Information Technology Reference
In-Depth Information
To insure that flow discontinuities are preserved, we use a regularized form of the
Frobenius norm of the Jacobian of
v
[27].
The term
E
1
ensures spatial regularity of the recreated frames. We use a smooth
form of the classical total variation ROF model [28]. It is especially useful when
the motion estimates are unreliable. Assuming nevertheless that we can reliably
estimate motion most of the time, this term should have a limited influence, by
means of giving it a small weight.
The detailed energy formulation we use thus is
λ
1
λ
2
2
)
dx
2
)
dx
E
(
u
,
v
)=
Ω
ψ
(
|
∇
u
|
+
Ω
ψ
(
|L
v
u
|
E
1
E
2
λ
3
ψ
2
)
dx
2
+
+
(
|
∇
3
v
1
|
|
∇
3
v
2
|
,
Du
=
u
0
(4)
Ω
E
3
t
)
T
where
Ω
is the entire image sequence domain,
∇
3
=(
∂
x
,
∂
y
,
∂
is the spatiotem-
poral gradient, and the
's are positive constants weighing the terms with re-
spect to each other.
v
1
and
v
2
are the
x
-and
y
-components of the flow field, i.e.
v
=(
v
1
,
v
2
)
T
. (In the implementation we use a double representation of the flow
field in the forward and backward directions respectively. In theory and in the con-
tinuous domain they are one and the same, but is split in practice—mainly due to
discretization.)
λ
(
s
2
)=
√
s
2
+
ψ
ε
2
is an approximation of the
|·|
function as the lat-
is a small positive constant (10
−
8
ter is non-differentiable at the origin.
ε
in our
implementation).
Splitting the energy (4) accordingly in an intensity and a flow part, we get this
energy to be minimized for the intensities
λ
s
λ
t
2
)
dx
E
i
(
u
)=
2
)
dx
Ω
ψ
(
|
∇
u
|
+
Ω
ψ
(
|L
v
u
|
,
Du
=
u
0
(5)
E
1
E
2
where
λ
s
=
λ
1
and
λ
t
=
λ
2
in (4). For the flow we need to minimize
λ
2
λ
3
ψ
2
)
dx
E
f
(
v
)=
2
)
dx
2
)+
Ω
ψ
(
|L
v
u
|
+
(
|
∇
v
1
|
ψ
(
|
∇
v
2
|
.
(6)
Ω
E
2
E
3
In order to improve quality, the BCA in
E
2
could be supplemented with the gradient
constancy assumption (GCA) proposed first by Brox et al. in [15] for optical flows
only. The GCA assumes that the spatial gradients remain constant along trajectories,
and can be written as
u
xx
u
xy
u
xy
u
yy
v
+
∇
u
t
= 0
.
