Image Processing Reference
In-Depth Information
9
Scales and Frequency Channels
Blurring a function by a linear filter is equivalent to suppressing high-frequency com-
ponents of its spectrum. Having void information at high frequencies suggests in turn
that there might be a redundancy in the representation which can be avoided. This
observation can be utilized in several ways in signal analysis, notwithstanding image
analysis, including the following:
1. The amount of data representing an image can be reduced, e.g., image compres-
sion.
2. The image can be efficiently filtered to contain only certain frequency compo-
nents, e.g., recognition of image contents.
3. The image can be resized, e.g., user interfaces and animation.
9.1 Spectral Effects of Down- and Up-Sampling
We consider first the band-limited 1D function f ( t ) to investigate the effects of up-
and down-sampling on F ( ω ), the spectrum.
Let the maximum frequency of the band-limited signal f be Ω 0 / 2 i.e., F ( ω )
is effectively zero outside of the interval [
, Ω 2 ]. Then f can be sampled with
the distance T 0 = 2 Ω 0 between the samples, without loss of information. What does
happen in the frequency domain when we sample f tighter and tighter using smaller
discretization steps than T 0 ? If information is not lost with the discretization step T 0 ,
then it will definitely not be lost when sampling f with a smaller sampling step, T ,
with T< 2 Ω 0 . Sampling in one domain with the sampling period T corresponds to a
periodization with the basic period of Ω = 2 T in the other domain.
Conversely, when periodizing F with a period Ω that is larger than the bandwidth
of the signal, we pad zeros after the highest frequency of the signal. This has the
effect that the part of the void band in which F is practically zero increases with
increased sampling rate, i.e., with smaller T . Both up- and down-sampling can be
achieved via continuous reconstruction as follows:
Ω 0
2
 
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