
the filter of the first stage. This gives the multiscale property with-
out the need for additional filter design. The proposed structure is
thus different from the separable NSWT. In particular, one band-
pass image is produced at each stage resulting in J + 1 redundancy.
By contrast, the NSWT produces three directional images at each
stage, resulting in 3J + 1 redundancy. The advantage of NSP is that
it is general and as a result, better filters can be obtained.
2.2. Nonsubsampled directional filter bank (NSDFB)
The directional filter bank of Cunha et al. (2006) is constructed
by combining critically-sampled two-channel fan filter banks and
resampling operations. The result is a tree-structured filter bank
that splits the 2-D frequency plane into directional wedges. A
shift-invariant directional expansion is obtained with a nonsub-
sampled DFB (NSDFB). The NSDFB is constructed by eliminating
the downsamplers and upsamplers in the DFB. This is done by
switching off the downsamplers/upsamplers in each two-channel
filter bank in the DFB tree structure and upsampling the filters
accordingly. This results in a tree composed of two-channel Non-
subsampled Filter Banks (NSFB). Fig. 3 illustrates a four channel
decomposition.
Note that in the second level, the upsampled fan filters U
i
(z
Q
),
i = 0, 1 have checker-board frequency support, and when com-
bined with the filters in the first level give the four directional
frequency decomposition shown in Fig. 3. The synthesis filter
bank is obtained similarly. Just like the critically-sampled direc-
tional filter bank, all filter banks in the nonsubsampled direc-
tional filter bank tree structure are obtained from a single NSFB
with fan filters. Moreover, each filter bank in the NSDFB tree
has the same computational complexity as that of the building-
block NSFB.
2.3. Combining the nonsubsampled pyramid and nonsubsampled
directional filter bank in the NSCT
The NSCT is constructed by combining the NSP and the NSDFB
as shown in Fig. 1(a). In constructing the NSCT, care must be taken
when applying the directional filters to the coarser scales of the
pyramid. Due to the tree-structure nature of the NSDFB, the direc-
tional response at the lower and upper frequencies suffers from ali-
asing which can be a problem in the upper stages of the pyramid.
This is illustrated in Fig. 4(a), where the passband region of the
directional filter is labeled as ‘‘Good” or ‘‘Bad.” Thus, we see that
Highpass
Subband
Image
Lowpass
subband
Bandpass
Directional
Subband
s
2
ω
)(
ππ ,
)(
ππ−− ,
1
ω
(a)
(b)
Highpass
Subband
Bandpass
Directional
Subband
s
Fig. 1. Nonsubsampled contourlet transform: (a) Nonsubsampled filter bank (NSFB) structure that implements the NSCT, (b) Idealized frequency partitioning obtained with
the NSCT structure.
Fig. 2. The NSP is a 2-D multiresolution expansion similar to the 1-D NSWT: (a) Three-stage pyramid decomposition. The lighter gray regions denote the aliasing caused by
upsampling, (b) Subbands on the 2-D frequency plane.
2084 P.-P. Niu et al. / Expert Systems with Applications 38 (2011) 2081–2098