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Factoring wavelet transforms into lifting steps. (English) Zbl 0913.42027
Given a signal $$(x_k)_{k \in Z}$$, we may split it into even $$x_e = (x_{2k})$$ and odd $$x_o = (x_{2k+1})$$ components. A lifting step is defined to be a transformation of the form $$x \mapsto (x_o, x_e - P(x_o))$$, where $$P(x_o)$$ is a linear predictor for the even component based on the odd component. (For instance, one could take $$P(x_o)_{2k} = (x_{2k-1} + x_{2k+1})/2$$). A dual lifting step is a transformation of the form $$(x_o,d) \mapsto (x_o + U(d), d)$$, where $$U(d)$$ is an update operator to the odd components based upon the detail $$d = x_e - P(x_o)$$. (For instance, one can take $$U(d)_{2k} = (d_{2k-1} + d_{2k+1})/4)$$. In this paper the authors show that any finite impulse response (FIR) filter bank or wavelet transform can be factored into a sequence of lifting steps and dual lifting steps, each of which is also given by an FIR filter. The method of proof relies upon the z-transform and then a factorization of a Laurent polynomial-valued matrix into elementary matrices by means of the Euclidean algorithm.

##### MSC:
 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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