Continuous wavelet transform

In mathematics, the continuous wavelet transform (CWT) is a formal (i.e., non-numerical) tool that provides an overcomplete representation of a signal by letting the translation and scale parameter of the wavelets vary continuously.
The continuous wavelet transform of a function formula_1 at a scale (a>0) formula_2 and translational value formula_3 is expressed by the following integral
Traditionally, this constant is called wavelet admissible constant. A wavelet whose admissible constant satisfies 
is called an admissible wavelet. An admissible wavelet implies that formula_6, so that an admissible wavelet must integrate to zero. To recover the original signal formula_1, the second inverse continuous wavelet transform can be exploited.
This inverse transform suggests that a wavelet should be defined as
where formula_10 is a window. Such defined wavelet can be called as an analyzing wavelet, because it admits to time-frequency analysis. An analyzing wavelet is unnecessary to be admissible.
Scale factor.
The scale factor formula_11 either dilates or compresses a signal. When the scale factor is relatively low, the signal is more contracted which in turn results in a more detailed resulting graph. However, the drawback is that low scale factor does not last for the entire duration of the signal. On the other hand, when the scale factor is high, the signal is stretched out which means that the resulting graph will be presented in less detail. Nevertheless, it usually lasts the entire duration of the signal.
Continuous wavelet transform properties.
In definition, the continuous wavelet transform is a convolution of the input data sequence with a set of functions generated by the mother wavelet. The convolution can be computed by using a fast Fourier transform (FFT) algorithm. Normally, the output formula_12 is a real valued function except when the mother wavelet is complex. A complex mother wavelet will convert the continuous wavelet transform to a complex valued function. The power spectrum of the continuous wavelet transform can be represented by formula_13 .
Applications of the wavelet transform.
One of the most popular applications of wavelet transform is image compression. The advantage of using wavelet-based coding in image compression is that it provides significant improvements in picture quality at higher compression ratios over conventional techniques. Since wavelet transform has the ability to decompose complex information and patterns into elementary forms, it is commonly used in acoustics processing and pattern recognition, but it has been also proposed as an instantaneous frequency estimator. Moreover, wavelet transforms can be applied to the following scientific research areas: edge and corner detection, partial differential equation solving, transient detection, filter design, electrocardiogram (ECG) analysis, texture analysis, business information analysis and gait analysis. Wavelet transforms can also be used in Electroencephalography (EEG) data analysis to identify epileptic spikes resulting from epilepsy. Wavelet transform has been also successfully used for the interpretation of time series of landslides. 
Continuous Wavelet Transform (CWT) is very efficient in determining the damping ratio of oscillating signals (e.g. identification of damping in dynamic systems). CWT is also very resistant to the noise in the signal.