Updated/made figure captions up to Fig. 7.
This commit is contained in:
j-hartling
285
main.tex
285
main.tex
@@ -67,12 +67,12 @@
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\newcommand{\thp}{T_{\text{HP}}} % Highpass filter adaptation interval
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% Math shorthands - Early representations:
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\newcommand{\raw}{x} % Placeholder input signal
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\newcommand{\filt}{\raw_{\text{filt}}} % Bandpass-filtered signal
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\newcommand{\env}{\raw_{\text{env}}} % Signal envelope
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\newcommand{\db}{\raw_{\text{log}}} % Logarithmically scaled signal
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\newcommand{\dbref}{\raw_{\text{ref}}} % Decibel reference intensity
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\newcommand{\adapt}{\raw_{\text{adapt}}} % Adapted signal
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\newcommand{\raw}{x_{\text{raw}}} % Placeholder input signal
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\newcommand{\filt}{x_{\text{filt}}} % Bandpass filtered signal
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\newcommand{\env}{x_{\text{env}}} % Signal envelope
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\newcommand{\db}{x_{\text{log}}} % Logarithmically scaled signal
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\newcommand{\dbref}{x_{\text{ref}}} % Decibel reference intensity
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\newcommand{\adapt}{x_{\text{adapt}}} % Adapted signal
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% Math shorthands - Kernel parameters:
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\newcommand{\kw}{\sigma} % Unspecific Gabor kernel width
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@@ -354,9 +354,8 @@ outlined in the following sections.
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\begin{figure}[!ht]
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\centering
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\includegraphics[width=\textwidth]{figures/fig_auditory_pathway.pdf}
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\caption{\textbf{Schematic organisation of the song recognition pathway in
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grasshoppers compared to the structure of the functional
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model pathway.}
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\caption{\textbf{Schematic organisation of the grasshopper song recognition
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pathway and structure of the functional model pathway.}
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\textbf{a}:~Simplified course of the pathway in the
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grasshopper, from the tympanal membrane over receptor
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neurons, local interneurons, and ascending neurons further
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@@ -365,12 +364,12 @@ outlined in the following sections.
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the three neuronal populations within the metathoracic
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ganglion.
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\textbf{c}:~Network representation of neuronal connectivity.
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\textbf{d}:~Flow diagram of the different signal
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representations and transformations along the model
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pathway. All representations are time-varying. 1st half:
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Preprocessing stage (one-dimensional). 2nd half: Feature
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extraction stage (high-dimensional).
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}
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\textbf{d}:~Flow diagram of consecutive signal
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representations~(boxes) and transformations~(arrows) along
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the model pathway. All representations are time-varying.
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1st half: Preprocessing stage~(one-dimensional
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representation). 2nd half: Feature extraction
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stage~(high-dimensional representation). }
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\label{fig:pathway}
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\end{figure}
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@@ -428,12 +427,12 @@ following feature extraction stage.
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\includegraphics[width=\textwidth]{figures/fig_pre_stages.pdf}
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\caption{\textbf{Representations of a song of \textit{O. rufipes} during
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the preprocessing stage.}
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\textbf{a}:~Bandpass-filtered tympanal signal.
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\textbf{b}:~Signal envelope.
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\textbf{c}:~Logarithmically scaled envelope.
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\textbf{d}:~Intensity-adapted envelope.
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\textbf{a}:~Bandpass filtered tympanal signal $\filt(t)$.
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\textbf{b}:~Signal envelope $\env(t)$.
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\textbf{c}:~Logarithmically compressed envelope $\db(t)$.
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\textbf{d}:~Intensity-adapted envelope $\adapt(t)$.
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}
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\label{fig:pre}
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\label{fig:stages_pre}
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\end{figure}
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\FloatBarrier
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@@ -543,14 +542,15 @@ can be read out by a simple linear classifier.
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\includegraphics[width=\textwidth]{figures/fig_feat_stages.pdf}
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\caption{\textbf{Representations of a song of \textit{O. rufipes} during
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the feature extraction stage.}
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Different colors indicate Gabor kernels with different
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lobe number $\kn$ and sign, with lighter colors for higher
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$\kn$~($1\,\leq\,\kn\,\leq\,4$; both $+$ and $-$ per $\kn$;
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two kernel widths $\kw$ of $4\,$ms and $32\,$ms per sign).
