New Families of Fourier Eigenfunctions for Steerable Filtering

Giuseppe Papari*, Patrizio Campisi, Nicolai Petkov

*Corresponding author for this work

    Research output: Contribution to journalArticleAcademicpeer-review

    11 Citations (Scopus)
    313 Downloads (Pure)

    Abstract

    A new diadic family of eigenfunctions of the 2-D Fourier transform has been discovered. Specifically, new wavelets are derived by steering the elongated Hermite-Gauss filters with respect to rotations, thus obtaining a natural generalization of the Laguerre-Gauss harmonics. Interestingly, these functions are also proportional to their 2-D Fourier transform. Their analytical expression is provided in a compact and treatable form, by means of a new ad hoc matrix notation in which the cases of even and odd orders of the Hermite polynomials are unified. Moreover, these functions can be efficiently implemented by means of a recursive formula that is derived in this paper. The proposed filters are applied to the problem of gradient estimation to improve the theoretical Canny tradeoff of position accuracy versus noise rejection that occurs in edge detection. Experimental results show considerable improvements in using the new wavelets over both isotropic Gaussian derivatives and other elongated steerable filters more recently introduced. Finally, being the proposed wavelets a set of Fourier eigenfunctions, they can be of interest in other fields of science, such as optics and quantum mechanics.

    Original languageEnglish
    Pages (from-to)2931-2943
    Number of pages13
    JournalIeee transactions on image processing
    Volume21
    Issue number6
    DOIs
    Publication statusPublished - Jun-2012

    Keywords

    • Filtering
    • series expansion methods
    • wavelets and fractals
    • LOCAL ORIENTATION ANALYSIS
    • CONTOUR-DETECTION
    • ACTIVE CONTOURS
    • WAVELET-DOMAIN
    • EDGE-DETECTION
    • EARLY VISION
    • GRADIENT
    • DESIGN
    • DECOMPOSITION
    • OPERATORS

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