The Euclidean algorithm for computing the greatest common divisor (gcd) of 2 integers is one of the oldest numerical recipes. The algorithm can be naturally extended to construction of gcd for (Laurent) polynomials, which finds important applications in signal processing when using the wavelet transforms implemented as filter banks. Wavelet filter banks can be efficiently implemented by using the ooncept of"lifting", as is done in one of the most notorious application of wavelet transform, the new image compression standard JPEG 2000. Finding the lifting steps requires the application of Euclid's Algorithm to two Laurent polynomials appearing in the polyphase matrix of the filter bank. The factorization in lifting steps is not unique, but traditionally was implemented in a particular way, here referred to as the ''traditional lifting scheme". In this article we show how to obtain in a systematic way many different lifting factorizations, by using the Euclidean Algorithm.
Euclid's algorithm, wavelet filter bank, integer to integer lifting scheme.
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