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The elementary theory of bivariate linear Diophantine equations overpolynomial rings is used to construct causal lifting factorizations for causaltwo-channel FIR perfect reconstruction filter banks and wavelet transforms. TheDiophantine approach generates causal factorizations satisfying certainpolynomial degree-reducing inequalities, enabling a new lifting factorizationstrategy called the \emph{Causal Complementation Algorithm}. This provides acausal, hence realizable, alternative to the noncausal lifting scheme developedby Daubechies and Sweldens using the Extended Euclidean Algorithm for Laurentpolynomials. The new approach replaces the Euclidean Algorithm with a slightgeneralization of polynomial division that ensures existence and uniqueness ofquotients whose remainders satisfy user-specified divisibility constraints. TheCausal Complementation Algorithm is shown to be more general than the causal(polynomial) version of the Euclidean Algorithm approach by generatingadditional causal lifting factorizations beyond those obtainable using thepolynomial Euclidean Algorithm.

Factoring Perfect Reconstruction Filter Banks into Causal Lifting Matrices: A Diophantine Approach