Graphics Reference
In-Depth Information
Figure 10.4.
A 2D filter kernel decomposed into two ID filters.
10.2 Gaussian Filter
The first image processing algorithm that we will investigate is the
Gaussian filter
.
This
filter is one of the most widely known, due to the wide range of applications that can use
it, and its relatively low computational cost. The filter itself is named after Carl Friedrich
Gauss, a brilliant German mathematician (1777-1885) who contributed to many different
areas. The term
Gaussian distribution
is often used to refer to a normal distribution in
statistics, which is where the name has become associated with an image processing filter.
The Gaussian filter produces a blurred version of the input image, and is hence re-
ferred to as a
low-pass filter
in signal processing terms. This means that the lower-fre-
quency content of the image (the portions that don't change rapidly) is mostly preserved,
while the higher-frequency content (sharp edges or transitions) is attenuated.
10.2.1 Theory
The filter receives its name from the use of the Gaussian function for creating the filter
kernel weights. The Gaussian function is shown in Equation (10.2) for two dimensional
inputs, where
x
and
y
are the input variables, and
σ
is a constant factor determined by the
developer(which will be clarified shortly). This equation has two main portions—the ex-
ponential, and a constant factor applied to it in the beginning of the equation. The exponent
in the exponential term calculates a measure of the distance that an input sample location
is from the origin by squaring both input parameters. Since these factors will always result
in positive values, the final sign of the exponent will always be negative. This results in the
exponential term always producing a value between 1 and 0, with 1 being produced when
x
and
y
are both zero, and the value falling off to 0 at some characteristic determined by the
a parameter. The constant term in the front of the equation simply scales the magnitude of
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