Wavelets are mathematical functions

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ABSTRACT

Wavelets are mathematical functions that cut up data into different frequency components, and then study each component with a resolution matched to its scale. They have advantages over traditional Fourier methods in analyzing physical situations where the signal contains discontinuities and sharp spikes. Wavelets were developed independently in the fields of mathematics, quantum physics, electrical engineering, and seismic geology. Interchanges between these fields during the last ten years have led to many new wavelet applications such as image compression, turbulence, human vision, radar, and earthquake prediction. This paper introduces wavelets to the interested technical person outside of the digital signal processing field. I describe the history of wavelets beginning with Fourier, compare wavelet transforms with Fourier transforms, state properties and other special aspects of wavelets, and

Finish with some interesting applications such as image compression, musical tones, and de-noising noisy data.

1. INTRODUCTION

A wavelet is a wave-like oscillation with amplitude that starts out at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one might see recorded Seismograph

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Or heart monitor. Generally, wavelets are purposefully crafted to have specific properties that make them useful for signal processing. Wavelets can be combined, using a "shift, multiply and sum" technique called convolution, with portions of an unknown signal to extract information from the unknown signal. Wavelets provide an alternative approach to traditional signal processing techniques such as Fourier analysis for breaking a signal up into its constituent parts. The driving impetus behind wavelet analysis is their property of being localised in time (space) as well as scale (frequency). This provides a time-scale map of a signal, enabling the extraction of features that vary in time. This makes wavelets an ideal tool for analysing signals of a transient or non-stationary nature.

2. HISTORY

The development of wavelets can be linked to several separate trains of thought, starting with Haar's work in the early 20th century. Notable contributions to wavelet theory can be attributed to Zweig's discovery of the continuous wavelet transform in 1975 (originally called the cochlear transform and discovered while studying the reaction of the ear to sound), Pierre Goupillaud, Grossmann and Morlet's formulation of what is now known as the CWT (1982), Jan-Olov Strömberg's early work on discrete wavelets (1983), Daubechies' orthogonal wavelets with compact support (1988), Mallat's multiresolution framework (1989), Nathalie Delprat's time-frequency interpretation of the CWT (1991), Newland's Harmonic wavelet transform (1993) and many others since.

  • First wavelet (Haar wavelet) by Alfred Haar (1909)
  • Since the 1970s: George Zweig, Jean Morlet, Alex Grossmann
  • Since the 1980s: Yves Meyer, Stéphane Mallat, Ingrid Daubechies, Ronald Coifman, Victor Wickerhauser

3. WAVELET THEORY

Wavelet theory is applicable to several subjects. All wavelet transforms may be considered forms of time-frequency representation for continuous-time (analog) signals and so are related to harmonic analysis. Almost all practically useful discrete wavelet transforms use discrete-time filter banks. These filter banks are called the wavelet and scaling coefficients in wavelets nomenclature. These filter banks may contain either finite impulse response (FIR) or infinite impulse response (IIR) filters. The wavelets forming a continuous wavelet transform (CWT) are subject to the uncertainty principle of Fourier analysis respective sampling theory: Given a signal with some event in it, one cannot assign simultaneously an exact time and frequency response scale to that event. The product of the uncertainties of time and frequency response scale has a lower bound. Thus, in the scale gram of a continuous wavelet transform of this signal, such an event marks an entire region in the time-scale plane, instead of just one point. Also, discrete wavelet bases may be considered in the context of other forms of the uncertainty principle. Wavelet transforms are broadly divided into three classes: continuous, discrete and multiresolution-based.

Above shown diagram shows all CWT (Continuous Wavelet), DWT (Discrete Wavelet). These all varies with the time and level and all graphs obtained are above shown.

4. WAVELET TRANSFORMS

There are a large number of wavelet transforms each suitable for different applications. For a full list see list of wavelet-related transforms but the common ones are listed below:

  • Continuous wavelet transform (CWT)
  • Discrete wavelet transform (DWT)
  • Fast wavelet transform (FWT)
  • Lifting scheme
  • Wavelet packet decomposition (WPD)
  • Stationary wavelet transform (SWT)

5. WAVELET PACKETS

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The wavelet transform is actually a subset of a far more versatile transform, the wavelet packet transform. Wavelet packets are particular linear combinations of wavelets. They form bases which retain many of the orthogonality, smoothness, and localization properties of their parent wavelets. The coefficients in the linear combinations are computed by a recursive algorithm making each newly computed wavelet packet coefficient sequence the root of its own analysis tree.

