Math.tex

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\begin{document}
A 2D gaussian filter is trivially converted into the generating vector by simply setting $y=0$ and normalizing, and this is done for the detection filters. This is not so easy for the other filter they use:

The spatial filter generation uses:
\begin{align}
    f(x,y)=a_1e^{-a_2(x^2+y^2)}+b_1e^{-b_2(x^2+y^2)}
\end{align}

As a sum of gaussians, we need to find a way to create two corresponding vectors.
$f(x,y)$ is used to create an $NxN$ matrix $F$ that will be the convolution kernel:
\begin{align}
    F_{i,j}=f(i-r,j-r),
\end{align}

with  $N=2r+1$ and $i,j$ spanning $[-r,r]$.
Note that $F$ is then normalized into $\tilde{F}$:
\begin{align}
\tilde{F}=F/S_F\\
S_F=\sum_i\sum_j F_{i,j}
\end{align}
This notation means every element of the matrix $F$ is divided by $S_F$.

We construct the correct vectors $g_1$ and $g_2$:
\begin{align}
g_1(x)=\sqrt{a_1}e^{-a_2x^2}\\
g_2(x)=\sqrt{b_1}e^{-b_2x^2}.
\end{align}

They have with the following property:
\begin{align}
F=g_1\otimes g_1+g_2\otimes g_2\\
F_{i,j}=g_1(j) * g_1(i) + g_2(j) * g_2(i),
\end{align}
with the crossed circle denoting the outer product.

Because $F$ is normalized, we need to find the normalization for our vectors too, characterized by:

\begin{align}
\tilde{F}=\tilde{G}= \frac{g_1\otimes g_1+g_2\otimes g_2}{S_G}.
\end{align}

By identification, we trivially have:
\begin{align}
S_G=S_F
\end{align}

We write the normalization:
\begin{align}
\tilde{g_1}\otimes \tilde{g_1} &=\frac{g_1\otimes g_1}{S(g_1)^2+S(g_2)^2}\\
\texttt{with:} \tilde{g_1} &=\alpha_1 g_1\\
\alpha g_1\otimes \alpha g_1 &=\frac{g_1\otimes g_1}{S(g_1)^2+S(g_2)^2}\\
\alpha^2 g_1\otimes g_1 &=\frac{g_1\otimes g_1}{S(g_1)^2+S(g_2)^2}\\
\texttt{by identification: } \alpha &=\frac{1}{\sqrt{S(g_1)^2+S(g_2)^2}}\\
\end{align}

So, we compute our vectors this way to create a separable convolution of the sum of gaussians.

Here in pseudocode for the convolution:
\begin{lstlisting}
    c1=convolve1D(image, alpha*g_1, alpha*g_1)
    c2=convolve1D(image, alpha*g_2, alpha*g_2)
    result=c1+c2.    
\end{lstlisting}

\end{document}