Local Amp. & Phase: 2D

Local 2d amplitude and phase have long been handled by windowed 2d Fourier transforms, and quadrature 2d filters, more or less linked to the concept of partial Hilbert transform, which raises several theoretical problems about directionality in the 2d space. After several attempts (In particular, the quaternionic signal by Bülow & Sommer), the Riesz transform have shown to be the most successful definition since 2002.

Proposed simultaneously by M. Felsberg and O. Larkin in 2001, the monogenic representation is the 2d extension of the analytic signal. It can be used to analyze the contours in greyscale images, based on a directional model. The advantage over previous methods is that it is based on optimal local orientation estimation, which makes it perfectly steerable (not biased by rotation of the 2d sample grid).

The Riesz Transform

The Riesz transform is defined as:

This operator locally acts as a Hilbert transform towards the main local direction of the signal under analysis. It so includes an optimal local orientation detector, very similar to the well known Gradient operator.

In practice, the Riesz transform is well approximated by sampling the above given Fourier response in the 2d FFT domain (like with the Hilbert transform). It outputs two 2d signals: the "x-" and "y-" components. In the above definition, the two components are embedded as real and imaginary part, but other algebraic embedding can be found, without any numerical difference.

The Riesz transform removes the DC component of the input signal and is usually applied jointly with a linear-phase isotropic band-pass filter.

The best 2d extension of the analytic signal turns out to be very similar to the classical 2d extension of the derivative - the Gradient operator. In practice, the Riesz transform norm and direction can be interpreted as an edge detector and a local main orientation measure.

It produces the well-known "double-response" around line-shaped contours, which are seen like two opposite edges. Gradient-like filters are optimal for edges whereas Laplacian-like filters are optimal for peaks and lines. For example, the isotropic band-pass filter used above has a well-centerd response on lines and a double-response around edges. We are going to see that combining both analyzes can provide a unified representation as optimal for lines as for edges.

Matlab script for the Riesz transform (no dependency).
'face4b.png': Test image.

Mathematical references about the Riesz transform:
* M. Riesz, "Sur les fonctions conjugées." Math. Zeitschr, 1927
* A. Calderón & A. Zygmund, "On the existence of certain singular integrals." Acta Math., 1952.
* E. Stein & G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces." Princeton Univ. Press, NJ, 1971
* M. Nabighian, "Toward a three-dimensional automatic interpretation of potential field..." Geophysics, 1984.

The Monogenic Signal

The monogenic signal has been proposed by Felsberg & Sommer in 2001 as the 2d generalization of the analytic signal. The Riesz transform replaces the Hilbert transform so as to redefine amplitude and phase. A simple way to compute the monogenic 2d amplitude and phase is to combine the band-pass version and the Riesz norm as a complex number:

The monogenic amplitude (modulus) performs optimal contour detection, with a well centered single response on both edges (bottom of the face) and lines (mouth and hair), as well as isotropic peaks (eyes). This unified handling of different contour shapes is called phase-invariant contour detection.

The phase information (argument) comes as a complementary information that classifies the detected contour. The phase equals 0 or π (red) on lines and ±π/2 (blue) on edges.

The Riesz direction comes as part of the monogenic model, by specifying the local spatial direction towards which the amplitude and phase concepts make sense.

Now that the amplitude and phase concepts are well defined for greyscale images, let us present our color extension.

Matlab script for the monogenic signal (no dependency).
'face4b.png': Test image.

References for the monogenic signal:
* Larkin et al., “Natural demodulation of two dimensional fringe patterns", J. Opt. Soc. Am., 2001.
* Felsberg & Sommer, “The monogenic signal”, IEEE Trans. Signal Process., 2001.

See also "AM-FM image model" and "Synchrosqueezing":
* A. Bovik. "Handbook of Image and Video Processing", Academic Press, 2000]
* M. Clausel et al. " The Monogenic Synchrosqueezed Wavelet Transform... ", Applied and Comp. Harmonic Analysis, 2015