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

)