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...nctions in a local neighborhood. We will 514 WIM SWELDENS show later how this can be seen as a special case of lifting. The lifting scheme can also be used to construct first generation wavelets; see =-=[105, 52]-=-. Although in this setting, the lifting will never come up with wavelets which could not have been found using the Cohen--Daubechies--Feauveau machinery in [29], it leads to two new insights: a custom...

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...rimal and dual lifting steps and a scaling. This theorem follows from a factorization well known to both electrical engineers (in linear system theory) as well as algebraists (in algebraic K-theory); =-=[9]-=- gives a self-contained proof as well as many examples coming from wavelet filter banks.. The proof is constructive; for a given wavelet transform, the filters used in the lifting steps can be found u...

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...s a general construction that is not limited to the Deslauriers-Dubuc family. Using the Euclidean algorithm, one can decompose any FIR wavelet transform into a sequence of prediction and update steps =-=[3]-=-. 3 Nonlinear Lifting The analysis lters used for wavelet compression applications typically correspond to fourth order polynomial predictors like the one described above. Such predictions work well i...

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... have been proposed that have the property of perfect reconstruction with minimum bit expansion. For example, a ladder network was introduced in [30]. More systematic results were summarized in [31], =-=[32]-=- with the name lifting scheme. The lifting structure enablessexible and fast biorthogonal transform, and it also allows lossless transform, making it a powerful building tool for wavelet transforms. I...

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...) works considerably better than (1) and (3), and this is the case we use when we refer to (2+2; 2) in Section 3. The next two transforms D4 and (9-7) [7] are built using the lifting factorization in =-=[8]-=-. To recover the usual normalization, a final scaling step would be needed (multiplying the lowpass coefficients by 1/K and highpass coefficients by K) for the D4 and (9-7), that can be implemented as...

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...ons, such models seem also adequate for describing transient signals. 3 An alternative algorithm, based on an initial polyphase decomposition and a factorization into a series of simple lifting steps =-=[27, 9]-=- may also be used, resulting in more efficient implementations. 4s2.3. Local trigonometric bases. It is well known that any square-integrable function on an interval may be represented by a Fourier se...

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...d scaled signal does not satisfy the constraint, we may choose in such a way that , given by (38), does satisfy this constraint. We refer to the work of Sweldens [23]–[25] and Daubechies and Sweldens =-=[38]-=- for more details. The update step in (38) gives rise to the diagrams depicted in Fig. 12. It is clear that the input signal can be reconstructed from and , since Thus, we arrive at the update lifting...