More Critically-Sampled Discrete Wavelet Transform Calculating wavelet coefficients at every possible scale is a fair amount of work, and it generates an awful lot of data. What if we choose only a subset of scales and positions at which to make our calculations? It turns out, rather remarkably, that if we choose scales and positions based on powers of two — so-called dyadic scales and positions — then our analysis will be much more efficient and just as accurate. We obtain such an analysis from the discrete wavelet transform DWT. An efficient way to implement this scheme using filters was developed in by Mallat see [Mal89] in References. The Mallat algorithm is in fact a classical scheme known in the signal processing community as a two-channel subband coder see page 1 of the book Wavelets and Filter Banks, by Strang and Nguyen [StrN96].
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This process is called decomposition or analysis. The other half of the story is how those components can be assembled back into the original signal without loss of information. This process is called reconstruction, or synthesis. The mathematical manipulation that effects synthesis is called the inverse discrete wavelet transform IDWT.
Where wavelet analysis involves filtering and downsampling, the wavelet reconstruction process consists of upsampling and filtering. Upsampling is the process of lengthening a signal component by inserting zeros between samples. The toolbox includes commands, like idwt and waverec , that perform single-level or multilevel reconstruction, respectively, on the components of 1-D signals. These commands have their 2-D and 3-D analogs, idwt2 , waverec2 , idwt3 , and waverec3.
Reconstruction Filters The filtering part of the reconstruction process also bears some discussion, because it is the choice of filters that is crucial in achieving perfect reconstruction of the original signal.
The downsampling of the signal components performed during the decomposition phase introduces a distortion called aliasing. A technical discussion of how to design these filters is available on page of the book Wavelets and Filter Banks, by Strang and Nguyen.
It is also possible to reconstruct the approximations and details themselves from their coefficient vectors. We pass the coefficient vector cA1 through the same process we used to reconstruct the original signal. However, instead of combining it with the level-one detail cD1, we feed in a vector of zeros in place of the detail coefficients vector: The process yields a reconstructed approximation A1, which has the same length as the original signal S and which is a real approximation of it.
Similarly, we can reconstruct the first-level detail D1, using the analogous process: The reconstructed details and approximations are true constituents of the original signal. Note that the coefficient vectors cA1 and cD1 — because they were produced by downsampling and are only half the length of the original signal — cannot directly be combined to reproduce the signal.
It is necessary to reconstruct the approximations and details before combining them. Extending this technique to the components of a multilevel analysis, we find that similar relationships hold for all the reconstructed signal constituents. That is, there are several ways to reassemble the original signal: Wavelets From Conjugate Mirror Filters In the section Reconstruction Filters , we spoke of the importance of choosing the right filters.
In fact, the choice of filters not only determines whether perfect reconstruction is possible, it also determines the shape of the wavelet we use to perform the analysis. To construct a wavelet of some practical utility, you seldom start by drawing a waveform. Instead, it usually makes more sense to design the appropriate quadrature mirror filters, and then use them to create the waveform. The filter coefficients can be obtained from the dbaux function.
By reversing the order of the scaling filter vector and multiplying every even element indexing from 1 by -1 , you obtain the high-pass filter. This relationship has profound implications. It means that you cannot choose just any shape, call it a wavelet, and perform an analysis. You are compelled to choose a shape determined by quadrature mirror decomposition filters.
There is an additional function associated with some, but not all, wavelets. The scaling function is very similar to the wavelet function. It is determined by the low-pass quadrature mirror filters, and thus is associated with the approximations of the wavelet decomposition. In the same way that iteratively upsampling and convolving the high-pass filter produces a shape approximating the wavelet function, iteratively upsampling and convolving the low-pass filter produces a shape approximating the scaling function.
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Wavelets and Filter Banks by T. Nguyen, Truong Nguyen and Gilbert Strang (1996, Hardcover)
Wavelets and Filter Banks