<s>
In	O
signal	O
processing	O
,	O
multidimensional	B-Algorithm
empirical	I-Algorithm
mode	I-Algorithm
decomposition	I-Algorithm
(	O
multidimensional	O
EMD	B-Algorithm
)	O
is	O
an	O
extension	O
of	O
the	O
one-dimensional	O
(	O
1-D	O
)	O
EMD	B-Algorithm
algorithm	O
to	O
a	O
signal	O
encompassing	O
multiple	O
dimensions	O
.	O
</s>
<s>
The	O
Hilbert	B-Algorithm
–	I-Algorithm
Huang	I-Algorithm
empirical	I-Algorithm
mode	I-Algorithm
decomposition	I-Algorithm
(	O
EMD	B-Algorithm
)	O
process	O
decomposes	O
a	O
signal	O
into	O
intrinsic	O
mode	O
functions	O
combined	O
with	O
the	O
Hilbert	B-Algorithm
spectral	I-Algorithm
analysis	I-Algorithm
,	O
known	O
as	O
the	O
Hilbert	B-Algorithm
–	I-Algorithm
Huang	I-Algorithm
transform	I-Algorithm
(	O
HHT	O
)	O
.	O
</s>
<s>
The	O
multidimensional	O
EMD	B-Algorithm
extends	O
the	O
1-D	O
EMD	B-Algorithm
algorithm	O
into	O
multiple-dimensional	O
signals	O
.	O
</s>
<s>
This	O
decomposition	O
can	O
be	O
applied	O
to	O
image	B-Algorithm
processing	I-Algorithm
,	O
audio	B-Algorithm
signal	I-Algorithm
processing	I-Algorithm
,	O
and	O
various	O
other	O
multidimensional	O
signals	O
.	O
</s>
<s>
Multidimensional	B-Algorithm
empirical	I-Algorithm
mode	I-Algorithm
decomposition	I-Algorithm
is	O
a	O
popular	O
method	O
because	O
of	O
its	O
applications	O
in	O
many	O
fields	O
,	O
such	O
as	O
texture	O
analysis	O
,	O
financial	O
applications	O
,	O
image	B-Algorithm
processing	I-Algorithm
,	O
ocean	O
engineering	O
,	O
seismic	O
research	O
,	O
etc	O
.	O
</s>
<s>
The	O
empirical	O
mode	O
decomposition	O
(	O
EMD	B-Algorithm
)	O
method	O
can	O
extract	O
global	O
structure	O
and	O
deal	O
with	O
fractal-like	O
signals	O
.	O
</s>
<s>
The	O
EMD	B-Algorithm
method	O
was	O
developed	O
so	O
that	O
data	O
can	O
be	O
examined	O
in	O
an	O
adaptive	O
time	O
–	O
frequency	O
–	O
amplitude	O
space	O
for	O
nonlinear	O
and	O
non-stationary	O
signals	O
.	O
</s>
<s>
The	O
EMD	B-Algorithm
method	O
decomposes	O
the	O
input	O
signal	O
into	O
several	O
intrinsic	O
mode	O
functions	O
(	O
IMF	O
)	O
and	O
a	O
residue	O
.	O
</s>
<s>
Although	O
adding	O
noise	O
may	O
result	O
in	O
a	O
smaller	O
signal-to-noise	O
ratio	O
,	O
the	O
added	O
white	O
noise	O
will	O
provide	O
a	O
uniform	O
reference	O
scale	O
distribution	O
to	O
facilitate	O
EMD	B-Algorithm
;	O
therefore	O
,	O
the	O
low	O
signal-noise	O
ratio	O
will	O
not	O
affect	O
the	O
decomposition	O
method	O
but	O
actually	O
enhances	O
it	O
by	O
avoiding	O
mode	O
mixing	O
.	O
</s>
<s>
The	O
dyadic	B-Algorithm
filter	O
bank	O
property	O
provides	O
a	O
control	O
on	O
the	O
periods	O
of	O
oscillations	O
contained	O
in	O
an	O
oscillatory	O
component	O
,	O
significantly	O
reducing	O
the	O
chance	O
of	O
scale	O
mixing	O
in	O
a	O
component	O
.	O
</s>
<s>
To	O
design	O
a	O
pseudo-BEMD	O
algorithm	O
the	O
key	O
step	O
is	O
to	O
translate	O
the	O
algorithm	O
of	O
the	O
1D	O
EMD	B-Algorithm
into	O
a	O
Bi-dimensional	O
Empirical	O
Mode	O
Decomposition	O
(	O
BEMD	O
)	O
and	O
further	O
extend	O
the	O
algorithm	O
to	O
three	O
or	O
more	O