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\textbf{a}:~Kernel-specific filter responses.
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\textbf{b}:~Binary responses.
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\textbf{c}:~Finalized features.
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}
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Different color shades indicate different types of Gabor
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kernels with specific lobe number $\kn$ and either $+$ or
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$-$ sign, sorted (dark to light) first by increasing $\kn$
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and then by sign~($1\,\leq\,\kn\,\leq\,4$; first $+$, then
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$-$ for each $\kn$; two kernel widths $\kw$ of $4\,$ms and
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$32\,$ms per type; 8 types, 16 kernels in total).
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\textbf{a}:~Kernel-specific filter responses $c_i(t)$.
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\textbf{b}:~Binary responses $b_i(t)$.
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\textbf{c}:~Finalized features $f_i(t)$.}
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\label{fig:stages_feat}
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\end{figure}
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\FloatBarrier
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@@ -576,29 +576,30 @@ specific operations involved, as outlined in the following sections.
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\begin{figure}[!ht]
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\centering
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\includegraphics[width=\textwidth]{figures/fig_invariance_rect_lp.pdf}
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\caption{\textbf{Intensity invariance by logarithmic compression and
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adaptation is restricted by the noise floor.}
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Synthetic input $\filt(t)$ consists of song component
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$\soc(t)$ scaled by $\sca$ with (\figc{} and \figd) or
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without (\figa{} and \figb) additive noise component
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$\noc(t)$. Input $\filt(t)$ is transformed into envelope
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$\env(t)$, logarithmically compressed envelope $\db(t)$,
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and intensity-adapted envelope $\adapt(t)$.
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\textbf{Left}:~$\env(t)$, $\db(t)$, and $\adapt(t)$ for
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different scales $\sca$.
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\textbf{Right}:~Ratios of the standard deviation of
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$\env(t)$, $\db(t)$, and $\adapt(t)$ relative to the
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respective reference standard deviation $\sigma_{\eta}$
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for input $\filt(t)=\noc(t)$.
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\figa{} and \figb:~Ideally, if $\filt(t)=\sca\cdot\soc(t)$, then
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$\adapt(t)$ is intensity-invariant across all $\sca$.
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\figc{} and \figd:~In practice, if
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$\filt(t)=\sca\cdot\soc(t)+\noc(t)$, the intensity
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invariance of $\adapt(t)$ is limited to sufficiently large
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$\sca$. Shaded area indicates saturation of $\adapt(t)$ at
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$95\,\%$ curve span.
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}
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\label{fig:inv_rect-lp}
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\caption{\textbf{Rectification and lowpass filtering improves SNR
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but does not contribute to intensity invariance.}
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Input $\raw(t)$ consists of song component $\soc(t)$ scaled by
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$\sca$ with optional noise component $\noc(t)$ and is
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successively transformed into tympanal signal $\filt(t)$ and
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envelope $\env(t)$. Different line styles indicate different
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cutoff frequencies $\fc$ of the lowpass filter extracting
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$\env(t)$.
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\textbf{Top}:~Example representations of $\filt(t)$ and
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$\env(t)$ for different $\sca$.
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\textbf{a}:~Noiseless case.
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\textbf{b}:~Noisy case.
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\textbf{Bottom}:~Intensity metrics over a range of $\sca$.
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\textbf{c}:~Noiseless case: Standard deviations $\sigma_x$ of
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$\filt(t)$ and $\env(t)$.
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\textbf{d}:~Noisy case: Ratios of $\sigma_x$ of $\filt(t)$ and
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$\env(t)$ to the respective reference standard deviation
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$\sigma_{\eta}$ for input $\raw(t)=\noc(t)$.
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\textbf{e}:~Ratios of $\sigma_x$ to $\sigma_{\eta}$ of
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$\env(t)$ as in \textbf{d} for different species (averaged
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over songs and recordings, see appendix
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Fig.\,\ref{fig:app_rect-lp}).
