It all started with a French mathematician physicist named Joseph Fourier. Fourier came up with the idea that any complex function could be represented as the weighted sum of simpler functions. These functions could be acquired from a prototype function. Consider the prototype function also known as a basis function to be like a building block. From this prototype function, the original function can be successfully obtained or approximated. Such approximations are very advantageous because they give us an insight into the analysis of complicated functions.
Fourier considered sinusoids as building blocks. These approximations gave us the spectral components present in the original signal. These Fourier approximations have been utilized in a number of fields that need signal processing or analysis. However, Fourier representations have one major limitation. Fourier approximations do not provide localization of time as well as frequency. In other words, it tells us what frequencies exist in the signal but does not tell us at what time instances these frequencies exist. Therefore, Fourier representation is not suitable for the analysis of non-stationary signals. Since, in such signals the frequency changes with time, it needs a time-frequency representation rather than just a frequency representation. This topic is discussed more elaborately later in this chapter.
In 1946, an improved version of Fourier transform was introduced by Dennis Gabor. It was known as the windowed Fourier Transform or the Short time Fourier transform (STFT). In STFT, the non-stationary signals were first segmented by utilizing a time-localized window and then the Fourier transform was calculated for each of these segments separately. As a result, the Short Time Fourier transform could provide a time frequency representation (TRF) of the signal.
Between 1940 and 1970, a number of scientists came up with different almost similar to the one devised by Gabor. The only dissimilarity between these TRFs was that each of them utilized different windowing functions. However, STFT and other TRFs had one major drawback. All of them used the same window to analyze the entire signal.
Wavelets were the solution to this drawback. Wavelets were first invented by Jean Morlet, a French geophysicist. At that time, he was working as a research engineer in Elf-Aquintine. He used these wavelets to find a solution for signal processing problems he encountered while extracting oil.
These wavelets eventually came to be known as Morlet wavelets. They maintained the same shape even when they we compressed, dilated or even when they were shifted in time. Similar to Morlet wavelets, other families of wavelets can also be formed from the mother wavelet by compressing, dilating or shifting in time.
Later, researchers found out that the shape of the mother wavelet significantly affects the accuracy of the approximation.
Morlet realized out that he could dissemble a wave into its wavelet components and then assemble them back into the original wave on his computer. But he was not satisfied with this. He wanted to find out if his method was mathematically plausible. After a few years, he found Alex Grossmann, with whom he worked for a year. Grossmann then confirmed that it was possible to reconstruct waves from their wavelet decompositions.
In fact, wavelet transforms turned out to work better than Fourier transforms because they are much less sensitive to small errors in the computation. An error or an unwise truncation of the Fourier coefficients can turn a smooth signal into a jumpy one or vice versa; wavelets avoid such disastrous consequences.
In 1984, Grossmann and Morlet's paper was published and their paper was the first to use the word "wavelet". Later in 1984, Yves Meyer widely recognized as one of the founders of wavelet theory, came to know about their work. He was the first person to find out that there was a connection between Morlet's wavelets and other mathematical wavelets invented earlier by Littlewood and Paley. Eventually, he found out that there were 16 separate rediscoveries of wavelet concept before Grossmann and Morlet's paper.
Meyer then invented a new wavelet which made wavelet transform as easy to work with and manipulate as fourier transform. This wavelet had a mathematical property called "orthogonality". It meant that information captured by a wavelet is not dependent on the information captured by another wavelet.
However, later it came to be known that orthonormal wavelets had already been discovered by J.O.Stromberg five years ago. It should also be noted that Alfred Haar was the first person to discover orthonormal wavelets. Haar wavelets are the simplest orthonormal wavelets which are of little use practically because of its poor frequency localization.
In 1986, Meyer's former student, Stephane Mallat linked the theory of wavelets to subband coding and quadrature mirror filters. Experts in the field of image processing were familiar to the idea of multi resolution analysis. Mallat showed that wavelets are implicit in the process of multi resolution analysis.
Wavelets became much simpler due to Mallat's work. You could perform wavelet analysis by knowing the formula of the mother wavelet. The process included simple operations of averaging groups of pixels together and taking their difference over and over again. Electrical engineers found wavelet very comfortable to work with because of their familiarity with terms like "filters", "high frequencies" and "low frequencies".
In 1987, Ingrid Daubechies revolutionized wavelet. He invented a new class of wavelet that were orthogonal and could be carried out using simple digital filtering ideas. These new wavelets were easy to program and were smooth. Signal processors could now separate digtal data into contributions of various scales. A simple, orthonormal transform was now available that could be rapidly calculated on modern digital computers.
