Snapshot Navigation in the Wavelet Domain
253
ˆ
r and r
∗ for different models. In addition to the pixel space based model M px and
the wavelet coefficients of level 1 one model M
1
wv , we also included 5 other models. We used two more wavelet models based on coefficients of level 2 (M
2
wv ) and
level 3 (M
3
wv ), as well as a model M g that performed a simple Gaussian low-pass
filtering on images, to see how lower frequency components influence robustness.
Furthermore, we used two different methods to extract the skyline height from
images (scanning from top - M st and scanning from bottom - M sb ) and calculated ξ st and ξ sb from this one-dimensional input, to see how much information
is provided by the shape of objects in front of the sky and the horizon.
Fig. 5. Overall accuracy of heading recovery for different models. Angular
error for different models (box labels) with increasing displacement (panels) from snapshot location as boxplots. Data is aggregated from 4 routes and displacements to either
side of the route. In each panel, boxes sharing letters do not have significantly different
medians. Boxes that do not share letters are significantly different (p-value < 0.05,
Kruskal-Wallis test). n = 139 for each condition. (Color figure online)
For each channel and model, we collected the respective ˆ
r and calculated
the angular error at the 139 locations per channel. We then determined the
median and the interquartile range (IQR) and visualised the distributions as
boxplots (Fig. 5). While IQR and median absolute angular error increased for
each model with increased displacement from the route, the wavelet based models
consistently show a lower IQR and median when compared to other models
(Fig. 5, red boxes). Indeed, with increased limitation to the lower frequency band
of the signal (as given by M
2
wv , M
3
wv and M g ) we observe increased angular
errors, indicating that high frequencies may carry important information about
the environment. It is noteworthy however, that M
2
wv performed similar to M
1
wv ,
offering the possibility of a simpler (lower resolution) option to decrease angular
errors. Across all experiments we observe that skyline height based models M st
and M sb ) are not robust when compared to raw pixel space. This is a strong
253
ˆ
r and r
∗ for different models. In addition to the pixel space based model M px and
the wavelet coefficients of level 1 one model M
1
wv , we also included 5 other models. We used two more wavelet models based on coefficients of level 2 (M
2
wv ) and
level 3 (M
3
wv ), as well as a model M g that performed a simple Gaussian low-pass
filtering on images, to see how lower frequency components influence robustness.
Furthermore, we used two different methods to extract the skyline height from
images (scanning from top - M st and scanning from bottom - M sb ) and calculated ξ st and ξ sb from this one-dimensional input, to see how much information
is provided by the shape of objects in front of the sky and the horizon.
Fig. 5. Overall accuracy of heading recovery for different models. Angular
error for different models (box labels) with increasing displacement (panels) from snapshot location as boxplots. Data is aggregated from 4 routes and displacements to either
side of the route. In each panel, boxes sharing letters do not have significantly different
medians. Boxes that do not share letters are significantly different (p-value < 0.05,
Kruskal-Wallis test). n = 139 for each condition. (Color figure online)
For each channel and model, we collected the respective ˆ
r and calculated
the angular error at the 139 locations per channel. We then determined the
median and the interquartile range (IQR) and visualised the distributions as
boxplots (Fig. 5). While IQR and median absolute angular error increased for
each model with increased displacement from the route, the wavelet based models
consistently show a lower IQR and median when compared to other models
(Fig. 5, red boxes). Indeed, with increased limitation to the lower frequency band
of the signal (as given by M
2
wv , M
3
wv and M g ) we observe increased angular
errors, indicating that high frequencies may carry important information about
the environment. It is noteworthy however, that M
2
wv performed similar to M
1
wv ,
offering the possibility of a simpler (lower resolution) option to decrease angular
errors. Across all experiments we observe that skyline height based models M st
and M sb ) are not robust when compared to raw pixel space. This is a strong