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3.4 The Primal-dual Relation
First, we describe a method to solve for β. To have
a unique solution for the DFC equation, A
†
must be
well-defined. However, by the way we defined the
constraints in the move-LP each block pair {a, b} has
two constraints
x
b
− x
a
≥ −p
ba
(14)
x
a
− x
b
≥ −p
ba
. (15)
To find a relation between β and move-LP con-
straints, we utilize the dual complementary slackness
conditions (Papadimitriou and Steiglitz, 1998). If the
complementary slackness conditions are satisfied, any
feasible solution x of the primal and β of the dual
problem are optimal. The dual complementary slack-
ness condition (DCSC) is given below.
Either β
ba
= 0 or A
ba
x = −p
ba
, (16)
where {b, a} is any site pair and A
ba
is the row for the
constraint associated with β
ba
. The complementary
slackness conditions are important in the design of
efficient approximation algorithms for complex prob-
lems. These algorithms are called primal-dual based
methods and they try to improve x and β by modi-
fying them in a way that more of their components
satisfy the complementary slackness conditions (Pa-
padimitriou and Steiglitz, 1998).
Our goal for applying the primal-dual method is
to re-design the energy function, which corresponds
to the Lagrangian of the primal problem. The La-
grangian is showed to be equivalent to the energy
function under the conditions on the Lagrangian mul-
tipliers given in Theorem 2. However, our choice of
the constraints were imposed for all sites pairing with
a site, independent of the discontinuity since the un-
known l is what we want to estimate. Fortunately,
ˆ
β reveals some information on the connectedness of
sites with their neighbors by the use of DCSC. DCSC
dictates that if
ˆ
β is non-zero, then the constraint is
binding and must be satisfied with equality. If the con-
straint had been relaxed, the interacting sites would
have chosen different labels. This implies that the two
sites are on different label segments. Hence, the con-
straint that is binding is conflicting with our smooth-
ness of l assumption in that locality, and should be
imposed less by decreasing its contribution in the en-
ergy. To this end, the Lagrangian multipliers must
be inversely related to
ˆ
β, for example by multiplying
with a function f(
ˆ
β) that is decreasing with
ˆ
β and has
range [0, 1]. Hence, by DCSC we propose to update
the β values as
β
DCSC
ba
= f(
ˆ
β
ba
)β
ba
, (17)
where
ˆ
β is a feasible solution of the dual problem
given in (13), and f is a decreasing function of
ˆ
β. Re-
placing Lagrangian multiplier β with β
DCSC
and sub-
stituting κ = 0 from the solution of the dual problem,
Lagrangian in (3) after some simplification becomes
E(v) =
∑
b∈B
D
b
(v
b
) + λ
∑
{b,a}∈N
f(
ˆ
β)V
b,a
(v
b
, v
a
). (18)
The above energy formulation does not isotropi-
cally enforce a smoothness constraint, but adapts the
weights of constraints in the energy with information
derived from the data via the dual problem. This will
enable us to obtain an energy function that is more
powerful to explain labeling across discontinuities.
4 EXPERIMENT RESULTS
We present experiment results using the proposed
method and compare our results with the state of the
art in the optical flow literature. Our implementation
of the proposed method is a hierarchical motion es-
timation algorithm that uses a full resolution and a
half resolution image produced by down-sampling the
full resolution image by two. For each half resolution
block, a motion search is performed to pick the two
best motion vectors to minimize a cost. The cost is
a sum of absolute deviation (SAD) based cost and all
motion vectors in a 2-D search window are evaluated
to find the minimum cost vectors. This way an ini-
tial motion-vector field v
0
is created and refined using
N standard-move iterations. Candidate vectors in re-
duced search window S is used to pick the best-two
standard move for each b. S consists of 18 vectors ob-
tained from a block’s and its eight-connectivityneigh-
borhoods’ best-two vectors. Smoothness cost is de-
rived from blocks in four-connectivity neighborhood
and weights of the neighboring blocks is adapted us-
ing information from the dual problem. Each block is
partitioned to quarter blocks so that a half-resolution
quarter-block matches with a full resolution block in
size. This will increase the reliability of the centered
motion search in full resolution. Also, partitioning
to quarter blocks increases the quality of half reso-
lution vectors in general, since with a smaller block
size block-based translational-motion model is less
problematic for rotation, zooming, and motion bound-
aries. After the partitioning, standard-move iterations
are again applied to refine further, before passing to
full resolution. Execution of our algorithm on a sam-
ple instance of the underlying problem is illustrated
in Figures 1 and 2. FR-FULL, FR-QUARTER, and
FR-QUARTER
2
images in Figures 1 and 2 have res-
olutions of 16x16, 8x8, 4x4 pixel blocks respectively.
ENERGY-MINIMIZATION BASED MOTION ESTIMATION USING ADAPTIVE SMOOTHNESS PRIORS
205