Approximation power of directionlets
In spite of the success of the standard wavelet transform (WT) in image processing, the efficiency of its representation is limited by the spatial isotropy of its basis functions built in only horizontal and vertical directions. One-dimensional (1-D) discontinuities in images (edges and contours), which are very important elements in visual perception, intersect too many wavelet basis functions and reduce the sparsity of the representation. To capture efficiently these anisotropic geometrical structures, a more complex multi-directional (M-DIR) and anisotropic transform is required. We present a new lattice-based perfect reconstruction and critically sampled anisotropic M-DIR WT (with the corresponding basis functions called directionlets) that retains the separable filtering and simple filter design from the standard two-dimensional (2-D) WT and imposes directional vanishing moments (DVM). Further-more, we show that this novel transform has non-linear approximation efficiency competitive to the other previously proposed over-sampled transform constructions.