6. WAVELETS IN MATLAB

Wavelet Toolbox software extends the MATLAB technical computing environment with graphical tools and command-line functions for developing wavelet-based algorithms for the analysis, synthesis, denoising, and compression of signals and images. Wavelet analysis provides more precise information about signal data than other signal analysis techniques, such as Fourier. The Wavelet Toolbox supports the interactive exploration of wavelet properties and applications. It is useful for speech and audio processing, image and video processing, biomedical imaging, and 1-D and 2-D applications in communications and geophysics.

7. WAVELETS VS FOURIER TRANSFORM

Each and every thing in this world comparable to it has some similarities and dissimilarities with that same is the case with the wavelets and Fourier transform. Wavelets can be compared with the Fourier transform on the basis of their similarities and dissimilarities which are explained as follows. Various kinds of similarities and dissimilarities of wavelets and Fourier transform are as follows.

7.1 SIMILARITIES BETWEEN FOURIER AND WAVELET TRANSFORMS

The fast Fourier transform (FFT) and the discrete wavelet transform (DWT) are both linear operations that generate a data structure that containssegments of various lengths, usually filling and transforming it into a different data vector of length. The mathematical properties of the matrices involved in the transforms are similar as well. The inverse transform matrix for both the FFT and the DWT is the transpose of the original. As a result, both transforms can be viewed as a rotation in function space to a different domain. For the FFT, this new domain contains basis functions that are sines and cosines. For the wavelet transform, this new domain contains more complicated basis functions called wavelets, mother wavelets, or analyzing wavelets. Both transforms have another similarity. The basic functions are localized in frequency, making mathematical tools such as power spectra (how much power is contained in a frequency interval) and scale grams (to be defined later) useful at picking out frequencies and calculating power distributions.

7.2 DISSIMILARITIES BETWEEN FOURIER AND WAVELET TRANSFORMS

The most interesting dissimilarity between these two kinds of transforms is that individual wavelet functions arelocalized in space.Fourier sine and cosine functions are not. This localization feature, along with wavelets' localization of frequency, makes many functions and operators using wavelets "sparse" when transformed into the wavelet domain. This sparseness, in turn, results in a number of useful applications such as data compression, detecting features in images, and removing noise from time series. One way to see the time-frequency resolution differences between the Fourier transform and the wavelet transform is to look at the basis function coverage of the time-frequency plane. The square wave window truncates the sine or cosine function to fit a window of a particular width. Because a single window is used for all frequencies in the WFT, the resolution of the analysis is the same at all locations in the time-frequency plane.

8. WAVELET APPLICATIONS

There are various kinds of applications in the field of wavelets which are as follows can be explained as follows

  • Computer and Human Vision
  • FBI Fingerprint Compression
  • Denoising Noisy Data
  • Musical Tones

8.1 COMPUTER AND HUMAN VISION

In the early 1980s, David Marr began work at MIT's Artificial Intelligence Laboratory on artificial vision for robots. He is an expert on the human visual system and his goal was to learn why the first attempts to construct a robot capable of understanding its surroundings were unsuccessful. Marr believed that it was important to establish scientific foundations for vision, and that while doing so; one must limit the scope of investigation by excluding everything that depends on training, culture, and so on, and focus on the mechanical or involuntary aspects of vision. This low-level vision is the part that enables us to recreate the three-dimensional organization of the physical world around us from the excitations that stimulate the retina. He then developed working algorithmic solutions to answer each of these questions. Marr's theory was that image processing in the human visual system has a complicated hierarchical structure that involves several layers of processing. At each processing level, the retinal system provides a visual representation that scales progressively in a geometrical manner. His arguments hinged on the detection of intensity changes. He theorized that intensity changes occur at different scales in an image, so that their optimal detection requires the use of operators of different sizes. He also theorized that sudden intensity changes produce a peak or trough in the first derivative of the image. These two hypotheses require that a vision filter have two characteristics: it should be a differential operator, and it should be capable of being tuned to act at any desired scale. Marr's operator was a wavelet that today is referred to as a "Marr wavelet."

8.2 FBI FINGERPRINT COMPRESSION

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Between 1924 and today, the US Federal Bureau of Investigation has collected about 30 million sets of fingerprints. The archive consists mainly of inked impressions on paper cards. Facsimile scans of the impressions are distributed among law enforcement agencies, but the digitization quality is often low. Because a number of jurisdictions are experimenting with digital storage of the prints, incompatibilities between data formats have recently become a problem. This problem led to a demand in the criminal justice community for a digitization and a compression standard. In 1993, the FBI's Criminal Justice Information Services Division developed standards for fingerprint digitization and compression in cooperation with the National Institute of Standards and Technology, Los Alamos National Laboratory, commercial vendors, and criminal justice communities. Let's put the data storage problem in perspective. Fingerprint images are digitized at a resolution of 500 pixels per inch with 256 levels of gray-scale information per pixel. A single fingerprint is about 700,000 pixels and needs about 0.6 Mbytes to store. A pair of hands, then, requires about 6 Mbytes of storage. So digitizing the FBI's current archive would result in about 200 terabytes of data. (Notice that at today's prices of about $900 per Gbyte for hard-disk storage, the cost of storing these uncompressed images would be about 200 million dollars.) Obviously, data compression is important to bring these numbers down.

