Color-Sift

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Formulas de color SIFT

Invarianza geométrica

$F(\vec{x})=F(T\vec{x})$

$L(\vec{x},t)=g(\vec{x},t)*f(\vec{x})$

Invarianza de Color

$E(\boldsymbol{\lambda}, \vec{x}) = e(\boldsymbol{\lambda}, \vec{x})(1- \boldsymbol{\rho} f(\vec{x})) ^2 R_\infty(\boldsymbol{\lambda}, \vec{x})+e(\boldsymbol{\lambda}, \vec{x})\boldsymbol{\rho} f(\vec{x})$

where λ is the wavelength and �x is a 2D vector which denotes the image position. e(λ, �x) denotes the illumination spectrum and ρf (�x) is the Fresnel reflectance at x. R∞(λ, �x) denotes the material reflectivity. E(λ, �x) represents the reflected spectrum in the viewing direction. This model is suitable for modelling non-transparent/nontranslucent materials. Some special cases can be derived from Eq. (3). For example, the Fresnel coefficient can be neglected for matte and dull surfaces. By assuming equal energy illumination, the spectral components of the source are constant over the wavelengthes and variable over the position, which is applicable for most of the practical cases.

$E(\boldsymbol{\lambda}, \vec{x}) = i(\vec{x})[\boldsymbol{\rho} f(\vec{x})+(1- \boldsymbol{\rho} f(\vec{x})) ^2 R_\infty(\boldsymbol{\lambda}, \vec{x})]$

By differentiating Eq. (4) with respect to λ, we get:

$E_\boldsymbol{\lambda}=i(\vec(x))(1-\boldsymbol{\rho} f(\vec{x})^2 \frac{\partial R_\infty(\boldsymbol{\lambda}, \vec{x})}{\partial \boldsymbol{\lambda}}$

and

$E_\boldsymbol{\lambda \lambda}=i(\vec(x))(1-\boldsymbol{\rho} f(\vec{x})^2 \frac{\partial^2 R_\infty(\boldsymbol{\lambda}, \vec{x})}{\partial \boldsymbol{\lambda}^2}$

By dividing Eq. (5) by Eq. (6), we get:

$H=\frac{E_\boldsymbol{\lambda}}{E_\boldsymbol{\lambda \lambda}}=\frac{\frac{\partial R_\infty(\boldsymbol{\lambda}, \vec{x})}{\partial \boldsymbol{\lambda}}}{\frac{\partial^2 R_\infty(\boldsymbol{\lambda}, \vec{x})}{\partial \boldsymbol{\lambda}^2}}=f(R_\infty(\boldsymbol{\lambda}, \vec{x}))$

Matriz resultante

\left (

     \begin{matrix} 
        0.06 & 0.63 & 0.27 \\
        0.3 & 0.04 & -0.35 \\
        0.34 & 0.6 & 0.17 
     \end{matrix}
  \right ).