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}
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\label{fig:rect-lp}
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\end{figure}
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\FloatBarrier
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@@ -634,7 +635,7 @@ space into an additive term, or offset, in logarithmic space
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\end{equation}
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which allows for its separation from $\soc(t)$ but introduces a scaling of
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$\noc(t)$ by the inverse of $\sca$. The subsequent
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highpass-filtering~(Eq.\,\ref{eq:highpass}) of $\db(t)$ can then be
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highpass filtering~(Eq.\,\ref{eq:highpass}) of $\db(t)$ can then be
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approximated as a subtraction of the local offset within a suitable time
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interval $0 \ll \thp < \frac{1}{\fc}$:
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% \begin{equation}
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@@ -675,29 +676,34 @@ the signal for reliable song recognition.
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\begin{figure}[!ht]
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\centering
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\includegraphics[width=\textwidth]{figures/fig_invariance_log_hp.pdf}
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\caption{\textbf{Intensity invariance by logarithmic compression and
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adaptation is restricted by the noise floor.}
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Synthetic input $\filt(t)$ consists of song component
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$\soc(t)$ scaled by $\sca$ with (\figc{} and \figd) or
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without (\figa{} and \figb) additive noise component
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$\noc(t)$. Input $\filt(t)$ is transformed into envelope
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$\env(t)$, logarithmically compressed envelope $\db(t)$,
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and intensity-adapted envelope $\adapt(t)$.
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\textbf{Left}:~$\env(t)$, $\db(t)$, and $\adapt(t)$ for
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different scales $\sca$.
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\textbf{Right}:~Ratios of the standard deviation of
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$\env(t)$, $\db(t)$, and $\adapt(t)$ relative to the
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respective reference standard deviation $\sigma_{\eta}$
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for input $\filt(t)=\noc(t)$.
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\figa{} and \figb:~Ideally, if $\filt(t)=\sca\cdot\soc(t)$, then
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$\adapt(t)$ is intensity-invariant across all $\sca$.
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\figc{} and \figd:~In practice, if
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$\filt(t)=\sca\cdot\soc(t)+\noc(t)$, the intensity
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invariance of $\adapt(t)$ is limited to sufficiently large
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$\sca$. Shaded area indicates saturation of $\adapt(t)$ at
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$95\,\%$ curve span.
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\caption{\textbf{Intensity invariance through logarithmic compression and
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adaptation is restricted by the noise floor and decreases
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SNR.}
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Input $\filt(t)$ consists of song component $\soc(t)$
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scaled by $\sca$ with optional noise component $\noc(t)$
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and is successively transformed into envelope $\env(t)$,
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logarithmically compressed envelope $\db(t)$, and
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intensity-adapted envelope $\adapt(t)$.
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\textbf{Top}:~Example representations of $\env(t)$,
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$\db(t)$, and $\adapt(t)$ for different $\sca$.
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\textbf{a}:~Noiseless case.
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\textbf{b}:~Noisy case.
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\textbf{Bottom}:~Intensity metrics over a range of $\sca$.
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\textbf{c}:~Noiseless case: Standard deviations $\sigma_x$
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of $\env(t)$, $\db(t)$, and $\adapt(t)$.
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\textbf{d}:~Noisy case: Ratios of $\sigma_x$ of $\env(t)$,
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$\db(t)$, and $\adapt(t)$ to the respective reference
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standard deviation $\sigma_{\eta}$ for input
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$\filt(t)=\noc(t)$. Shaded areas indicate $5\,\%$ (dark
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grey) and $95\,\%$ (light grey) curve span for
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$\adapt(t)$.
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\textbf{e}:~Ratios of $\sigma_x$ to $\sigma_{\eta}$ of
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$\adapt(t)$ as in \textbf{d} for different species
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(averaged over songs and recordings, see appendix
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Fig\,\ref{fig:app_log-hp_curves}). Dots indicate $95\,\%$
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curve span per species.
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}
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\label{fig:inv_log-hp}
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\label{fig:log-hp}
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\end{figure}
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\FloatBarrier
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@@ -706,40 +712,93 @@ the signal for reliable song recognition.
