Residual MultiSmoothlets Smoothlets, as one of the adaptive geometrical multi resolution methods, are efficient in the approximatb1ion of images with various edges. However, when multiple edges present in blocks, smoothlets become less efficient due to its requirement in further partitioning the blocks. To overcome this problem, multismoothlets have been developed, which could efficiently represent multiple edges with a vector of smoothlets. However, the improvement in terms of the quality of the reconstructed image by the existing implementations of serial and parallel multismoothlets is relatively small. In this paper, we propose a new residual multismoothlets method by iteratively calculating the smoothlets at the original image and multiple levels of residual images to obtain the multismoothlets. Experimental results demonstrate that the proposed method can effectively improve the quality of the reconstructed image in image approximation compared with the existing multismoothlets.

The edge driven oriented wavelet transform : An anisotropic multidirectional representation with oriented lifting scheme In spite of its success, the standard 2D discrete wavelet transform (2DDWT) is not completely adapted to represent image entities like edges or oriented textures. Indeed the DWT is limited by the spatial isotropy of its basis functions that can not take advantage of edges regularity and moreover, direction edge that is neither vertical or horizontal is represented using many of these wavelet basis functions which does mean that DWT does not provide a sparse representation for such discontinuities. Several representations have been proposed to overcome this lack. Some of them deal with more orientations while introducing redundancy (e.g. ridgelets, curvelets, contourlets) and their implementations are not trivial or require 2D non separable filtering. We present two oriented liftingbased schemes using separable filtering, lead by edge extraction, and inspired from bandelets and curved wavelets. An image is decomposed into a quadtree according to the edge elements orientation. For each leaf, a wavelet transform is performed along the most regular orientation, and then along its orthogonal direction. Different adapted filters may be used for these two directions in order to achieve anisotropic filtering. Our method permits also a perfect reconstruction and a critical sampling.