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j-hartling
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\newlabel{eq:toy_highpass_pure}{{13}{13}{}{}{}}
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\newlabel{eq:toy_env_noise}{{14}{13}{}{}{}}
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\newlabel{eq:toy_log_noise}{{15}{13}{}{}{}}
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\newlabel{eq:toy_highpass_noise}{{16}{13}{}{}{}}
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\newlabel{eq:toy_highpass_noise}{{16}{14}{}{}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces \textbf {Intensity invariance through logarithmic compression and adaptation is restricted by the noise floor and decreases SNR.} Input $x_{\text {filt}}(t)$ consists of song component $s(t)$ scaled by $\alpha $ with optional noise component $\eta (t)$ and is successively transformed into envelope $x_{\text {env}}(t)$, logarithmically compressed envelope $x_{\text {log}}(t)$, and intensity-adapted envelope $x_{\text {adapt}}(t)$. \textbf {Top}:~Example representations of $x_{\text {env}}(t)$, $x_{\text {log}}(t)$, and $x_{\text {adapt}}(t)$ for different $\alpha $. \textbf {a}:~Noiseless case. \textbf {b}:~Noisy case. \textbf {Bottom}:~Intensity metrics over a range of $\alpha $. \textbf {c}:~Noiseless case: Standard deviations $\sigma _x$ of $x_{\text {env}}(t)$, $x_{\text {log}}(t)$, and $x_{\text {adapt}}(t)$. \textbf {d}:~Noisy case: Ratios of $\sigma _x$ of $x_{\text {env}}(t)$, $x_{\text {log}}(t)$, and $x_{\text {adapt}}(t)$ to the respective reference standard deviation $\sigma _{\eta }$ for input $x_{\text {filt}}(t)=\eta (t)$. Shaded areas indicate $5\,\%$ (dark grey) and $95\,\%$ (light grey) curve span for $x_{\text {adapt}}(t)$. \textbf {e}:~Ratios of $\sigma _x$ to $\sigma _{\eta }$ of $x_{\text {adapt}}(t)$ as in \textbf {d} for different species (averaged over songs and recordings, see appendix Fig\,\ref {fig:app_log-hp_curves}). Dots indicate $95\,\%$ curve span per species. }}{15}{}\protected@file@percent }
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\newlabel{fig:log-hp}{{5}{15}{}{}{}}
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\@writefile{toc}{\contentsline {subsection}{\numberline {3.3}Thresholding nonlinearity \& temporal averaging}{16}{}\protected@file@percent }
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\newlabel{eq:pdf_split}{{17}{16}{}{}{}}
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\newlabel{eq:feat_avg}{{18}{16}{}{}{}}
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\newlabel{eq:feat_prop}{{19}{16}{}{}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces \textbf {Intensity invariance through thresholding and temporal averaging is mediated by the interaction of threshold value and noise floor.} Input $x_{\text {adapt}}(t)$ consists of song component $s(t)$ scaled by $\alpha $ with optional noise component $\eta (t)$ and is transformed into single kernel response $c(t)$, binary response $b(t)$, and feature $f(t)$. Different color shades indicate different threshold values $\Theta $ (multiples of reference standard deviation $\sigma _{\eta }$ of $c(t)$ for input $x_{\text {adapt}}(t)=\eta (t)$, with darker colors for higher $\Theta $). \textbf {Left}:~Noisy case: Example representations of $x_{\text {adapt}}(t)$ as well as $c(t)$, $b(t)$, and $f(t)$ for different $\alpha $. \textbf {a}:~$x_{\text {adapt}}(t)$ with kernel $k(t)$ in black. \textbf {b\,-\,d}: $c(t)$, $b(t)$, and $f(t)$ based on the same $x_{\text {adapt}}(t)$ from \textbf {a} but with different $\Theta $. \textbf {Right}:~Average value $\mu _f$ of $f(t)$ for each $\Theta $ from \textbf {b\,-\,d}, once for the noisy case (solid lines) and once for the noiseless case (dotted lines). Dots indicate $95\,\%$ curve span (noisy case). \textbf {e}:~$\mu _f$ over a range of $\alpha $. \textbf {f}:~$\mu _f$ over the standard deviation of noisy input $x_{\text {adapt}}$ corresponding to the values of $\alpha $ shown in \textbf {e}. }}{18}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces \textbf {Intensity invariance through thresholding and temporal averaging is mediated by the interaction of threshold value and noise floor.