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Erschienen in:
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2016 | OriginalPaper | Buchkapitel

14. Discrete Wavelet Transform (DWT)

verfasst von : Silvia Maria Alessio

Erschienen in: Digital Signal Processing and Spectral Analysis for Scientists

Verlag: Springer International Publishing

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Abstract

The wavelet transform can be seen as a wavelet-based expansion (decomposition) of a finite-energy signal. In the discrete wavelet transform (DWT), economy in the representation of the signal and possibility of perfect signal reconstruction (PR) are crucial. The simplest formulation of the DWT problem includes two types of basis functions for the expansion: the scaling and wavelet functions. We will see how an ideal, infinite-length but finite-energy signal can be decomposed from the point of view function spaces, and how this decomposition can be obtained using a two-channel digital filter bank. The description of a fast wavelet decomposition/reconstruction algorithm will lead us to the practical implementation of the DWT in the real-life case of a finite-length, sampled input signal, as well as to the properties of PR filters, which are strictly related to the scaling and wavelet functions. After allowing for the necessary conditions that the filters of the bank must satisfy, primarily biorthogonality or orthogonality, a number of degrees of freedom remain available to design different wavelet systems suited for different purposes. A real-world example of signal DWT decomposition will be provided. The chapter ends with an appendix in which the various wavelet systems used for the CWT and/or DWT are reviewed.

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Vorheriges Kapitel Non-stationary Spectral Analysis
Nächstes Kapitel De-noising and Compression by Wavelets
Fußnoten
1
Note that \(\mathrm {L}^1(\mathbb {R})\) is more restrictive than \(\mathrm {L}^2(\mathbb {R})\), because an absolutely integrable function is also square-integrable, but the converse is not necessarily true.
 
2
To be precise, we must remark that while the continuous wavelets involved in the CWT are subject to the uncertainty principle of Fourier analysis, the discrete wavelet systems involved in the DWT, often defined through the associated digital filters, do not. However, the DWT does share the MRA property of all forms of wavelet transform: discrete wavelet bases can be shown to possess the MRA property by introducing the concept of nested spanned function spaces (Sect. 14.3).
 
3
There is an interesting correspondence between wavelet notation and musical notation. In a musical score, each note specifies a frequency and a position in time by its vertical and horizontal placements, respectively, in a way that closely resembles a wavelet signal representation, except that it has fractional jumps in frequency. An article written in 1994 by G. Strang for a nontechnical audience (American Scientist, vol. 82, pp. 250–255) clearly explains this similarity.
 
4
The signals we are considering are real, and in the DWT they are normally expanded using real functions. For this reason, from now on we will use the notation which is appropriate for real functions, thus writing the inner product without any conjugation sign.
 
5
The prototype (unscaled) wavelet function is what we called the “mother wavelet” in the previous chapter. The prototype scaling function is sometimes referred to as the “father wavelet”. Note that in order to simplify the notation, here we dropped the subscript adopted in the previous chapter to indicate the mother wavelet: we wrote \(\psi (\theta )\) instead of \(\psi _0(\theta )\).
 
6
For \(j<0\), the contrary would be true: only coarser features could be represented, and \(V_j\) would be narrower than \(V_0\).
 
7
Note that in this equation we should actually write \(\eta =\theta /a\) in place of \(\theta \), since the larger scale is unspecified. However, the one presented here is the standard way in which the relation is written in literature.
 
8
Note that the choice \(j_0=0\) is arbitrary, and we might as well decide to chose a smaller starting scale—a larger degree of detail, e.g. \(j_0=10\), \(a=2^{-10}\)—and write
$$\begin{aligned} \mathrm {L}^2(\mathbb {R})= V_{10} \oplus W_{10} \oplus W_{11} \oplus W_{12} \ldots , \end{aligned}$$
or we might prefer a larger starting scale—a smaller degree of detail, e.g. \(j_0=-5\), \(a=2^{5}\)—and write
$$\begin{aligned} \mathrm {L}^2(\mathbb {R})= V_{-5} \oplus W_{-5} \oplus W_{-4} \oplus W_{-3} \ldots . \end{aligned}$$
We could even start from \(j_0=-\infty \), i.e., from an infinitely large scale: since \(V_{-\infty } ={0}\), we would then write
$$\begin{aligned} \mathrm {L}^2(\mathbb {R})= \cdots \oplus W_{-2} \oplus W_{-1} \oplus W_0 \oplus W_{1} \oplus W_{2} \oplus \cdots . \end{aligned}$$
In this way we would eliminate the scaling function and would get an expansion of the signal on the basis of wavelets solely. In the following discussion we will set \(j_0=0\), unless explicitly stated otherwise.
 
