WAVELET TRANSFORMS

By B.V.Divya

FOURIER TRANSFORM

A periodic function is expressed as an infinite sum of periodic complex exponential functions. X(f)= x(t).e^-2 ft dt It is widely used frequency transformation technique. FT cannot distinguish between stationary and nonstationary signals which creates problem during reconstruction of signal.

Stationary signal

Non-Stationary signal

SHORT TIME FOURIER TRANSFORM

It uses a window function of a predefined width and moves it along the entire signal. The transform is a function of time and frequency (unlike, FT which is a function of frequency only), the transform would be two dimensional. The STFT is almost similar to FT, it considers a part of the non-stationary signal where the frequency is same as stationary and applies FT to it.

The equation of STFT is given as -j2Tft STFT (td, f) = [ x(t) y w( t- td )]y e dt

DISADVANTAGE OF STFTUnchanged window size. Narrow windows give good time resolution but poor frequency resolution. Wide windows give good frequency resolution but poor time resolution. It only gives the information about what spectral band exist in a time interval but does not give information about the individual frequencies.

WAVELET TRANSFORM

Wavelet transform is capable of providing the time and frequency information simultaneously, hence giving a time-frequency representation of the signal. Higher frequencies are better resolved in time, and lower frequencies are better resolved in frequency. So wavelet transforms gives a variable resolution

DIFFERENCE B/W STFT AND WAVELET TRANSFORM

The Fourier transforms of the windowed signals are not taken, and therefore single peak will be seen corresponding to a sinusoid, i.e., negative frequencies are not computed. The width of the window is changed as the transform is computed for every single spectral component, which is probably the most significant characteristic of the wavelet transform.

WAVELETS-THEORY

The term wavelet means a small wave. Wavelets are functions defined over a finite interval and having an average value of zero. The basic idea of the wavelet transform is to represent any arbitrary function f(t) as a superposition of a set of such wavelets or basis function. These basis functions are derived from a single prototype called mother wavelet.

CONTINUOUS WAVELET TRANSFORMThe continuous wavelet transform is defined as follows:

The transformed signal is a function of two variables, tau and s, the translation and scale parameter respectively. Psi(t) is the transforming function, and it is called the mother wavelet.

COMPUTATION OF THE CWTThe signal to be analyzed is taken. The mother wavelet is chosen and the computation is begun with s=1.The CWT is computed for all values of s. The wavelet is placed at the beginning of the signal at the point which corresponds to time=0. The wavelet is multiplied with the signal and multiplied over all the times. The result is then multiplied by the constant 1/sqrt(s). The above step normalizes the energy,so that the transformed signal has the same energy at every scale.

Cont..The wavelet at scale s =1 is then shifted to the right by and the above steps are repeated until the wavelet reaches the end of the signal.

APPLICATIONSDigital Image processing

Image and data compression Biometrics and Forensic services

Medical Imaging

Disease Diagnosis Wavelet denoising

Image Coding Multi resolution image display Geo-information exchange Invisible water marking schemes

CONCLUSION

Since the time and frequency resolutions can be achieved together,it has wide applications in image compression research. Computational complexity is also less compared to other transformation techniques.