We propose an approach to construct a family of two-dimensional compactlysupported real-valued symmetric quincunx tight framelets $\{\phi;\psi_1,\psi_2,\psi_3\}$ in $L_2(R^2)$ with arbitrarily high orders of vanishingmoments. Such symmetric quincunx tight framelets are associated with quincunxtight framelet filter banks $\{a;b_1,b_2,b_3\}$ having increasing orders ofvanishing moments and enjoying the additional double canonical properties: \[b_1(k_1,k_2)=(-1)^{1+k_1+k_2} a(1-k_1,-k_2), b_3(k_1,k_2)=(-1)^{1+k_1+k_2}b_2(1-k_1,-k_2). \] For a low-pass filter $a$ which is not a quincunxorthonormal wavelet filter, we show that a quincunx tight framelet filter bank$\{a;b_1,\ldots,b_L\}$ with $b_1$ taking the above canonical form must have$L\ge 3$ high-pass filters. Thus, our family of symmetric double canonicalquincunx tight framelets has the minimum number of generators. Numericalcalculation indicates that this family of symmetric double canonical quincunxtight framelets can be arbitrarily smooth. Using one-dimensional filters havinglinear-phase moments, in this paper we also provide a second approach toconstruct multiple canonical quincunx tight framelets with symmetry. Inparticular, the second approach yields a family of $6$-multiple canonicalreal-valued quincunx tight framelets in $L_2(R^2)$ and a family of doublecanonical complex-valued quincunx tight framelets in $L_2(R^2)$ such that bothof them have symmetry and arbitrarily increasing orders of smoothness andvanishing moments. Several examples are provided to illustrate our generalconstruction and theoretical results on canonical quincunx tight framelets in$L_2(R^2)$ with symmetry, high vanishing moments, and smoothness. Symmetricquincunx tight framelets constructed by both approaches in this paper are ofparticular interest for their applications in computer graphics and imageprocessing.
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