Image Reconstruction from Fourier Data Using Sparsity of Edges

Gabriel Wasserman, Rick Archibald, Anne Gelb

    Research output: Contribution to journalArticle

    7 Citations (Scopus)

    Abstract

    Data of piecewise smooth images are sometimes acquired as Fourier samples. Standard reconstruction techniques yield the Gibbs phenomenon, causing spurious oscillations at jump discontinuities and an overall reduced rate of convergence to first order away from the jumps. Filtering is an inexpensive way to improve the rate of convergence away from the discontinuities, but it has the adverse side effect of blurring the approximation at the jump locations. On the flip side, high resolution post processing algorithms are often computationally cost prohibitive and also require explicit knowledge of all jump locations. Recent convex optimization algorithms using $$l^1$$l1 regularization exploit the expected sparsity of some features of the image. Wavelets or finite differences are often used to generate the corresponding sparsifying transform and work well for piecewise constant images. They are less useful when there is more variation in the image, however. In this paper we develop a convex optimization algorithm that exploits the sparsity in the edges of the underlying image. We use the polynomial annihilation edge detection method to generate the corresponding sparsifying transform. Our method successfully reduces the Gibbs phenomenon with only minimal blurring at the discontinuities while retaining a high rate of convergence in smooth regions.

    Original languageEnglish (US)
    Pages (from-to)533-552
    Number of pages20
    JournalJournal of Scientific Computing
    Volume65
    Issue number2
    DOIs
    StatePublished - Dec 18 2014

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    Image Reconstruction
    Image reconstruction
    Sparsity
    Convex optimization
    Jump
    Gibbs Phenomenon
    Discontinuity
    Rate of Convergence
    Convex Optimization
    Edge detection
    Optimization Algorithm
    Transform
    Polynomials
    Edge Detection
    Flip
    Annihilation
    Post-processing
    Processing
    Regularization
    Finite Difference

    Keywords

    • Convex optimization
    • Edge detection
    • Fourier data
    • l regularization
    • Polynomial annihilation

    ASJC Scopus subject areas

    • Software
    • Computational Theory and Mathematics
    • Theoretical Computer Science
    • Engineering(all)

    Cite this

    Image Reconstruction from Fourier Data Using Sparsity of Edges. / Wasserman, Gabriel; Archibald, Rick; Gelb, Anne.

    In: Journal of Scientific Computing, Vol. 65, No. 2, 18.12.2014, p. 533-552.

    Research output: Contribution to journalArticle

    Wasserman, Gabriel ; Archibald, Rick ; Gelb, Anne. / Image Reconstruction from Fourier Data Using Sparsity of Edges. In: Journal of Scientific Computing. 2014 ; Vol. 65, No. 2. pp. 533-552.
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