Super-specific Gabor stuff.

main.tex

@@ -72,6 +72,8 @@
 \newcommand{\kpi}{\kp_i} % Specific Gabor kernel phase
 \newcommand{\kni}{\kn_i} % Specific Gabor kernel lobe number
 \newcommand{\ksi}{\ks_i} % Specific Gabor kernel sign
+\newcommand{\rh}{\text{RH}} % Relative Gaussian height for FWRH
+\newcommand{\fwrh}{\text{FWRH}} % Gaussian full-width at relative height
 
 % Math shorthands - Threshold nonlinearity:
 \newcommand{\thr}{\Theta_i} % Step function threshold value
@@ -415,25 +417,38 @@ matching, can be modelled as a convolution
     \label{eq:conv}
 \end{equation}
 of the intensity-adapted envelope $\adapt(t)$ with a kernel $k_i(t)$ per
-ascending neuron. We used Gabor kernels as basis functions for creating
-different preferred patterns. An arbitrary one-dimensional, real Gabor kernel
-is generated by multiplication of a Gaussian envelope and a sinusoidal carrier
+ascending neuron. We use Gabor kernels as basis functions for creating
+different template patterns. An arbitrary one-dimensional, real Gabor kernel is
+generated by multiplication of a Gaussian envelope and a sinusoidal carrier
 \begin{equation}
     k_i(t,\,\kwi,\,\kfi,\,\kpi)\,=\,e^{-\frac{t^{2}}{2{\kwi}^{2}}}\,\cdot\,\sin(\kfi\,t\,+\,\kpi), \qquad \kfi\,=\,2\pi f_{sin}
     \label{eq:gabor}
 \end{equation}
 with Gaussian standard deviation or kernel width $\kwi$, carrier frequency
-$\kfi$, and carrier phase $\kpi$. Different combinations of $\kwi$, $\kfi$, and
-$\kpi$ result in Gabor kernels with different lobe number $\kni$ and sign
-$\ksi$. If the function space is constrained to only include mirror- or
-point-symmetric Gabor kernels, frequency $\kf$ is related to lobe number $\kn$
-by
+$\kfi$, and carrier phase $\kpi$. Different combinations of $\kw$, $\kf$, and
+$\kp$ result in Gabor kernels with different lobe number $\kn$, which is the
+number of half-periods of the carrier that fit under the Gaussian envelope
+within reasonable limits of attenuation. These limits are a matter of
+definition, since the Gaussian function never fully decays to zero. A good
+measure is the Gaussian full-width at relative height, which can be calculated
+as
+\begin{equation}
+    \fwrh(\kw,\,\rh)\,=\,2\,\cdot\,\sqrt{-2\,\ln \rh}\cdot\,\kw, \qquad \rh\,\in\,(0,\,1]
+\end{equation}
+With this, an appropriate carrier frequency $\kf$ for obtaining a Gabor kernel
+with width $\kw$ and a desired lobe number $\kn$ can be approximated as 
+\begin{equation}
+    \kf(\kn,\,\fwrh)\,=\,\frac{0.5\,\cdot\,\kn\,+\,0.5}{\fwrh}
+\end{equation}
+We restrict the Gabor kernels to be either even functions~(mirror-symmetric,
+uneven $\kn$) or odd functions~(point-symmetric, even $\kn$). Under this
+condition, phase $\kp$ is related to lobe number $\kn$ by
 \begin{equation}
     \kp(\kn,\,\ks)\,=\,0.5\,\cdot\,(1\,-\,\text{mod}[\kn,\,2]\,+\,\ks)
+    \label{eq:gabor_phase}
 \end{equation}
 which results in the specific phase values shown in
-Table\,\mbox{\ref{tab:gabor_phases}}.
-
+Table\,\mbox{\ref{tab:gabor_phase}}.
 \FloatBarrier
 \begin{table}[!ht]
     \centering
@@ -447,14 +462,10 @@ Table\,\mbox{\ref{tab:gabor_phases}}.
         -1 & $-\pi\,/\,2$ & $0$\\
         \hline
     \end{tabular}
-    \label{tab:gabor_phases}
+    \label{tab:gabor_phase}
 \end{table}
 \FloatBarrier
-In order to create a Gabor kernel with a specific lobe number $\kn$ and kernel
-width $\kw$, frequency $\kf$ has to be set to
-\begin{equation}
-    \kf(\kn,\,\kw)\,=\,\frac{\kn}{2\pi\,\kw}
-\end{equation}
+
 
 \textbf{Stage-specific processing steps and functional approximations:}