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too by Gauss-Seidel relaxation. Only the discretization of
L
v
u
is somewhat non
standard. It uses the OFC/BCA approximation
L
≈
u
(
x
+
v
,
t
+ 1)
−
u
(
x
,
t
)
v
u
which gives rise to a numerical intensity diffusion along trajectories with correction
for the potential trajectories divergence. Details can be found in [29].
2.2.2
Algorithm
Before detailing our frame doubling algorithm, here it is in overview (leaving out
the special initialization at the top level):
At each level from the top, coarse to fine, for
k = levels
until
k=1
1. Calculate the forward and backward flows,
v
0
and
v
0
, of the resized origi-
nal input sequence
u
0
,
k
minimizing (6) with/without
E
2
from (7).
2. Initialize new frames:
u
(
x
,
t
,
k
)=
resize
[
u
(
x
,
t
,
k
1)] in the domain
D
.
3. Initialize forward and backward flows of new frames:
v
(
x
,
t
,
k
)=
resize
[
v
(
x
,
t
,
k
−
−
1)] in the domain
D
.
4. Calculate the flows
v
f
and
v
b
of the output sequence
u
minimizing (6)
with/without
E
2
from (7).
5. Calculate new frames in
u
|
D
by minimizing (5).
In our multiresolution settings, on each level
k
of the pyramid, we first compute
the forward and backward flows,
v
0
and
v
0
,ofthe
original input sequence u
0
(re-
sized to the size of the current level), minimizing (6) (including
E
2
from (7) to give
the option of using GCA or not) with the resized input sequence
u
0
simply replac-
ing
u
.(
E
f
(
v
) is minimized over the domain
K
instead of over
.) This is to have
a highly reliable anchor point flow when calculating the flows
v
f
and
v
b
of the full
output sequence. At the given level of the pyramid,
k
, we then initialize intensities
and the flows of the new frames by resizing the intensities and flows calculated at
the above coarser level
k
+ 1. Then we calculate the flows from these initializations
by minimizing either (6) or (7) (w/o GCA). Next we calculate
u
at level
k
by mini-
mizing the energy (5) knowing
v
f
and
v
b
and using the resized intensities from level
k
+ 1 as initialization of
u
in the new frames, just as when calculating
v
f
Ω
and
v
b
.The
resizing function (
resize
)usedisgivenin[31].
The use of a multiresolution schemes is considered essential when doing varia-
tional flow calculations. In TSR, calculating both flow and intensities at each level
solves the hen-egg problem of what comes first in a new frame: The flow or the
intensities. Thus we iteratively improve first one and then the other to get simulta-
neous computations and optimize our solution using a small scale factor between
levels to get optimal initializations.
