  We consider the construction of orthonormal directional wavelet bases in the
multi-resolution analysis (MRA) framework with quincunx dilation downsampling.
We show that the Parseval frame property in MRA is equivalent to the identity
summation and shift cancellation conditions on M functions, which essentially
characterize the scaling (father) function and all directional (mother)
wavelets. Based on these two conditions, we further derive sufficient
conditions for orthonormal bases and build a family of quasishearlet
orthonormal bases, that has the same frequency support as that of the least
redundant shearlet system. In addition, we study the limitation of our proposed
bases design due to the shift cancellation conditions.
