  We construct directional wavelet systems that will enable building efficient
signal representation schemes with good direction selectivity. In particular,
we focus on wavelet bases with dyadic quincunx subsampling. In our previous
work, We show that the supports of orthonormal wavelets in our framework are
discontinuous in the frequency domain, yet this irregularity constraint can be
avoided in frames, even with redundancy factor less than 2. In this paper, we
focus on the extension of orthonormal wavelets to biorthogonal wavelets and
show that the same obstruction of regularity as in orthonormal schemes exists
in biorthogonal schemes. In addition, we provide a numerical algorithm for
biorthogonal wavelets construction where the dual wavelets can be optimized,
though at the cost of deteriorating the primal wavelets due to the intrinsic
irregularity of biorthogonal schemes.
