Definition of a Discrete Color Monogenic Wavelet Transform - Advances in Heuristic Signal Processing and Applications - page 344

which implies a correct data restoration of one scale to the other, and

π/ 2 )H(ω) π/ 2 )G(ω) H(ω + + G(ω + = 0 (15.34)

which represents the compensation of recovery effects introduced by the downsam-
pling.
Because of decimation, the Mallat's decomposition is completely time variant.
Moreover, it is difficult to define a 2D filter bank associated with one 2D isotropic
bandpass component at each scale.
A simple way to obtain a time-invariant and isotropic system is to compute all
the integer shifts of the signal. This algorithm was named algorithm “à trous”[ 16 ]
and its link with the Mallat's algorithm is discussed in [ 23 ]. Because the decom-
position is not decimated, filters are dilated between each projection. Therefore, in
the signal case, each wavelets' scale has the same number of points as the original
signal. For the scale L , these N points correspond to 2 L different decompositions
obtained with the decimated transform using all the circulant shifts of the signal.
These decompositions, each one composed of N/ 2 L points, are intertwined.
The algorithm “à trous” presents many advantages:

• A simpler filter selection . Condition ( 15.34 ), which was required for a perfect

reconstruction, is no longer necessary because coefficients are no longer down-
sampled.

• Knowledge of all wavelets' coefficients . Coefficients removed during the down-

sampling are not necessary for a perfect reconstruction, but they may contain
information and are necessary to obtain a time invariant decomposition.

In this work, we propose a Monogenic Wavelet transform using the decomposi-
tion “à trous”. For a perfect reconstruction, the algorithm “à trous” requires that the
filters verify condition ( 15.33 ). A lot of works (for example, [ 4 ]) propose defining
the highpass decomposition filter G(ω) as

G(ω) = 1 − H(ω) (15.35)

where H(ω) is the lowpass filter, and the reconstruction filters are defined as

H(ω) = G(ω) =

1 . (15.36)

It is easy to verify that filters defined in Eqs. ( 15.35 ) and ( 15.36 ) satisfy
Eq. ( 15.33 ) for all H(ω) .
From Eq. ( 15.35 ), wavelets' coefficients are simply computed by the difference
between two successive smoothed sequences, and the reconstruction is the sum of
all wavelets' scales, plus the smoothed signal at the coarsest scale.
The generalization to the 2D case is done by the application of the lowpass fil-
ter along the two directions. Wavelets' coefficients are computed by the difference
between two successive smoothed sequences and consequently are associated to
isotropic bandpass.