Journal of Mathematical Imaging and Vision

, Volume 39, Issue 3, pp 210–229 | Cite as

Two Classes of Elliptic Discrete Fourier Transforms: Properties and Examples

  • Artyom M. Grigoryan


This paper analyzes the block structure of the matrix of the N-point discrete Fourier transform (DFT) in the real space R 2N . Each block of this matrix corresponds to the Givens transformation, or elementary rotation describing the multiplications by twiddle coefficients. Such rotations around the circle can be substituted by other kinds of rotations, for instance rotations around ellipses, while reserving the block-wise representation of the matrix and main properties of the DFT. To show that, we present two classes of the elliptic discrete Fourier transforms (EDFT), that are defined by different types of the Nth roots of the identity matrix 2×2, whose groups of motion move points around different ellipses. These two classes (the N-block EDFT of types I and II) are parameterized and exist for any order N. Properties and examples of application of the proposed elliptic EDFTs in signal and image processing are given.


Fourier analysis Discrete Fourier transform Signal and image processing 


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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Electrical and Computer EngineeringThe University of Texas at San AntonioSan AntonioUSA

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