Symmetric Canonical Quincunx Tight Framelets with High Vanishing Moments and Smoothness

In this paper, we propose an approach to construct a family of two-dimensional compactly supported realvalued symmetric quincunx tight framelets {φ;ψ1, ψ2, ψ3} in L2(R) with arbitrarily high orders of vanishing moments. Such symmetric quincunx tight framelets are associated with quincunx tight framelet filter banks {a; b1, b2, b3} having increasing orders of vanishing moments and enjoying the additional double canonical properties: b1(k1, k2) = (−1)12a(1− k1,−k2), b3(k1, k2) = (−1)12b2(1− k1,−k2), ∀ k1, k2 ∈ Z. Moreover, the supports of all the high-pass filters b1, b2, b3 are no larger than that of the low-pass filter a. For a low-pass filter a which is not a quincunx orthonormal wavelet filter, we show that a quincunx tight framelet filter bank {a; b1, . . . , bL} with b1 taking the above canonical form must have L ≥ 3 high-pass filters. Thus, our family of symmetric double canonical quincunx tight framelets has the minimum number of generators. Numerical calculation indicates that this family of symmetric double canonical quincunx tight framelets can be arbitrarily smooth. Using one-dimensional filters having linear-phase moments, in this paper we also provide a second approach to construct multiple canonical quincunx tight framelets with symmetry. In particular, the second approach yields a family of 6-multiple canonical real-valued quincunx tight framelets in L2(R ) and a family of double canonical complex-valued quincunx tight framelets in L2(R ) such that both of them have symmetry and arbitrarily increasing orders of smoothness and vanishing moments. Several examples are provided to illustrate our general construction and theoretical results on canonical quincunx tight framelets in L2(R ) with symmetry, high vanishing moments, and smoothness. Symmetric quincunx tight framelets constructed by both approaches in this paper are of particular interest for their applications in computer graphics and image processing due to their polynomial preserving property, full symmetry, short support, and high smoothness and vanishing moments.