Multidimensional multichannel FIR deconvolution using Gröbner bases

Jianping Zhou, Minh N. Do

Research output: Contribution to journalArticlepeer-review


We present a new method for general multidimensional multichannel deconvolution with finite impulse response (FIR) convolution and deconvolution filters using Gröbner bases. Previous work formulates the problem of multichannel FIR deconvolution as the construction of a left inverse of the convolution matrix, which is solved by numerical linear algebra. However, this approach requires the prior information of the support of deconvolution filters. Using algebraic geometry and Gröbner bases, we find necessary and sufficient conditions for the existence of exact deconvolution FIR filters and propose simple algorithms to find these deconvolution filters. The main contribution of our work is to extend the previous Gröbner basis results on multidimensional multichannel deconvolution for polynomial or causal filters to general FIR filters. The proposed algorithms obtain a set of FIR deconvolution filters with a small number of nonzero coefficients (a desirable feature in the impulsive noise environment) and do not require the prior information of the support. Moreover, we provide a complete characterization of all exact deconvolution FIR filters, from which good FIR deconvolution filters under the additive white noise environment are found. Simulation results show that our approaches achieve good results under different noise settings.

Original languageEnglish (US)
Pages (from-to)2998-3007
Number of pages10
JournalIEEE Transactions on Image Processing
Issue number10
StatePublished - Oct 2006


  • Algebraic geometry
  • Deconvolution
  • Exact deconvolution
  • Finite impulse response (FIR)
  • Gröbner bases
  • Multichannel
  • Multidimensional
  • Multivariate
  • Nullstellensatz

ASJC Scopus subject areas

  • Electrical and Electronic Engineering
  • Computer Graphics and Computer-Aided Design
  • Software
  • Theoretical Computer Science
  • Computational Theory and Mathematics
  • Computer Vision and Pattern Recognition


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