Biomedical Engineering Reference
In-Depth Information
Solving this system for the variables
u
k+1
= u
k
I
x
(I
x
u
k
+ I
y
v
k
+ I
z
w
k
+ I
t
)
2
+ I
x
+ I
y
+ I
z
v
k+1
= v
k
I
y
(I
x
u
k
+ I
y
v
k
+ I
z
w
k
+ I
t
)
2
+ I
x
+ I
y
+ I
z
(8.7)
w
k+1
= w
k
I
z
(I
x
u
k
+ I
y
v
k
+ I
z
w
k
+ I
t
)
2
+ I
x
+ I
y
+ I
z
where the superscript k denotes the iteration number.
Advantages of the Horn{Schunck algorithm include that it yields a high
density of flow vectors; i.e., the flow information missing in inner parts of
homogeneous objects is filled in from the motion boundaries. However, it is
more sensitive to noise than local methods [8],[46].
8.6 Bruhn optical flow
It is obviously advantageous to combine both types of algorithms to get an
algorithm which is robust in presence of noise and yields dense vector fields.
Bruhn [8] has presented a mathematical framework which allows combinations
of both Lucas{Kanade as well as Horn{Schunck algorithms. Dening:
0
@
1
A
0
@
1
A
and
u
v
w
1
I
x
I
y
I
z
I
t
V =
; rI =
jrVj
2
= jruj
2
+ jrvj
2
+ jrwj
2
we can rewrite the Lucas{Kanade algorithm as minimization of a function
f
LK
:
f
LK
= V
T
(k (rIrI
T
))V
= k (I
x
u + I
y
v + I
z
w + I
t
)
2
(8.8)
where k is a weighting function which is convolved with the image data. Min-
imization of Equation (8.8) means @
u
f
LK
= 0; @
v
f
LK
= 0; @
w
f
LK
= 0.
Similarly the Horn{Schunck algorithm can be rewritten as minimization
of a function f
HS
:
Z
(V
T
rIrI
T
V + jrVj
2
)dxdydz:
f
HS
=
(8.9)
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