The Daubechies wavelets have surprising features like connections with the fractal theory. If their graph is viewed under magnification, characteristics jagged wiggles can be seen, no matter how strong the magnification is. This exquisite complexity of detail means, there is no simple formula for these wavelets. They are ungainly and asymmetric. But the most important thing is they work. The Daubechies wavelets turned theory into a practical tool that is simple to program and can be utilized by any scientist with some basic mathematical training.
2 FOURIER ANALYSIS
2.1 FOURIER TRANSFORM
Any function can be represented as a summation of a series of sine and cosine terms of varying frequency. In theory, this summation can consist of an infinite number of sine and cosine terms. In other words, any time varying data can be transformed into another domain called the frequency domain. Fourier transform is found to be very useful in a number of applications particularly signal processing.
2.2 FREQUENCY DOMAIN
The basic question that needs to be asked is why it is necessary to transform a signal into its frequency domain. This is necessary because the time domain does not contain all the information present in the signal. Therefore, it is necessary to transform a signal into its frequency domain to obtain further information that is not available in the time domain. Basically, the frequency domain shows the spectral components that are present in the signal.
Consider an image represented in the frequency domain. If it has high frequencies in the frequency domain then it suggests that the image has sharp details or edges.
For example, if an image represented in frequency domain has high frequencies, then it means that the image has sharp edges or details. Frequency domain has a number of advantages for image processing. Filtering operations are performed faster and it also helps in separating noise from a signal. Such operations would be very difficult to perform in the time domain.
Fourier transform can be also used to eliminate periodic noise from an image. Consider a Xerox with a few grayish spots in a pattern. Now, if this image is converted to its frequency domain, these grayish spots will present as bright spots. If we remove all the spots using a filter and apply inverse Fourier transform, we will get back the original image free of spots. Hence in this way we can improve the quality of the image.
wavelet 1.JPG
(a). A dirty Xeroxed image (b). Representation of the xeroxed image in spectral domain.
(c). After eliminating all the bright spots (star like) from the image
(d). Reconstruction of the image using Inverse Fourier Transform
2.3 MATHEMATICAL BASICS
To successfully understand Fourier Transform, it is imperative to be well versed with the mathematical basics discussed in this section.
2.3.1 COMPLEX NUMBERS
The following is the representation of a complex number
,
where
and both are real numbers,
is the real part & is the imaginary part
There are many ways to represent a complex number. One such way is its representation as rectangular coordinate. In this way, the complex numbers basically rest on a plane. The horizontal axis and vertical axis are referred to as the real axis and the imaginary axis respectively. Hence, a complex number is represented as on the two dimension plane.
It can also be represented as a polar coordinate, i.e. in terms of distance from origin (magnitude), and angle it makes with the positive real axis (angle).
2.3.2 EULER'S FORMULA
The following is the Euler's formula:
=2.71828
is the angle. It can take any real value.
Complex number as a polar coordinate :
is its distance from the origin the magnitude of the polar form of complex number
is its angle with positive real axis
2.4 CONTINUOUS FOURIER TRANSFORM
Let be a continuous function of variable . The following is its Fourier Transform
'' - frequency variable
The following Euler's equation will make the summation of sines and cosines more clear
The inverse Fourier transform of F can be obtained from the following equation:
Here, and are known as Fourier transform pairs. These pairs exist only when is integrable and continuous.
fourier example.JPG
A simple function and its spectrum(Fourier)
2.5 DISCRETE FOURIER TRANSFORM
In discrete Fourier transform, finite sample points are chosen and from these points the FT of the is estimated. These sampled points should be typical of what the signal looks like at all other times. Moreover, discrete Fourier transform has symmetry properties similar to the symmetry properties of continuous Fourier transform
When N discrete samples of the function are sampled in uniform steps,
for = 0,1,2,…,
for =0,1,2…
2.6 FAST FOURIER TRANSFORM
The discrete Fourier transform of a function can be calculated using a computer, but this method is inefficient. The DFT is computationally intensive. The computation of the equations mentioned in the previous section requires a lot of complex additions & multiplications. The number of these complex calculations is proportional to. For the calculation of every, each of the terms from are used. Hence, a total of are computed.
