IEEE Geoscience and Remote Sensing Magazine - September 2013 - 13
j -1
j
H eq, l (z) =
H
j
eq, h
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Normalized Frequency
Figure 4. Equivalent filters frequency responses obtained from
8-tap Daubechies orthogonal wavelets [34].
r
~y
HH
LH
HH
HL
LL
HL
r
HH
LH
(z) = = % H 0 (z )G $ H 1 (z
HH
-r
(a)
Lowpass
Subband
x(n)
Bandpass
Directional
Subbands
Bandpass
Directional
Subbands
m
2m
~x
0
-r
% H 0 (z 2 ),
m =0
j -2
(b)
2 j -1
),
(22)
m =0
where the subscripts l and h refer to the approximation (low
pass) and detail or wavelet (bandpass and high pass) signals, whereas j denotes the level of the decomposition. An
example of the equivalent filters frequency responses, relative to a four level decomposition, is shown in Fig. 4.
September 2013
1
Magnitude
modeling in the transformed domain, otherwise the spatial
adaptivity may get lost in favor of the scale adaptivity.
The wavelet analysis provides a multiresolution representation of continuous and discrete-time signals and
images [35]. For discrete-time signals, the classical maximally decimated wavelet decomposition is implemented
by filtering the input signal with a low pass filter H 0 (z) and
a high pass filter H 1 (z) and downsampling each output by
a factor two. The synthesis of the signal is obtained with a
scheme symmetrical to that of the analysis stage, i.e., by
upsampling the coefficients of the decomposition and by
low pass and high pass filtering. Analysis and synthesis filters are designed in order to obtain the perfect reconstruction of the signal and by using different constraints (e.g.,
orthogonal or biorthogonal decomposition, linear phase
filters). Applying the same decomposition to the low pass
channel output yields a two-level wavelet transform: such
a scheme can be iterated in a dyadic fashion to generate a
multilevel decomposition. The analysis and synthesis stages
of a two-level decomposition are depicted in Fig. 3-(a).
In several image processing applications, e.g., compression, the DWT is particularly appealing since it compacts
energy in few coefficients. However, for most of the tasks
concerning images, the use of an undecimated discrete
wavelet transform (UDWT) is more appropriate thanks
to the shift-invariance property. UDWT is also referred
to as stationary WT (SWT) [37], [38], as opposite to Mallat's octave (dyadic) wavelet decomposition DWT [35],
which is maximally, or critically, decimated. The rationale
for working in the UDWT domain is that in DWT, when
coefficients are changed, e.g., thresholded or shrunk,
the constructive aliasing terms between two adjacent subbands are no longer canceled during the synthesis stage,
thereby resulting in the onset of structured artifacts [39].
As to the construction of the UDWT, it can be shown
that if we omit the downsamplers from the analysis stage
and the upsamplers from the synthesis stage, then the
perfect reconstruction property can still be achieved. The
relative scheme for a two-level decomposition is depicted
in Fig. 3-(b). In the block diagram, by applying the noble
identities [40], the downsamplers (upsamplers) have been
shifted towards the output (input) of the analysis (synthesis) stage. Eliminating these elements yields the UDWT. As
a consequence, the coefficients in the transform domain
can be obtained by filtering the original signal by means of
the following equivalent transfer functions:
ieee Geoscience and remote sensing magazine
Figure 5. Frequency splitting from a single-level separable DWT (a),
obtained by low pass (L) and high pass (H) filtering along the rows
and the columns (LL, HL, LH, and HH denote all possible combination); in (b), the splitting obtained from the nonsubsampled Laplacian
pyramid decomposition (on the left) and the nonsubsampled directional filter banks (on the right) composing the contourlet transform.
13
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