Wavelet transform

In mathematics, a wavelet series is a representation of a square-integrable function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform.

Definition

A function 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.
The Hilbert basis is constructed as the family of functions by means of dyadic translations and dilations of,
for integers.
If, under the standard inner product on,
this family is orthonormal, then it is an orthonormal system:
where is the Kronecker delta.
Completeness is satisfied if every function may be expanded in the basis as
with convergence of the series understood to be convergence in norm. Such a representation of is known as a wavelet series. This implies that an orthonormal wavelet is self-dual.
The integral wavelet transform is the integral transform defined as
The wavelet coefficients are then given by
Here, is called the binary dilation or dyadic dilation, and is the binary or dyadic position.

Principle

The fundamental idea of wavelet transforms is that the transformation should allow only changes in time extension, but not shape, imposing a restriction on choosing suitable basis functions. Changes in the time extension are expected to conform to the corresponding analysis frequency of the basis function. Based on the uncertainty principle of signal processing,
where represents time and angular frequency.
The higher the required resolution in time, the lower the resolution in frequency has to be. The larger the extension of the analysis windows is chosen, the larger is the value of.
When is large

Bad time resolution
Good frequency resolution
Low frequency, large scaling factor

When is small

Good time resolution
Bad frequency resolution
High frequency, small scaling factor

In other words, the basis function can be regarded as an impulse response of a system with which the function has been filtered. 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. The difference in time resolution at ascending frequencies for the Fourier transform and the wavelet transform is shown below. Note however, that the frequency resolution is decreasing for increasing frequencies while the temporal resolution increases. This consequence of the Fourier uncertainty principle is not correctly displayed in the Figure.
This shows that wavelet transformation is good in time resolution of high frequencies, while for slowly varying functions, the frequency resolution is remarkable.
Another example: The analysis of three superposed sinusoidal signals with STFT and wavelet-transformation.

Wavelet compression

Wavelet compression is a form of data compression well suited for image compression. Notable implementations are JPEG 2000, DjVu and ECW for still images, JPEG XS, CineForm, and the BBC's Dirac. The goal is to store image data in as little space as possible in a file. Wavelet compression can be either lossless or lossy.

Method

First a wavelet transform is applied. This produces as many coefficients as there are pixels in the image. These coefficients can then be compressed more easily because the information is statistically concentrated in just a few coefficients. This principle is called transform coding. After that, the coefficients are quantized and the quantized values are entropy encoded and/or run length encoded.
A few 1D and 2D applications of wavelet compression use a technique called "wavelet footprints".

Evaluation

Requirement for image compression

For most natural images, the spectrum density of lower frequency is higher. As a result, information of the low frequency signal is generally preserved, while the information in the detail signal is discarded. From the perspective of image compression and reconstruction, a wavelet should meet the following criteria while performing image compression:

Being able to transform more original image into the reference signal.
Highest fidelity reconstruction based on the reference signal.
Should not lead to artifacts in the image reconstructed from the reference signal alone.
Requirement for shift variance and ringing behavior

Wavelet image compression system involves filters and decimation, so it can be described as a linear shift-variant system. A typical wavelet transformation diagram is displayed below:
The transformation system contains two analysis filters, a decimation process, an interpolation process, and two synthesis filters. The compression and reconstruction system generally involves low frequency components, which is the analysis filters for image compression and the synthesis filters for reconstruction. To evaluate such system, we can input an impulse and observe its reconstruction ; The optimal wavelet are those who bring minimum shift variance and sidelobe to. Even though wavelet with strict shift variance is not realistic, it is possible to select wavelet with only slight shift variance. For example, we can compare the shift variance of two filters:

		Length	Filter coefficients	Regularity
Wavelet filter 1	H0	9	.852699,.377402, -.110624, -.023849,.037828	1.068
Wavelet filter 1	G0	7	.788486,.418092, -.040689, -.064539	1.701
Wavelet filter 2	H0	6	.788486,.047699, -.129078	0.701
Wavelet filter 2	G0	10	.615051,.133389, -.067237,.006989,.018914	2.068

By observing the impulse responses of the two filters, we can conclude that the second filter is less sensitive to the input location.
Another important issue for image compression and reconstruction is the system's oscillatory behavior, which might lead to severe undesired artifacts in the reconstructed image. 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.

Derivation of impulse response

As mentioned earlier, impulse response can be used to evaluate the image compression/reconstruction system.
For the input sequence, the reference signal after one level of decomposition is goes through decimation by a factor of two, while is a low pass filter. Similarly, the next reference signal is obtained by goes through decimation by a factor of two. After L levels of decomposition, the analysis response is obtained by retaining one out of every samples:.
On the other hand, to reconstruct the signal x, we can consider a reference signal. If the detail signals are equal to zero for, then the reference signal at the previous stage is, which is obtained by interpolating and convoluting with. Similarly, the procedure is iterated to obtain the reference signal at stage. After L iterations, the synthesis impulse response is calculated:, which relates the reference signal and the reconstructed signal.
To obtain the overall L level analysis/synthesis system, the analysis and synthesis responses are combined as below:
Finally, the peak to first sidelobe ratio and the average second sidelobe of the overall impulse response can be used to evaluate the wavelet image compression performance.
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.

Limitations

While wavelet transforms offer theoretical advantages, their practical limitations have effectively limited wavelet compression to analyzing localized changes and transient signals. Despite decades of research, wavelet-based compression systems for common multimedia like audio and video do not consistently match the efficiency and perceptual quality of current Discrete Cosine Transform-based systems.
For one-dimensional data like audio or ECGs, wavelets excel at representing and compressing transient signals—sudden, isolated events such as a drum hit in music or the sharp peaks in a heart rhythm. For example, the discrete wavelet transform has been successfully applied for the compression of electrocardiograph signals. However, for smooth, periodic signals, which make up much of typical audio, harmonic analysis in the frequency domain with Fourier-related transforms achieve better compression and sound quality. 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, however allows the addition of a hybrid wavelet filter bank for improved reproduction of transients.
For higher-dimensional data, wavelet compression faces significant challenges. In video, for instance, modern compression techniques such as intra coding and motion compensation and mixed and dynamic block sizes become incredibly complex with wavelets because of their overlapping nature. This complexity translates to more processing power and slower speed, making them less practical for widespread use. Furthermore, while wavelets might score well on traditional measures such as PSNR, DCT blocks create a perception of sharpness that wavelets often lack, requiring higher bitrates to achieve similar subjective quality.