  The elementary theory of bivariate linear Diophantine equations over
polynomial rings is used to construct causal lifting factorizations for causal
two-channel FIR perfect reconstruction filter banks and wavelet transforms. The
Diophantine approach generates causal factorizations satisfying certain
polynomial degree-reducing inequalities, enabling a new lifting factorization
strategy called the Causal Complementation Algorithm. This provides a causal,
hence realizable, alternative to the noncausal lifting scheme developed by
Daubechies and Sweldens using the Extended Euclidean Algorithm for Laurent
polynomials. The new approach replaces the Euclidean Algorithm with a slight
generalization of polynomial division that ensures existence and uniqueness of
quotients whose remainders satisfy user-specified divisibility constraints. The
Causal Complementation Algorithm is shown to be more general than the causal
(polynomial) version of the Euclidean Algorithm approach by generating
additional causal lifting factorizations beyond those obtainable using the
polynomial Euclidean Algorithm.
