Wavelet transform

is called an orthonormal wavelet if it can be used to define a Hilbert basis, that is, a complete orthonormal system for the Hilbert space of square-integrable functions on the real line.

Changes in the time extension are expected to conform to the corresponding analysis frequency of the basis function.

The transformed signal provides information about the time and the frequency.

Therefore, wavelet-transformation contains information similar to the short-time-Fourier-transformation, but with additional special properties of the wavelets, which show up at the resolution in time at higher analysis frequencies of the basis function.

This consequence of the Fourier uncertainty principle is not correctly displayed in the Figure.

The goal is to store image data in as little space as possible in a file.

[5] Using a wavelet transform, the wavelet compression methods are adequate for representing transients, such as percussion sounds in audio, or high-frequency components in two-dimensional images, for example an image of stars on a night sky.

This means that the transient elements of a data signal can be represented by a smaller amount of information than would be the case if some other transform, such as the more widespread discrete cosine transform, had been used.

Discrete wavelet transform has been successfully applied for the compression of electrocardiograph (ECG) signals[6] In this work, the high correlation between the corresponding wavelet coefficients of signals of successive cardiac cycles is utilized employing linear prediction.

But smooth, periodic signals are better compressed using other methods, particularly traditional harmonic analysis in the frequency domain with Fourier-related transforms.

Compressing data that has both transient and periodic characteristics may be done with hybrid techniques that use wavelets along with traditional harmonic analysis.

For example, the Vorbis audio codec primarily uses the modified discrete cosine transform to compress audio (which is generally smooth and periodic), however allows the addition of a hybrid wavelet filter bank for improved reproduction of transients.

This produces as many coefficients as there are pixels in the image (i.e., there is no compression yet since it is only a transform).

[8][9] For most natural images, the spectrum density of lower frequency is higher.

The compression and reconstruction system generally involves low frequency components, which is the analysis filters

; The optimal wavelet are those who bring minimum shift variance and sidelobe to

To achieve this, the wavelet filters should have a large peak to sidelobe ratio.

So far we have discussed about one-dimension transformation of the image compression system.

This issue can be extended to two dimension, while a more general term - shiftable multiscale transforms - is proposed.

[12] As mentioned earlier, impulse response can be used to evaluate the image compression/reconstruction system.

After L levels of decomposition (and decimation), the analysis response is obtained by retaining one out of every

Similarly, the procedure is iterated to obtain the reference signal

To obtain the overall L level analysis/synthesis system, the analysis and synthesis responses are combined as below:

Wavelets have some slight benefits over Fourier transforms in reducing computations when examining specific frequencies.

However, they are rarely more sensitive, and indeed, the common Morlet wavelet is mathematically identical to a short-time Fourier transform using a Gaussian window function.

[13] The exception is when searching for signals of a known, non-sinusoidal shape (e.g., heartbeats); in that case, using matched wavelets can outperform standard STFT/Morlet analyses.

For instance, signal processing of accelerations for gait analysis,[15] for fault detection,[16] for the analysis of seasonal displacements of landslides,[17] for design of low power pacemakers and also in ultra-wideband (UWB) wireless communications.

represents time shift factor and as already mentioned in this context, the wavelet-transformation corresponds to a convolution of a function

With this the following approach of implementation results into: For processing temporal signals in real time, it is essential that the wavelet filters do not access signal values from the future as well as that minimal temporal latencies can be obtained.

Time-causal wavelets representations have been developed by Szu et al[23] and Lindeberg,[24] with the latter method also involving a memory-efficient time-recursive implementation.