Abstract
The 2D Quaternionic Fourier Transform (QFT), applied to a real 2D image, produces an invertible quaternionic spectrum. If we conserve uniquely the first quadrant of this spectrum, it is possible, after inverse transformation, to obtain, not the original image, but a 2D quaternion image, which generalize in 2D the classical notion of 1D analytical image. From this quaternion image, we compute the corresponding correlation product, then, by applying the direct QFT, we obtain the 4D Wigner-Ville distribution of this analytical signal. With reference to the shift variables ?1 , ?2 used for the computation of the correlation product, we obtain a local quaternion Wigner-Ville distribution spectrum.
Highlights
The most common method of analysis of the frequency content of an n-D real signal is the classical complex Fourier transform
The Fourier transforms have been widely used in signal and image processing. ever since the discovery of the Fast Fourier Transform in 1965 (Cooley-Tukey algorithm) which made the computation of Discrete Fourier Transform feasible using a computer
The analytic signal is a complex extension of a 1D signal that is based upon the Hilbert transform; it was introduced to signal theory by Gabor in 1946 (Ref 3).This representation gives access to the instantaneous amplitude and phase
Summary
The most common method of analysis of the frequency content of an n-D real signal is the classical complex Fourier transform. The Fourier transforms have been widely used in signal and image processing. Based on the concept of quaternion the quaternion Fourier transform (QFT) has been introduced by Ell (Ref 1) and implemented by Pei (Ref 2) with conventional 2D Fourier transform. The analytic signal is a complex extension of a 1D signal that is based upon the Hilbert transform; it was introduced to signal theory by Gabor in 1946 (Ref 3).This representation gives access to the instantaneous amplitude and phase. Several attempts to generalize the analytic signal to two dimensions have been reported in the literature, based on the properties of Hilbert and Riesz transforms (Ref 4)
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