is approximated by a sum of sines and
cosines. Wavelet analysis is designed to better handle non-periodic functions,
because of the focus on the time intervals where functions are
defined.
Wavelets have many applications in image processing. For example, the FBI uses wavelets to compress fingerprint images. JPEG2000 is also wavelet-based.
Wavelet analysis is a modification of Fourier analysis, where
functions other than sine and cosine are used as the basis functions.
In Fourier analysis, the goal is to decompose a function by thinking of
it as a combination of trigonometric functions with different
frequencies. This approach is very useful when dealing with
periodic functions. However, if we wish to analyze non-periodic
functions, then the focus on frequencies can lead to poor results. An
example can be found in Figure 1, where the characteristic function on
the time interval is approximated by a sum of sines and
cosines. Wavelet analysis is designed to better handle non-periodic functions,
because of the focus on the time intervals where functions are
defined.
Figure 1: An Example from Fourier analysis.
Figure 2 is a graph of the scaling function known as D4. Scaling functions play a role similar to the sine and cosine functions of Fourier analysis in that other wavelets in a wavelet family are generated from the scaling function. The scaling function is also called the father wavelet.
Figure 2: The D4 Scaling Function.
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Over the past several years at the GVSU REU, students have used wavelets to solve interesting applied problems. For example, two students improved the method of Lyu et al. to determine the difference between authentic and forged handwriting. The problem of detecting airplanes in aerial photographs led to interesting results. In 2003, students used wavelets to help break CAPTCHAs. Wavelets were also applied for the design of position codes, which are used for "smart pens". All of these projects used wavelet-based filters. In each case, filtering is done for basically two reasons: information rearrangement and information reduction. Select the pictures on the right in order to learn more about these projects. |
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Industry and the military are interested in wavelets and other multiresolution methods to tackle problems in machine vision, character or shape recognition, data hiding, measurement, motion detection, image enhancement, and surface analysis. There is also an increasing use of wavelets in medicine, biology, and art. These applications are several possibilities for research in wavelets at the 2008 REU at GVSU.
More information on wavelets can be found at the Discovering Wavelets web site.
