# Coherence Enhancement Based on Recursive Anisotropic Scale-Space with Adaptive Kernels
## 2 Preliminary  
Let \(I\): \(\Omega\mapsto\mathbb{R}\) be an image whose spatial domain is denoted by \(\Omega\). The conventional Gaussian scale-space is constructed by solving the heat equation defined by  
\[ \begin{cases}\frac{\partial u(x,t)}{\partial t}=\Delta u(x,t)&:x\in\Omega,t>0\\ u(x,0)=I(x)&:x\in\Omega\end{cases}  \tag{1}\]  
where \(t\geq 0\) is a scale parameter, \(u\colon\Omega\times\mathbb{R}^{+}\mapsto\mathbb{R}\) represents the solution of the heat equation, and \(\mathbb{R}^{+}\) denotes the non-negative temporal domain. The initial condition \(u(x,0)\) of the heat equation for constructing the Gaussian scale-space is defined by the given image \(I\). Such a construction of the Gaussian scale-space based on the heat equation is often useful for characterizing the global properties of objects present in an image. However, the isotropic diffusion process in the construction of scale-space leads to the dislocation of critical points and undesirable geometrical distortion of the objects. Thus, algorithms that consider the local geometrical properties of image features in the determination of the degree and direction of the diffusion process were developed. A nonlinear diffusion algorithm was proposed to avoid the blurring and localization limitations that stem from the linear filtering [25]. An inhomogeneous process is applied to reduce the diffusivity at the locationswith larger contrast. A typical measure of contrast is the norm of the image gradient \(\|\
abla u\|_{2}^{2}\). The diffusion process is modified to consider the local image structure characterized by the image gradient in the following diffusion equation:  
\[\frac{\partial u(x,t)}{\partial t}=\text{div}\left(g(\|\
abla u(x,t)\|_{2}^{2} )\
abla u(x,t)\right) \tag{2}\]  
where the diffusion is controlled by the function \(g\colon\mathbb{R}^{+}\mapsto\mathbb{R}^{+}\) defined by  
\[g(s^{2})=\frac{\lambda^{2}}{\lambda^{2}+s^{2}}. \tag{3}\]  
\(\lambda>0\) is a control parameter that determines the degree of diffusion. The diffusivity function \(g\) is designed to preserve significant geometrical image features such as edges and corners. It leads to an anisotropic behavior in which blurring occurs preferentially along linear features such as edges while blurring across edges is suppressed. The diffusivity function \(g\) determines the degree of diffusion based on the local image features using a scalar value, but it is desirable to also consider the orientation of the diffusion in addition to its degree. A more sophisticated diffusion process that considers a vector-valued structure tensor \(D\) to determine the direction of the diffusion process based on Gaussian smoothing is defined by  
\[D(x)=R(x)^{T} \begin{bmatrix}\lambda_{1}&0\\ 0&\lambda_{2}\end{bmatrix} R(x).\]  
Here, \(R\) is the rotation matrix defining the local coordinate system \((\lambda_{1},\lambda_{2})\) aligned with the gradient vector defined by  
\[R(x)=\frac{1}{N(x)} \begin{bmatrix}\frac{\partial I_{\sigma}(x)}{\partial x_{1} }&-\frac{\partial I_{\sigma}(x)}{\partial x_{2}}\\ \frac{\partial I_{\sigma}(x)}{\partial x_{2}}&\frac{\partial I_{\sigma}(x)}{ \partial x_{1}}\end{bmatrix} \]  
where \(x=(x_{1},x_{2})\) denotes the coordinates in the image domain \(x\in\Omega\) and \(I_{\sigma}\) denotes the blurred image with a Gaussian kernel \(K_{\sigma}\) defined by  
\[I_{\sigma}(x)=(K_{\sigma}*I)(x). \tag{4}\]  
Here, \(K_{\sigma}\) is a Gaussian kernel with the standard deviation \(\sigma\), \(*\) denotes the convolution operator, and \(N\) is defined by  
\[N(x)=\sqrt{\left(\frac{I_{\sigma}(x)}{\partial x_{1}}\right)^{2}+\left(\frac{I _{\sigma}(x)}{\partial x_{2}}\right)^{2}}.\]  
The structure tensor matrix \(S_{0}\) is defined by  
\[S_{0}(\
abla I_{\sigma}):=\
abla I_{\sigma}\otimes\
abla I_{\sigma}=\
abla I _{\sigma}\
abla I_{\sigma}^{T} \tag{5}\]  
where \(\otimes\) denotes the outer product operator and \({}^{T}\) denotes the transpose operator.