Image Processing Reference
In-Depth Information
σ
R
thus controls the
amplitude
of the edge,
range weight of the pixel. The value of
and its corresponding weight.
If we denote the bilateral filtered image by
I
BF
(
x
,
y
)
, then the filtering operation
in discrete domain is given by Eq. (
3.1
).
G
R
I
)
I
)
1
I
BF
(
x
,
y
)
=
(
x
−˜
x
,
y
−˜
y
)
G
(
x
,
y
)
−
I
(
˜
x
,
˜
y
(
˜
x
,
˜
y
σ
σ
S
W
(
x
,
y
)
˜
x
y
˜
(3.1)
with,
exp
x
2
y
2
+
G
σ
S
(
x
,
y
)
=
−
2
S
2
σ
exp
2
ζ
G
σ
R
(ζ )
=
−
(3.2)
2
R
2
σ
G
σ
S
(
G
σ
R
I
)
,
(
,
)
=
−˜
,
−˜
)
(
,
)
−
(
˜
,
˜
W
x
y
x
x
y
y
x
y
I
x
y
(3.3)
x
y
where
(
˜
x
,
˜
y
)
refers to the neighborhood of the pixel location
(
x
,
y
)
.Theterm
W
(
x
,
y
)
is the normalization factor. For the 2-D spatial kernel
G
σ
S
can
be chosen based on the desired low pass characteristics. Similarly, for the 1-D range
kernel
G
S
, the spread value
σ
σ
R
can be set as per the desired definition of what
pixel difference constitutes an edge. In the case of an amplification or attenuation
of image intensities, the
R
, the spread value
σ
σ
R
needs to be appropriately scaled to maintain the results.
However, due to the range filter component, the kernel of the bilateral filter needs to
be recomputed for every pixel. The filter, thus, turns out to be computationally quite
expensive.
The bilateral filter contains a range filter in addition to the conventional spa-
tial filter kernel. The advantage of this additional kernel over the traditional spatial
Gaussian filter can be illustrated through a simple example. Consider Fig.
3.1
a that
shows the 3-D representation of a step edge corrupted with an additive Gaussian
noise. The bilateral filtered output of this noisy image (edge) smoothens the regions
on the both sides of the edge, however produces a minimal degradation to the edge
itself which can be observed from Fig.
3.1
b. On the other hand, the conventional low
pass filter smoothens the edge along with smoothening of other parts of the image
as it can be seen from Fig.
3.1
c.
The computational complexity is an important limitation of the conventional
implementation of bilateral filter which is of the order of
n
2
[175]. The bilateral
filter involves a space-variant range kernel, which has to be re-computed for every
pixel. However, in the past decade, several fast implementations of bilateral filter have
been suggested. Paris and Durand have developed a signal processing-based imple-
mentation involving downsampling in the space and intensity which accelerates the
computation by reducing the complexity to the order of
O(
)
[126]. Another
fast implementation using a box spatial kernel has been proposed by Weiss [187].
O(
n
log
n
)
