How Haar Wavelet Deals with 2D signals

Haar wavelet deals 2D signals in two ways

a) Standard Decomposition:
 
 

b) Non Standard Decomposition

in both cases, button right component is average component and when moving from top left to right component is moving higher frequency signal component to lower component and so is it when moving top left to button. 

Continuous Wavelet Transform

Transformation of any function into its wavelet transform constitutes following simple steps:

1. take a section of the function and compare it to wavelet (mother wavelet)
2. calculate correlation coefficient
3. shift the wavelet to right
    * repeat the steps 1 and 2.
4. scale (stretch ) wavelet function
    * repeat steps 1 to 3.
5. Repeat the steps 1 to 4 for all scales.


Admissibility of Wavelets

To be classified a function as wavelet function some mathematical criteria must be satisfied and the criteria is admissibility condition. Two major admissibility conditions are:

1. Wavelet function should be of finite energy.
     $$ E= \int_{-\infty}^\infty |\psi(t)|^2 dt < \infty $$

2. Wavelet function should be of non zero frequency components that is zero mean function.
     Fourier transform of wavelet is given as:
    $$ \psi(f) = \int_{-\infty}^\infty \psi(t) e^{-2 \pi f t}dt $$
   then admissibility constant Cg is given as:

   $$ Cg = \int_0 ^\infty \frac{|\psi(f)|^2}f df < \infty $$
   This implies that wavelet has no zero frequency component ie. zero frequency component should not be exist.

Why transform is Wavelet ?

In Fourier Transform, signal is decomposed as the constituent of sine and cosine functions losing time information about the signal signal that is FT can not give the time of occurrence for the particular frequency sine or cosine signal.

Where as in Wavelet Transform, in Continuous Wavelet Transform analysis function is correlated to small segment of signal to be analysed translating wavelet function to calculated wavelet coefficient and again wavelet is dilated and then again correlated to segments of signal to be analysed translating the dilated wavelet along the time axis. This seems that small segment is formed which is wavelet of the signal to be analysed.

In Discrete Wavelet Transform, signal is passed through bank of analysis filters which separates the signal of particular frequency band signal that is signal of particular band of frequency is separated which is small wavelet of the signal to be analysed. Thus this kind of analysis is sub band coding or multi resolutional analysis.

De-Noising a Signal with WT

Wavelet Transform is also one of the most useful tool to remove noise from the communication signal. Significant de-noising can be obtained with higher level of approximation coefficients in wavelets.

Fig. 1

In the above fig first plot is original signal with noise when sample area is 3000 and the signal is decomposed with wavelet transform and then reconstructed. As we go downwards smoother signal is reconstructed. Second Last plot is reconstructed plot with approximation lavel-5.

In addition to that, wavelet transform can be used for compression techniques which can give good compression with lossy or lossless type compression. 

Discrete Wavelet analysis with an example

MRA of a signal is equivalently decomposing signal into average information and detail informations of the signal to be analysed. Discrete signal is decomposed into approximations and details passing through analysis bank of filters.

A random lececcum signal is analysed and its analysing signals are obtained as shown in the fig. below

fig.1

In the above plot, first plot is the original signal to be analysed which is analysed in three stage decomposition. plots 2,3 and 4 are details of the signal and plot 5th is approximation of the signal. The analysis is done with matlab routines available for one dimensional analysis for one dimensional signal. 

If informations in plots 2,3,4 and 5 are passed through reconstruction filters that is synthesis bank of filters then original signal plot in 1st can be obtained.

fig. 2

Decomposition stages are as shown above in fig.2. A3 is approximation and D1, D2, D3 are details of the signal decomposed and corresponding plots are as shown in Fig 1. Each signal after decomposition is decimated by 2 to keep bit length same applying Nyquist criteria.

Similarly 5 stage decomposition with 5 details are as follow in fig 3:

fig. 3

Multi Resolutional Analysis (MRA)

In STFT, time resolution and frequency resolution is same for particular analysis as windowing with is constant for a particular analysis, but in case of wavelet transform, scaling function changes the frequency resolution and window width correspondingly.

Continuous Wavelet Transform is give as:
$$ S(a, b)= \int f(t) \psi{(\frac{t-a}b}) dt $$

$$ \text{ changing value of b in } \psi{(\frac{t-a}b}) \text{ gives the different frequency band which means signal analysis in instant} $$ $$ \text{ of a with }\frac{1}{ b} \text{ frequency band, that is multiresolutional analysis. } $$

Similarly for digital signal, multi-resolutional analysis is performed by passing signals through analysis filters, where filters composed of high pass filters, band pass filters and low pass filters.

Out of them, output of high pass filters are to be considered as details of the signals and out put of low pass filter is to be considered as approximation. Above structural diagram briefly describe the MRA of discrete signals. d0, d1, d2.... represent details and A0 represents approximation.

