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Here, we propose to extend the 1D MFT enhanced by Li (1997) to 2 dimensions. The idea
consists on decomposing the two-dimensional signal using a Gaussian filter
(, , )
nm
Gk   defined as:
Gk
(, ,
 
)
G k
(, )
G k
(,
)
nm
1
n
2
m
2
2
(15)
k
k
n
n
e
e
n
n
Where  and  are variable center angular frequencies (or wavenumbers) of the
respective filters
Gk  and
1 (, )
Gk  . The bandwidths
2 (,
)
 and
 of both filters are
n
m
1
2
calculated as above (Eq. 14).
4. Application to simulated 2D-mBm paths
In this section, the suggested estimators of the local regularity are tested on synthetic 2D-
mBm paths whose lengths are 256 x 256, generated using the kriging method (Barrière,
2007). Three types of Hölder function H are chosen:
bilinear :
Hxx
,

0.8
0.6
xx
112
12
0.4
logistic :
Hxx
,

0.7
212
1exp 20
x
0.5
2
3
periodic :
Hxx
,

0.5
0.3 sin 2
x
cos
x
312
1
2
2
The regularity functions and the simulated 2D-mBm paths corresponding to the three
theoretical H functions are presented in Figure 2. The larger H value, the smoother the
modeled surface.
Using the three algorithms, we have estimated H maps. For the first wavelet-based
algorithm, we use the Morlet wavelet with  0x = 0y = 8.9443, while for 2D MFT, the selected
parameters for the two-dimensional Gaussian filters are:
- The shaping factor  = 40,
- The minimal central wavenumber of the filter ξ min min = 2.10 -3 rad/m,
- The maximal central wavenumber of the filter ξ max max = 2.10 0.5 rad/m,
- The number of the central wavenumbers of the filter N= 100,
- The sampling rate is selected as 0.1524m.
The H maps obtained by the three estimators, presented in Figure 3, show that the regularity
estimated by the first wavelet-based algorithm using the Morlet wavelet are better than that
calculated by the second algorithm with the Mexican hat. In addition, the suggested 2D MFT
provides the best estimations of the regularity maps with the least errors. For this reason, we
retain only this estimator in the following. It can be also remarked that all the used
algorithms yield large absolute values of the estimation error in the limits of the analyzed
2D-mBms.
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