Quasi-tight framelets with high vanishing moments derived from arbitrary refinable functions

Abstract

Construction of multivariate tight framelets is known to be a challenging problem because it is linked to the difficult problem on sum of squares of multivariate polynomials in real algebraic geometry. Multivariate dual framelets with vanishing moments generalize tight framelets and are not easy to be constructed either, since their construction is related to syzygy modules and factorization of multivariate polynomials. On the other hand, 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 ϕ∈L2(Rd) be an arbitrary compactly supported real-valued M-refinable function with a general dilation matrix M and ϕˆ(0)=1 such that its underlying real-valued 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 ϕ a directional compactly supported real-valued quasi-tight M-framelet in L2(Rd) associated with a directional quasi-tight M-framelet filter bank, each of whose high-pass filters has one vanishing moment and only two nonzero coefficients. If in addition all the coefficients of its low-pass filter are nonnegative, then such a quasi-tight M-framelet becomes a directional tight M-framelet in L2(Rd). Furthermore, we show by a constructive algorithm that we can always derive from the arbitrary M-refinable function ϕ a compactly supported quasi-tight M-framelet in L2(Rd) 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.