Biomedical Engineering Reference
In-Depth Information
Combining both functions gives us a hybrid local-global function:
Z
(kV
T
(rIrI
T
)V + jrVj
2
)dxdydz:
f
LG
=
(8.10)
To avoid heavy penalization of outliers a square root function can be used
s
1 +
s
2
i
i
(s
2
) = 2
i
;i 2 1; 2
(8.11)
where
i
is a scaling factor [8]. With this improvement the function f
LG
becomes convex in s and thus has a unique solution:
div(
0
2
(S)ru)
1
0
1
(D)I
x
(I
x
u + I
y
v + I
z
w + I
t
) = 0
div(
0
2
(S)rv)
1
0
1
(D)I
y
(I
x
u + I
y
v + I
z
w + I
t
) = 0
div(
0
2
(S)rw)
1
0
1
(D)I
z
(I
x
u + I
y
v + I
z
w + I
t
) = 0
with D = kV
T
(rIrI
T
)V and S = jruj
2
+ jrvj
2
+ jrwj
2
.
The algorithmic solution is then given by:
X
0
2i
+
0
2j
2
I
x
i
I
y
i
v
i
+I
z
i
w
i
+I
t
i
h
2
u
x
j
0
1i
j2N
u
k+
i
=
X
0
2i
+
0
2j
2
h
2
I
x
i
+
0
1i
j2N
X
0
2i
+
0
2j
2
Iy
i
I
x
i
u
i
+I
z
i
w
i
+I
t
i
h
2
v
y
j
0
1i
j2N
v
k+
i
=
X
0
2i
+
0
2j
2
h
2
I
y
i
+
0
1i
j2N
X
0
2i
+
0
2j
2
h
2
I
z
i
I
x
i
u
i
+I
y
i
v
i
+I
t
i
w
z
j
0
1i
j2N
w
k+
i
=
X
0
2i
+
0
2j
2
h
2
I
z
i
+
0
1i
j2N
with
1
0
1i
(D) =
q
(8.12)
(I
x
i
u
i
+I
y
i
v
i
+I
z
i
w
i
+I
t
i
)
2
2
i
1 +
and
1
q
1 +
jru
i
j
2
+jrv
i
j
2
+jrw
i
j
2
2
i
0
2i
(S) =
(8.13)
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