Image Processing Reference
In-Depth Information
NN
Σ Σ
v
v
2
i
=
( - )
u
NFP
(8.5)
u
,
v
u
=1
=1
These measures are shown for the three Brodatz textures in Code
8.1
. In a way, they are
like the shape descriptions in the previous chapter: the measures should be the same for the
same object and should differ for a different one. Here, the texture measures are actually
different for each of the textures. Perhaps the detail in the French canvas, Code
8.1
(a),
could be made to give a closer measure to that of the full resolution, Code
8.1
(b), by using
the frequency scaling property of the Fourier transform, discussed in Section 2.6.3. The
beach sand clearly gives a different set of measures from the other two, Code
8.1
(c). In
fact, the beach sand in Code
8.1
(c) would appear to be more similar to the French canvas
in Code
8.1
(b), since the inertia and energy measures are much closer than those for Code
8.1
(a) (only the entropy measure in Code
8.1
(a) is closest to Code
8.1
(b)). This is consistent
with the images: each of the beach sand and French canvas has a large proportion of higher
frequency information, since each is a finer texture than that of the detail in the French
canvas.
entropy(FD20)=-253.11
entropy(FD21)=-196.84
entropy(FD29)=-310.61
inertia(FD20)=5.55·10
5
inertia(FD21)=6.86·10
5
inertia(FD29)=6.38·10
5
energy(FD20)=5.41
energy(FD21)=7.49
energy(FD29)=12.37
(a) French canvas (detail)
(b) French canvas
(c) Beach sand
Code 8.1
Measures of the Fourier transforms of the three Brodatz textures
By Fourier analysis, the measures are inherently
position
-invariant. Clearly, the entropy,
inertia and energy are relatively immune to
rotation
, since order is not important in their
calculation. Also, the measures can be made
scale
invariant, as a consequence of the
frequency scaling property of the Fourier transform. Finally, the measurements (by virtue
of the normalisation process) are inherently invariant to linear changes in
illumination
.
Naturally, the descriptions will be subject to noise. In order to handle large data sets we
need a larger set of measurements (larger than the three given here) in order to better
discriminate between different textures. Other measures can include:
1.
the energy in the major peak;
2.
the Laplacian of the major peak;
3.
the largest horizontal frequency;
4.
the largest vertical frequency.
Amongst others, these are elements of Liu's features (Liu, 1990) chosen in a way aimed to
give Fourier transform-based measurements good performance in noisy conditions.
Naturally, there are many other transforms and these can confer different attributes in
analysis. The wavelet transform is very popular since it allows for localisation in time and
frequency (Laine, 1993) and (Lu, 1997). Other approaches use the Gabor wavelet (Bovik,
1990), (Jain, 1991) and (Daugman, 1993), as introduced in Section 2.7.3. One comparison
