Definition of a Discrete Color Monogenic Wavelet Transform - Advances in Heuristic Signal Processing and Applications - page 329

Fig. 15.1 Felsberg's monogenic signal associated to a narrow-band signal s . Orientation θ is
shown modulo π for visual convenience. Phase values of small coefficients have no sense so they
are replaced by black pixels

The monogenic signal is composed of the three following features:

s 2

2 ,

Amplitude: A = +| R s |

(15.3)

Orientation: θ = arg { R s }∈[− π ; π [ ,

1D Phase: ϕ = arg { s + j | R s |} ∈ [ 0 ; π ] .

A monogenic signal analysis is illustrated on Fig. 15.1 . Felsberg shows a direct link
between the angles θ and ϕ and the geometric local structure of s . The signal is
expressed like an “ A -strong” 1D structure with orientation θ . ϕ is analogous to the
1D local phase and indicates if the structure is a line or an edge. A direct drawback
is that intrinsically 2D structures are not handled.
From a signal processing viewpoint, the AM/FM representation provided by an
analytic signal is accordingly well suited for narrow-band signals. That is why it
seems natural to embed it in a multiresolution transform that performs subband
decomposition. We now present the monogenic analysis proposed in [ 27 ].

15.2.3 Monogenic Multiresolution

The first proposition of analytic wavelets is for the 1D scalar case with the Dual-
tree Complex Wavelet Transform ( WT) in 1999 [ 22 ]. It is a 1D discrete scheme

consisting of two parallel filterbanks which filters are linked by Hilbert transforms.
In fact, the Hilbert transforms are approximate because of discrete constraints. This
method allows near shift-invariance of wavelet coefficients (shift-variance is a fa-
mous problem of classical wavelets).
In 2004, a Quaternion Wavelet Transform (QWT) [ 3 , 8 ] based on the quaternionic
analytic signal of [ 6 ] is proposed for grayscale images. The quaternionic signal is a
2D generalization of the analytic signal that is prior to and maybe less convincing
than the monogenic signal.
Finally, in 2009 a Monogenic Wavelet Transform was proposed in [ 27 ]. This
representation—specially defined for 2D signals—is a great theoretic improvement
of the complex and quaternion wavelets.
It provides 3-vector valued monogenic subbands consisting of a rotation-
covariant magnitude and this new 2D phase . The proposition of [ 27 ] consists of one

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