} Input $x_{\text {adapt}}(t)$ consists of song component $s(t)$ scaled by $\alpha $ with optional noise component $\eta (t)$ and is transformed into single kernel response $c(t)$, binary response $b(t)$, and feature $f(t)$. Different color shades indicate different threshold values $\Theta $ (multiples of reference standard deviation $\sigma _{\eta }$ of $c(t)$ for input $x_{\text {adapt}}(t)=\eta (t)$, with darker colors for higher $\Theta $). \textbf {Left}:~Noisy case: Example representations of $x_{\text {adapt}}(t)$ as well as $c(t)$, $b(t)$, and $f(t)$ for different $\alpha $. \textbf {a}:~$x_{\text {adapt}}(t)$ with kernel $k(t)$ in black. \textbf {b\,-\,d}: $c(t)$, $b(t)$, and $f(t)$ based on the same $x_{\text {adapt}}(t)$ from \textbf {a} but with different $\Theta $. \textbf {Right}:~Average value $\mu _f$ of $f(t)$ for each $\Theta $ from \textbf {b\,-\,d}. Dots indicate $95\,\%$ curve span (noisy case). \textbf {e}:~$\mu _f$ over a range of $\alpha $, once for the noisy case (solid lines) and once for the noiseless case (dotted lines). \textbf {f}:~Noisy case: $\mu _f$ over the standard deviation of input $x_{\text {adapt}}$ corresponding to the values of $\alpha $ shown in \textbf {e}. Shaded area indicates standard deviations that would be capped in the output $x_{\text {adapt}}(t)$ of the previous transformation pair (see Fig.\,\ref {fig:log-hp}cd). }}{18}{}\protected@file@percent }
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\newlabel{fig:thresh-lp_single}{{6}{18}{}{}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces \textbf {Feature representation of different species-specific songs saturates at different points in feature space.} Same input and processing as in Fig.\,\ref {fig:thresh-lp_single} but with three different kernels $k_i$, each with a single kernel-specific threshold value $\Theta _i=0.5\cdot \sigma _{\eta _i}$. \textbf {a}:~Examples of species-specific grasshopper songs. \textbf {Middle}:~Average value $\mu _{f_i}$ of each feature $f_i(t)$ over $\alpha $ per species (averaged over songs and recordings, see appendix Figs.\,\ref {fig:app_thresh-lp_pure} and \ref {fig:app_thresh-lp_noise}). Different color shades indicate different kernels $k_i$. Dots indicate $95\,\%$ curve span per $k_i$. \textbf {b}:~Noiseless case. \textbf {c}:~Noisy case. \textbf {Bottom}:~2D feature spaces spanned by each pair of $f_i(t)$. Each trajectory corresponds to a species-specific combination of $\mu _{f_i}$ that develops with $\alpha $ (colorbars). Horizontal dashes in the colorbar indicate $5\,\%$ (dark grey) and $95\,\%$ (light grey) curve span of the norm across all three $\mu _{f_i}$ per species. \textbf {d}:~Noiseless case. \textbf {e}:~Noisy case. Shaded areas indicate the average minimum $\mu _{f_i}$ across all species-specific trajectories. }}{19}{}\protected@file@percent }
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\newlabel{fig:thresh-lp_species}{{7}{19}{}{}{}}
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[0] Config.pm:307> INFO - This is Biber 2.19
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[36] biber:340> INFO - === Di Mai 5, 2026, 15:25:39
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@@ -610,16 +610,16 @@ This effect is more pronounced for lower $\fc$ of the lowpass filter and is
|
||||
presumably caused by the attenuation of high-frequency components in the
|
||||
signal, which are more prominent in the noise component $\noc(t)$ than in the
|
||||
song component $\soc(t)$. The effect also appears relatively consistent across
|
||||
different species, although small variations based on different song structures
|
||||
and distributions exist~(Fig.\,\ref{fig:rect-lp}e). In summary, the standard
|
||||
deviation of $\env(t)$ has never been observed to transition into a saturation
|
||||
regime for larger $\sca$ but rather continues to increase proportionally to
|
||||
$\sca$ for all tested $\fc$, in both the noiseless and the noisy case and
|
||||
across different species. Consequently, the combination of rectification and
|
||||
lowpass filtering does not contribute to intensity invariance. However, this
|
||||
transformation pair does improve the SNR of $\env(t)$ relative to $\filt(t)$
|
||||
and thus provides subsequent processing stages with a more robust input
|
||||
representation and higher input SNR.
|
||||
different species, although small variations exist~(Fig.\,\ref{fig:rect-lp}e)
|
||||
that are presumably based on different song structures and frequency spectra.
|
||||
In summary, the standard deviation of $\env(t)$ has never been observed to
|
||||
transition into a saturation regime for larger $\sca$ but rather continues to
|
||||
increase proportionally to $\sca$ for all tested $\fc$, in both the noiseless
|
||||
and the noisy case and across different species. Consequently, the combination
|
||||
of rectification and lowpass filtering does not contribute to intensity
|
||||
invariance. However, this transformation pair does improve the SNR of $\env(t)$
|
||||
relative to $\filt(t)$ and thus provides subsequent processing stages with a
|
||||
more robust input representation and higher input SNR.
