  We propose a 2D generalization to the $M$-band case of the dual-tree
decomposition structure (initially proposed by N. Kingsbury and further
investigated by I. Selesnick) based on a Hilbert pair of wavelets. We
particularly address (\textit{i}) the construction of the dual basis and
(\textit{ii}) the resulting directional analysis. We also revisit the necessary
pre-processing stage in the $M$-band case. While several reconstructions are
possible because of the redundancy of the representation, we propose a new
optimal signal reconstruction technique, which minimizes potential estimation
errors. The effectiveness of the proposed $M$-band decomposition is
demonstrated via denoising comparisons on several image types (natural,
texture, seismics), with various $M$-band wavelets and thresholding strategies.
Significant improvements in terms of both overall noise reduction and direction
preservation are observed.
