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Engineering presentation of the stochastic interpolation framework and its applications

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Abstract

The paper is an engineering exposition of the Stochastic Interpolation Framework, a novel mathematical approach to data regularization, which recovers a function from input data that is a representation of this data. The framework is an area-based method that comprises a two-step procedure: de-convolution and convolution, involving row-stochastic matrices. Varying the extent of convolution with respect to de-convolution in the framework obtains a gamut of functional recovery ranging from interpolation to approximation, to peak sharpening. Construction of the row stochastic matrices is achieved by means of a mollifier, a positive function which serves as the generator of the row space of these matrices. The properties of the recovered function will depend on the choice of this mollifier. For example, only if the mollifier is differentiable so is the recovered function, and the framework can obtain derivatives anywhere in the domain. The mollifier can be a probability distribution function. Thus, the framework connects interpolation to statistical analysis. Two novel applications in image analysis illustrate the potential of the framework for security applications: as an alternative method of lossy image compression, and as an alternative method to zoom-up an image.

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Notes

  1. Throughout, SI is discussed as being applied in one space dimension x or as space products to address images, but truly multi-dimensional versions of SI are available.

  2. Jet Propulsion Laboratory reported some years ago that their electronic nose suffered from baseline drift.

References

  • Botcher A, Grudsky SM (2000) Toeplitz matrices, asymptotic linear algebra, and functional analysis. Birkhauser, Basel

  • Howard D, Kolibal J (2007) Image analysis by means of the stochastic matrix method of function recovery. In: 2007 ECSIS Symposium BLISS 2007. Edinburgh, Scotland, pp 97–101

  • Howard D, Kolibal J (2008) Stochastic interpolation as an approach to data regularization. In: Proceedings of the electromagnetic remote sensing defence technology centre 5th annual technical conference, b23, Edinburgh, 2008. [Online]. Available from: http://www.emrsdtc.com/conferences/2008/confer_material_session_b.htm. Accessed: 29 January 2009

  • Howard D, Roberts SC (2001) Genetic programming solution of the convection-diffusion equation. In: Spector L et al (ed) Proceedings of the genetic and evolutionary computation conference (GECCO-2001). San Francisco, California, USA, 7–11 July 2001. Morgan Kaufmann, San Francisco, pp 34–41

  • Howard D, Brezulianu A, Kolibal J (2009) Genetic programming of the stochastic interpolation framework: convection–diffusion equation. Soft Comput. doi:10.1007/s00500-009-0520-3

  • Kolibal J, Fang S (2000) Integral kernel/frequency shift algorithms for image data compression. In: Proceedings of the southern conference on computing. The University of Southern Mississippi, 26–28 October

  • Kolibal J, Howard D (2006a) Maldi-tof baseline drift removal using stochastic Bernstein approximation. EURASIP Journal on Applied Signal Processing, Special Issue on Advanced Signal Processing Techniques for Bioinformatics, 2006:1–9

  • Kolibal J, Howard D (2006b) The novel stochastic Bernstein method of functional approximation. In: First NASA/ESA Conference on adaptive hardware and systems, 2006. AHS 2006, pp 97–100, Istanbul, Turkey, 15–18 June 2006. IEEE Press

  • Lorentz G (1986) Bernstein polynomials. Chelsea, New York

    MATH  Google Scholar 

  • Runge C (1901) Uber empirische funktionen und die interpolation zwischen aquidistanten ordinaten. Zeitschrift für Mathematik und Physik 46:224–243

    Google Scholar 

  • Trench WF (1964) An algorithm for the inversion of finite Toeplitz matrices. J Soc Ind Appl Math 12(3):515–522

    Article  MATH  MathSciNet  Google Scholar 

  • Tseng H-W, Chang C-C (2004) Steganography using jpeg-compressed images. In: Proceedings of the fourth international conference on computer and information technology, pp 12–17. IEEE Computer Society, 14–16 September 2004

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Howard, D., Kolibal, J. & Brezulianu, A. Engineering presentation of the stochastic interpolation framework and its applications. Soft Comput 15, 79–87 (2011). https://doi.org/10.1007/s00500-009-0517-y

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