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S. Meyer et al.
approximation matrix c a (L
∗ ) to 0. The remaining coefficients are then shaped
into a representation matrix C of size P × K where P =
H
2 L−1 and k =
W
2 L−1 ,
consisting of 4 coefficient block matrices 0, c v (L
∗ ), c h (L
∗ ), c d (L
∗ ) of size
P
2 ×
K
2
representing the orientational coefficients.
C(L
∗ ) =
0 c v (L
∗ )
c h (L
∗ ) c d (L
∗ )
(2)
For each block matrix, the 1% largest absolute coefficients are determined and
the magnitude of the lowest of these is selected as a threshold: Every coefficient
in the block that has smaller absolute value is set to zero. The resulting RIDF
for any L
∗ is then given as:
ξ wv (C X , C Y , r) =
1
P ∗ K
P
p
K
k
(C X [p, k] − C
r
Y [p, k])
(3)
where C X and C
r
Y are the coefficient matrices for the view at the origin and
the view at the displaced location with orientation r, respectively. It is worth
mentioning that rotation is applied to the image before coefficient extraction. It
can also easily be seen that ξ wv can be applied in an analogous way to each coefficient block matrix in an isolated fashion, which allows one to analyse the RIDF
of each set of coefficients separately. In our experiments we used models with
L = 1, L = 2 and L = 3 (filter width 2, 4 and 8 pixel respectively) as models with
higher values for L performed poorly in pilot studies. The DWT is performed
with MatLab’s Wavelet Toolbox.
In order to get an understanding of how much accuracy is lost when discarding the approximation matrix, we additionally used a model M gauss that
would use a Gaussian filter on an image with a filter width of σ = 5 (determined
experimentally), limiting the input signal to its lower frequency band. In addition, we compare our results to models which use the height of the skyline, a
feature ants can use to navigate [9,21], and which can be viewed as a low spatial
frequency signal composed of oriented UV-contrast edges, which the ant visual
system is tuned to [7,19,20,22,27,28]. Our final two models, therefore, consist
of two approaches for extracting the skyline height in our virtual environment.
M st scans an image columnwise from the top until it encounters a pixel that is
not sky coloured. The skyline value S[w] at a specific image column w is then
given by the difference between height H and the index h of the encountered
non-sky pixel. Repeating this procedure for each column of the image will result
in a vector S of size 1 × W . In an analogous way, M sb scans an image columnwise from the bottom up, until it encounters a pixel that is sky coloured. The
resulting RIDF ξ s (S X , S Y ) is given as the mean squared error between S X and
S Y .
Simulated Environment: AntWorld. Experiments are conducted in a virtual
environment (AntWorld, Fig. 1), which was reconstructed from an ant field-site
in Spain [17]. Lacking major landmarks, the world contains open areas filled with
S. Meyer et al.
approximation matrix c a (L
∗ ) to 0. The remaining coefficients are then shaped
into a representation matrix C of size P × K where P =
H
2 L−1 and k =
W
2 L−1 ,
consisting of 4 coefficient block matrices 0, c v (L
∗ ), c h (L
∗ ), c d (L
∗ ) of size
P
2 ×
K
2
representing the orientational coefficients.
C(L
∗ ) =
0 c v (L
∗ )
c h (L
∗ ) c d (L
∗ )
(2)
For each block matrix, the 1% largest absolute coefficients are determined and
the magnitude of the lowest of these is selected as a threshold: Every coefficient
in the block that has smaller absolute value is set to zero. The resulting RIDF
for any L
∗ is then given as:
ξ wv (C X , C Y , r) =
1
P ∗ K
P
p
K
k
(C X [p, k] − C
r
Y [p, k])
(3)
where C X and C
r
Y are the coefficient matrices for the view at the origin and
the view at the displaced location with orientation r, respectively. It is worth
mentioning that rotation is applied to the image before coefficient extraction. It
can also easily be seen that ξ wv can be applied in an analogous way to each coefficient block matrix in an isolated fashion, which allows one to analyse the RIDF
of each set of coefficients separately. In our experiments we used models with
L = 1, L = 2 and L = 3 (filter width 2, 4 and 8 pixel respectively) as models with
higher values for L performed poorly in pilot studies. The DWT is performed
with MatLab’s Wavelet Toolbox.
In order to get an understanding of how much accuracy is lost when discarding the approximation matrix, we additionally used a model M gauss that
would use a Gaussian filter on an image with a filter width of σ = 5 (determined
experimentally), limiting the input signal to its lower frequency band. In addition, we compare our results to models which use the height of the skyline, a
feature ants can use to navigate [9,21], and which can be viewed as a low spatial
frequency signal composed of oriented UV-contrast edges, which the ant visual
system is tuned to [7,19,20,22,27,28]. Our final two models, therefore, consist
of two approaches for extracting the skyline height in our virtual environment.
M st scans an image columnwise from the top until it encounters a pixel that is
not sky coloured. The skyline value S[w] at a specific image column w is then
given by the difference between height H and the index h of the encountered
non-sky pixel. Repeating this procedure for each column of the image will result
in a vector S of size 1 × W . In an analogous way, M sb scans an image columnwise from the bottom up, until it encounters a pixel that is sky coloured. The
resulting RIDF ξ s (S X , S Y ) is given as the mean squared error between S X and
S Y .
Simulated Environment: AntWorld. Experiments are conducted in a virtual
environment (AntWorld, Fig. 1), which was reconstructed from an ant field-site
in Spain [17]. Lacking major landmarks, the world contains open areas filled with