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On Globally Optimal Local Modeling: From Moving Least Squares to Over-parametrization

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Innovations for Shape Analysis

Part of the book series: Mathematics and Visualization ((MATHVISUAL))

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Abstract

This paper discusses a variational methodology, which involves locally modeling of data from noisy samples, combined with global model parameter regularization. We show that this methodology encompasses many previously proposed algorithms, from the celebrated moving least squares methods to the globally optimal over-parametrization methods recently published for smoothing and optic flow estimation. However, the unified look at the range of problems and methods previously considered also suggests a wealth of novel global functionals and local modeling possibilities. Specifically, we show that a new non-local variational functional provided by this methodology greatly improves robustness and accuracy in local model recovery compared to previous methods. The proposed methodology may be viewed as a basis for a general framework for addressing a variety of common problem domains in signal and image processing and analysis, such as denoising, adaptive smoothing, reconstruction and segmentation.

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Acknowledgements

This research was supported by the Israel Science foundation (ISF) grant no. 1551/09.

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Authors and Affiliations

  1. Technion - Israel Institute of Technology, 2000, Haifa, Israel

    Shachar Shem-Tov, Guy Rosman, Ron Kimmel & Alfred M. Bruckstein

  2. Rafael, Haifa, Israel

    Gilad Adiv

Authors
  1. Shachar Shem-Tov
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  2. Guy Rosman
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  3. Gilad Adiv
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  4. Ron Kimmel
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  5. Alfred M. Bruckstein
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Corresponding author

Correspondence to Shachar Shem-Tov .

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Editors and Affiliations

  1. , Inst. for Appl. Math.&Scient.Comp., Brandenburg Technical University, Platz der Deutschen Einheit 1, Cottbus, 03046, Germany

    Michael Breuß

  2. Department of Computer Science, Technion-Israel Institute of Technology, Haifa, 32000, Israel

    Alfred Bruckstein

  3. School of Electrical &, Computer Engineering, National Technical University of Athens, Athens, 15773, Greece

    Petros Maragos

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Shem-Tov, S., Rosman, G., Adiv, G., Kimmel, R., Bruckstein, A.M. (2013). On Globally Optimal Local Modeling: From Moving Least Squares to Over-parametrization. In: Breuß, M., Bruckstein, A., Maragos, P. (eds) Innovations for Shape Analysis. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34141-0_17

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