Definition of a Discrete Color Monogenic Wavelet Transform - Advances in Heuristic Signal Processing and Applications - page 338

Fig. 15.6 The color
monogenic wavelet transform
( γ = 3)

We have first to compute the 6 subband decompositions. The amplitude, phase,
and color axis can be retrieved with Eq. ( 15.28 ), and the orientation with Eq. ( 15.26 ).
Figure 15.6 illustrates the multiscale color monogenic features obtained from our
color monogenic wavelet transform.

15.4.4 Algorithm Discussion

In this subsection, let us give some practical remarks. The monogenic analysis
is basically defined in a continuous framework. Constraints related to filterbank
design—perfect reconstruction, small redundancy—conflict with desirable proper-
ties of isotropy and rotation invariance. The important bridge to discrete implemen-
tation and use in practical applications is tenuous at best. The choice that is made
by Unser et al. [ 27 ] is to provide the 'minimally-redundant wavelet counterpart of
Felsberg's monogenic signal'. The presented color monogenic wavelet transform is
by extension in the same spirit. Since filters cannot be exactly isotropic, the analysis
is expected to mildly favor some directions. In addition, the subbands are highly
subsampled (yet not 'critically' since the number of coefficients is larger than the
number of pixels), implying that the phase data is varying fast with respect to sam-
pling.
In the last part of this chapter, we believe that such a signal processing tool must
be studied from a discrete viewpoint. This last part presents a new approach that
aims at representing color information with a discrete color Monogenic Wavelet
transform based on discrete Monogenic transform.
We address this issue by introducing a scheme that uses a discrete Radon trans-
form based on discrete geometry. The advantage is that extensions to filterbanks and
to higher dimensions are facilitated, thanks to the perfect invertibility and computa-
tional simplicity of the used Radon algorithm.
The analysis now presented is based on two facts:

• There is a fundamental link between the monogenic framework and the Radon

transform [ 18 ];

• 'True' discrete counterparts of Radon transforms have already been defined (for

example, in [ 7 ]).