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[[File:PhasePortraitRules.png|800px]]
 
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In our case this rules can be used as features to describe the ROI

Revision as of 13:01, 20 September 2015

Contents

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 obatin 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

We obtain 2 values of a \lambda because it has a square root and it mean that we must with space real and complex. In the descriptor we use both and obtain 6 values and we aggregate the 3 statistics of the trace and the determinant

Rules

  • When det(A) < 0 is saddle point.
  • When det(A) > 0 is possible a node or a spiral
  • It is a spiral if Trace(A)^2 - 4 det(A) < 0
  • It is a node if Trace(A)^2 - 4 det(A) > 0
  • It is a stable if Trace(A) < 0
  • It is a unstable if Trace(A) > 0

PhasePortraitRules.png

In our case this rules can be used as features to describe the ROI

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