Definition of a Discrete Color Monogenic Wavelet Transform - Advances in Heuristic Signal Processing and Applications - page 331

15.3 First Extension of Color Monogenic Wavelet

Our first proposition of color Monogenic Wavelet combines a fundamental gener-
alization of the monogenic signal to color with the monogenic wavelets described
above. The challenge is to avoid the classical marginal definition that would be ap-
plying a grayscale monogenic transform on each of the three color channels of a
color image.

15.3.1 The Color Monogenic Signal

Starting from Felsberg's approach that is originally expressed in the geometric al-
gebra of

3 , the extension proposed in [ 10 ] is written in the geometric algebra

R

5 . By simply increasing the dimension, we can embed each color channel
along a different axis, and the original equation associated with the monogenic
signal from Felsberg involving a 3D Laplace operator can be generalized in 5D.
Then, the system can be simplified by splitting it into three systems with a 3D
Laplace equation, reducing to an application of Felsberg's condition to each color
channel. Instead of naively applying the Riesz transform to each color channel,
this fundamental generalization carries out the following color monogenic signal:
s A = (s R ,s G ,s B ,s r 1 ,s r 2 ) where s r 1 and s r 2 are the Riesz transforms applied to
s R + s G + s B [ 10 ]. Now, that the color extension of Felsberg's monogenic signal is
defined, let us construct the color extension of the monogenic wavelets.

of R

15.3.2 The Color Monogenic Wavelet Transform

We can now define a wavelet transform whose subbands are color monogenic sig-
nals. We can simply use the transforms presented above by applying the primary one
on each color channel and the Riesz part on the sum of the three. The five related
color wavelets forming one color monogenic wavelet ψ A are:

⎛
⎝

⎞
⎠ ,

⎛
⎝

⎞
⎠ ,

⎛
⎝

⎞
⎠ ,

ψ
0
0

0
ψ
0

0
0
ψ

ψ R = G = B =

(15.7)

⎛
⎝

⎞
⎠

⎛
⎝

⎞
⎠

y

x

∗ ∗

ψ ψ

3 3

2 π x 2 π x

y

x

∗ ψ ∗ ψ

ψ r 1 = , r 2 = , (15.8)

3 3

2 π x 2 π x

y

x

∗ ψ ∗ ψ

3 3

2 π x 2 π x

ψ A = (ψ R ,ψ G ,ψ B ,ψ r 1 ,ψ r 2 ). (15.9)