Deconvolution and Regularization with Toeplitz Matrices

    Research output: Contribution to journalJournal articleResearchpeer-review


    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.
    Original languageEnglish
    JournalNumerical Algorithms
    Issue number4
    Pages (from-to)323-378
    Publication statusPublished - 2002

    Fingerprint Dive into the research topics of 'Deconvolution and Regularization with Toeplitz Matrices'. Together they form a unique fingerprint.

    Cite this