9
This statement is not in contradiction with what we said in the previous chapter about the frequency response of the filters associated with complex analytic wavelets. Those wavelets are used for CWT only and do not have a corresponding scaling function. They actually act as passband filters.
 
10
A B-spline is a piecewise polynomial function in one independent variable, exhibiting knots or break-points. The number of internal knots is equal to the degree of the polynomial if there are no knot multiplicities. A B-spline is a continuous function at the knots. For any given set of knots, the B-spline is unique, hence the name, B being short for Basis.
 
11
Returning for a moment to Fig. 14.10, at this point we may wonder if we would be in a position to proceed further to a more accurate level-8 approximation in \(V_{-1}\). The answer is no; in the example of Fig. 14.10, the signal length is \(N=120\), and after repeated downsampling operations, the coarsest-resolution approximation and detail coefficients (\(c_7[k],\,d_7[k]\)) are made by just one sample each. The decomposition was pushed up to the maximum possible level. The remaining detail coefficients up to \(d_1\) are progressively longer, since they represent less downsampled signals. Now, suppose we had \(N=240\) samples. We would then be able to attain \(J=8\), and (\(c_7[k],\,d_7[k]\)) would have one sample each. However, in Fig. 14.10 we would see level 1 in \(V_7\), since \(J-j=8-1=7\), and so on, up to level 8 in \(V_0\): we would always label the last subspace as \(V_0\).
 
12
Note that the coefficient vectors \(c_1[k]\) and \(d_1[k]\) cannot be directly combined to reproduce the signal. The coefficients are produced by downsampling and are only half the length of the original signal. It is necessary to reconstruct the approximations and details before combining them.
 
13
The word “trous” means “holes” in English.
 
14
Note that that here we are presenting a definition of regularity of the scaling filter, not of the scaling function or of the wavelet.
 
15
Daubechies (1992) defined two classes of wavelets, via criteria that select a particular scaling filter. One criterion leads to “extremal phase” (minimum phase) Daubechies wavelets, i.e., the ones illustrated here. Another criterion leads to “least asymmetric” Daubechies wavelets, also called “symlets”.
 
16
Note that in the previous chapter we gave a different and less general definition of the complex Morlet wavelet,
$$\begin{aligned} \psi _0(\theta ) = \frac{1}{^4\sqrt{\pi }} {\mathrm {e}}^{{\mathrm {j}}\omega _0 \theta } {\mathrm {e}}^{-\frac{\theta ^2}{2}}, \end{aligned}$$
that corresponds to fixed values of \(T_p\) and \(f_c\). More precisely, it corresponds to \(T_p=2\) and \(f_c=2 \pi /\omega _0\). In the standard expression of the Gaussian probability density function, we would have \(T_p=2\sigma ^2\), where \(\sigma \) is the standard deviation of the Gaussian distribution. Thus, \(T_p=2\) means \(\sigma =1\). Note, however, that the constant factor \(1/\sqrt{\pi T_p}\), that for \(T_p=2\) becomes \(1/\sqrt{2\pi }\), does not coincide with the factor \(1/{^4\sqrt{\pi }}\).
 
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Metadaten
Titel
Discrete Wavelet Transform (DWT)
verfasst von
Silvia Maria Alessio
Copyright-Jahr
2016
Buch
Digital Signal Processing and Spectral Analysis for Scientists

Print ISBN: 978-3-319-25466-1

Electronic ISBN: 978-3-319-25468-5

Copyright-Jahr: 2016

DOI
https://doi.org/10.1007/978-3-319-25468-5_14

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