Search

Smooth orthogonal projections with good localization properties were originally studied in the wavelet literature as a way to both understand and generalize the construction of smooth wavelet bases on L 2 (ℝ) . Smoothness plays a critical role in the construction of wavelet bases and their generalizations as it is instrumental to achieve excellent approximation properties. In this paper, we extend the construction of smooth orthogonal projections to higher dimensions, a challenging problem in general for which relatively few results are found in the literature. Our investigation is motivated by the study of multidimensional nonseparable multiscale systems such as shearlets. Using our new class of smooth orthogonal projections, we construct new smooth Parseval frames of shearlets in L 2 (ℝ 2) and L 2 (ℝ 3) . [ABSTRACT FROM AUTHOR]

Edges and surface boundaries are often the most relevant features in images and multidimensional data. It is well known that multiscale methods including wavelets and their more sophisticated multidimensional siblings offer a powerful tool for the analysis and detection of such sets. Among such methods, the continuous shearlet transform has been especially successful. This method combines anisotropic scaling and directional sensitivity controlled by shear transformations in order to precisely identify not only the location of edges and boundary points but also edge orientation and corner points. In this paper, we show that this framework can be made even more flexible by controlling the scaling parameter of the anisotropic dilation matrix and considering non-parabolic scaling. We prove that, using 'higher-than-parabolic' scaling, the modified shearlet transform is also able to estimate the degree of local flatness of an edge or surface boundary, providing more detailed information about the geometry of edge and boundary points. [ABSTRACT FROM AUTHOR]

This paper introduces a new decomposition of the 3D X-ray transform based on the shearlet representation, a multiscale directional representation which is optimally efficient in handling 3D data containing edge singularities. Using this decomposition, we derive a highly effective reconstruction algorithm yielding a near-optimal rate of convergence in estimating piecewise smooth objects from 3D X-ray tomographic data which are corrupted by white Gaussian noise. This algorithm is achieved by applying a thresholding scheme on the 3D shearlet transform coefficients of the noisy data which, for a given noise level ε, can be tuned so that the estimator attains the essentially optimal mean square error rate O(log( ε) ε), as ε→0. This is the first published result to achieve this type of error estimate, outperforming methods based on Wavelet-Vaguelettes decomposition and on SVD, which can only achieve MSE rates of O( ε) and O( ε), respectively. [ABSTRACT FROM AUTHOR]