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.
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