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2.2 Multifractional Brownian motion
Multifractional Brownian motion (mBm) was introduced by Peltier and Lévy-Véhél (1995),
and Benassi
et al.
(1997) by allowing
H
to vary over time. Even if no longer stationary nor
self-similar compared to the fBm, the mBm presents the advantage to be very flexible since
the function
H
(
t
) can model phenomena whose sample paths display a time changing
regularity.
For a continuous function
2
, the isotropic multifractional Brownian field
Hx R
:
R
H
W
is a centered Gaussian field with a covariance function
Hx Hy
Hx
Hy
Hx Hy
EW
xW
y
x
y
x y
(3)
Hx
Hy
Identically to the 2D-fBm, the local regularity of the 2D-mBm paths is measured by means of
the pointwise Hölder exponent. For a differentiable function
H
, the relation
is demonstrated, almost surely, for all
2
xR
xHx
.
B
3. Regularity analysis using the wavelet transform
3.1 Two- dimensional continuous wavelet transform
The two-dimensional continuous wavelet transform (2D- CWT) of a signal
sxy
is given
,
by (Chui, 1992; Holschneider, 1995):
yb
1
xb
y
x
S a b
(,
, )
b
s x y g
(, )
,
dx dy
(4)
xy
a
a
a
where
g x y
is the analyzing wavelet, "
a
" is the scale parameter, "
b
x
" and "
b
y
" are the
respective translations according to X-axis and Y-axis. The symbol "
―
" denotes the complex
conjugate.
Let
,
sxy
be a self-affine fractal surface. It satisfies then the relation:
,
H
sxy
(,
)
. (,
sxy
)
(5)
with the Hurst exponent
H
and a positive factor . If
s
is a stochastic process, the two sides
of the relation follow the same law.
Let
us
define
the
function
s
0
,
(,)
x y
in
h
i t
(,
xy
)
by:
xy
00
. This function also satisfies the self-affine property
described by (Mandelbrot, 1977, 1982; Feder, 1988; Vicsek 1989; Edgard, 1990) :
s
(,)
x y
sx
(
x yy
,
) (,)
sx y
xy
,
0
0
00
H
s
(
x
,
y
)
s
(
x y
,
)
(6)
xy
,
xy
,
00
00
This property is reflected by the 2D-CWT provided that the analyzing wavelet decreases
swiftly enough to zero and has enough vanishing moments (Holschneider, 1995):
1
Hx y
(,
)
2
(7)
00
Sax
(
,
by
,
b
)
Saxbyb
( ,
,
)
,
0
0
x
0
y
0
x
0
y
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