Definition of a Discrete Color Monogenic Wavelet Transform - Advances in Heuristic Signal Processing and Applications - page 345

Fig. 15.9 Principle of the undecimated monogenic Wavelet transform

The scaling function is the B 3 -Spline, and the associated filter H is

1
16 z − 2

1
4 z − 1

3
8 +

1
4 z

1
16 z 2 .

H(z) = + + + (15.37)

The limitation of this definition is that filters are not quadrature mirror filters.
A consequence is the nonorthogonal decomposition, with a correlation between
scales.
And finally, to compute the second “Riesz part”, we apply on each bandpass fil-
tering signal the process previously presented (Discrete Radon transform, Hilbert
transform, normalization, and inverse Radon transform). An example of the princi-
ple of the decomposition is given in Fig. 15.9 .
This last part shows that a discrete monogenic analysis can be performed in the
Radon domain by using existing Radon transform algorithms. The improvement
over the classical FFT-based method may become significant in more developed
processes like wavelet transforms. The existing discrete Radon transform is a good
tool having exact reconstruction and computational simplicity, thanks to discrete
geometry.

15.6 Conclusion

In this chapter, we introduce and analyze some numerical color extensions of the
monogenic wavelet transform. This new transform is a geometric non-marginal