However, this number can be significantly reduced. Proper decomposition of the following equation
can reduce the number of addition & multiplication operations and make the number proportional to . This decomposition process is called the fast Fourier transform algorithm. There are a many ways using which this algorithm can be implemented but, discussing these algorithms is outside the scope of this project.
2.7 LIMITATION OF FOURIER TRANSFORM
Fourier transform is suitable for analyzing stationary signals i.e. signals that are periodic in nature. However, for analyzing non-stationary signals like biomedical signals and speech signal a time frequency representation is preferable and Fourier transform does not give such a representation.
In stationary signals, the spectral components that exist in the signal are present all the time i.e. they are present for the entire duration of the signal. But, in non-stationary signals this does not happen, the spectral components are not present for the entire duration of the signal, they change continuously. Since, Fourier transform does not localize spectral components in time; it is not suitable for analyzing non stationary signals. However, if one wants to know only the frequency components
present in the signal then Fourier transform could be used.
2.8 SHORT TIME FOURIER TRANSFORM (WINDOWED FOURIER TRANSFORM)
The STFT is a modified version of Fourier transform. First, the non stationary signals are broken down into small segments, which are considered to be stationary locally. Second, the fast Fourier transform (FFT) is applied to each of the segments. Then, STFT of (a non-stationary signal) is obtained by multiplying (window function), centered at with the signal, with the signal. Finally, this produces a modified signal.
The following is the energy density of the spectrum at time :
From this, a different spectrum is obtained at every time. These different spectra are grouped together and this gives the TIME FREQUENCY distribution. This is called the spectrogram of the signal. Before we go into the details of the limitation posed by STFT, we need to study the Heisenberg's Uncertainty Principle. According to this principle, we cannot determine the exact TFR of a signal. In other words, it is not possible to determine exactly at what time the spectral components in a signal exist. It is only possible to determine at what time intervals a certain band of frequencies exist. Now, this poses a resolution problem. In STFT the problem is that the width of the window is fixed. A small window gives good time resolution and poor frequency resolution. On the other hand, a large window gives a good frequency resolution and poor time resolution.
The wavelet transform (continuous) was introduced to overcome the resolution problem posed by STFT. But, before we discuss wavelet transform in detail we will first study WAVELETS in detail.
stft tfr.JPG
Time frequency plane for STFT
3 WAVELET ANALYSIS
The anaIysis of non-stationary signaIs using waveIet transforms gives superior results than FT or STFT. Signal aspects like discontinuities and breakdown points are revealed successfully by wavelet analysis while other transforms like FT and STFT may fail to do the same. Multi resolution analysis (MRA) can be performed using WT.
3.1 MULTIRESOLUTION ANALYSIS
As I have discussed in the previous section, the resolution problems are caused by Heisenberg's uncertainty principle. This resolution problem exists in alI analysis techniques. In STFT, a steady time frequency resolution is used. But, WT uses multi resolutions analysis (MRA). MRA analyses the signal at different frequencies with different resolutions. It is designed in such a way that at high frequencies it gives good time resolution and poor frequency resolution and at low frequencies it gives goof frequency resolution and poor time resolution. This approach is suitabIe for almost all non stationary signals because generally in non stationary signals high frequencies exist for a short period and low signals are present for longer durations. Therefore, this method is an extremely valuable tool for signal anaIysis.
Wavelet dia.JPG
Time Frequency plane for Wavelet transform
3.2 WAVELETS
A 'wavelet' function is a smaIl wave and it has its energy concentrated in time. It has an undulating wavelike characteristic and also posses the capacity to do time and frequency analysis of non stationary signals together
In order to be called a wavelet, the analyzing function must satisfy the following properties:
WaveIet must have finite energy
If is a wavelet and its FT is , then the following must be true
According to the above condition, must be equal to zero. This condition is called the admissibility constant. The vaIue of depends on the wavelet that is selected.
It is a necessity that the FT is real and that it is absent for negative frequencies.
3.3 DIFFERENT COMMUNITIES OF WAVELETS
There are various instances of functions that can be used for MRA of the signal. These are known as wavelets. Few of them are:
Classical waveIets are the dyadic dilates and translates of a single function.
Wavelet packets are an extension of the classical wavelets which gives basis function that has finer frequency localization. But this is a costlier transform than the previous one.
Local trigonometric bases: The main purpose here is to use the sins & cosines that are defined in finite intervals. In addition to this they should be associated with an uncomplicated but extremely powerful technique to smoothly combine the basis functions at the boundaries.
