Quasi-tight Framelets with Directionality or High Vanishing Moments Derived from Arbitrary Refinable Functions

Construction of multivariate tight framelets is known to be a challenging problem. Multivariate dual framelets with vanishing moments generalize tight framelets and are not easy to be constructed either. Compactly supported multivariate framelets with directionality or high vanishing moments are of interest and importance in both theory and applications. In this paper we introduce the notion of a quasi-tight framelet, which is a dual framelet, but behaves almost like a tight framelet. Let $\phi\in L_2(R^d)$ be an arbitrary compactly supported $M$-refinable function such that its underlying low-pass filter satisfies the basic sum rule. We first constructively prove by a step-by-step algorithm that we can always easily derive from the arbitrary $M$-refinable function $\phi$ a directional compactly supported quasi-tight $M$-framelet in $L_2(R^d)$ associated with a directional quasi-tight $M$-framelet filter bank, each of whose high-pass filters has only two nonzero coefficients with opposite signs. If in addition all the coefficients of its low-pass filter are nonnegative, such a quasi-tight $M$-framelet becomes a directional tight $M$-framelet in $L_2(R^d)$. Furthermore, we show by a constructive algorithm that we can always derive from the arbitrary $M$-refinable function $\phi$ a compactly supported quasi-tight $M$-framelet in $L_2(R^d)$ with the highest possible order of vanishing moments. We shall also present a result on quasi-tight framelets whose associated high-pass filters are purely differencing filters with the highest order of vanishing moments. Several examples will be provided to illustrate our main theoretical results and algorithms in this paper.