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\begin{figure}[!ht]
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\centering
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\includegraphics[width=\textwidth]{figures/fig_invariance_thresh_lp_single.pdf}
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\caption{\textbf{Intensity invariance by thresholding and temporal
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averaging depends on both the threshold value and the
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noise floor.}
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Synthetic input $\adapt(t)$ consists of song component
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$\soc(t)$ scaled by $\sca$ with additive noise component
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$\noc(t)$. Input $\adapt(t)$ is transformed into kernel
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response $c_i(t)$, binary response $b_i(t)$, and feature
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$f_i(t)$. Threshold value $\thr$ is set to multiples of
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the reference standard deviation $\sigma_{\eta}$ of $c_i(t)$ for input
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$\adapt(t)=\noc(t)$. Darker colors correspond to higher
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$\thr$.
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\textbf{Left}:~$\adapt(t)$, $c_i(t)$, $b_i(t)$, and
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$f_i(t)$ for different scales $\sca$ and threshold values
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$\thr$. Left-most column is is the pure-noise reference.
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\textbf{Right}:~Average value of $f_i(t)$ during the song
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for the different $\thr$.
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\figa:~Input $\adapt(t)$.
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\figb-\figd:~$c_i(t)$, $b_i(t)$, and $f_i(t)$ for the
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different $\thr$ based on the same $\adapt(t)$ from
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\figa{}.
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\fige:~Average value of $f_i(t)$ during the song for
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the different $\thr$ in \figb{}-\figd.
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\caption{\textbf{Intensity invariance through thresholding and temporal
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averaging is mediated by the interaction of threshold
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value and noise floor.}
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Input $\adapt(t)$ consists of song component $\soc(t)$
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scaled by $\sca$ with optional noise component $\noc(t)$
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and is transformed into single kernel response $c(t)$,
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binary response $b(t)$, and feature $f(t)$. Different
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color shades indicate different threshold values $\Theta$
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(multiples of reference standard deviation $\sigma_{\eta}$
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of $c(t)$ for input $\adapt(t)=\noc(t)$, with darker
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colors for higher $\Theta$).
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\textbf{Left}:~Noisy case: Example representations of
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$\adapt(t)$ as well as $c(t)$, $b(t)$, and $f(t)$ for
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different $\sca$.
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\textbf{a}:~$\adapt(t)$ with kernel $k(t)$ in black.
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\textbf{b\,-\,d}: $c(t)$, $b(t)$, and $f(t)$ based on the
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same $\adapt(t)$ from \textbf{a} but with different
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$\Theta$.
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\textbf{Right}:~Average value $\mu_f$ of $f(t)$ for each
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$\Theta$ from \textbf{b\,-\,d}, once for the noisy case
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(solid lines) and once for the noiseless case (dotted
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lines). Dots indicate $95\,\%$ curve span (noisy case).
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\textbf{e}:~$\mu_f$ over a range of $\sca$.
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\textbf{f}:~$\mu_f$ over the standard deviation of noisy
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input $\adapt$ corresponding to the values of $\sca$ shown
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in \textbf{e}.
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% Why plot noiseless case over SD of noisy input? Omit?
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}
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\label{fig:inv_thresh-lp_single}
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\label{fig:thresh-lp_single}
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\end{figure}
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\FloatBarrier
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% \caption{\textbf{Rectification and lowpass filtering improves SNR
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% but does not contribute to intensity invariance.}
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% Input $\raw(t)$ consists of song component $\soc(t)$ scaled by
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% $\sca$ with optional noise component $\noc(t)$ and is
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% successively transformed into tympanal signal $\filt(t)$ and
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% envelope $\env(t)$. Different line styles indicate different
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% cutoff frequencies $\fc$ of the lowpass filter extracting
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% $\env(t)$.
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% \textbf{Top}:~Example representations of $\filt(t)$ and
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% $\env(t)$ for different $\sca$.
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% \textbf{a}:~Noiseless case.
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% \textbf{b}:~Noisy case.
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% \textbf{Bottom}:~Intensity metrics over a range of $\sca$.
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% \textbf{c}:~Noiseless case: Standard deviations of $\filt(t)$
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% and $\env(t)$.