Multi wavelets: In place of applying a single function to translate and dilate, a number of wavelet functions are used for producing basis functions.
Second generation wavelets: The concept of dilation and translation is completely eliminated here. This provides us with additional flexibility that can be utilized to produce wavelets adapted to irreguIar samples.
3.4 Different FAMILIES OF WAVALETS WITHIN WAVELET COMMUNITIES
Wavelets are basically grouped into different families depending on the features they posses. Few wavelets are for DWT and a few other wavelets are for CWT.
Given below are a few families which are a part of the "classical" community of wavelets:
Wavelets for CWT(Gaussian, MorIet, Mexican Hat)
Daubechies MaxfIat wavelets
SymIets
CoifIets
BiorthognaI spIine wavelets
Complex wavelets
CONTINUOUS WAVELET TRANSFORM
The CWT is represented as
The above is a transformed signal that is a function of the translational and scale parameters and respectively. is the mother wavelet and indicates that for complex wavelets the complex conjugate is utilized.
The signal energy is normalized at every scale by dividing the wavelet coefficients by . This is done so that the wavelets possess the same energy at each scale.
The mother wavelet is compressed and expanded by altering the scale parameter . The alteration in scale parameter varies the central frequency as well as the window length. Therefore, in place of the frequency the scale represents the outcome of the wavelet analysis. The translation parameter indicates the position of the wavelet in time, by varying the wavelet can be repositioned over the signal. For constant and varying rows of the time scale plane are filled. For constant and varying the columns of the time scale plane are filled.
Like FT and STFT, the wavelet transform is also reversible. The following is the equation for the inverse continuous wavelet transform (ICWT):
At each scale, the wavelet function has a central frequency. This frequency is inversely proportional to the scale. A Iarge scale corresponds to low frequency, providing overall information about the signal. SmaIl scale corresponds to higher frequency, providing detailed information about the signal. In other words, scale parameter >1 expands the signal whereas scale a<1 condenses the signal.
3.5 DISCRETE WAVELET TRANSFORM
Although the discretized continuous wavelet transform enables the computation of the continuous wavelet transform by computers, it is not a true discrete transform. As a matter of fact, the wavelet series is simply a sampled version of the CWT, and the information it provides is highly redundant as far as the reconstruction of the signal is concerned. This redundancy, on the other hand, requires a significant amount of computation time and resources. The discrete wavelet transform (DWT), on the other hand, provides sufficient information both for analysis and synthesis of the original signal, with a significant reduction in the computation time. The DWT is considerably easier to implement when compared to the CWT.
3.6 THE SUBBAND CODING AND MULTIRESOLUTION ANALYSIS
The main idea is the same as it is in the CWT. A time-scale representation of a digital signal is obtained using digital filtering techniques. Recall that the CWT is a correlation between a wavelet at different scales and the signal with the scale (or the frequency) being used as a measure of similarity. The continuous wavelet transform was computed by changing the scale of the analysis window, shifting the window in time, multiplying by the signal, and integrating over all times. In the discrete case, filters of different cutoff frequencies are used to analyze the signal at different scales. The signal is passed through a series of high pass filters to analyze the high frequencies, and it is passed through a series of low pass filters to analyze the low frequencies.
The resolution of the signal, which is a measure of the amount of detail information in the signal, is changed by the filtering operations, and the scale is changed by up sampling and down sampling (sub-sampling) operations. Sub-sampling a signal corresponds to reducing the sampling rate, or removing some of the samples of the signal. For example, sub sampling by two refers to dropping every other sample of the signal. Sub sampling by a factor n reduces the number of samples in the signal n times.
Up sampling a signal corresponds to increasing the sampling rate of a signal by adding new samples to the signal. For example, up sampling by two refers to adding a new sample, usually a zero or an interpolated value, between every two samples of the signal. Up sampling a signal by a factor of n increases the number of samples in the signal by a factor of n.
Although it is not the only possible choice, DWT coefficients are usually sampled from the CWT on a dyadic grid. Since the signal is a discrete time function, the terms function and sequence will be used interchangeably in the following discussion. This sequence will be denoted by x[n], where n is an integer.
The procedure starts with passing this signal (sequence) through a half band digital low pass filter with impulse response h[n]. Filtering a signal corresponds to the mathematical operation of convolution of the signal with the impulse response of the filter. The convolution operation in discrete time is defined as follows:
A half band low pass filter removes all frequencies that are above half of the highest frequency in the signal. For example, if a signal has a maximum of 1000 Hz component, then half band low pass filtering removes all the frequencies above 500 Hz.
