Undecimated Dual-Tree Complex Wavelet Transforms

Two undecimated forms of the Dual Tree Complex Wavelet Transform (DT-CWT) are introduced and their application to image denoising is described. These undecimated transforms extend the DT-CWT through the removal of downsampling of the filter outputs together with upsampling of the filters in a similar structure to the Undecimated Discrete Wavelet Transform (UDWT). Both the developed transforms offer exact translational invariance, improved scale-to-scale coefficient correlation together with the directional selectivity of the DT-CWT. Additionally, within each of these developed transforms, the subbands are of a consistent size. They therefore benefit from a direct one-to-one relationship between co-located coefficients at all scales. This is an important relationship that can be exploited within applications such as denoising, image fusion and segmentation. The enhanced properties of the transforms have been exploited within a bivariate shrinkage denoising application, demonstrating quantitative improvements in denoising results compared to the DT-CWT. The two novel transforms together with the DT-CWT offer a trade off between denoising performance, computational efficiency and memory requirements.