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% \textbf{d}:~Noisy case: Ratios of standard deviations of
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% $\filt(t)$ and $\env(t)$ to the respective reference standard
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% deviation for input $\raw(t)=\noc(t)$.
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% \textbf{e}:~Ratios of standard deviations of $\env(t)$ as in
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% \textbf{b} for different species (averaged over songs and
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% recordings, see appendix Fig.\,\ref{fig:app_rect-lp}).
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% }
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\begin{figure}[!ht]
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\centering
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\includegraphics[width=\textwidth]{figures/fig_invariance_thresh_lp_species.pdf}
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\caption{\textbf{Feature representation of different species-specific songs
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saturates at different points in feature space.}
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Same input and processing as in
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Fig.\,\ref{fig:thresh-lp_single} but with three different
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kernels $k_i$, each with a single kernel-specific
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threshold value $\thr=0.5\cdot\sigma_{\eta_i}$.
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\textbf{a}:~Examples of species-specific grasshopper
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songs.
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\textbf{Middle}:~Average value $\mu_{f_i}$ of each feature
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$f_i(t)$ over $\sca$ per species (averaged over songs and
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recordings, see appendix
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Figs.\,\ref{fig:app_thresh-lp_pure} and
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\ref{fig:app_thresh-lp_noise}). Different color shades
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indicate different kernels $k_i$. Dots indicate $95\,\%$
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curve span per $k_i$.
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\textbf{b}:~Noiseless case.
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\textbf{c}:~Noisy case.
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\textbf{Bottom}:~2D feature spaces spanned by each pair of
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$f_i(t)$. Each trajectory corresponds to a
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species-specific combination of $\mu_{f_i}$ that develops
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with $\sca$ (colorbars). Horizontal dashes in the colorbar
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indicate $5\,\%$ (dark grey) and $95\,\%$ (light grey)
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curve span of the norm across all three $\mu_{f_i}$ per
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species.
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\textbf{d}:~Noiseless case.
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\textbf{e}:~Noisy case. Shaded areas
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}
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\label{fig:inv_thresh-lp_species}
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\label{fig:thresh-lp_species}
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\end{figure}
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\FloatBarrier
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@@ -749,7 +808,7 @@ the signal for reliable song recognition.
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\caption{\textbf{Step-wise emergence of intensity invariant song
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representation along the model pathway.}
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}
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\label{fig:inv_full}
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\label{fig:pipeline_full}
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\end{figure}
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\FloatBarrier
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@@ -759,7 +818,7 @@ the signal for reliable song recognition.
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\caption{\textbf{Step-wise emergence of intensity invariant song
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representation along the model pathway.}
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}
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\label{fig:inv_short}
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\label{fig:pipeline_short}
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\end{figure}
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\FloatBarrier
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@@ -768,7 +827,7 @@ the signal for reliable song recognition.
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\includegraphics[width=\textwidth]{figures/fig_features_cross_species.pdf}
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\caption{\textbf{Inter- and intraspecific feature variability.}
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}
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\label{fig:cross_species}
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\label{fig:feat_cross_species}
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\end{figure}
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\FloatBarrier
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@@ -778,7 +837,7 @@ the signal for reliable song recognition.
