[Theory][Gaussian] 2D Gaussian function and derivatives

The 2D Gaussian function is given by:

\[ G = \frac{1}{2\pi\sigma^2}e^{-\frac{(x^2 + y^2)}{2\sigma^2}} \]

The first derivatives:

\[ G_x = -\frac{x}{2\pi\sigma^4}e^{-\frac{(x^2 + y^2)}{2\sigma^2}} \]

\[ G_y = -\frac{y}{2\pi\sigma^4}e^{-\frac{(x^2 + y^2)}{2\sigma^2}} \]

The second derivatives:

\[ G_{x^2} = \frac{x^2-\sigma^2}{2\pi\sigma^6}e^{-\frac{(x^2 + y^2)}{2\sigma^2}} \]

\[ G_{y^2} = \frac{y^2-\sigma^2}{2\pi\sigma^6}e^{-\frac{(x^2 + y^2)}{2\sigma^2}} \]

\[ G_{xy} = \frac{xy}{2\pi\sigma^6}e^{-\frac{(x^2 + y^2)}{2\sigma^2}} \]

The third derivatives:

\[ G_{x^3} = \frac{3\sigma^2x-x^3}{2\pi\sigma^8}e^{-\frac{(x^2 + y^2)}{2\sigma^2}} \]

\[ G_{y^3} = \frac{3\sigma^2y-y^3}{2\pi\sigma^8}e^{-\frac{(x^2 + y^2)}{2\sigma^2}} \]

\[ G_{xxy} = \frac{y(\sigma^2 - x^2)}{2\pi\sigma^8}e^{-\frac{(x^2 + y^2)}{2\sigma^2}} \]

\[ G_{xyy} = \frac{x(\sigma^2 - y^2)}{2\pi\sigma^8}e^{-\frac{(x^2 + y^2)}{2\sigma^2}} \]

The fourth derivatives:

\[ G_{x^4} = \frac{x^4 - 6\sigma^2x^2 + 3\sigma^4}{2\pi\sigma^{10}}e^{-\frac{(x^2 + y^2)}{2\sigma^2}} \]

\[ G_{y^4} = \frac{y^4 - 6\sigma^2y^2 + 3\sigma^4}{2\pi\sigma^{10}}e^{-\frac{(x^2 + y^2)}{2\sigma^2}} \]

\[ G_{x^3y} = \frac{xy(x^2-3\sigma^2)}{2\pi\sigma^{10}}e^{-\frac{(x^2 + y^2)}{2\sigma^2}} \]

\[ G_{xy^3} = \frac{xy(y^2-3\sigma^2)}{2\pi\sigma^{10}}e^{-\frac{(x^2 + y^2)}{2\sigma^2}} \]

\[ G_{x^2y^2} = \frac{(x^2-\sigma^2)(y^2-\sigma^2)}{2\pi\sigma^{10}}e^{-\frac{(x^2 + y^2)}{2\sigma^2}} \]

Laplacian of Gaussian:

\[ LoG = L_{x^2}+L_{y^2} = \frac{x^2 + y^2 - 2\sigma^2}{2\pi\sigma^6}e^{-\frac{(x^2+y^2)}{2\sigma^2}} \]