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.

翻译：本文利用多项式环上二元线性丢番图方程的基本理论，为因果双通道FIR完美重构滤波器组及小波变换构建因果提升分解。该丢番图方法生成的因果分解满足特定的多项式降阶不等式，从而形成一种称为“因果补全算法”的新型提升分解策略。这为Daubechies和Sweldens使用洛朗多项式扩展欧几里得算法开发的非因果提升方案提供了一种因果性（即可实现）的替代方案。新方法将欧几里得算法替换为多项式除法的轻度推广，该推广确保满足用户指定可除性约束的余数所对应的商存在且唯一。通过生成超出多项式欧几里得算法可获得范围的额外因果提升分解，因果补全算法被证明比欧几里得算法方法的因果（多项式）版本更具一般性。