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Understanding Wavelets

Wavelets are oscillations that have amplitude and are similar in movement to a wave. Consider an ultrasound machine that measures the rise and fall of a heartbeat and the line that it produces and you will have a visual or mental picture of a wavelet. Wavelets are used in engineering and mathematics as a form of signal processing; however, they are not limited to one form. Wavelets may be used to measure a variety of different signals, such as sound, electrocardiograms, sensory data, images, and more. Any signal that is transmitted either digitally or through analog creates wavelets that may be documented. Additionally, wavelets may be manipulated through a process known as convolution. Convolution is a mathematical operation that can alter the form of wavelets by taking away aspect or adding to them.

Wavelets are important to scientists and mathematicians as they may be used as a source form which information is extracted or derived. Wavelets can be analyzed and broken down to create a clearer image of a photo or to detect sound that is barely audible in a file. The applications in which wavelets may be used are vast and the science is used in multiple industries. Wavelets are a relatively new scientific branch with the theory’s origins tracing back to Alfred Haar who in 1909 used a series of mathematical equations that would later be referred to as the Haar wavelet. The function is also known as the simplest wavelet. Exploration with wavelets continued in the 40s, when Dennis Gabor worked with what are known as the Garbor functions or the Gabor atoms. Gabor theorized producing sound with a granular system. Wavelet theory made great advances in the 1970s when George Zweig discovered the continuous wavelet transform. Essentially, wavelets may be considered as signals that have been broken down into smaller, compartmentalized signal components that allow scientist to examine their construction.

When describing wavelets terms include the mother and the daughter wavelet. The mother wavelet is the mother wavelet is one that has a finite length, while the daughter wavelets are those that are scaled or copies. In addition to mother and daughter wavelets, they may also be classified as Continuous Wavelet Transforms or (CWTs) and Discrete Wavelet Transforms (DWTs), Fast Wavelet Transform (FWT), Wavelet Packet Decomposition (WPD), Stationary Wavelet Transform (SWT), and Fractional Fourier Transform (FrFT). This is not an extensive list but shows some of the variations of wavelet transformations. Their families or associations may also define wavelets. These include scaling filter, scaling function, and wavelet function. Wavelets provide many benefits and advantages to those in a multiple fields.

As pertaining to signal processing applications, wavelets provide a number of uses; these include compression applications, edge detection, graphics, and numerical analysis. Compression applications regarding signal processing applications involves removing some of the least important or least significant aspects of the signal, and then keeping the significant aspect intact. The wavelet can be transformed into a low and high frequency part, decomposing the wavelet. 

Edge detection is an application that enables the wavelet such as the Haar basis to be compressed with the edges detected. This is often used in the scientific process of fingerprint compression. As with compression applications, both high and low, frequencies can be detected and low frequency parts may be discarded if desirable. There are basic two uses for wavelets as pertaining to graphics. These include wavelet radiosity and curve and surface representations.