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We use the more compact form
u
= 0. It will improve the quality of the flow
to add it to
E
f
(
v
) in (6), but as shown in [29], the added complexity only pays off
in a minimal quality improvement if it is added to the intensity energy
E
i
(
u
) in (5).
Adding the GCA to
E
2
for the flow in
E
f
(
v
),gives
E
2
(
v
)=
L
v
∇
2
)
dx
2
+
λ
2
ψ
(
|L
V
u
|
γ
|L
V
∇
u
|
(7)
Ω
where
= 0 in our implementation, we
are back at (6) and thus we are able to test both with and without GCA for the flow
energy in one joint implementation.
γ
is a positive constant weight. If we set
γ
2.2
Implementation of TSR: Frame Doubling
To test our ideas we have chosen to implement a frame rate doubler, but implement-
ing solutions for other conversion rates would be easy. Frame doubling here means
that the projection
D
forgets every second frame. We may decompose the domain
Ω
Ω
\
as the domain of
known frames K
and its complement
K
. The constraint
Du
=
u
0
becomes
u
|
K
=
u
0
.
2.2.1
Euler-Lagrange Equations and Their Solvers
To minimize the intensity and flow energies given in (5), (6) and (7) we derive and
solve the associated Euler-Lagrange equations. Let us start with the flow energy
minimization: After exchanging the
E
2
-term of the flow energy in (6) with the
E
2
-
term from (7) to incorporate to option of using GCA, the flow Euler-Lagrange equa-
tion is derived. It is implemented numerically along the lines given by Brox et al. in
[15] and by Lauze in [22, 30] and minimized iteratively by repeated linearizations
of it, each linear equation being solved by a Gauss-Seidel solver.
Details on the computation of the gradient of the intensity energy (5) can be found
in [22] (details on discretization in [24]), and we here recall the final result:
∇
u
E
i
=
−
λ
s
∇
2
·
(
A
(
u
)
∇
u
)
−
λ
t
∇
3
·
(
B
(
u
)(
L
v
u
)
V
)
(8)
where
V
=(
v
T
,
1)
T
is the spatiotemporal extension of
v
,
∇
2
·
is the 2-dimensional
divergence operator, while
∇
3
·
is the 3-dimensional one and the coefficients
A
(
u
)
and
B
(
u
) are, respectively
ψ
(
2
)
,
ψ
(
2
)
.
A
(
u
)=
|
∇
u
|
B
(
u
)=
|L
v
u
|
In order to solve (8) numerically, we again use a fixed point approach: At each fixed
point iteration,
A
(
u
) and
B
(
u
) are computed from the estimated values of
u
and
v
and thus frozen. Equation (8) then becomes linear. It is discretized and solved here
