Definition of a Discrete Color Monogenic Wavelet Transform - Advances in Heuristic Signal Processing and Applications - page 330

real-valued “primary” wavelet transform in parallel with an associated complex-
valued wavelet transform. Both transforms are linked by the Riesz transform so
they carry out a multiresolution monogenic analysis. We end up with 3-vector coef-
ficients forming subbands that are monogenic.

15.2.3.1 Primary Transform

The primary transform is real-valued and relies on a dyadic pyramid decomposition
tied to a wavelet frame. Only one 2D wavelet is needed and the dyadic downsam-
pling is done only at the low frequency branch; leading to a redundancy of 4:3. The
scaling function ϕ γ is defined in the Fourier domain:

sin 2 ω 2 ) γ 2 ( 4 ( sin 2 ω 2 + sin 2 ω 2 )

3 sin 2 ω 2

8

−

ϕ γ ←→

(15.4)

ω

γ

and the mother wavelet ψ is defined as

) γ 2 ϕ 2 γ ( 2 x )

ψ( x ) = ( − (15.5)

where is the Laplacian operator and ϕ γ is a cardinal polyharmonic spline of order
γ and spans the space of those splines with its integer shifts. It also generates—as
a scaling function—a valid multiresolution analysis. This particular construction
is made by an extension of a wavelet basis (non-redundant) related to a critically-
sampled filterbank. In addition, a specific subband regression algorithm is used at
the synthesis side. The construction is fully described in [ 28 ].

15.2.3.2 The Monogenic Transform

The second “Riesz part” transform is a complex-valued extension of the primary
one with the Riesz transform:

x
2 π

ψ( x ) j y

2 π

ψ( x ) . ψ =−

3 ∗

+

3 ∗

(15.6)

x x

It can be shown that it generates a valid wavelet basis and that it can be extended
to the pyramid described above. The joint consideration of both transforms form
monogenic subbands from which the amplitude and phase can be extracted for an
overall redundancy of 4:1.
In [ 27 ], a demonstration of AM/FM analysis is done with fine orientation estima-
tion and gives very good results in terms of coherency and accuracy. Accordingly,
this tool may be rather used for analysis tasks than processing. We propose now to
describe a first generalization for color images.