The Color Wave

The monogenic framework provides local amplitude and phase of 2d "grey" signals, based on a directional model implying local orientation analysis. How can we exploit it for color images? A channelwise approach would give three different amplitude and phase values, which is irrelevant a priori.

Our proposition is to start by studying color oscillations in the context of 1d signal processing. We end up with an elliptical model which elegantly generalizes the amplitude and phase while adding complementary color features. The 2d counterpart (for images) is treated in the next chapter.

All technical details can be found in this article:

Soulard & Carré, Elliptical Monogenic Wavelets... IEEE TSP 2015

The Color Analytic Signal

What is a color oscillation?

Let us assume a vector valued sinusoid which channels are parametrized by an independent amplitude, an independent phase, and a fixed frequency:

On top of the figure are plotted the three separate sinusoids, for red, green and blue channels. When these three waves are considered as coordinates of a path in the 3d color space, the signal turns out to draw an ellipse (see figure). The originality of our proposition is to re-think amplitude and phase concepts by considering the color oscillation as a global elliptical and periodic path.

Matlab code for the "ellipse figure".

The amplitude can then be redefined by the ellipse's size:

The phase is re-interpreted as the relative position of the maximum with respect to the origin. When the signal is elliptical, the maximum can be translated to the apogee. The new color phase is then defined as the angle formed by the path between the origin and the ellipse's apogee.

Given the "color Fourier number"

it can be shown that this definition of the color phase implies:

In the greyscale case, the path oscillates linearly along the grey axis, instead of a true ellipse. In the general case, the path has to be described with complementary features, to specify the ellipse's shape and orientation. These features typically convey colorimetric information.

The first one is the linearity (measuring the difference between the major and minor axes):

This feature encodes the complexity of the color pattern (λ~1 for linear ellipses, λ~0 for round ellipses).

Finally, the ellipse position is encoded through 3 angles (α1,α2,α3), analogous to spherical coordinates. The first 2 angles form the color axis (α1,α2), coinciding with the ellipse's major axis, pointing towards two particular opposite colors at the apogees. This axis will be useful to analyze the color of contours. The third angle α3 encodes the orientation of the ellipse's minor axis (irrelevant when λ~1).

Thanks to this complete description of the ellipse, the 6 original parameters of the color wave (the "color Fourier number") can be re-written as a function of its 6 features:

This rewriting of the "color Fourier number" can be seen as a conversion of the channel-wise amplitude and phase features given in the first place. The classical features can be identified in the first part of the equation. The second part - in brackets - is due to the increasing of the dimension, with the new color features.

The color analytic signal can now be simply defined as the time-varying version of the "color Fourier number". Given a color signal s(t), the analytic extension is defined as:

The real part is the input signal s(t) and the imaginary part is given by the channelwise Hilbert transform. Time-varying elliptical features can be computed from the given equations (see the .pdf article for details), and the signal can be re-written as a function of them. The amplitude and phase envelopes of an example color signal are given on the last figure. The plotted phase values are according to our perception. For example, the green and pink bars are clearly perceived as the "color maxima" within their pattern, which properly corresponds to phase values of 0 or π.

Let us now try to settle the elliptical model into the monogenic framework, to finally deal with color images.

Matlab script for the color analytic signal.
Dependencies (ellipse conversion and utility).

* We proposed a slightly different model in [Soulard IEEE TIP 2013] limited to a color axis concept (no ellipse).
* The elliptical model is highly due to [J. Lilly "Modulated oscillations in three dimensions" IEEE TSP 2011]
* A 2-valued "oscillating path" was studied in [Le Bihan & Sangwine "Quaternionic spectral analysis of non stationary improper complex signals” Proc. ICCA9 2011]
* A color monogenic signal was proposed in [Demarcq et al. "The color monogenic signal... JMIV 2011]