Undecimated Dual-Tree Complex Wavelet Transforms

• Proposed transforms have exact translational invariance.
• Coefficients have one-to-one cross scale relationships.
• Improved results for two example applications.
• Matlab code available at: www.bristol.ac.uk/vi-lab/projects/udtcwt.

Two undecimated forms of the Dual Tree Complex Wavelet Transform (DT-CWT) are introduced together with their application to image denoising and robust feature extraction. These undecimated transforms extend the DT-CWT through the removal of downsampling of filter outputs together with upsampling of the complex filter pairs in a similar structure to the Undecimated Discrete Wavelet Transform (UDWT).Both developed transforms offer exact translational invariance, improved scale-to-scale coefficient correlation together with the directional selectivity of the DT-CWT. Additionally, within each developed transform, the subbands are of a consistent size. They therefore benefit from a direct one-to-one relationship between co-located coefficients at all scales and therefore this offers consistent phase relationships across scales. These advantages can be exploited within applications such as denoising, image fusion, segmentation and robust feature extraction. The results of two example applications (bivariate shrinkage denoising and robust feature extraction) demonstrate objective and subjective improvements over the DT-CWT. The two novel transforms together with the DT-CWT offer a trade-off between denoising performance, computational efficiency and memory requirements.