Frequency Analysis on Ellipses: Elliptic Fourier Transform

Abstract

The discrete Fourier transform (DFT) is widely used in data compression, pattern recognition, and image reconstruction. The study of properties of the DFT shows that this transform can be generalized and used mere effectively for solving many problems in such areas as signal and image processing. The traditional N-point DFT is defined as the decomposition of the signal by N roots of the unit, which are on the unit circle. On different stages of the DFT, the data of the signal are multiplied by the exponential factors, or rotated around the circles. This paper presents a novel concept of the DFT by considering the block-wise representation of the transform in the real space, which can be generalized to obtain new methods in spectral analysis. For that, the new concept of the N-point elliptic discrete Fourier transform (EDFT) is introduced, whose basic 2×2 transformations that are not Given transformations, but rather rotations around ellipses. The elliptic transformation is therefore defined by different Nth roots of the identity matrix 2×2, matrix whose groups of motion move a point around the ellipses. The main properties of the EDFT are described and examples are provided. The EDFT preserves main properties of the DFT, such as shifting and energy preservation among others. The EDFT distinguishes the carrying frequencies of the signal in both real and imaginary parts better than the DFT. It also has a simple inverse matrix. It is parameterized and includes the DFT, as a partial case. The preliminary results show that by using different parameters, the EDFT can be used effectively for solving many problems in signal and image processing field, which includes image enhancement, filtration, encryption and others.