Digital Signal Processing Reference
In-Depth Information
log
R
−
1
/
2
F
R
i
,(
n
)
R
−
1
/
2
log
R
−
1
/
2
R
i
,(
n
)
R
−
1
/
2
n
N
=
0 with
=
0
i
=
1
i
=
1
(9.46)
The isobarycentric flow could be approximated on
HPD
(
n
) manifold when all
matrices are closed to each other by using following approximation:
N
k
=
i
log
R
−
1
/
2
R
k
,
n
R
−
1
/
2
i
,
n
F
R
i
,(
n
)
=
e
ε
i
,
n
R
1
/
2
i
,
n
R
1
/
2
i
,
n
R
i
,
n
+
1
=
(9.47)
If we note
n
R
k
,
n
−
R
i
,
n
R
−
1
/
2
R
−
1
/
2
i
R
k
,
n
R
−
1
/
2
R
−
1
/
2
i
=
I
+
=
I
+
T
i
,
k
(9.48)
,
n
i
,
n
,
i
,
n
By using approximation of
log
(
·
)
and
exp
(
·
)
:
1
2
G
2
1
3
G
3
1
2
H
2
1
3
H
3
(
+
)
=
−
+
−···
)
=
+
+
+
+···
log
I
G
G
and exp
(
H
I
H
!
We are then in the same case than in Euclidean space:
⎡
⎛
⎞
⎤
⎦
R
1
/
2
i
N
n
R
k
,
n
−
R
i
,
n
R
−
1
/
2
R
1
/
2
i
R
−
1
/
2
i
⎣
I
⎝
⎠
R
i
,
n
+
1
=
+
ε
,
n
,
i
,
n
,
n
k
=
i
N
R
k
,
n
−
R
i
,
n
R
i
,
n
+
1
=
R
i
,
n
+
ε
(9.49)
k
=
i
In this case, barycentric flow convergence is obvious. To study convergence of
barycentric flow when all matrices are not closed to each other, consideration on
curvature should be studied in the more general framework of potential theory.
9.7 Fourier Heat Equation Flow on 1D Graph
of HPD(n) Matrices
We can replace Median computation by anisotropic diffusion. In normed vector space
in
1D
, if we note
u
n
=
(
ˆ
u
n
+
1
+
u
n
−
1
) /
2, Fourier diffusion Equation is given by:
2
u
x
2
u
n
,
t
−
u
n
,
t
=
(
1
−
ρ)
·
u
n
,
t
+
ρ
·
u
n
,
t
∂
u
∂
t
=
∂
2
∇
t
∇
⇒
u
n
,
t
+
1
=
u
n
,
t
+
=
u
n
,
t
◦
ρ
u
n
,
t
(9.50)
x
2
∂
By analogy, we can define diffusion equation on a
1D
graph of
HPD
(
n
)
by:
