# Systematic Evaluation on Speckle Suppression Methods in Examination of Ultrasound Breast Images
## 2 Filtering Methods
### PDE-Based Filter
#### 2.4.3 Total Variation Filter  
TV Filter is also a filter method according to variational method, which minimizes a specific energy function to perform de-noising of an image. It performs as a diffusion processing. The diffusion direction of TV filter vertical to the gradient direction of image, and the direction parallel to gradient direction of image have a diffusion coefficient zero, thus it can both de-noise and protect the details of edges. The TV Filter was introduced by [PERSON] et al. (2005). Nowadays, TV used in de-speckling for medical ultrasound images has gained an excellent effect [PERSON] et al. (2005). It has attracted the attention of many researchers. The total variation of an image is the integral of the absolute gradient of an image. It has been proposed that an image with excessive details and noises can have high total variation. Minimizing the total variation of an image can remove the noise whilst preserving important edges. As a result, the process is equivalent to the following total variation minimization problem:  
\[\begin{array}{l}\text{minimize}\int\limits_{\Omega}\sqrt{I_{x}^{2}+I_{y}^{ 2}}\text{d}x\text{d}y\\ \\ \text{with}\int\limits_{\Omega}I_{f}\text{d}x\text{d}y=\int\limits_{\Omega}I_ {g}\text{d}x\text{d}y\\ \\ \text{and}\int\limits_{\Omega}\frac{1}{2}(I_{f}-I_{g})^{2}\text{d}x\text{d}y= \sigma^{2}\end{array} \tag{23}\]  
where \(I_{f}\) is the desired image, and \(\Omega\) denotes as the region of image. \(I_{x}\) and \(I_{y}\) represent the two vectors of the 2-D image \(I_{f}\) about the gradient, and \(\sigma\) is the standard deviation of GWN.  
The first constrain in Equation (24) ensures that the volume covered by the surface of the image remains unchanged during the minimization process. The second constrain in Equation (25) can be derived by the estimated variance of noise as important priori information.  
With the Lagrange multiplier \(\beta\), the minimization problem can be interpreted as  
\[\begin{array}{l}\text{minimize}\int\limits_{\Omega}\sqrt{I_{x}^{2}+I_{y}^ {2}}+\beta(I_{f}-I_{g})^{2}\text{d}x\text{d}y\\ \\ \end{array} \tag{24}\]  
The corresponding Euler-Lagrange equation is  
\[0=\frac{\partial}{\partial x}\left(\frac{I_{x}}{\sqrt{I_{x}^{2}+I_{y}^{2}}} \right)+\frac{\partial}{\partial y}\left(\frac{I_{y}}{\sqrt{I_{x}^{2}+I_{y}^{ 2}}}\right)-\beta(I_{f}-I_{g}) \tag{25}\]  
with  
\[\frac{\partial I_{f}}{\partial n}=0\text{ on }\partial\Omega \tag{26}\]  
where \(\partial\Omega\) is the boundary of \(\Omega\). Using the gradient projection method, we get \[\frac{\partial I_{f}}{\partial t}=\frac{\partial}{\partial x}\left(\frac{I_{x}}{ \sqrt{I_{x}^{2}+I_{y}^{2}}}\right)+\frac{\partial}{\partial y}\left(\frac{I_{y }}{\sqrt{I_{x}^{2}+I_{y}^{2}}}\right)-\beta(I_{f}-I_{g}) \tag{29}\]  
This is a forward time evolution process, i.e., \(t>0\). When the process evolves to a steady state Equation (28), the recovered image is obtained. The TV Filter is efficient at simultaneously preserving edge and smoothing noise. It keeps a strong diffusion in the homogeneous region where the image gradient is relatively small.