In mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) 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. [1] [2] [3] [4] [5]
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.
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, 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 f 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.
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 (, where is ordinary 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,
When is small
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 is a form of data compression well suited for image compression (sometimes also video compression and audio 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. [6]
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 [7] In this work, the high correlation between the corresponding wavelet coefficients of signals of successive cardiac cycles is utilized employing linear prediction.
Wavelet compression is not effective for all kinds of data. Wavelet compression handles transient signals well. 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. [8]
See Diary Of An x264 Developer: The problems with wavelets (2010) for discussion of practical issues of current methods using wavelets for video compression.
First a wavelet transform is applied. 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). 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". [9] [10]
For most natural images, the spectrum density of lower frequency is higher. [11] As a result, information of the low frequency signal (reference 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:
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 low pass filter and a high pass filter ), a decimation process, an interpolation process, and two synthesis filters ( and ). 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: [12]
Length | Filter coefficients | Regularity | ||
---|---|---|---|---|
Wavelet filter 1 | H0 | 9 | .852699, .377402, -.110624, -.023849, .037828 | 1.068 |
G0 | 7 | .788486, .418092, -.040689, -.064539 | 1.701 | |
Wavelet filter 2 | H0 | 6 | .788486, .047699, -.129078 | 0.701 |
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 (i.e. it is less shift variant).
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. [13]
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 (and decimation), the analysis response is obtained by retaining one out of every samples: .
On the other hand, to reconstruct the signal x(n), we can consider a reference signal . If the detail signals are equal to zero for , then the reference signal at the previous stage ( 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.
Transform | Representation | Input |
---|---|---|
Fourier transform | : frequency | |
Time–frequency analysis | time; frequency | |
Wavelet transform | scaling ; time shift factor |
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. [14] 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. [15]
The wavelet transform can provide us with the frequency of the signals and the time associated to those frequencies, making it very convenient for its application in numerous fields. For instance, signal processing of accelerations for gait analysis, [16] for fault detection, [17] for design of low power pacemakers and also in ultra-wideband (UWB) wireless communications. [18] [19] [20]
Leading to wavelets of the form, the discrete formula for the basis wavelet:
Such discrete wavelets can be used for the transformation:
where is scaling factor, represents time shift factor
and as already mentioned in this context, the wavelet-transformation corresponds to a convolution of a function and a wavelet-function. A convolution can be implemented as a multiplication in the frequency domain. 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.
Synchro-squeezed transform can significantly enhance temporal and frequency resolution of time-frequency representation obtained using conventional wavelet transform. [25] [26]
In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer.
The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known. In other words, the more accurately one property is measured, the less accurately the other property can be known.
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the number and direction of its pulses. Wavelets are imbued with specific properties that make them useful for signal processing.
In mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be represented in terms of an orthonormal basis. The Haar sequence is now recognised as the first known wavelet basis and is extensively used as a teaching example.
In physics, engineering and mathematics, the Fourier transform (FT) is an integral transform that converts a function into a form that describes the frequencies present in the original function. The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made the Fourier transform is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches.
In mathematics, the Morlet wavelet is a wavelet composed of a complex exponential (carrier) multiplied by a Gaussian window (envelope). This wavelet is closely related to human perception, both hearing and vision.
Fourier optics is the study of classical optics using Fourier transforms (FTs), in which the waveform being considered is regarded as made up of a combination, or superposition, of plane waves. It has some parallels to the Huygens–Fresnel principle, in which the wavefront is regarded as being made up of a combination of spherical wavefronts whose sum is the wavefront being studied. A key difference is that Fourier optics considers the plane waves to be natural modes of the propagation medium, as opposed to Huygens–Fresnel, where the spherical waves originate in the physical medium.
In mathematics, the continuous wavelet transform (CWT) is a formal tool that provides an overcomplete representation of a signal by letting the translation and scale parameter of the wavelets vary continuously.
In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal resolution: it captures both frequency and location information.
