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An alternative to using the basic two-band tree-structured filter bank is a lattice-structured filter bank. Because of the relationship betweenthe scaling filter $h\left(n\right)$ and the wavelet filter ${h}_{1}\left(n\right)$ given in [link] , some of the calculation can be done together with a significant savings in arithmetic. This is developed in Chapter: Calculation of the Discrete Wavelet Transform [link] .
Still another approach to the calculation of discrete wavelet transforms and to the calculations of the scaling functions and wavelets themselvesis called “lifting." [link] , [link] Although it is related to several other schemes [link] , [link] , [link] , [link] , this idea was first explained by Wim Sweldens as a time-domainconstruction based on interpolation [link] . Lifting does not use Fourier methods and can be applied to more general problems(e.g., nonuniform sampling) than the approach in this chapter. It was first applied to the biorthogonal system [link] and then extended to orthogonal systems [link] . The application of lifting to biorthogonal is introduced in Section: Biorthogonal Wavelet Systems later in this book. Implementations based on lifting also achievethe same improvement in arithmetic efficiency as the lattice structure do.
The development of wavelet decomposition and the DWT has thus far been in terms of multiresolution where the higher scale wavelet components areconsidered the “detail" on a lower scale signal or image. This is indeed a powerful point of view and an accurate model for many signals and images,but there are other cases where the components of a composite signal at different scales and/or time are independent or, at least, not details ofeach other. If you think of a musical score as a wavelet decomposition, the higher frequency notes are not details on a lower frequency note; theyare independent notes. This second point of view is more one of the time-frequency or time-scale analysis methods [link] , [link] , [link] , [link] , [link] and may be better developed with wavelet packets (see Section: Wavelet Packets ), M-band wavelets (see Section: Multiplicity-M (M-band) Scaling Functions and Wavelets ), or a redundant representation (see Section: Overcomplete Representations, Frames, Redundant Transforms, and Adaptive Bases ), but would still be implemented by some sort of filter bank.
Unlike the Fourier series, the DWT can be formulated as a periodic or a nonperiodic transform. Up until now, we have considered a nonperiodicseries expansion [link] over $-\infty <t<\infty $ with the calculations made by the filter banks being an on-going string ofcoefficients at each of the scales. If the input to the filter bank has a certain rate, the output at the next lower scale will be twosequences, one of scaling function coefficients ${c}_{j-1,k-1}$ and one of wavelet coefficients ${d}_{j-1,k-1}$ , each, after down-sampling, being at half the rate of the input. At the next lower scale, the sameprocess is done on the scaling coefficients to give a total output of three strings, one at half rate and two at quarter rate. In otherwords, the calculation of the wavelet transform coefficients is a multirate filter bank producing sequences of coefficients at differentrates but with the average number at any stage being the same. This approach can be applied to any signal, finite or infinite in length,periodic or nonperiodic. Note that while the average output rate is the same as the average input rate, the number of output coefficientsis greater than the number of input coefficients because the length of the output of convolution is greater than the length of the input.
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