Image Processing Reference
In-Depth Information
basis functions. Discrete signals can map better into collections of binary components
rather than sinusoidal ones. These collections (or sequences) of binary data are called
sequency components and form the basis functions of the
Walsh transform
(Walsh, 1923).
This has found wide application in the interpretation of digital signals, though it is less
widely used in image processing. The Karhunen-Loéve transform (Karhunen, 1947) (Loéve,
1948) (also called the
Hotelling
transform from which it was derived, or more popularly
Principal Components Analysis
(Jain, 1989) is a way of analysing (statistical) data to
reduce it to those data which are
informative
, discarding those which are not.
2.8
Applications using frequency domain properties
Filtering is a major use of Fourier transforms, particularly because we can understand an
image, and how to process it, much better in the frequency domain. An analogy is the use
of a graphic equaliser to control the way music sounds. In images, if we want to remove
high frequency information (like the hiss on sound) then we can filter, or remove, it by
inspecting the Fourier transform. If we retain low frequency components, then we implement
a
low-pass filter
. The low-pass filter describes the area in which we retain spectral components,
the size of the area dictates the range of frequencies retained, and is known as the filter's
bandwidth
. If we retain components within a circular region centred on the d.c. component,
and inverse Fourier transform the filtered transform then the resulting image will be
blurred
. Higher spatial frequencies exist at the sharp
edges
of features, so removing them
causes blurring. But the amount of fluctuation is reduced too; any high frequency noise
will be removed in the filtered image.
The implementation of a low-pass filter which retains frequency components within a circle
of specified radius is the function
low_filter
, given in Code
2.5
. This operator assumes
that the radius and centre co-ordinates of the circle are specified prior to its use. Points within
the circle remain unaltered, whereas those outside the circle are set to zero, black.
low_filter(pic) := for y
∈
0..rows(pic)-1
for x
∈
0..cols(pic)-1
filtered
y,x
←
2
2
if y-
rows(pic)
2
+x-
cols(pic)
2
pic
y,x
2
-radius 0
0 otherwise
filtered
Code 2.5
Implementing low-pass filtering
When applied to an image we obtain a low-pass filtered version. In application to an
image of a face, the low spatial frequencies are the ones which change slowly as reflected
in the resulting, blurred image, Figure
2.25
(a). The high frequency components have been
removed as shown in the transform, Figure
2.25
(b). The radius of the circle controls how
much of the original image is retained. In this case, the radius is 10 pixels (and the image
resolution is 256 ×
256). If a larger circle were to be used, more of the high frequency