|
||||
|
||||
\begin{figure}[!ht]
|
||||
\centering
|
||||
@@ -880,7 +880,21 @@ the SNR of $\adapt(t)$ are much less understood and likely relate to properties
|
||||
of the signal, whereas the SNR of $f(t)$ depends on the choice of $\Theta$ and
|
||||
can be more directly manipulated by the system.
|
||||
|
||||
Finally,
|
||||
Finally, the effects of thresholding and temporal averaging must be seen in the
|
||||
context of the previous transformation pair of logarithmic compression and
|
||||
adaptation.
|
||||
|
||||
Finally, the question remains whether the intensity-invariant output $\adapt(t)$
|
||||
of the previous transformation pair allows feature
|
||||
|
||||
Finally, the output $\adapt(t)$ of the previous transformation
|
||||
pair~(Fig.\,\ref{fig:log-hp}cd) can be related to the input $\adapt(t)$ of the
|
||||
current transformation pair by plotting the values of $f(t)$ over the standard
|
||||
deviation of input $\adapt(t)$ instead of
|
||||
$\sca$~(Fig.\,\ref{fig:thresh-lp_single}f). This is relevant because, unlike
|
||||
$\sca$, the standard deviation of $\adapt(t)$ is capped to a maximum value of
|
||||
around 10\,dB by the previous transformation pair~(Fig.\,\ref{fig:log-hp}cd)
|
||||
|
||||
|
||||
\begin{figure}[!ht]
|
||||
\centering
|
||||
@@ -904,14 +918,17 @@ Finally,
|
||||
same $\adapt(t)$ from \textbf{a} but with different
|
||||
$\Theta$.
|
||||
\textbf{Right}:~Average value $\mu_f$ of $f(t)$ for each
|
||||
$\Theta$ from \textbf{b\,-\,d}, once for the noisy case
|
||||
(solid lines) and once for the noiseless case (dotted
|
||||
lines). Dots indicate $95\,\%$ curve span (noisy case).
|
||||
\textbf{e}:~$\mu_f$ over a range of $\sca$.
|
||||
\textbf{f}:~$\mu_f$ over the standard deviation of noisy
|
||||
input $\adapt$ corresponding to the values of $\sca$ shown
|
||||
in \textbf{e}.
|
||||
% Why plot noiseless case over SD of noisy input? Omit?
|
||||
$\Theta$ from \textbf{b\,-\,d}. Dots indicate $95\,\%$
|
||||
curve span (noisy case).
|
||||
\textbf{e}:~$\mu_f$ over a range of $\sca$, once for the
|
||||
noisy case (solid lines) and once for the noiseless case
|
||||
(dotted lines).
|
||||
\textbf{f}:~Noisy case: $\mu_f$ over the standard
|
||||
deviation of input $\adapt$ corresponding to the values of
|
||||
$\sca$ shown in \textbf{e}. Shaded area indicates standard
|
||||
deviations that would be capped in the output $\adapt(t)$
|
||||
of the previous transformation pair (see
|
||||
Fig.\,\ref{fig:log-hp}cd).
|
||||
}
|
||||
\label{fig:thresh-lp_single}
|
||||
\end{figure}
|
||||
|
||||
@@ -266,6 +266,12 @@ leg_kwargs = dict(
|
||||
handlelength=1.5,
|
||||
columnspacing=1,
|
||||
)
|
||||
cap_kwargs = dict(
|
||||
color='k',
|
||||
alpha=0.5,
|
||||
lw=0,
|
||||
zorder=5,
|
||||
)
|
||||
plateau_settings = dict(
|
||||
low=0.05,
|
||||
high=0.95,
|
||||
@@ -293,6 +299,12 @@ kern_specs = np.array([
|
||||
[3, 0.002],
|
||||
])[np.array([1])]
|
||||
|
||||
# PREPARATION:
|
||||
|
||||
# Get saturation level of invariant envelope from log-hp analysis:
|
||||
inv_path = search_files(example_file, dir='../data/inv/log_hp/')[0]
|
||||
sigma_cap = load_data(inv_path, files='measure_inv')[0]['measure_inv'][-1]
|
||||
|
||||
# EXECUTION:
|
||||
print(f'Processing {data_path}')
|
||||
|
||||
@@ -444,31 +456,45 @@ for i in range(thresh_rel.size):
|
||||
ind = get_saturation(noise_data['measure_feat'][:, i], **plateau_settings)[1]
|
||||
saturation_inds.append(ind)
|
||||
|
||||
# Plot analysis results:
|
||||