dimensions	O
which	O
is	O
similar	O
to	O
the	O
BEMD	O
by	O
extending	O
the	O
procedure	O
on	O
successive	O
dimensions	O
.	O
</s>
<s>
The	O
first	O
row	O
of	O
the	O
matrix	O
RX	O
(	O
m	O
,	O
i	O
,	O
j	O
)	O
is	O
the	O
mth	O
EMD	B-Algorithm
component	O
decomposed	O
from	O
the	O
first	O
row	O
of	O
the	O
matrix	O
X	O
(	O
i	O
,	O
j	O
)	O
.	O
</s>
<s>
The	O
second	O
row	O
of	O
the	O
matrix	O
RX	O
(	O
m	O
,	O
i	O
,	O
j	O
)	O
is	O
the	O
mth	O
EMD	B-Algorithm
component	O
decomposed	O
from	O
the	O
second	O
row	O
of	O
the	O
matrix	O
X	O
(	O
i	O
,	O
j	O
)	O
,	O
and	O
so	O
on	O
.	O
</s>
<s>
Finally	O
,	O
the	O
2D	O
decomposition	O
will	O
result	O
into	O
m×	O
n	O
matrices	O
which	O
are	O
the	O
2D	O
EMD	B-Algorithm
components	O
of	O
the	O
original	O
data	O
X(i,j )	O
.	O
</s>
<s>
where	O
each	O
element	O
in	O
the	O
matrix	O
CRX	O
is	O
an	O
i	O
×	O
j	O
sub-matrix	O
representing	O
a	O
2D	O
EMD	B-Algorithm
decomposed	O
component	O
.	O
</s>
<s>
Following	O
the	O
convention	O
of	O
1D	O
EMD	B-Algorithm
,	O
the	O
last	O
component	O
of	O
the	O
complete	O
2D	O
components	O
is	O
called	O
residue	O
.	O
</s>
<s>
This	O
new	O
approach	O
is	O
based	O
on	O
separating	O
the	O
original	O
data	O
into	O
one-dimensional	O
slices	O
,	O
then	O
applying	O
ensemble	O
EMD	B-Algorithm
to	O
each	O
one-dimensional	O
slice	O
.	O
</s>
<s>
It	O
employs	O
1D	O
curve	O
fitting	O
in	O
the	O
sifting	O
process	O
of	O
each	O
dimension	O
,	O
and	O
has	O
no	O
difficulty	O
as	O
encountered	O
in	O
the	O
2D	O
EMD	B-Algorithm
algorithms	O
using	O
surface	O
fitting	O
,	O
which	O
has	O
the	O
problem	O
of	O
determining	O
the	O
saddle	O
point	O
as	O
a	O
local	O
maximum	O
or	O
minimum	O
.	O
</s>
<s>
Sifting	O
scheme	O
for	O
pseudo-BEMD	O
is	O
like	O
the	O
1D	O
sifting	O
where	O
the	O
local	O
mean	O
of	O
the	O
standard	O
EMD	B-Algorithm
is	O
replaced	O
by	O
the	O
mean	O
of	O
multivariate	O
envelope	O
curves	O
.	O
</s>
<s>
Hence	O
,	O
it	O
could	O
exceed	O
the	O
computation	O
capacity	O
for	O
a	O
Geo-Physical	O
data	O
processing	O
system	O
when	O
the	O
number	O
of	O
EMD	B-Algorithm
in	O
the	O
algorithm	O
is	O
large	O
.	O
</s>
<s>
A	O
data	O
compression	O
method	O
that	O
uses	O
principal	B-Application
component	I-Application
analysis	I-Application
(	O
PCA	O
)	O
/empirical	O
orthogonal	O
function	O
(	O
EOF	O
)	O
analysis	O
or	O
principal	O
oscillation	O
pattern	O
analysis	O
is	O
used	O
to	O
compress	O
data	O
.	O
</s>
<s>
In	O
statistics	O
,	O
EOF	O
analysis	O
is	O
known	O
as	O
principal	B-Application
component	I-Application
analysis	I-Application
(	O
PCA	O
)	O
.	O
</s>
<s>
The	O
time	O
series	O
of	O
each	O
mode	O
(	O
aka	O
,	O
principle	B-Application
components	I-Application
)	O
are	O
determined	O
by	O
projecting	O
the	O
derived	O
eigen	O
vectors	O
onto	O
the	O
spatially	O
weighted	O
anomalies	O
.	O
</s>
<s>
By	O
construction	O
,	O
the	O
EOF	O
patterns	O
and	O
the	O
principal	B-Application
components	I-Application
are	O
independent	O
.	O
</s>
<s>
where	O
is	O
the	O
th	O
principal	B-Application
component	I-Application
and	O
the	O
