Multiresolution Monogenic Signal Analysis Using the Riesz–Laplace Wavelet Transform

The monogenic signal is the natural 2D counterpart of the 1D analytic signal. We propose to transpose the concept to the wavelet domain by considering a complexified version of the Riesz transform which has the remarkable property of mapping a real-valued (primary) wavelet basis of L
₂
(R
²
) into a complex one. The Riesz operator is also steerable in the sense that it give access to the Hilbert transform of the signal along any orientation. Having set those foundations, we specify a primary polyharmonic spline wavelet basis of L
₂
(R
²
) that involves a single Mexican-hat-like mother wavelet (Laplacian of a B-spline). The important point is that our primary wavelets are quasi-isotropic: they behave like multiscale versions of the fractional Laplace operator from which they are derived, which ensures steerability. We propose to pair these real-valued basis functions with their complex Riesz counterparts to specify a multiresolution monogenic signal analysis. This yields a representation where each wavelet index is associated with a local orientation, an amplitude and a phase. We give a corresponding wavelet-domain method for estimating the underlying instantaneous frequency. We also provide a mechanism for improving the shift and rotation-invariance of the wavelet decomposition and show how to implement the transform efficiently using perfect-reconstruction filterbanks. We illustrate the specific feature-extraction capabilities of the representation and present novel examples of wavelet-domain processing; in particular, a robust, tensor-based analysis of directional image patterns, the demodulation of interferograms, and the reconstruction of digital holograms.