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Multiscale approximation with hierarchical radial basis functions networks.

Stefano Ferrari1, Mauro Maggioni, N Alberto Borghese

  • 1Department of Information Technology, University of Milan, 26013 Crema (CR), Italy.

IEEE Transactions on Neural Networks
|September 25, 2004
PubMed
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A novel hierarchical radial basis function (HRBF) network offers efficient, self-organizing multiscale data approximation. This neural model excels at reconstructing 3-D models from noisy data, outperforming traditional methods.

Area of Science:

  • Artificial Intelligence
  • Machine Learning
  • Computational Mathematics

Background:

  • Radial Basis Function (RBF) networks are established tools for function approximation.
  • Existing RBF networks often lack efficient multiscale processing capabilities.
  • Multiresolution analysis (MRA) using wavelets is a common approach for handling data at different scales.

Purpose of the Study:

  • To introduce a novel, self-organizing, multiscale version of an RBF network, termed the hierarchical radial basis function (HRBF) network.
  • To demonstrate the HRBF network's ability to achieve uniform residual error and adaptively allocate resources based on data complexity.
  • To evaluate the HRBF network's performance in reconstructing 3-D models from noisy range data.

Main Methods:

  • The HRBF network is constructed using hierarchical layers with Gaussian grids at decreasing scales.

Related Experiment Videos

  • Units are adaptively inserted based on local error thresholds, ensuring efficient resource allocation.
  • Harmonic analysis is employed to understand the network's approximation properties and relation to wavelet-based MRA.

    • Main Results:

    • The HRBF network demonstrates efficient, quasi-real-time construction through local operations without data iteration.
    • Harmonic analysis confirms that HRBF networks utilize Riesz bases and possess strong asymptotic approximation properties.
    • Extensive application to 3-D model reconstruction from noisy range data shows superior denoising and multiscale reconstruction quality compared to MRA.

    Conclusions:

    • The hierarchical radial basis function (HRBF) network is a powerful, self-organizing neural model for multiscale approximation.
    • HRBF networks offer significant advantages in denoising and reconstructing complex data, particularly 3-D models from noisy inputs.
    • The HRBF network provides an effective alternative to traditional wavelet-based multiresolution analysis for various approximation tasks.