Stransform as a time–frequency distribution was developed in 1994 for analyzing geophysics data. In this way, the S transform is a generalization of the short-time Fourier transform (STFT), extending the continuous wavelet transform and overcoming some of its disadvantages. For one, modulation sinusoids are fixed with respect to the time axis; this localizes the scalable Gaussian window dilations and translations in S transform. Moreover, the S transform doesn't have a cross-term problem and yields a better signal clarity than Gabor transform. However, the S transform has its own disadvantages: the clarity is worse than Wigner distribution function and Cohen's class distribution function.
In mathematics, Fourier–Bessel series is a particular kind of generalized Fourier series based on Bessel functions.
Continuous wavelets of compact support alpha can be built, which are related to the beta distribution. The process is derived from probability distributions using blur derivative. These new wavelets have just one cycle, so they are termed unicycle wavelets. They can be viewed as a soft variety of Haar wavelets whose shape is fine-tuned by two parameters and . Closed-form expressions for beta wavelets and scale functions as well as their spectra are derived. Their importance is due to the Central Limit Theorem by Gnedenko and Kolmogorov applied for compactly supported signals.
In functional analysis, the Shannon wavelet is a decomposition that is defined by signal analysis by ideal bandpass filters. Shannon wavelet may be either of real or complex type.
Overcompleteness is a concept from linear algebra that is widely used in mathematics, computer science, engineering, and statistics. It was introduced by R. J. Duffin and A. C. Schaeffer in 1952.
Fractional wavelet transform (FRWT) is a generalization of the classical wavelet transform (WT). This transform is proposed in order to rectify the limitations of the WT and the fractional Fourier transform (FRFT). The FRWT inherits the advantages of multiresolution analysis of the WT and has the capability of signal representations in the fractional domain which is similar to the FRFT.
In applied mathematical analysis, shearlets are a multiscale framework which allows efficient encoding of anisotropic features in multivariate problem classes. Originally, shearlets were introduced in 2006 for the analysis and sparse approximation of functions . They are a natural extension of wavelets, to accommodate the fact that multivariate functions are typically governed by anisotropic features such as edges in images, since wavelets, as isotropic objects, are not capable of capturing such phenomena.
Gabor wavelets are wavelets invented by Dennis Gabor using complex functions constructed to serve as a basis for Fourier transforms in information theory applications. They are very similar to Morlet wavelets. They are also closely related to Gabor filters. The important property of the wavelet is that it minimizes the product of its standard deviations in the time and frequency domain. Put another way, the uncertainty in information carried by this wavelet is minimized. However they have the downside of being non-orthogonal, so efficient decomposition into the basis is difficult. Since their inception, various applications have appeared, from image processing to analyzing neurons in the human visual system.
In mathematics, in functional analysis, several different wavelets are known by the name Poisson wavelet. In one context, the term "Poisson wavelet" is used to denote a family of wavelets labeled by the set of positive integers, the members of which are associated with the Poisson probability distribution. These wavelets were first defined and studied by Karlene A. Kosanovich, Allan R. Moser and Michael J. Piovoso in 1995–96. In another context, the term refers to a certain wavelet which involves a form of the Poisson integral kernel. In still another context, the terminology is used to describe a family of complex wavelets indexed by positive integers which are connected with the derivatives of the Poisson integral kernel.
For discrete aperture antennas in which the element spacing is greater than a half wavelength, a spatial aliasing effect allows plane waves incident to the array from visible angles other than the desired direction to be coherently added, causing grating lobes. Grating lobes are undesirable and identical to the main lobe. The perceived difference seen in the grating lobes is because of the radiation pattern of non-isotropic antenna elements, which effects main and grating lobes differently. For isotropic antenna elements, the main and grating lobes are identical.
Multiresolution Fourier Transform is an integral fourier transform that represents a specific wavelet-like transform with a fully scalable modulated window, but not all possible translations.
Vorbis I is a forward-adaptive monolithic transform CODEC based on the Modified Discrete Cosine Transform. The codec is structured to allow addition of a hybrid wavelet filterbank in Vorbis II to offer better transient response and reproduction using a transform better suited to localized time events.