transform the speckle noise into additive noise. The two filter functions are utilized to improve edge information by enhancing contrast and smooths the additive noise of the transformed image. Figure 2 shows the output of the enhanced homomorphic filter at the BW-HP and GLP filter stages. It is clear that the image in Figure 2b has an increased intensity because the low frequency signal is attenuated and the image in Figure 2c is smoothed by the GLP filter. The sum of these two signals is the final transformed noisy image. An ultrasound image \(x_{\Re}(i,j)\) can be represented as sparse in the gradient domain. We thus define here a difference signal. A pixel-based TV regularization can be performed on the transformed image for more effective denoising. The horizontal and vertical difference matrices are  
This convex constrained problem can be transformed into an unconstrained optimization problem using the Lagrange multiplier method [51]:  
\[H_{B}(u,v)=\frac{1}{1+\left[D_{0}/\sqrt{u^{2}+v^{2}}\right]^{2}}, \tag{3}\]0  
Using the unconstrained problem, we are able to combine a regularization term, which is weighted by parameter \(\tau>0\) and a quadratic data-fidelity term. Equation (10) is not ready for use yet since we do not know the sparsity dictionary \(A\). Therefore, we use the following approach where the dictionary, the sparse representation coefficient vector \(\alpha\), and the image vector \(x_{\Re}\) are estimated altogether. The overall optimized discrete sparse model proposed in this paper, for denoising the ultrasound image, can be written as  
\[H_{B}(u,v)=\frac{1}{1+\left[D_{0}/\sqrt{u^{2}+v^{2}}\right]^{2}}, \tag{3}\]1  
where \(R_{ij}\) is an operation that extracts a square image patch from the transformed image \(t\) located at the \(i,j\) pixels of the image. The notation \(\|.\|_{1}\) is used to imply the \(l_{1}\) norm, which is the sum of the absolute values of the argument signal, which in this case is the difference signal \(Vx_{R}\). There are two positive parameters \(\lambda\) and \(\tau\) used to balance the contribution of different terms. In Equation (11), the first and second terms are the TV regularization norm and the sparse representation prior. Optimization in Equation (11) seeks to find a solution with which each patch of the recovered image can be represented by a dictionary matrix with sparse coefficient \(\alpha\) in the sense of a bounded error. The \(l_{0}\) norm gives the sparsity constraint which controls the sparsity coefficients of any small image patch. As mentioned in Related Work Section 2.2, there is a sparse coding stage that utilizes the KSVD iterative process. In the first stage, sparse coding is performed assuming fixed \(x_{\Re}\) and \(A\). In the second stage, dictionary \(A\) is updated to minimize using known sparse coefficients \(\alpha\) and \(x_{\Re}\). The sparse coefficients \(\widehat{\alpha}_{ij}\) are computed using the OMP method [52] because of its efficiency and simplicity. [PERSON] et al. [29] showed that learning a dictionary trained from good quality image patches and noisy images results in better performance. In this paper, we use two approaches to train the dictionary.