  We provide a detailed analysis of the obstruction (studied first by S. Durand
and later by R. Yin and one of us) in the construction of multidirectional
wavelet orthonormal bases corresponding to any admissible frequency partition
in the framework of subband filtering with non-uniform subsampling. To
contextualize our analysis, we build, in particular, multidirectional
alias-free hexagonal wavelet bases and low-redundancy frames with optimal
spatial decay. In addition, we show that a 2D cutting lemma can be used to
subdivide the obtained wavelet systems in higher frequency rings so as to
generate bases or frames that satisfy the ``parabolic scaling'' law enjoyed by
curvelets and shearlets. Numerical experiments on high bit-rate image
compression are conducted to illustrate the potential of the proposed systems.
