Discrete integer Fourier transform in real space: elliptic Fourier transform

Artyom M. Grigoryan; Merughan M. Grigoryan

doi:10.1117/12.804543

10 February 2009 Discrete integer Fourier transform in real space: elliptic Fourier transform

Artyom M. Grigoryan, Merughan M. Grigoryan

Proceedings Volume 7245, Image Processing: Algorithms and Systems VII; 724502 (2009) https://doi.org/10.1117/12.804543
Event: IS&T/SPIE Electronic Imaging, 2009, San Jose, California, United States

Abstract

The concept of the N-point DFT is generalized, by considering it in the real space (not complex). The multiplication by twiddle coefficients is considered in matrix form; as the Givens transformation. Such block-wise representation of the matrix of the DFT is effective. The transformation which is called the T-generated N-block discrete transform, or N-block T-GDT is introduced. For each N-block T-GDT, the inner product is defined, with respect to which the rows (and columns) of the matrices X are orthogonal. By using different parameterized matrices T, we define metrics in the real space of vectors. The selection of the parameters can be done among only the integer numbers, which leads to integer-valued metric. We also propose a new representation of the discrete Fourier transform in the real space R^2N. This representation is not integer, and is based on the matrix C (2x2) which is not a rotation, but a root of the unit matrix. The point (1, 0) is not moving around the unite circle by the group of motion generated by C, but along the perimeter of an ellipse. The N-block C-GDT is therefore called the N-block elliptic FT (EFT). These orthogonal transformations are parameterized; their properties are described and examples are given.

Citation Download Citation

Artyom M. Grigoryan and Merughan M. Grigoryan "Discrete integer Fourier transform in real space: elliptic Fourier transform", Proc. SPIE 7245, Image Processing: Algorithms and Systems VII, 724502 (10 February 2009); https://doi.org/10.1117/12.804543

ACCESS THE FULL ARTICLE

INSTITUTIONAL
Select your institution to access the SPIE Digital Library.

SELECT YOUR INSTITUTION

PERSONAL
Sign in with your SPIE account to access your personal subscriptions or to use specific features such as save to my library, sign up for alerts, save searches, etc.

PERSONAL SIGN IN

No SPIE Account? Create one

PURCHASE THIS CONTENT

SUBSCRIBE TO DIGITAL LIBRARY

50 downloads per 1-year subscription

Members: $195

Non-members: $335 ADD TO CART

25 downloads per 1 - year subscription

Members: $145

Non-members: $250 ADD TO CART

PURCHASE SINGLE ARTICLE

Includes PDF, HTML & Video, when available