In this paper we provide a reconstruction algorithm for piecewise-smooth functions with a priori known smoothness and a number of discontinuities, from their Fourier coefficients, possessing the maximal possible asymp...
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In this paper we provide a reconstruction algorithm for piecewise-smooth functions with a priori known smoothness and a number of discontinuities, from their Fourier coefficients, possessing the maximal possible asymptotic rate of convergence-including the positions of the discontinuities and the pointwise values of the function. This algorithm is a modification of our earlier method, which is in turn based on the algebraic method of K. Eckhoff proposed in the 1990s. The key ingredient of the new algorithm is to use a different set of Eckhoff's equations for reconstructing the location of each discontinuity. Instead of consecutive Fourier samples, we propose to use a "decimated" set which is evenly spread throughout the spectrum.
In this study, the authors proposed a novel regularisation model for resolution enhancement of clean or noisy single image based on the total generalised variation (TGV) and Shearlet transform. The proposed model has ...
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In this study, the authors proposed a novel regularisation model for resolution enhancement of clean or noisy single image based on the total generalised variation (TGV) and Shearlet transform. The proposed model has two main contributions. Firstly, different from models with total variation regularisation, which assume that images consist of piecewise-constant areas, the author's TGV-based model is aware of higher-order smoothness, thus eliminates the staircase-like artefacts. Secondly, various image features including edges and fine details can be preserved by their model. This is nature since the Shearlets mathematically provide an optimally sparse approximation for the class of piecewise-smooth functions with rich geometric information. Moreover, to solve the proposed model, an efficient numerical scheme is explicitly developed based on the Nesterov's algorithm. A series of numerical experiments validate the effectiveness of the proposed method.
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