Image Processing Reference
In-Depth Information
(a) French canvas (detail)
(b) French canvas
(c) Beach sand
Figure 8.3
Fourier transforms of the three Brodatz textures
FP
= FFT(
P
) (8.1)
where
FP
u
,
v
and
P
x
,
y
are the transform and pixel data, respectively. One clear advantage of
the Fourier transform is that it possesses shift invariance (Section 2.6.1): the transform of
a bit of (large and uniform) cloth will be the same, whatever segment we inspect. This is
consistent with the observation that phase is of little use in Fourier-based texture systems
(Pratt, 1992), so the modulus of the transform (its magnitude) is usually used. The transform
is of the same size as the image, even though conjugate symmetry of the transform implies
that we do not need to use all its components as measurements. As such we can
filter
the
Fourier transform (Section 2.8) so as to select those frequency components deemed to be
of interest to a particular application. Alternatively, it is convenient to collect the magnitude
transform data in different ways to achieve a reduced set of measurements. First though the
transform data can be normalised by the sum of the squared values of each magnitude
component (excepting the zero-frequency components, those for
u
= 0 and
v
= 0), so that
the magnitude data is invariant to linear shifts in illumination to obtain normalised Fourier
coefficients
NFP
as
|
FP
|
v
u
,
NFP
=
(8.2)
u
,
v
2
|
FP
|
v
u
,
(
v
u
0)
(
0)
Alternatively, histogram equalisation (Section 3.3.3) can provide such invariance but is
more complicated than using Equation 8.2. The spectral data can then be described by the
entropy
,
h
, as
NN
Σ Σ
v
h
=
NFP
log(
NFP
)
(8.3)
u
,
v
u
,
v
u
=1
=1
or by their
energy
,
e
, as
NN
Σ Σ
v
2
e
=
(
NFP
)
(8.4)
v
u
,
u
=1
=1
Another measure is their
inertia
,
i
, defined as