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\caption{\textbf{Step-wise emergence of intensity invariant song
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representation along the model pathway.}
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}
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\label{fig:inv_field}
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\label{fig:pipeline_field}
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\end{figure}
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\FloatBarrier
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@@ -936,7 +995,7 @@ initiation of one behavior over another is categorical (e.g. approach/stay)
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\includegraphics[width=\textwidth]{figures/fig_noise_env_sd_conversion_appendix.pdf}
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\caption{\textbf{}
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}
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\label{}
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\label{fig:app_env-sd}
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\end{figure}
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\FloatBarrier
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@@ -945,7 +1004,7 @@ initiation of one behavior over another is categorical (e.g. approach/stay)
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\includegraphics[width=\textwidth]{figures/fig_invariance_rect-lp_appendix.pdf}
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\caption{\textbf{}
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}
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\label{}
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\label{fig:app_rect-lp}
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\end{figure}
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\FloatBarrier
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@@ -954,7 +1013,7 @@ initiation of one behavior over another is categorical (e.g. approach/stay)
|
||||
\includegraphics[width=\textwidth]{figures/fig_invariance_log-hp_appendix.pdf}
|
||||
\caption{\textbf{}
|
||||
}
|
||||
\label{}
|
||||
\label{fig:app_log-hp_curves}
|
||||
\end{figure}
|
||||
\FloatBarrier
|
||||
|
||||
@@ -963,7 +1022,7 @@ initiation of one behavior over another is categorical (e.g. approach/stay)
|
||||
\includegraphics[width=\textwidth]{figures/fig_saturation_log-hp_appendix.pdf}
|
||||
\caption{\textbf{}
|
||||
}
|
||||
\label{}
|
||||
\label{fig:app_log-hp_saturation}
|
||||
\end{figure}
|
||||
\FloatBarrier
|
||||
|
||||
@@ -972,7 +1031,7 @@ initiation of one behavior over another is categorical (e.g. approach/stay)
|
||||
\includegraphics[width=\textwidth]{figures/fig_invariance_thresh-lp_pure_appendix.pdf}
|
||||
\caption{\textbf{}
|
||||
}
|
||||
\label{}
|
||||
\label{fig:app_thresh-lp_pure}
|
||||
\end{figure}
|
||||
\FloatBarrier
|
||||
|
||||
@@ -981,7 +1040,7 @@ initiation of one behavior over another is categorical (e.g. approach/stay)
|
||||
\includegraphics[width=\textwidth]{figures/fig_invariance_thresh-lp_noise_appendix.pdf}
|
||||
\caption{\textbf{}
|
||||
}
|
||||
\label{}
|
||||
\label{fig:app_thresh-lp_noise}
|
||||
\end{figure}
|
||||
\FloatBarrier
|
||||
|
||||
@@ -990,7 +1049,7 @@ initiation of one behavior over another is categorical (e.g. approach/stay)
|
||||
\includegraphics[width=\textwidth]{figures/fig_kernel_sd_perc_thresh_lp_appendix.pdf}
|
||||
\caption{\textbf{}
|
||||
}
|
||||
\label{}
|
||||
\label{fig:app_thresh-lp_kern-sd}
|
||||
\end{figure}
|
||||
\FloatBarrier
|
||||
|
||||
@@ -999,7 +1058,7 @@ initiation of one behavior over another is categorical (e.g. approach/stay)
|
||||
\includegraphics[width=\textwidth]{figures/fig_kernel_sd_perc_full_appendix.pdf}
|
||||
\caption{\textbf{}
|
||||
}
|
||||
\label{}
|
||||
\label{fig:app_full_kern-sd}
|
||||
\end{figure}
|
||||
\FloatBarrier
|
||||
|
||||
@@ -1008,7 +1067,7 @@ initiation of one behavior over another is categorical (e.g. approach/stay)
|
||||
\includegraphics[width=\textwidth]{figures/fig_kernel_sd_perc_short_appendix.pdf}
|
||||
\caption{\textbf{}
|
||||
}
|
||||
\label{}
|
||||
\label{fig:app_short_kern-sd}
|
||||
\end{figure}
|
||||
\FloatBarrier
|
||||
|
||||
@@ -1017,7 +1076,7 @@ initiation of one behavior over another is categorical (e.g. approach/stay)
|
||||
\includegraphics[width=\textwidth]{figures/fig_kernel_sd_perc_field_appendix.pdf}
|
||||
\caption{\textbf{}
|
||||
}
|
||||
\label{}
|
||||
\label{fig:app_field_kern-sd}
|
||||
\end{figure}
|
||||
\FloatBarrier
|
||||
|
||||
@@ -1027,7 +1086,7 @@ initiation of one behavior over another is categorical (e.g. approach/stay)
|
||||
\includegraphics[width=\textwidth]{figures/fig_invariance_cross_species_thresh_appendix.pdf}
|
||||
\caption{\textbf{}
|
||||
}
|
||||
\label{}
|
||||
\label{fig:app_cross_species_thresh}
|
||||
\end{figure}
|
||||
\FloatBarrier
|
||||
|
||||
|
||||
Reference in New Issue
Block a user