Definition of a Discrete Color Monogenic Wavelet Transform - Advances in Heuristic Signal Processing and Applications - page 328

15.2 Analytical Signal and 2D Generalization

This section recalls existing definitions around the analytic signal and the mono-
genic signal. The multiscale aspect will be presented through an overview of ex-
isting analytic wavelets followed by a more detailed description of the monogenic
wavelets by Unser et al. [ 27 ].

15.2.1 Analytic Signal (1D)

An analytic signal s A is a multi-component signal associated to a real signal s to be
analyzed. The definition is well known in the 1D case where s A (t) = s(t) + j (h ∗

1

s)(t) is the complex signal made of s and its Hilbert transform (with h(t)

πt ).
The polar form of the 1D analytic signal provides an AM/FM representation
of s with

=

arg (s A ) the instantaneous
phase . This classical tool can be found in many signal processing topics and is
used in communications, for example. The growing interest in this tool within the
image community is due to an alternative interpretation of amplitude, phase, and
frequency in terms of a local geometric shape. We can interpret the phase in terms
of a signal shape. Such a link between a 2D phase and local geometric structures of
images would be very attractive in image processing. That is why there were several
attempts to generalize it for 2D signals; and among them the monogenic signal [ 12 ]
seems the most advanced since it is rotation invariant.

| s A | being the amplitude envelope and ϕ =

15.2.2 Monogenic Signal (2D)

We here review the key points of the fundamental construction of the monogenic
signal, which will be necessary to understand the color extension. The 2D extension
of the analytic signal has been defined in several ways [ 6 , 14 , 15 ]. We are interested
in the monogenic signal [ 14 ] because this is rotation invariant and its generalization
is according to both fundamental definition and signal interpretation. Given a 2D
real (scalar) signal s , the associated monogenic signal s M is 3-vector valued (instead
of complex-valued in the 1D case) and must be taken in spherical coordinates:

⎡ ⎤ ⎡ ⎤

s
{ R

A cos ϕ
A sin ϕ cos θ
A sin ϕ sin θ

⎣ ⎦ = ⎣ ⎦

s M = s } (15.1)

{ R s }

where R s is the complex-valued Riesz transform of s :

p.v. τ 1 +

j τ 2
2 π τ

ω 2 − j ω 1

ω

F
←→

{ R s } ( x ) =

3 s( x

− τ )d τ ˆ s( ω ). (15.2)