Numerical Algorithms

, Volume 29, Issue 4, pp 323–378

Deconvolution and Regularization with Toeplitz Matrices

  • Per Christian Hansen

DOI: 10.1023/A:1015222829062

Cite this article as:
Hansen, P.C. Numerical Algorithms (2002) 29: 323. doi:10.1023/A:1015222829062


By deconvolution we mean the solution of a linear first-kind integral equation with a convolution-type kernel, i.e., a kernel that depends only on the difference between the two independent variables. Deconvolution problems are special cases of linear first-kind Fredholm integral equations, whose treatment requires the use of regularization methods. The corresponding computational problem takes the form of structured matrix problem with a Toeplitz or block Toeplitz coefficient matrix. The aim of this paper is to present a tutorial survey of numerical algorithms for the practical treatment of these discretized deconvolution problems, with emphasis on methods that take the special structure of the matrix into account. Wherever possible, analogies to classical DFT-based deconvolution problems are drawn. Among other things, we present direct methods for regularization with Toeplitz matrices, and we show how Toeplitz matrix–vector products are computed by means of FFT, being useful in iterative methods. We also introduce the Kronecker product and show how it is used in the discretization and solution of 2-D deconvolution problems whose variables separate.

deconvolution regularization Toeplitz matrix Kronecker product SVD analysis image deblurring 

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Per Christian Hansen
    • 1
  1. 1.Department of Mathematical ModellingTechnical University of DenmarkLyngbyDenmark