Sparse Image Representation by Directionlets

In spite of the success of the standard wavelet transform (WT) in image processing in recent years, the efficiency and sparsity of its representation is limited by the spatial symmetry and separability of its basis functions built in the horizontal and vertical directions. One-dimensional (1-D) discontinuities in images (edges or contours) that are very important elements in visual perception, intersect too many wavelet basis functions and lead to a non-sparse representation. To capture efficiently these elongated structures characterized by geometrical regularity along different directions (not only the horizontal and vertical), a more complex multi-directional (M-DIR) and asymmetric transform is required. We present a lattice-based perfect reconstruction and critically sampled asymmetric M-DIR WT. The transform retains the separable filtering and subsampling and the simplicity of computations and filter design from the standard two-dimensional (2-D) WT, unlike what is the case for some other directional transform constructions (e.g. curvelets, contourlets or edgelets). The corresponding asymmetric basis functions, called directionlets, have directional vanishing moments (DVM) along any two directions with rational slopes, which allows for a sparser representation of elongated and oriented features. As a consequence of the improved sparsity, directionlets provide an efficient tool for non-linear approximation (NLA) of images, significantly outperforming the standard 2-D WT. Furthermore, directionlets combined with wavelet-based image compression methods lead to a gain in the performance in terms of both the mean-square error and visual quality, especially at the low bit-rate compression, while having the same complexity. Finally, a shift-invariant non-subsampled version of directionlets is successfully implemented in image interpolation, where critical sampling is not a key requirement.