# Systematic Evaluation on Speckle Suppression Methods in Examination of Ultrasound Breast Images
## 2 Filtering Methods
### PDE-Based Filter
#### 2.4.2 Weickert J Diffusion Filter  
WJD filter is also a kind of filter based on anisotropic diffusion theory. During the diffusion process, it chooses a large diffusion coefficient at the direction vertical to the edge direction while a small diffusion coefficient at the direction parallel to the edge direction to protect the local details of image. WJD filter is a modified filter from PMD, aiming to realize the anisotropy diffusion in the discontinuous edge of image. [PERSON] et al. [24] replaced the diffusion coefficient \(c(x,y,t)\) in PMD with a second order structure tensor \(D\). This change greatly improves the diffusion model in de-noising effect, and it has been used for de-speckling medical ultrasound image [7]  
\[\frac{\partial I_{g}}{\partial t}=div(D\
abla I_{g}) \tag{17}\]  
where \(D\in R^{2\times 2}\) is a symmetry positive semi-definite diffusion tensor. The diffusion tensor \(D\) can be represented by its eigenvalues and eigenvectors  
\[D=(w_{1}\;w_{2})\Bigg{(}\begin{array}{cc}\lambda_{1}&0\\ 0&\lambda_{2}\end{array}\Bigg{)}(w_{1}\;w_{2})^{T} \tag{18}\]  
where \(\lambda_{1}\) and \(\lambda_{2}\) are the eigenvalues, and \(w_{1}\) and \(w_{2}\) are the eigenvectors.  
\(D\) can be described as a multi-scale structure matrix [46] which takes the following form  
\[J_{\rho}(\
abla I_{\sigma})=\left[\begin{array}{cc}J_{11}&J_{12}\\ J_{21}&J_{22}\end{array}\right]=K_{\rho}\times(\
abla I_{\sigma}\cdot\
abla I _{\sigma}^{T}) \tag{19}\]  
where \(I_{\sigma}\) is denoted as the homogeneous region of the pre-smoothed image of contaminated image \(I_{g}\), and the pre-smoothing is to prevent mistaking speckles for edges by a Gaussian function with standard deviation \(\sigma\). \(K_{\rho}\) is a Gaussian kernel with standard deviation \(\rho\). The matrix \(\
abla I_{\sigma}\cdot\
abla I_{\sigma}^{T}\) is further smoothed by the Gaussian kernel \(K_{\rho}\). Subsequently, eigenvalue decomposition is performed on \(J_{\rho}(\
abla I_{\sigma})\) to obtain \(w_{1}\) and \(w_{2}\).  
\[J_{\rho}(\
abla I_{\sigma})=(w_{1}\;w_{2})\Bigg{(}\begin{array}{cc}\lambda _{1}&0\\ 0&\lambda_{2}\end{array}\Bigg{)}(w_{1}\;w_{2})^{T} \tag{20}\]  
To enhance the image local structures, [PERSON] [24] redesigned the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) in two models as below  
Coherence model 1:  
\[\begin{split}\lambda_{1}&=\alpha\\ \lambda_{2}&=\left\{\begin{array}{cc}\alpha&\text{if }\lambda_{1}= \lambda_{2}\\ \alpha+(1-\alpha)\exp\left(\frac{-C}{(\lambda_{1}-\lambda_{2})^{2m}}\right)& \text{else}\end{array}\right.\end{split} \tag{21}\]Coherence model 2:  
\[\begin{array}{l}\lambda_{1}=1-\exp(\frac{-C}{(|\
abla I_{\sigma}|/K)^{m}})\\ \lambda_{2}=1\end{array} \tag{22}\]  
where \(\alpha\) is often a small positive number, \(m\) is a real number, \(C\) is a constant greater than zero, and \(K\) is a constant controlling the sensitivity to image gradient. Via \(\lambda_{1}\) the vector along the edge direction is obtained, and via \(\lambda_{2}\) the vector vertical to the edge direction is obtained. Model 2 is the function of the value of gradient, thus in order to consider the impact of the value and direction of gradient on the diffusion more clear, we prefer to choose Model 2 as the algorithm evaluated in the experiments.