Signal and Data Processing

Parabolic scaling and anisotropic dilation form the core of famous multi-resolution transformations such as curvelet and shearlet, which are widely used in signal processing applications like denoising. These non-adaptive geometrical wavelets are commonly used to extract structures and geometrical features of multi-dimensional signals and preserve them in noise removal treatments. In discrete setups, it is shown that shearlets can outperform other rivals since in addition to scaling, they are formed by shear operator which can fully remain on integer grid. However, the redundancy of multidimensional shearlet transform exponentially grows with respect to the number of dimensions which in turn leads to the exponential computational and space complexity. This, seriously limits the applicability of shearlet transform in higher dimensions. In contrast, separable transforms process each dimension of data independent of other dimensions which result in missing the informative relations among different dimensions of the data.
Therefore, in this paper a modified discrete shearlet transform is proposed which can overcome the redundancy and complexity issues of the classical transform. It makes a better tradeoff between completeness of the analysis achieved by processing full relations among dimensions on one hand and the redundancy and computational complexity of the resulting transform on the other hand. In fact, how dilation matrix is decomposed and block diagonalized, gives a tuning parameter for the amount of inter dimension analysis which may be used to control computation complexity and also redundancy of the resultant transform.
In the context of video denoising, three different decompositions are proposed for 3x3 dilation matrix. In each block diagonalization of this dilation matrix, one dimension is separated and the other two constitute a 2D shearlet transform. The three block shearlet transforms are computed for the input data up to three levels and the resultant coefficients are treated with automatically adjusted thresholds. The output is obtained via an aggregation mechanism which combine the result of reconstruction of these three transforms. Using experiments on standard set of videos at different levels of noise, we show that the proposed approach can get very near to the quality of full 3D shearlet analysis while it keeps the computational complexity (time and space) comparable to the 2D shearlet transform.