Super-specific Gabor stuff.

This commit is contained in:
j-hartling 2026-02-11 16:07:18 +01:00
parent 015a3032c1
commit d56ba575ad
9 changed files with 2543 additions and 59 deletions

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@ -221,21 +221,22 @@
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@ -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:}