A dual-tree complex wavelet transform with improved orthogonality and symmetry properties

We present a new form of the dual-tree complex wavelet transform (DT CWT) with improved orthogonality and symmetry properties. Beyond level 1, the previous form used alternate odd-length and even-length bi-orthogonal filter pairs in the two halves of the dual-tree, whereas the new form employs a single design of even-length filter with asymmetric coefficients. These are similar to the Daubechies orthonormal filters, but designed with the additional constraint that the filter group delay should be approximately one quarter of the sample period. The filters in the two trees are just the time-reverse of each other, as are the analysis and reconstruction filters. This leads to a transform, which can use shorter filters, which is orthonormal beyond level 1, and in which the two trees are very closely matched and have a more symmetric sub-sampling structure, but which preserves the key DT CWT advantages of approximate shift-invariance and good directional selectivity in multiple dimensions.