Wavelet-Based Image Compression - Wavelet Background
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Transcript of Wavelet-Based Image Compression - Wavelet Background
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FOUNDATIONS OF WAVELET ANALYSIS
Wavelet theory uses a two-dimensional expansion set to characterize and give a
time-frequency localization of a one-dimensional signal. Founded on the same principles of
Fourier theory, the wavelet transform calculates inner products of a signal with a set of basis
functions to find coefficients that represent the signal. Since this is a linear system, the signal
can be reconstructed by a weighted sum of the basis functions. In contrast to the
one-dimensional Fourier basis localized in only frequency, the wavelet basis is
two-dimensional - localized in both frequency and time. A signal's energy, therefore, is usually
well represented by just a few wavelet expansion coefficients.
WAVELET SYSTEM CHARACTERISTICS
Generated from a single scaling function or wavelet by scaling and translation1.
Satisfies multiresolution conditions. That is, if the basic functions are made half as wide
and translated in steps half as wide, they will represent a larger class of signals exactly or
give a better approximation of any signal
2.
Efficient calculation of the DWT. Lower resolution coefficients can be calculated from
higher resolution coefficients by a tree-structured algorithm known as a filter bank.
3.
Discrete Wavelet Transform
where the two-dimensional set of coefficients aj,k is the DWT of f(t).
TRANSLATION AND SCALING
As the index k changes, the location and scaling of the wavelet moves along the time
axis. As the index j changes, the shape of the wavelet changes in scale. As the scale
becomes finer (j larger), the time steps become smaller. Both the narrower wavelet and the
smaller steps allow a representation of greater detail or resolution.
THE SCALING FUNCTION
In order to use the idea of multiresolution, a scaling function j(t) is used to define the
wavelet function.
A two-dimensional family of functions is generated from the basic scaling function by
scaling and translation.
velet-Based Image Compression - Wavelet Background http://www.clear.rice.edu/elec301/Projects00/wavelet_image_comp/ba...
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The spans of the various scaling functions are nested, and after some linear algebra the
recursive scaling function can be rewritten as:
where the coefficients h(n) are a sequence of real or complex numbers called the scaling
function coefficients and the root two maintains the norm of the scaling function.
The wavelet function can then be represented by a weighted sum of shifted scaling
functions
for some set of coefficients h1(n).
DISPLAYING THE DWT
There are five displays that show the various characteristics of the DWT well:
Time-domain. This, however, gives no information about frequency or scale.1.
Three-dimensional plot c(k) and dj(k) over thej, kplane.2.
Time functions fj(t) at each scale by summing overkso that3.
Time function fk(t) at each translation by summing overkso that4.
Partitioning the time-scale plane as if the time translation index and scale index were
continuous variables. (a.k.a. tiling the time-frequency plane)
5.
EFFECTIVENESS OF WAVELET ANALYSIS
Wavelet analysis produces several important benefits, particularly for image compression.
First, an unconditional basis causes the size of the expansion coefficients to drop off with j
and k for many signals. Since wavelet expansion also allows a more accurate local description
velet-Based Image Compression - Wavelet Background http://www.clear.rice.edu/elec301/Projects00/wavelet_image_comp/ba...
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and separation of signal characteristics, the DWT is very efficient for compression. Second, an
infinity of different wavelets creates a flexibility to design wavelets to fit individual applications.
[back to main page]
velet-Based Image Compression - Wavelet Background http://www.clear.rice.edu/elec301/Projects00/wavelet_image_comp/ba...
r 3 16/01/2013 10:06