| Image
Processing Articles
Index |
 |
| Brief
Description |
| The
Gaussian smoothing
operator is a 2-D
convolution operator
that is used to `blur'
images and remove
detail and noise.
In this sense it is
similar to the mean
filter, but it uses
a different kernel
that represents the
shape of a Gaussian
(`bell-shaped') hump.
This kernel has some
special properties
which are detailed
below. |
|
| How
It Works |
|
| The
Gaussian distribution
in 1-D has the form: |
|
 |
|
where is
the standard deviation
of the distribution.
We have also assumed
that the distribution
has a mean of zero
(i.e. it is centered
on the line x=0).
The distribution is
illustrated in following
diagram. |
|
 |
|
| In
2-D, an isotropic
(i.e. circularly symmetric)
Gaussian has the form: |
| |
 |
|
| This
distribution is shown
in Figure 2. |
|
 |
|
| The
idea of Gaussian smoothing
is to use this 2-D
distribution as a
`point-spread' function,
and this is achieved
by convolution. Since
the image is stored
as a collection of
discrete pixels we
need to produce a
discrete approximation
to the Gaussian function
before we can perform
the convolution. In
theory, the Gaussian
distribution is non-zero
everywhere, which
would require an infinitely
large convolution
kernel, but in practice
it is effectively
zero more than about
three standard deviations
from the mean, and
so we can truncate
the kernel at this
point. Figure 3 shows
a suitable integer-valued
convolution kernel
that approximates
a Gaussian with a
of 1.0. |
|
 |
| Once
a suitable kernel
has been calculated,
then the Gaussian
smoothing can be performed
using standard convolution
methods. The convolution
can in fact be performed
fairly quickly since
the equation for the
2-D isotropic Gaussian
shown above is separable
into x and y components.
Thus the 2-D convolution
can be performed by
first convolving with
a 1-D Gaussian in
the x direction, and
then convolving with
another 1-D Gaussian
in the y direction.
(The Gaussian is in
fact the only completely
circularly symmetric
operator which can
be decomposed in such
a way.) Figure 4 shows
the 1-D x component
kernel that would
be used to produce
the full kernel shown
in Figure 3 (after
scaling by 273, rounding
and truncating one
row of pixels around
the boundary because
they mostly have the
value 0. This reduces
the 7x7 matrix to
the 5x5 shown above.).
The y component is
exactly the same but
is oriented vertically. |
 |
|
| A
further way to compute
a Gaussian smoothing
with a large standard
deviation is to convolve
an image several times
with a smaller Gaussian.
While this is computationally
complex, it can have
applicability if the
processing is carried
out using a hardware
pipeline. |
|
| The
Gaussian filter not
only has utility in
engineering applications.
It is also attracting
attention from computational
biologists because
it has been attributed
with some amount of
biological plausibility,
e.g. some cells in
the visual pathways
of the brain often
have an approximately
Gaussian response. |
|
| Basic
Working: |
| The
effect of a Gaussian
Smoothing on the Image: |
|
 |
|
| The
result after convolution
is as: |
|
 |
|
| Sample
Project |
|
| The
GUI of the whole Project
is as: |
|
 |
|
| The
project is a part
of the series of the
image processing articles
written just for the
prosperity and help
for the students searching
for Image Processing
free stuff. |
References:
E. Davies Machine
Vision: Theory, Algorithms
and Practicalities,
Academic Press, 1990,
pp 42 - 44.
R. Gonzalez and R.
Woods Digital Image
Processing, Addison-Wesley
Publishing Company,
1992, p 191.
R. Haralick and L.
Shapiro Computer and
Robot Vision, Addison-Wesley
Publishing Company,
1992, Vol. 1, Chap.
7.
B. Horn Robot Vision,
MIT Press, 1986, Chap.
8.
D. Vernon Machine
Vision, Prentice-Hall,
1991, pp 59 - 61,
214. |
|
Download
Project Files
 |
|
| Image
Processing Articles
Index |
