Graphics Reference
In-Depth Information
Low-pass filter:
A filter that cuts off all frequencies above a threshold and lets
those below it pass through.
High-pass filter:
This is the opposite of a low-pass filter. A filter that cuts off all
frequencies below a threshold and lets those above it pass through.
Band-pass filter:
A filter that cuts off all frequencies outside of a range of
frequencies.
We have come across these filters in our earlier discussion. We can use Figure 21.9
to help us understand a typical definition-to-display path of a signal. The original
signal is shown in (a). Some equally spaced pulses and the sampled signal are shown
in (c) and (e), respectively. The assumption is that we can get our hands on only the
sampled signal. (If we really “had” the original signal, then none of this work we are
describing would be necessary.) At this point, our goal is to reconstruct the original
signal. To accomplish this one chooses a suitable “reconstruction” filter and convolves
it with the sampled signal. Next, our goal is to display the resulting signal S. The
display may have some finitary constraints, such as a monitor that can only show
something at a finite number of pixels. For this reason, one typically uses another
filter, perhaps a low-pass filter, to eliminate frequencies that one cannot display. One
convolves the signal S with such a filter. Finally, this signal is sampled and recon-
structed with another “reconstruction” filter to produce our displayed signal. For a
good discussion of this process see [Glas95].
21.9
Wavelets
Digital image processing has two components. One deals with the representation or
analysis of a function. That is what the Fourier transform was about. It represented
a function in terms of their frequency content. The other part of digital image pro-
cessing is concerned with putting it all back together again and reconstructing a func-
tion from its inherent frequencies. So the Fourier transform computed the Fourier
coefficients, which gave us the representation or analysis of the function, and then
the Fourier series reconstructed the function using these Fourier coefficients. One
problem with the Fourier transform though is that it is based on the sine and cosine
functions that do
not
have compact support. That is why, if we want to represent a
function that
has
compact support, its representation in the frequency domain will
not also be a function with compact support but involve an infinite number of fre-
quencies. We saw that in the case of the box function and the impulse function. This
is where wavelets come in.
This section will only give a brief glimpse of wavelets. We shall make no attempt
to give a formal definition of what they are but simply discuss them in the context of
examples. Wavelets are functions that can have compact support. By analyzing func-
tions using wavelets, one can get a better handle on those functions with compact
support. Additionally, they give one the means of specifying the amount of detail one
wants from the analysis based on the number of samples taken. There is no unique
set of wavelet basis functions. We briefly describe the Haar basis and some of its prop-
erties. For simplicity we restrict ourselves to analyzing functions defined on [0,1].
Define functions f, f
j,i
, Y , Y
j,i
:
R
Æ
R
by
