Image processing with complex wavelets

TLDR

Wavelet transforms, particularly the discrete wavelet transform (DWT), are widely used for multi‑resolution image processing, but achieving shift invariance and directional selectivity has been challenging, especially for complex wavelet transforms (CWT) with perfect reconstruction. This study proposes the dual‑tree complex wavelet transform (CWT) as a solution to these limitations, aiming to provide shift‑invariant, directionally selective processing for applications such as motion estimation, denoising, texture analysis, synthesis, and object segmentation. The dual‑tree CWT achieves perfect reconstruction by employing two parallel DWT trees, thereby avoiding the shape similarity constraints that cause severe shift dependence in conventional DWTs and enabling directionally selective filters for diagonal features. Experimental results demonstrate that the dual‑tree CWT eliminates shift dependence, offers superior directional selectivity, and improves performance across motion estimation, denoising, texture analysis, synthesis, and segmentation tasks compared to standard DWT approaches.

Abstract

We first review how wavelets may be used for multi–resolution image processing, describing the filter–bank implementation of the discrete wavelet transform (DWT) and how it may be extended via separable filtering for processing images and other multi–dimensional signals. We then show that the condition for inversion of the DWT (perfect reconstruction) forces many commonly used wavelets to be similar in shape, and that this shape produces severe shift dependence (variation of DWT coefficient energy at any given scale with shift of the input signal). It is also shown that separable filtering with the DWT prevents the transform from providing directionally selective filters for diagonal image features. Complex wavelets can provide both shift invariance and good directional selectivity, with only modest increases in signal redundancy and computation load. However, development of a complex wavelet transform (CWT) with perfect reconstruction and good filter characteristics has proved difficult until recently. We now propose the dual–tree CWT as a solution to this problem, yielding a transform with attractive properties for a range of signal and image processing applications, including motion estimation, denoising, texture analysis and synthesis, and object segmentation.