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An optimisation of Gaussian mixture models for integer processing units

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Abstract

This paper investigates sub-integer implementations of the adaptive Gaussian mixture model (GMM) for background/foreground segmentation to allow the deployment of the method on low cost/low power processors that lack Floating Point Unit. We propose two novel integer computer arithmetic techniques to update Gaussian parameters. Specifically, the mean value and the variance of each Gaussian are updated by a redefined and generalized “round” operation that emulates the original updating rules for a large set of learning rates. Weights are represented by counters that are updated following stochastic rules to allow a wider range of learning rates and the weight trend is approximated by a line or a staircase. We demonstrate that the memory footprint and computational cost of GMM are significantly reduced, without significantly affecting the performance of background/foreground segmentation.

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Authors and Affiliations

  1. TeCIP Institute, Scuola Superiore Sant’Anna, via Moruzzi 1, 56124, Pisa, Italy

    Claudio Salvadori

  2. National Inter-University Consortium for Telecommunications, Pisa, Italy

    Matteo Petracca

  3. The Institute of Electronics, Communications and Information Technology (ECIT), Queens University of Belfast, Belfast, UK

    Jesus Martinez del Rincon

  4. Department of Informatic Engineering, Universidad de Santiago de Chile, Santiago de Chile, Chile

    Sergio A. Velastin

  5. Digital Imaging Research Centre, Kingston University, London, UK

    Dimitrios Makris

Authors
  1. Claudio Salvadori
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  2. Matteo Petracca
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  3. Jesus Martinez del Rincon
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  4. Sergio A. Velastin
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  5. Dimitrios Makris
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Corresponding author

Correspondence to Claudio Salvadori.

Appendix

Appendix

1.1 Proof from Sect. 3.5.1

A uniformly distributed random number \({\bf X} \in [0, \rm max]\) is characterized by the cumulative distribution function in Eq. 48 and the probability density function in Eq. 49.

$$ F_X(x) = P({\bf X}\leq x) = {{x}\over {\rm max}} $$
(48)
$$ f_X(x) = {{\partial}\over {\partial x}} F_X(x) = 1 $$
(49)

On the other hand, if we consider the value P(X n  > T X ) and we use the probability properties, the following relation can be obtained:

$$ P({\bf X} > T_X) = 1 - F_X(x) = 1 - {{T_X}\over {\rm max}} $$
(50)

From the assumption in Eqs. 38, 39 is true.

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Salvadori, C., Petracca, M., del Rincon, J.M. et al. An optimisation of Gaussian mixture models for integer processing units. J Real-Time Image Proc 13, 273–289 (2017). https://doi.org/10.1007/s11554-014-0402-5

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  • DOI: https://doi.org/10.1007/s11554-014-0402-5

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