th	O
empirical	O
orthogonal	O
function	O
(	O
EOF	O
)	O
pattern	O
and	O
K	O
is	O
the	O
smaller	O
one	O
of	O
M	O
and	O
N	O
.	O
PC	O
and	O
EOFs	O
are	O
often	O
obtained	O
by	O
solving	O
the	O
eigenvalue/eigenvector	O
problem	O
of	O
either	O
temporal	O
co-variance	O
matrix	O
or	O
spatial	O
co-variance	O
matrix	O
based	O
on	O
which	O
dimension	O
is	O
smaller	O
.	O
</s>
<s>
If	O
the	O
data	O
subjected	O
to	O
PCA/EOF	O
analysis	O
is	O
all	O
white	O
noise	O
,	O
all	O
eigenvalues	O
are	O
theoretically	O
equal	O
and	O
there	O
is	O
no	O
preferred	O
vector	O
direction	O
for	O
the	O
principal	B-Application
component	I-Application
in	O
PCA/EOF	O
space	O
.	O
</s>
<s>
In	O
this	O
compressed	O
computation	O
,	O
we	O
have	O
used	O
the	O
approximate	O
dyadic	B-Algorithm
filter	O
bank	O
properties	O
of	O
EMD/EEMD	O
.	O
</s>
<s>
Another	O
advantage	O
of	O
using	O
the	O
MEEMD	O
is	O
that	O
the	O
mode	O
mixing	O
is	O
reduced	O
significantly	O
due	O
to	O
the	O
function	O
of	O
the	O
EEMD.The	O
denoising	O
and	O
detrending	O
strategy	O
can	O
be	O
used	O
for	O
image	B-Algorithm
processing	I-Algorithm
to	O
enhance	O
an	O
image	O
and	O
similarly	O
it	O
could	O
be	O
applied	O
to	O
Audio	O
Signals	O
to	O
remove	O
corrupted	O
data	O
in	O
speech	O
.	O
</s>
<s>
It	O
does	O
not	O
incur	O
any	O
communication	O
or	O
synchronization	O
between	O
the	O
threads	O
until	O
the	O
results	O
are	O
merged	O
since	O
the	O
execution	O
of	O
each	O
EMD	B-Algorithm
or	O
EEMD	O
is	O
independent	O
.	O
</s>
<s>
In	O
the	O
GPU	O
CUDA	O
implementation	O
,	O
each	O
EMD	B-Algorithm
,	O
is	O
mapped	O
to	O
a	O
thread	O
.	O
</s>
<s>
The	O
Partial	O
Differential	O
Equation-Based	O
Multidimensional	B-Algorithm
Empirical	I-Algorithm
Mode	I-Algorithm
Decomposition	I-Algorithm
(	O
PDE-based	O
MEMD	O
)	O
approach	O
is	O
a	O
way	O
to	O
improve	O
and	O
overcome	O
the	O
difficulties	O
of	O
mean-envelope	O
estimation	O
of	O
a	O
signal	O
from	O
the	O
traditional	O
EMD	B-Algorithm
.	O
</s>
<s>
In	O
order	O
to	O
perform	O
multidimensional	O
EMD	B-Algorithm
,	O
we	O
need	O
to	O
extend	O
the	O
1-D	O
PDE-based	O
sifting	O
process	O
to	O
2-D	O
space	O
as	O
shown	O
by	O
the	O
steps	O
below	O
.	O
</s>
<s>
Here	O
,	O
we	O
take	O
2-D	O
PDE-based	O
EMD	B-Algorithm
as	O
an	O
example	O
.	O
</s>
<s>
Finally	O
,	O
we	O
can	O
use	O
the	O
alternating	B-Algorithm
direction	I-Algorithm
implicit	I-Algorithm
(	O
ADI	O
)	O
scheme	O
.	O
</s>
<s>
Also	O
,	O
this	O
method	O
is	O
narrowly	O
restricted	O
to	O
be	O
use	O
on	O
texture	O
analysis	O
and	O
image	B-Algorithm
processing	I-Algorithm
.	O
</s>
<s>
In	O
the	O
second	O
part	O
,	O
the	O
PDE-based	O
MEMD	O
and	O
FAMEMD	O
can	O
be	O
implemented	O
on	O
audio	O
processing	O
,	O
image	B-Algorithm
processing	I-Algorithm
and	O
texture	O
analysis	O
.	O
</s>
<s>
Finally	O
,	O
the	O
BPBEMD	O
method	O
has	O
good	O
performance	O
for	O
image	B-Algorithm
processing	I-Algorithm
and	O
texture	O
analysis	O
due	O
to	O
its	O
property	O
to	O
solve	O
the	O
extension	O
boundary	O
problems	O
in	O
recent	O
techniques	O
.	O
</s>
