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A well-posed multiscale regularization scheme for digital image denoising

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Prasath V.
International Journal of Applied Mathematics and Computer Science
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2011
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tom 21
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nr 4
769-777
EN
We propose an edge adaptive digital image denoising and restoration scheme based on space dependent regularization. Traditional gradient based schemes use an edge map computed from gradients alone to drive the regularization. This may lead to the oversmoothing of the input image, and noise along edges can be amplified. To avoid these drawbacks, we make use of a multiscale descriptor given by a contextual edge detector obtained from local variances. Using a smooth transition from the computed edges, the proposed scheme removes noise in flat regions and preserves edges without oscillations. By incorporating a space dependent adaptive regularization parameter, image smoothing is driven along probable edges and not across them. The well-posedness of the corresponding minimization problem is proved in the space of functions of bounded variation. The corresponding gradient descent scheme is implemented and further numerical results illustrate the advantages of using the adaptive parameter in the regularization scheme. Compared with similar edge preserving regularization schemes, the proposed adaptive weight based scheme provides a better multiscale edge map, which in turn produces better restoration.
2
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Application of the partitioning method to specific Toeplitz matrices

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Stanimirović P., Miladinović M., Stojanović I., Miljković S.
International Journal of Applied Mathematics and Computer Science
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2013
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tom 23
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nr 4
809-821
EN
We propose an adaptation of the partitioning method for determination of the Moore-Penrose inverse of a matrix augmented by a block-column matrix. A simplified implementation of the partitioning method on specific Toeplitz matrices is obtained. The idea for observing this type of Toeplitz matrices lies in the fact that they appear in the linear motion blur models in which blurring matrices (representing the convolution kernels) are known in advance. The advantage of the introduced method is a significant reduction in the computational time required to calculate the Moore-Penrose inverse of specific Toeplitz matrices of an arbitrary size. The method is implemented in MATLAB, and illustrative examples are presented.
3
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An efficient algorithm for adaptive total variation based image decomposition and restoration

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Liu X., Huang L.
International Journal of Applied Mathematics and Computer Science
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2014
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tom 24
|
nr 2
405-415
EN
With the aim to better preserve sharp edges and important structure features in the recovered image, this article researches an improved adaptive total variation regularization and H −1 norm fidelity based strategy for image decomposition and restoration. Computationally, for minimizing the proposed energy functional, we investigate an efficient numerical algorithm-the split Bregman method, and briefly prove its convergence. In addition, comparisons are also made with the classical OSV (Osher-Sole-Vese) model (Osher et al., 2003) and the TV-Gabor model (Aujol et al., 2006), in terms of the edge-preserving capability and the recovered results. Numerical experiments markedly demonstrate that our novel scheme yields significantly better outcomes in image decomposition and denoising than the existing models.
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