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A Framework for Moving Least Squares Method with Total Variation Minimizing Regularization

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Abstract

In this paper, we propose a computational framework to incorporate regularization terms used in regularity based variational methods into least squares based methods. In the regularity based variational approach, the image is a result of the competition between the fidelity term and a regularity term, while in the least squares based approach the image is computed as a minimizer to a constrained least squares problem. The total variation minimizing denoising scheme is an exemplary scheme of the former approach with the total variation term as the regularity term, while the moving least squares method is an exemplary scheme of the latter approach. Both approaches have appeared in the literature of image processing independently. By putting schemes from both approaches into a single framework, the resulting scheme benefits from the advantageous properties of both parties. As an example, in this paper, we propose a new denoising scheme, where the total variation minimizing term is adopted by the moving least squares method. The proposed scheme is based on splitting methods, since they make it possible to express the minimization problem as a linear system. In this paper, we employed the split Bregman scheme for its simplicity. The resulting denoising scheme overcomes the drawbacks of both schemes, i.e., the staircase artifact in the total variation minimizing based denoising and the noisy artifact in the moving least squares based denoising method. The proposed computational framework can be utilized to put various combinations of both approaches with different properties together.

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Acknowledgements

This work was supported by the Basic Science Research Programs 2012R1A1A2004518 (J. Yoon), 2010-0011689 (Y. Lee), 2010-0006567 (S. Lee), and the Priority Research Centers Program 2009-0093827 (Y. Lee and J. Yoon) through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology.

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Authors and Affiliations

  1. Institute of Mathematical Sciences, Ewha W. Univ., Seoul, 120-750, South Korea

    Yeon Ju Lee

  2. Dept. Software Engineering, Dongseo Univ., Busan, 617-716, South Korea

    Sukho Lee

  3. Department of Mathematics, Ewha W. Univ., Seoul, 120-750, South Korea

    Jungho Yoon

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  1. Yeon Ju Lee
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Lee, Y.J., Lee, S. & Yoon, J. A Framework for Moving Least Squares Method with Total Variation Minimizing Regularization. J Math Imaging Vis 48, 566–582 (2014). https://doi.org/10.1007/s10851-013-0428-5

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