Image Compression Using Wavelets
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Abstract
Images require substantial storage and transmission resources, thus image compression is advantageous
to reduce these requirements. The report covers some background of wavelet analysis, data compression
and how wavelets have been and can be used for image compression. An investigation into the process
and problems involved with image compression was made and the results of this investigation are
discussed. It was discovered that thresholding was had an extremely important influence of
compression results so suggested thresholding strategies are given along with further lines of research
that could be undertaken.
Introduction
Often signals we wish to process are in the time-domain, but in order to process them more easily other
information, such as frequency, is required. Mathematical transforms translate the information of signals
into different representations. For example, the Fourier transform converts a signal between the time
and frequency domains, such that the frequencies of a signal can be seen. However the Fourier
transform cannot provide information on which frequencies occur at specific times in the signal as time
and frequency are viewed independently. To solve this problem the Short Term Fourier Transform
(STFT) introduced the idea of windows through which different parts of a signal are viewed. For a
given window in time the frequencies can be viewed. However Heisenburgs Uncertainty Principle
states that as the resolution of the signal improves in the time domain, by zooming on different sections,
the frequency resolution gets worse. Ideally, a method of multiresolution is needed, which allows
certain parts of the signal to be resolved well in time, and other parts to be resolved well in frequency.
The power and magic of wavelet analysis is exactly this multiresolution.
Images contain large amounts of information that requires much storage space, large transmission
bandwidths and long transmission times. Therefore it is advantageous to compress the image by storing
only the essential information needed to reconstruct the image. An image can be thought of as a matrix
of pixel (or intensity) values. In order to compress the image, redundancies must be exploited, for
example, areas where there is little or no change between pixel values. Therefore images having large
areas of uniform colour will have large redundancies, and conversely images that have frequent and
large changes in colour will be less redundant and harder to compress.
Background
The Need for Wavelets
Often signals we wish to process are in the time-domain, but in order to process them more easily other
information, such as frequency, is required. A good analogy for this idea is given by Hubbard[4], p14.
The analogy cites the problem of multiplying two roman numerals. In order to do this calculation we
would find it easier to first translate the numerals in to our number system, and then translate the answer
back into a roman numeral. The result is the same, but taking the detour into an alternative number
system made the process easier and quicker. Similarly we can take a detour into frequency space to
analysis or process a signal.
Multiresolution and Wavelets
The power of Wavelets comes from the use of multiresolution. Rather than examining entire signals
through the same window, different parts of the wave are viewed through different size windows (or
resolutions). High frequency parts of the signal use a small window to give good time resolution, low
frequency parts use a big window to get good frequency information.
An important thing to note is that the ’windows’ have equal area even though the height and width may
vary in wavelet analysis. The area of the window is controlled by Heisenberg’s Uncertainty principle,
as frequency resolution gets bigger the time resolution must get smaller.
Sampling and the Discrete Wavelet Series
In order for the Wavelet transforms to be calculated using computers the data must be discretised. A
continuous signal can be sampled so that a value is recorded after a discrete time interval, if the Nyquist
sampling rate is used then no information should be lost. With Fourier Transforms and STFT’s the
sampling rate is uniform but with wavelets the sampling rate can be changed when the scale changes.
Higher scales will have a smaller sampling rate. According to Nyquist Sampling theory.
Conservation and Compaction of Energy
An important property of wavelet analysis is the conservation of energy. Energy is defined as the sum
of the squares of the values. So the energy of an image is the sum of the squares of the pixel values, the
energy in the wavelet transform of an image is the sum of the squares of the transform coefficients.
During wavelet analysis the energy of a signal is divided between approximation and details signals but
the total energy does not change. During compression however, energy is lost because thresholding
changes the coefficient values and hence the compressed version contains less energy.
The compaction of energy describes how much energy has been compacted into the approximation
signal during wavelet analysis. Compaction will occur wherever the magnitudes of the detail
coefficients are significantly smaller than those of the approximation coefficients. Compaction is
important when compressing signals because the more energy that has been compacted into the
approximation signal the less energy can be lost during compression.