Local Amp. & Phase: 1D
Our research makes an intensive use of local amplitude and phase concepts, coming from the fundamental signal processing theory. While the Fourier transform provides this information for the whole time domain, the analytic signal presented here provides instantaneous Fourier-like analysis. This concept will be extended to the world of color images in the subsequent chapters.
The Hilbert Transform
The Hilbert transform - building block of the analytic signal - is written as a principal value integral having a well defined Fourier domain expression (see figure). It acts as a pure π/2 phase shifting filter with unitary gain (excepted that it removes the DC component). The obtained phase shifted version of the input signal shows a similar oscillating behavior, but 90° out of phase.
In practice, the Hilbert transform is numerically approximated by discretizing its Fourier response in the FFT domain, which is generally a very good approximation (see Matlab code below).
The Analytic Signal
The analytic signal is defined by the complex valued extension of a signal made of itself as the real part and its Hilbert transform as its imaginary part.
The complex modulus and argument of the analytic signal convey two fundamental physical quantities: the amplitude envelope and the local phase ("instantaneous"). These are classically used for AM-FM demodulation, where the instantaneous frequency is obtained by differentiating the phase.
The famous Gabor wavelets process the analytic signal in different frequency bands, to perform multiscale local amplitude and phase analysis. In general, this kind of analysis is meaningful for band-limited signals (oscillating), which suggests the combination with fiterbanks or wavelet transforms.
Considering this tool with a view towards images, the correspondance between the phase value and the signal's local shape is of great interest.
Matlab script for the Hilbert transform and the analytic signal (no dependency).
References: * J. Ville, “Théorie et applications de la notion de signal analytique”, Cables et transmission, 1948. * D. Gabor, “Theory of communication”, J. Inst. Elec. Eng., 1946.