Symmetric tight framelet filter banks with three high-pass filters

In this paper we continue our investigation of symmetric tight framelet filter banks (STFFBs) with a minimum number of generators in [6]. In particular, we shall systematically study STFFBs with three high-pass filters which are derived from the oblique extension principle. To our best knowledge, except the papers [1] and [10], there are no other papers in the literature so far systematically studying this problem. In this paper we show that there are two different types, called type I and type II, of STFFBs with three high-pass filters. Then we provide a detailed analysis and a complete algorithm to obtain all type I STFFBs with three high-pass filters. Our results not only significantly generalize the results in [1] and [10], but also help us answer several unresolved problems on STFFBs. Based on [6], we also propose an algorithm to construct all type II STFFBs with three high-pass filters and with the shortest possible filter supports. Several examples are given to illustrate the results and algorithms in this paper.