Digital Signal Processing Reference
In-Depth Information
1.
the neighborhood contains two or more differently moving objects.
2. the ground truth motion is not smooth. Consider an extreme case of
the center of a spinning wheel;
motions within that area are clearly
not equal.
3. the neighborhood is too small and SSD has multiple minimal solutions
Consider the block inside a homogeneous region.
For block matching, there exists a trade-off in neighborhood size. Is-
sue #3 can be solved by increasing the neighborhood size. However, a
larger neighborhood size also increases the probability that issues #1
and #2 are problematic. Solutions to this trade-off involve the integra-
tion of high-level knowledge and is beyond the scope of block matching
algorithms.
METHOD 2: HORN AND SCHUNCK ITERATIVE
MOTION FIELD OPTIMIZATION
To find the motion field via calculus of variations, we formulate the
calculation of the motion field of a given frame as an optimization prob-
lem.
The formulation in this section is used in the original work by Horn
and Schunck [Horn and Schunck, 1981]. We use the optical flow con-
straint as one optimization criterion and we translate a secondary con-
straint of smoothness as another. The linear combination of the two
criteria yield the function to be optimized. The error for the optical
flow constraint can be quantified from Eq. 2.2:
(2.5)
where
u
(
x
,
y
) and
v
(
x
,
y
) are the x and y components of motion field over
a single frame at
time
t
o
,
respectively.
E
flow
underconstrains
the
prob-
lem;
we
add smoothness
component
as the
magnitude of the gradients
of u and v.
2
2
2
(
)
2
( )
( )
( )
∂
u
-
∂
+
∂
v
+
∂
v
-
∂
-
∂
-
∂
(2.6)
E
smooth
(
x,y
)
=
+
∂
x
y
x
y
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