Image Processing Reference
In-Depth Information
Fig. 16.7. The function relating the dissimialrity to the central coefficient of the butterfly filter
We still need to define the filters for different orientations
θ
. This is done by rotat-
ing the mask defined for the horizontal direction
θ
=0and redistributing the weights
for matching the grid of an image. The filters are computed for a fixed number of di-
rections (eight here) and are stored in the memory, i.e., a lookup table.Starting from
the coarsest level
l
=
L
, the boundary refinement procedure can be enumerated as
follows:
1. Project down the labels, i.e.,
Γ
(
r
k
,l
−
1) =
Γ
(
r
k
/
2
,l
) and define the boundary
n
and variance (
σ
n
)
2
for all classes.
2. For each boundary pixel at level
l
compute the dominant direction
θ
using Eqs.
(16.10) and (16.11). Determine the two classes
A
, and
B
on both sides of the
boundary. Choose the corresponding butterfly filter, and propagate its identity to
the corresponding pixels at level
l
region
β
at level
l
−
l
. Compute the mean
μ
−
1.
3. For each pixel
r
k
∈
β
, apply the filter corresponding to the current position to
(all components of) the feature vectors
f
(
r
k
,l
1). If a feature-vector participat-
ing to averaging is outside of
β
, take the vector of its corresponding prototype.
The left and right halves of the filters are applied separately, reducing the risk
of smoothing across the boundaries. Two responses are obtained,
h
A
(
r
k
,l
−
−
1)
and
h
B
(
r
k
,l
1). This smoothing is repeated a certain number of times found
empirically (four in the examples) using a small filter size (3
−
3 here). This
is equivalent to use a large filter size in one iteration, but it is computationally
faster.
4. For each pixel
r
k
, compute the four distances between the two filter responses
h
A
,
h
B
×
A
,
B
and the prototypes
μ
μ
corresponding to the classes
A, B
on each
A
−
h
A
A
−
h
B
B
−
h
A
B
side of the boundary region, i.e.,
μ
,
μ
,
μ
,
μ
−