Wavelets for Department of Hydrology and Meteorology

Department of Hydrology and Meteorology, Nepal has inaccurate weather prediction. That is because of inaccurate analysis of weather data collected. If the data temperature, humidity etc is analysed accurately using proper analysis tool, DHM Nepal would have more accurate prediction.

Wavelet analysis is emerging tool for the weather data analysis as weather data is non-stationary. Being weather data aperiodic and some kinds of shift in change in weather situations, Wavelet will be best tool for the analysis which can give information about breakdown points, discontinuities etc. which can give the shift in weather situation as well with number of changing weather occurrence. 

Video streaming and Coding

Visual Psychology
There are two type of visual psychologies, high number of sequencial image frames per second in streaming video and visual range of radio frequencies. Optimum numbers of image frames at least 25 frames per second is acceptable for video streaming. These sampled image frames are processed for transmission and storing. For better compression, sequencial frames are differentiated and only difference signal can be processed.

This optimum sampling of video streaming reduces the data rate per second for transmission of video stream.

Coding
Image frames can be coded with lower number of data bits per pixel. Sequencially differentiated image frames can be coded with lower difference with lower number of data bits per pixel and higher difference with higher data bits assuming lower difference with low information content and higher difference with higher information content.

Further coded data bits could be compressed using discrete wavelet transform, where wavelet function is chosen in such a way higher number of transformation coefficients equals to zero or tends to zero. Thus better compression can be achieved.

Telemedicine with Wavelet Synthesis

Biomedical signal like ECG, MRI etc are with important information content signals. If these biomedical signals are to be analysed and transmitted, careful transformation is required and for transmission, there should be no information loss during the transmission and if the signal is supposed to be transmitted for remote area(telemedicine), biomedical signals should be transmitted through low bandwidth channels considering bandwidth cost without information loss.

For this purpose, wavelet could be better transformation for analysis and compression method for signal data transmission through low bandwidth channels. Being better lossless data compression using wavelet transformation, data can be transmitted through low bandwidth channel and being better analysis for real signal using wavelets, biomedical signals can be better analysed. 

Signal and Bandwidth

In terms of channel bandwidth cost, Communication cost is controlled by government's rules and regulation. But a researcher can make an effort to reduce the effective communication cost by reducing signal bandwidth. If signal bandwidth is reduced by any algorithm then more numbers of communication signals can be transmitted through same channel with rated bandwidth cost. Lager the number of communication signal transmitted bandwidth cost will be distributed over all communication signal carrier, effectively reducing the communication cost.

Signal bandwidth can be reduced either by compressing data signal or by effective channel coding reducing the number of data bit with information signal. Thus to reduce the signal bandwidth, a engineer can work either in the area of source coding or in the area of channel coding.

Fourier Transform

Fourier Transform is well developed transformation theory with lots of approximation. In Fourier Transform, signals in time domain is transformed to frequency domain which can give frequency information about signal occurred without time information that is Fourier Transformation can not give time range or time of occurrence of particular signal or band of signals.

To get the time information, STFT is introduced with approximation in signal resolution because of width of windowing function. Because of windowing function some of important information in the signal may be lost if the information content frequency signal is out of integration range of window function.If the windowing function has small width it will have good time resolution but poor frequency resolution and have poor time resolution and good frequency resolution at large windowing function width as stated in Heisenberg Principle.
To overcome these resolution problems of varying windowing function, wavelet transform is introduced which maps the time signal into frequency-time domain. Because of changing window size of analysis function, Wavelet Transform gives good frequency resolution at lower frequency range and good time resolution at high frequency range.

Discrete Cosine Transform (DCT)

In Fourier Transform basis function is exponential function where signal to be transformed is convoluted with exponential function.

Discrete Cosine Transform is another transform where basis function is cosine function. Energy compaction property of DCT is most useful property because of which data compression is possible with DCT. Because of single analysis function, compression ratio can be increased only with setting transformation coefficient to zero with loss of information. But there are infinite number of analysis in WT so optimum selection of wavelet function maximize the compression ratio with optimum information loss.

Wavelet Transform

Wavelet is the newly emerging transformation theory, where you transform real signals into frequency and time domain at the same time so that the time range of occurrence for some band signals could be found. In this transformation, translated and scaled wavelet function is convoluted to the signal to be analysed.

Wavelet transform of impulse function at t=500 with haar wavelet as basis function.

Wavelet technology could be best available compression technologies with complete reconstruction of original signal with no information loss. Being always lossy compression with DCT compression technology, wavelet transform can be used for lossless compression. In any compression technologies, larger the number of transformation coefficients near to zero higher the compression ration is. Lager the number of coefficient made zero which are near to zero, higher the information is lost and the compression is lossy.