After passing the signal through a half band low pass filter, half of the samples can be eliminated according to the Nyquist's rule, since the signal now has a highest frequency of /2 radians instead of radians. Simply discarding every other sample will subsample the signal by two, and the signal will then have half the number of points. The scale of the signal is now doubled. Note that the low pass filtering removes the high frequency information, but leaves the scale unchanged. Only the sub-sampling process changes the scale. Resolution, on the other hand, is related to the amount of information in the signal, and therefore, it is affected by the filtering operations. Half band low-pass filtering removes half of the frequencies, which can be interpreted as losing half of the information. Therefore, the resolution is halved after the filtering operation. Note, however, the subsampling operation after filtering does not affect the resolution, since removing half of the spectral components from the signal makes half the number of samples redundant anyway. Half the samples can be discarded without any loss of information. In summary, the lowpass filtering halves the resolution, but leaves the scale unchanged. The signal is then subsampled by 2 since half of the number of samples is redundant. This doubles the scale.
This procedure can be mathematically expressed as
The following discussion is about how DFT is computed. The DWT analyzes the signal at different frequency bands with different resolutions by decomposing the signal into a coarse approximation and detail information. DWT employs two sets of functions, called scaling functions and wavelet functions, which are associated with low pass and high pass filters, respectively. The decomposition of the signal into different frequency bands is simply obtained by successive high pass and low pass filtering of the time domain signal. The original signal x[n] is first passed through a half band high pass filter g[n] and a low pass filter h[n]. After the filtering, half of the samples can be eliminated according to the Nyquist's rule, since the signal now has a highest frequency of p /2 radians instead of p. The signal can therefore be sub-sampled by 2, simply by discarding every other sample. This constitutes one level of decomposition and can mathematically be expressed as follows:
In the above equations and are outputs of highpass and lowpass filters, respectivelyThis decomposition dives the time resolution into two since only half the number of samples now characterizes the entire signal. However, this operation doubles the frequency resolution, since the frequency band of signal now spans only half the previous frequency band, effectively reducing the uncertainty in the frequency by half.
3.7 WAVELET TRANSFORM VS FOURIER TRANSFORM
The wavelet transform is often compared with the Fourier transform. Fourier transform is a powerful tool for analyzing the components of a stationary signal (a stationary signal is a signal where there is no change in the properties of signal). For example, the Fourier transform is a powerful tool for processing signals that are composed of some combination of sine and cosine signals (sinusoids).
The Fourier transform is less useful in analyzing non-stationary signal (a non-stationary signal is a signal where there is change in the properties of signal). Wavelet transforms allow the components of a non-stationary signal to be analyzed. Wavelets also allow filters to be constructed for stationary and non-stationary signals. The Fourier transform shows up in a remarkable number of areas outside of classic signal processing. Even taking this into account, we think that it is safe to say that the mathematics of wavelets is much larger than that of the Fourier transform. In fact, the mathematics of wavelets encompasses the Fourier transform. The size of wavelet theory is matched by the size of the application area. Initial wavelet applications involved signal processing and filtering. However, wavelets have been applied in many other areas including non-linear regression and compression
The main difference is that wavelets are well localized in both time and frequency domain whereas the standard Fourier transform is only localized in frequency domain. The Short-time Fourier transform (STFT) is also time and frequency localized but there are issues with the frequency time resolution and wavelets often give a better signal representation using Multi resolution analysis.
4 APPLICATIONS OF WAVELET TRANSFORMS
Wavelets are a powerful statistical tool which can be used for a wide range of
applications, namely
Signal processing
Data compression
Smoothing and image denoising
Fingerprint verification
DNA analysis, protein analysis
Analysis of biomedical signals like EEG,ECG,EMG
Finance, for detecting the properties of quick variation of values
Speech recognition
Computer graphics and multifractal analysis
Molecular dynamics, astrophysics, optics, turbulence and quantum mechanics
Wavelets have been used successfully in other areas of geophysical study. Orthonormal wavelets, for instance, have been applied to the study of atmospheric layer turbulence. In one study by J.F. Howell and L. Mahrt, turbulence measurements were taken over a nine-hour period and analyzed using wavelet decomposition. In another study by Brunet and Collineau, turbulence data recorded
over a corn crop was analyzed using the wavelet transform. Wavelets have also been used to analyze seafloor bathymetry or the topography of the ocean floor. In one study by Sarah Little, the use of wavelet analysis revealed patterns, trends, and structures that may be overlooked in raw data.