8.3 DENOISING NOISY DATA

In diverse fields from planetary science to molecular spectroscopy, scientists are faced with the problem of recovering a true signal from incomplete, indirect or noisy data. Can wavelets help solve this problem? The answer is certainly "yes," through a technique called wavelet shrinkage and thresholding methods that David Donoho has worked on for several years. The technique works in the following way. When you decompose a data set using wavelets, you use filters that act as averaging filters and others that produce details. Some of the resulting wavelet coefficients correspond to details in the data set. If the details are small, they might be omitted without substantially affecting the main features of the data set. The idea of thresholding, then, is to set to zero all coefficients that are less than a particular threshold. These coefficients are used in an inverse wavelet transformation to reconstruct the data set. Figure 6 is a pair of "before" and "after" illustrations of a nuclear magnetic resonance (NMR) signal. The signal is transformed, threshold and inverse-transformed. The technique is a significant step forward in handling noisy data because the denoising is carried out without smoothing out the sharp structures. The result is cleaned-up signal that still shows important details.

Fig.8.3.1 displays an image created by Donoho of Ingrid Daubechies (an active researcher in wavelet analysis and the inventor of smooth orthonormal wavelets of compact support), and then several close-up images of her eye: an original, an image with noise added, and finally denoised image. To denoise the image, Donoho:

  • transformed the image to the wavelet domain using Coiflets with three vanishing moments,
  • applied a threshold at two standard deviations, and
  • Inverse-transformed the image to the signal domain.

8.4 MUSICAL TONES

Victor Wickerhauser has suggested that wavelet packets could be useful in sound synthesis. His idea is that a single wavelet packet generator could replace a large number of oscillators. Through experimentation, a musician could determine combinations of wave packets that produce especially interesting sounds. Wickerhauser feels that sound synthesis is a natural use of wavelets. Say one wishes to approximate the sound of a musical instrument. A sample of the notes produced by the instrument could be decomposed into its wavelet packet coefficients. Reproducing the note would then require reloading those coefficients into a wavelet packet generator and playing back the result. Transient characteristics such as attack and decay- roughly, the intensity variations of how the sound starts and ends- could be controlled separately (for example, with envelope generators), or by using longer wave packets and encoding those properties as well into each note. Any of these processes could be controlled in real time, for example, by a keyboard. Notice that the musical instrument could just as well be a human voice, and the notes words or phonemes.

A wavelet-packet-based music synthesizer could store many complex sounds efficiently because

  • wavelet packet coefficients, like wavelet coefficients, are mostly very small for digital samples of smooth signals; and
  • Discarding coefficients below a predetermined cutoff introduces only small errors when we are compressing the data for smooth signals.

Similarly, a wave packet-based speech synthesizer could be used to reconstruct highly compressed speech signals. Figure 8.4.1 illustrates a wavelet musical tone or toneburst.

9. ADVANTAGES OF WAVELET TRANSFORMATION

Advantages of wavelet transformation are as follows which are discussed below.

  • Space and Time Efficiency (Low Complexity of DWT).
  • Generality & Adaptability (Different Basis and Wavelet Functions).
  • Multiresolution Properties (Hierarchical Representation & Manipulation).
  • Adaptability of the Transformation (Different Basis Functions allow different Properties of the Transformation)
  • Transformation is Hierarchical (Multiresolution - Properties)
  • Transformation is Loss-Free
  • Efficiency of the Transformation (Linear Time and Space Complexity for Orthogonal Wavelets)
  • Generality of the Transformation (Generalization of other Transformations)
  • CONCLUSION AND FUTURE SCOPE

    Most of basic wavelet theory has been done. The mathematics has been worked out in excruciating detail and wavelet theory is now in the refinement stage. The refinement stage involves generalizations and extensions of wavelets, such as extending wavelet packet techniques. The future of wavelets lies in the as-yet uncharted territory ofapplications.Wavelet techniques have not been thoroughly worked out in applications such as practical data analysis, where for example discretely sampled time-series data might need to be analyzed. Such applications offer exciting avenues for exploration. Basically after working on this term paper we came to know about the concept of the wavelets its relation with the Fourier transform its advantages in residing world.

    REFERENCES

    1. www.yahoo.com (a really friendly guide to wavelets).
    2. www.google.com (wavelets ppt.).
    3. www.wikipedia.com (wavelets).
    4. www.google.com (Seminar Report on wavelets by ROBI POLIKAR)
    5. www.google.com (applications of wavelets).