Book Chapter

Sparse Image Representation by Directionlets

Despite the success of the standard wavelet transform (WT) in image processing in recent years, the efficiency and sparsity of its representation are limited by the spatial symmetry and separability of its basis functions built in the horizontal and vertical directions. One-dimensional discontinuities in images (edges or contours), which are 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 multidirectional (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 (2D) WT, unlike the case for some other directional transform constructions (e.g., curvelets, contourlets, or edgelets). The corresponding asymmetric basis functions, called direction-lets, have directional vanishing moments 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 nonlinear approximation of images, significantly outperforming the standard 2D WT. Furthermore, directionlets combined with wavelet-based image compression methods lead to a gain in performance in terms of both the mean square error and visual quality, especially at low bit-rate compression, while retaining 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.

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