for ax, x in zip([alpha_ax, sigma_ax], [scales, noise_data['measure_inv']]):
|
||||
# Plot pure-song analysis results:
|
||||
handles = ax.plot(x, pure_data['measure_feat'], lw=lw['big'], ls='dotted')
|
||||
[h.set_color(c) for h, c in zip(handles, shaded['feat'])]
|
||||
# Plot pure-song analysis results over alpha:
|
||||
handles = alpha_ax.plot(scales, pure_data['measure_feat'], lw=lw['big'], ls='dotted')
|
||||
[h.set_color(c) for h, c in zip(handles, shaded['feat'])]
|
||||
|
||||
# Plot noise-song analysis results:
|
||||
handles = ax.plot(x, noise_data['measure_feat'], lw=lw['big'])
|
||||
[h.set_color(c) for h, c in zip(handles, shaded['feat'])]
|
||||
# Plot noise-song analysis results over alpha:
|
||||
handles = alpha_ax.plot(scales, noise_data['measure_feat'], lw=lw['big'])
|
||||
[h.set_color(c) for h, c in zip(handles, shaded['feat'])]
|
||||
|
||||
# Indicate threshold-specific saturation:
|
||||
for i, ind in enumerate(saturation_inds):
|
||||
color = shaded['feat'][i]
|
||||
ax.plot(x[ind], 0, c='w', alpha=1, zorder=5.5, **plateau_dot_kwargs,
|
||||
transform=ax.get_xaxis_transform())
|
||||
ax.plot(x[ind], 0, mfc=color, mec='k', alpha=0.75, zorder=6,
|
||||
**plateau_dot_kwargs, transform=ax.get_xaxis_transform())
|
||||
ax.vlines(x[ind], ax.get_ylim()[0], noise_data['measure_feat'][ind, i],
|
||||
color=color, **plateau_line_kwargs)
|
||||
# Indicate threshold-specific saturation:
|
||||
for i, ind in enumerate(saturation_inds):
|
||||
color = shaded['feat'][i]
|
||||
alpha_ax.plot(scales[ind], 0, c='w', alpha=1, zorder=5.5, **plateau_dot_kwargs,
|
||||
transform=alpha_ax.get_xaxis_transform())
|
||||
alpha_ax.plot(scales[ind], 0, mfc=color, mec='k', alpha=0.75, zorder=6,
|
||||
**plateau_dot_kwargs, transform=alpha_ax.get_xaxis_transform())
|
||||
alpha_ax.vlines(scales[ind], alpha_ax.get_ylim()[0], noise_data['measure_feat'][ind, i],
|
||||
color=color, **plateau_line_kwargs)
|
||||
|
||||
# Add proxy legend:
|
||||
if ax == alpha_ax:
|
||||
h1 = ax.plot([], [], c='k', lw=lw['big'], label='$\\alpha\\cdot s(t) + \\eta(t)$')[0]
|
||||
h2 = ax.plot([], [], c='k', lw=lw['big'], ls='dotted', label='$\\alpha\\cdot s(t)$')[0]
|
||||
ax.legend(handles=[h1, h2], **leg_kwargs)
|
||||
# Add proxy legend:
|
||||
h1 = alpha_ax.plot([], [], c='k', lw=lw['big'], label='$\\alpha\\cdot s(t) + \\eta(t)$')[0]
|
||||
h2 = alpha_ax.plot([], [], c='k', lw=lw['big'], ls='dotted', label='$\\alpha\\cdot s(t)$')[0]
|
||||
alpha_ax.legend(handles=[h1, h2], **leg_kwargs)
|
||||
|
||||
# Plot noise-song analysis results over sigma:
|
||||
handles = sigma_ax.plot(noise_data['measure_inv'], noise_data['measure_feat'], lw=lw['big'])
|
||||
[h.set_color(c) for h, c in zip(handles, shaded['feat'])]
|
||||
|
||||
# Indicate threshold-specific saturation:
|
||||
for i, ind in enumerate(saturation_inds):
|
||||
color = shaded['feat'][i]
|
||||
sigma_ax.plot(scales[ind], 0, c='w', alpha=1, zorder=5.5, **plateau_dot_kwargs,
|
||||
transform=sigma_ax.get_xaxis_transform())
|
||||
sigma_ax.plot(scales[ind], 0, mfc=color, mec='k', alpha=0.75, zorder=6,
|
||||
**plateau_dot_kwargs, transform=sigma_ax.get_xaxis_transform())
|
||||
sigma_ax.vlines(scales[ind], sigma_ax.get_ylim()[0], noise_data['measure_feat'][ind, i],
|
||||
color=color, **plateau_line_kwargs)
|
||||
|
||||
# Indicate sigma range capped by log-hp mechanism:
|
||||
sigma_ax.axvspan(sigma_cap, sigma_ax.get_xlim()[1], **cap_kwargs)
|
||||
|
||||
if save_path is not None:
|
||||
fig.savefig(save_path)
|
||||
|
||||
Reference in New Issue
Block a user