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is proposed. Similarly to α(p) the new exponent here-
after called β(p) is parametrized by the size r of the
neighbourhood. At each point p ∈I, with coordinate
(i, j), a window w : I([i −r, i + r], [ j −r, j + r]) is ex-
tracted and mapped onto a cloud of points C(w), sat-
isfying:
(i, j, k) ∈ C(w) ↔ w(i, j) = k. (8)
In the following, each point p ∈ C(w) is dilated by
a sphere B(p, d), centred at p with radius d and the
dilated volume is computed through
V (d) =
[
p∈C(w)
B(p, d). (9)
The local H
¨
older exponent β(p) is computed by
β(p) = lim
d→0
logV (d)
logd
, (10)
and this limit is estimated by fitting a straight line to
the curve log d ×logV (d) and computing the corre-
sponding slope for each point in the image. Finally,
the exponent is locally assigned to the pixel p. Figure
1 illustrates the steps involved in the dilation process
of the pixels within w.
In order to verify how well β(p) estimates the
H
¨
older exponent and consequently the multifractal
spectrum of a grey-level image, Figure 2 compares
the proposed f (β) with the method in (Xu et al.,
2009) over a synthetic multifractal texture. Figure
2(a) shows the image generated by the Meakin model
(Meakin, 1987). Figure 2(b) exhibits f (α) curve ob-
tained through (Xu et al., 2009) and the proposed one
compared to the theoretical spectrum, obtained by a
procedure described in (Chhabra et al., 1989). Figure
2(c) shows a histogram of the average error when fit-
ting a straight line to the log−log curve at each point
p.
The f (α) and f (β) curves have a similar shape to
that of the expected curve, although the experimen-
tal values obtained depart from the theoretical values,
due to the data being a discrete image. In the partic-
ular case of f (β) this is also caused by the sparsity
of the spheres in the space. An isolated point that is
dilated and suddenly becomes connected to other di-
lated points has its volume raised very quickly leading
to overestimated values of the exponent. However, the
logd ×logV (d) curve in the proposed method showed
the best fit to a straight line than logµ(B(p, r)) ×logr
in (Xu et al., 2009), as illustrated by the smaller er-
ror in the histogram. Such behaviour is due largely
to the smoother growing of the dilation volume when
the radius is gradually increased.
Finally, Figure 3 illustrates the obtained H
¨
older
exponents for a texture of example from the real-
world and the respective multifractal spectrum. Here,
even the shape of the spectrum curve is different from
those found in well-defined multifractals like in the
Meakin model, which again can be explained by the
limited number of scales allowed by the image do-
main. Another important remark here is that the val-
ues of α (or β) are expected to stay between 2 and
3, since they are basically the fractal dimension of an
object immersed in the three-dimensional topological
space . However, here the sparsity of spheres in more
discontinuous regions of the image causes the expo-
nent to extrapolate such interval, as previously ex-
plained. Such extrapolation is also common in other
applied works on multifractals, like (Xu et al., 2009;
Meakin, 1987; Bedford, 1989)and several others.
Following (Xu et al., 2009), this exponent is also
computed not only on the intensity image, but on the
image gradient, and on its Laplacian. Therefore, three
variants of β(p) are defined:
β
I
(p) = lim
d→0
log
S
p∈C(w)
B(p, d)
logd
, (11)
β
G
(p) = lim
d→0
log
S
p∈C(|∇w|)
B(p, d)
logd
, (12)
and
β
L
(p) = lim
d→0
log
S
p∈C(|∇
2
w|)
B(p, d)
logd
, (13)
From equations 5 and 6, three MFS vectors are
obtained (M
β
I
, M
β
G
and M
β
L
) and they are concatenated
to be used as the texture descriptors. The descriptors
based on β(p) were also interchanged and merged
with those from α(p) to obtain even more precise
classification.
5 EXPERIMENTS
Some parameters in the proposed method were empir-
ically chosen. Thus to estimate the H
¨
older exponent
by the Bouligand-Minkowski method a maximum di-
lation radius of 5 and a 5 ×5 neighbourhood were em-
ployed at each pixel. Moreover, the values of β were
limited within the interval [1, 4] and sampled into 33
bins, so that each M
β
I
, M
β
G
and M
β
L
is a vector of 33
real values.
The proposed method was used for the classifica-
tion of Brodatz textures (Brodatz, 1966). The first
40 images (D1-D40) at a resolution of 512x512 pix-
els were used. All images were divided into 16 non-
overlapping windows giving rise to a final set of 40
classes with 16 images in each class. The Brodatz
collection is made of a set of texture images contain-
ing a wide variety of levels of regularity, granularity,
MultifractalTextureAnalysisusingaDilation-basedHölderExponent
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