Also, the use of methods like local oracles allowed for separation of data in regions of interest.
Several other geophysical applications such as analysis of marine seismic data and characterization of hydraulic conductivity distributions have also been used. The usefulness of wavelets in data analysis is clear, particularly in the field of geophysics, where large and cumbersome data sets abound. Studies such as the atmospheric layer turbulence and corn crop turbulence have further shown the
proficiency of wavelets in the analysis of time-dependent data.
4.1 COMPUTER AND HUMAN VISION
In early 1980's, David Marr began work at MIT's Artificial Intelligence Laboratory on artificial vision of robots. He is an expert on the human visual system and his goal was to learn why his first attempts to construct a robot capable of understanding its surroundings were unsuccessful.
He believed that it was important to establish scientific foundations for vision, and during this process one must limit the scope of investigation by excluding everything that depends on training, culture 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.
Marr was curious to know how it was possible to define the contours of objects from the variations of their light intensity. He was also curious to know how it is possible to sense depth and how movement is sensed.
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".
4.2 FBI FINGERPRINT COMPRESSION
Between 1924 and today, the US Federal Bureau of Investigation has collected more than 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 Gigabyte 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.
Fig5
An FBI digitized left thumb print. The one on the left is the original and the one on the right is reconstructed from a 26:1 compression.
4.3 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. Wavelets can solve this problem through a technique called wavelet shrinkage and thresholding methods.
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, thresholded 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.
Fig6
Before and after illustrations of a nuclear magnetic resonance signal. The original signal is at the top and the denoised signal at the bottom.
5 REASON FOR USING WAVELET TRANSFORM FOR ANALYSING BIOMEDICAL SIGNALS
Biomedical signals are generally non-stationary with significant events (e.g. heartbeats) characterized by both their time location and frequency content. Consequently the frequency analysis of these biomedical signals using Fourier techniques is fundamentally unsatisfactory since they are based upon modeling the signal as a linear combination of sinusoids extending throughout the duration of the signal. Fourier analysis is only good at determining what frequencies are present (i.e. it provides good frequency discrimination), but poor at pinpointing when these frequencies occur (i.e. it has poor time localization). This situation can be improved by windowing the basis functions of the Fourier transform so that the sinusoids used to model the signal under analysis no longer extend across the entire signal. Using such windowing produces the Short Time Fourier transform. The result of the windowing process is that STFT is able to locate events in time (since the basis functions are now restricted in time) but, due to the uncertainty principle, has poorer frequency resolution.
Use of the STFT allows a plot of the frequency content of a signal against time. The spectrogram is a valuable tool for the analysis of non-stationary signals. However, the use of a fixed length window means that the resolution of the spectrogram, in both time and frequency, is fixed. This is often referred to as the time-frequency plane being tiled by fixed size rectangular windows.
The wavelet transform (WT) has many similarities with the STFT but is fundamentally different in that its basis function (the wavelets) is not of fixed length. Rather the wavelets are derived by stretching (in time) a base wavelet to give the full wavelet basis consisting of expanded versions of the base function. This repeated stretching to produce the wavelet basis obviously produces functions with decreasing frequency thus allowing the frequency content of the signal to be analyzed. However, the increase in length of the expanded wavelets hinders them from being able to localize the precise timing of events within the signal. This feature of the WT shows in the tiling (which is related to time frequency resolution) of time-frequency plane by providing good time localization at high frequencies and improved frequency discrimination at lower frequencies at the expense of poorer time localization. Consequently, the tiles at low frequencies are shorter and wider (in frequency and time respectively) than the taller and narrower ones at high frequencies. This feature of WT enables them to analyze biomedical signals to a large extent.
6 CONCLUSION
The rest of the project (to be completed in the next two months) is going to discuss biomedical signals like EEG, ECG, EMG in detail. EEG signal will be analyzed using wavelet transform using MATLAB. This normal EEG signal is going to be compared with other EEG signals with disturbed brain activities like sleep disorders and epilepsy. These signals will be compared and analyzed. Finally, the results and analysis will be noted down.
7 LIST OF FIGURES
Image explaining the advantages of Fourier transform(page 5)
A simple function and its fourier spectrum(page 7)
Time Frequency plane for STFT(page 10)
Time Frequency plane for WT(page 11)
Finger print image showing data compression(page 18)
Denoising of a Signal (page 19)
