Image Processing Reference
In-Depth Information
between Gabor wavelets and tree- and pyramidal-structured wavelets suggested that Gabor
has the greater descriptional ability, but a penalty of greater computational complexity
(Pichler, 1996). There has also been renewed resurgence of interest in Markov random
fields (Gimmel' farb, 1996) and (Wu, 1996). Others, such as the Walsh transform (where
the basis functions are 1s and 0s) appear yet to await application in texture description, no
doubt due to basic properties. In fact, a recent survey (Randen, 2000) includes use of
Fourier, wavelet and discrete cosine transforms (Section 2.7.1) for texture characterisation.
These approaches are structural in nature: an image is viewed in terms of a transform
applied to a whole image as such exposing its
structure
. This is like the dictionary definition
of an arrangement of parts. Another part of the dictionary definition concerned
detail
: this
can of course be exposed by analysis of the high frequency components but these can be
prone to noise. An alternative way to analyse the detail is to consider the
statistics
of an
image.
8.3.3
Statistical approaches
The most famous statistical approach is the
co-occurrence matrix
. This was the result of
the first approach to describe, and then classify, image texture (Haralick, 1973). It remains
popular today, by virtue of good performance. The co-occurrence matrix contains elements
that are counts of the number of pixel pairs for specific brightness levels, when separated
by some distance and at some relative inclination. For brightness levels
b
1 and
b
2 the co-
occurrence matrix
C
is
N
N
Σ Σ
C
=
(
P
= 1)
b
(
P
= 2)
b
(8.6)
bb
1,
2
xy
,
xy
,
x
=1
y
=1
where the
x
co-ordinate
x
is the offset given by the specified distance
d
and inclination
by
x
′ =
x
+
d
cos(θ
)
(
d
∈ 1, max(
d
)) ∧ (θ
∈ 0, 2π
)
(8.7)
and the
y
co-ordinate
y
′ is
) (8.8)
When Equation 8.6 is applied to an image, we obtain a square, symmetric, matrix whose
dimensions equal the number of grey levels in the picture. The co-occurrence matrices for
the three Brodatz textures of Figure
8.2
are shown in Figure
8.4
. In the co-occurrence
matrix generation, the maximum distance was 1 pixel and the directions were set to select
the four nearest neighbours of each point. Now the result for the two samples of French
canvas, Figures
8.4
(a) and (b), appear to be much more similar and quite different to the
co-occurrence matrix for sand, Figure
8.4
(c). As such, the co-occurrence matrix looks like
it can better expose the underlying nature of texture than can the Fourier description. This
is because the co-occurrence measures spatial relationships between brightness, as opposed
to frequency content. This clearly gives alternative results. To generate results faster, the
number of grey levels can be reduced by brightness scaling of the whole image, reducing
the dimensions of the co-occurrence matrix, but this reduces discriminatory ability.
These matrices have been achieved by the implementation in Code
8.2
. The subroutine
tex_cc
generates the co-occurrence matrix of an image
im
given a maximum distance
d
and a number of directions
dirs
. If
d
and
dirs
are set to 1 and 4, respectively (as was
y
=
y
+
d
sin(
)
(
d
1, max(
d
))
(
0, 2
