|
ile Texture
able
ectively, se-
composition
detector. A
n of wavelet
Mallat,1989).
ecomposable
iral detector
(5)
(6)
(7)
irst
sider such a
c wavelet
n6)) (8)
The corresponding family of wavelet function is
v (x- xy -y,.0- of T2222) (9)
o o
For practical application, (9) is discretized as
v, mon) e T2220) (10)
where c €R,0 e[0, z],m.n, j eZ.
3.2 Multiscale Decomposition of Texture Image
Let a=2/, multiscale decomposition in Q direction
through wavelet transform is then defined by
W, f(x,.y,,0) » f| Gy) w, x — x, y — y, Odxdy (11)
Here the Mallat's multiresolution decomposition al-
gorithm (Mallat 1989) is employed for our purpose
of multiscale decomposition.
3.3 Multiscale Textural Primitive Planes
After multiscale decomposition, we define the
decomposable value, amplitude, phase angle and
standard deviation etc. as textural primitives, which
consist of the basis for computing textural features.
3.4 Nonlinearity
Each texture primitive plane is subjected to a
nonlinear transformation, we use the following
bounded nonlinearity
(12)
where «a is a constant. Nonlinear transformation is
computed in a window a,(x,y).
3.5 Computing Textural Features
After nonlinear transformation of the textural
primitive planes, we computer standard deviations
from the decomposable value, amplitude, or average
absolute deviations from the standard deviation,
phase angle in overlapping window nxn through
edge-preserving and noise-smoothing procedure
EEE Es
(Jixian Zhang, 1994; Jixian Zhang and Deren
Li ,1995) as textural features. We can also compute
local fractal dimension, textural density as texture
measures.
4. FRACTAL FEATURE IN MULTISCALE
TEXTURE ANALYSIS
Fractal dimension is a powerful feature of texture
for the description of its coarseness and complexity
which may integrate some measures described by
other methods. Textural image can be regarded as a
process of fractional Brownian motion (fBm), fBm
is described by the scalar parameter H, which is
related to fractal dimension D=3-H.
In 1-dimensional space, Flandrin (Flandrin, 1992)
showed that for any j, the wavelet coefficients of
fBm give rise to time sequences which are self-
similar and stationary under orthonormal wavelet
decomposition.
E[W, (n)W, (m)| = fu (n-m)q"adj (13)
where
W.(n)- Q^) ^ ^W, (nr) 4,(1,7) (14)
A (a, 9) 2 J2 [7 w())wCat - 2dt (15)
W,(n) are the wavelet coefficients for scale 2',y(7)
is the basic wavelet.
According to above theorem, fractal feature in our
multiscale texture analysis is computed through
following two methods.
4.1 General Fractal dimension
We compute general fractal dimension as the result
of all used scales, let n=m, from (10) we get follow-
ing equation:
Var(W, (n)) i ano: y" sCQ'yem (16)
where
Var(W,(n)) - EV, (mW, (n), | VG)» -[;r,0r|d ae
It follows that
log, (Var (Ww, (n))) =(2H+1)j+ constant (17)
1001
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
