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Multidimensional multichannel FIR deconvolution using Gröbner bases.

Jianping Zhou1, Minh N Do

  • 1Department of Electrical and Computer Engineering and the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA. jianping@ti.com

IEEE Transactions on Image Processing : a Publication of the IEEE Signal Processing Society
|October 7, 2006
PubMed
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This study introduces a novel Gröbner basis method for multidimensional multichannel deconvolution. The approach effectively identifies exact finite impulse response (FIR) deconvolution filters without needing prior support information.

Area of Science:

  • Signal Processing
  • Algebraic Geometry
  • Multidimensional Systems

Background:

  • Multichannel deconvolution with finite impulse response (FIR) filters is crucial in signal processing.
  • Existing numerical linear algebra methods require prior knowledge of deconvolution filter support.
  • Previous algebraic geometry approaches were limited to polynomial or causal filters.

Purpose of the Study:

  • To develop a new method for general multidimensional multichannel deconvolution using Gröbner bases.
  • To extend existing Gröbner basis techniques to handle general FIR deconvolution filters.
  • To provide algorithms for finding exact FIR deconvolution filters without prior support information.

Main Methods:

  • Utilized algebraic geometry and Gröbner bases to analyze FIR deconvolution.

Related Experiment Videos

  • Derived necessary and sufficient conditions for the existence of exact deconvolution FIR filters.
  • Developed algorithms to compute these deconvolution filters.

    • Main Results:

    • Successfully extended Gröbner basis methods to general multidimensional multichannel FIR deconvolution.
    • Proposed algorithms yield FIR deconvolution filters with minimal non-zero coefficients.
    • Provided a complete characterization of all exact deconvolution FIR filters, enabling selection of optimal filters for additive white noise environments.

    Conclusions:

    • The Gröbner basis approach offers a robust framework for FIR deconvolution.
    • The developed algorithms efficiently find exact deconvolution filters, applicable in noisy environments.
    • Simulation results validate the effectiveness of the proposed methods across various noise conditions.