  We propose an approach to construct a family of two-dimensional compactly
supported real-valued symmetric quincunx tight framelets $\{\phi;
\psi_1,\psi_2,\psi_3\}$ in $L_2(R^2)$ with arbitrarily high orders of vanishing
moments. Such symmetric quincunx tight framelets are associated with quincunx
tight framelet filter banks $\{a;b_1,b_2,b_3\}$ having increasing orders of
vanishing 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 quincunx
orthonormal 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 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 $L_2(R^2)$ and a family of double
canonical complex-valued quincunx tight framelets in $L_2(R^2)$ 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
$L_2(R^2)$ 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.
