Relative phase in dual tree shearlets

• We examine the relative phase originally generated from PDTDFB.
• The relative phase model in PDTDFB is not accurate enough.
• Properties of the optimal transform for relative phase are described.
• We propose a new transform named dual tree shearlets which is suitable for relative phase.
• The dual tree shearlets are highly directional sensitive and computational efficient.

The relative phase is an efficient approach to exploit the phase information of complex wavelet coefficients. However, the relative phase original generated from the pyramidal dual tree directional filter bank (PDTDFB) has three defaults. Firstly, its texture retrieval performance does not simultaneously improve in general as the scale increases. Secondly, it is not accurate enough when the directional subbands are not uniformly downsampled along row and column. Thirdly, its 2n number of directions are not optimal. In this paper, we propose a new multiscale and multidirection transform for relative phase, named as dual tree shearlets. The transform is based on the discrete shearlet transform, but a dual tree Laplacian pyramid is adopted to create a real-imaginary pair structure for deriving phase information under multiscale framework. The dual tree shearlets have the properties of uniform downsampled subbands; higher directional sensitivity and the 2-D Hilbert transform relationship between two channels like the dual tree complex wavelet transform (DTCWT). The numerical experiments presented in this paper demonstrate that the relative phase of our proposed method outperforms that of the PDTDFB in texture retrieval application both in terms of performance and computational efficiency. The results show that relative phase of the dual tree shearlets amends the above mentioned defaults.