Digital Signal Processing Reference
In-Depth Information
(a)
(b)
FIGURE 10.1
Lena (256
256, 8 bpp) image and its three-scale wavelet transform, where the gray level corre-
sponds to the magnitudes of wavelet coefficients.
×
tremendous research activities. In particular, the discrete wavelet transform
(DWT) can provide a more favorable representation that helps us develop ef-
ficient image modeling and processing techniques. For example, Figure 10.1
shows the Lena image and its three-scale DWT, which demonstrates a
compact
,
joint spatial frequency
, and
multiscale
image representation. First, the compact
property of DWT indicates that most image energy can be compacted onto
a few wavelet coefficients with large magnitudes. At the same time, most
coefficients are very small. This compact property allows us to capture the
key characteristics of an image from those large wavelet coefficients. Second,
the joint spatial-frequency representation results in two evident properties of
wavelet coefficient distribution, i.e.,
interscale persistence
and
intrascale clus-
tering
. These two observations inspire many researchers to develop various
statistical models for image restoration, compression, and classification, etc.
Third, the multiscale representation of DWT is also useful in many image
processing applications, such as progressive image transmission, embedded
image compression, multiresolution image analysis, etc.
Another important mathematical tool of statistical modeling discussed in
this chapter is the hidden Markov model (HMM). The theory of HMMs was
originally developed in the 1960s.
4
HMMs have earned popularity mainly
from their successful application to speech recognition. A recent review of
the HMMs theory can be found in Reference 5. An HMM model has a finite
set of
states
, each of which is associated with a (generally multidimensional)
probability distribution. Transitions among the states are governed by a set of
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