Descriptor Proposed JFMR

File:DivergenceGradient.png > Descriptor Proposed JFMR > ROIs Relevants JFMR > Descriptor Proposed JFMR

Definition

Given a vector field $\vec{V}=(u,v)$, where $u$ and $v$ represent the components of the winds of each of the annotations.

The vector field representation allow to observe the behavior of the winds in a determined region, these behavior are represented as:

• Vortex
• Confluence
• Difluence
• Saddle Point

The meteorologists annotated the different configurations that it can present in a period of time and a level determined, this process was done with an annotations tool that allow to select this configurations.

Model Mathematic

Given a vector field $\vec{V}$, a way to classify this configurations is through the use of phase portrait, this method is used in the dynamic system to understand the behavior of functions of 2 or more variables. In computer vision is used in the detection corner, mammography and others.

This method allows to construct a equation system in which these can calculate the determinant and the trace.

$A = \begin{bmatrix} \frac{\partial U}{\partial x} & \frac{\partial U}{\partial y} \\ \frac{\partial V}{\partial x} & \frac{\partial V}{\partial y} \\ \end{bmatrix}$

The trace and the determinant indicate as is the behavior of the system. To calculate the trace and determinant with their:

$Trace(A) = \frac{\partial U}{\partial x} + \frac{\partial V}{\partial y}$

$det(A) = \left(\frac{\partial U}{\partial x} \frac{\partial V}{\partial y}\right) - \left(\frac{\partial U}{\partial y} \frac{\partial V}{\partial x} \right)$

Implementation

The implementation of a descriptor that use the phase portrait depends of the components $u$ and $v$ obtained of the simulation that allow to construct the equation system, although needed the gradient of $u$ and $v$ to construct the system. With the system, we can calculate the solution of the system through the equation characteristic $det(A-I\lambda) = 0$ where I is the identify matrix of the size of A, at expand the equation obtains: $\lambda^2-Trace(A)\lambda+det(A) = 0$ when we solve this equation have the next:

$\lambda_1,\lambda_2 = 0.5 (Trace(A) \pm \sqrt{Trace(A)^2 - 4det(A)})$

The descriptor is calculated using moments statistics in the trace and the determinant because is necessary to have a only value of these components, we use the mean, standard desviation and the variance, we use these 3 statistics