Digital Signal Processing Reference
In-Depth Information
c
j+1
c
j
c
j
h
h
2
2
w
j+1
-
c
j+1
h
2
c
j
+
w
j+1
Figure 3.14. Laplacian pyramid transform. (top) Analy-
sis, passage from
c
j
to the detail, and smooth coeffi-
cients
w
j
+1
and
c
j
+1
. (bottom) Usual reconstruction.
where
P
+
is the projection on the positive orthant. The last iterative scheme can
also be interpreted as an alternating projections onto convex sets (POCS) algorithm
(Starck et al. 2007). It has proven very effective at many tasks, such as image ap-
proximation and restoration, when using the UWT (Starck et al. 2007).
3.7 PYRAMIDAL WAVELET TRANSFORM
3.7.1 The Laplacian Pyramid
The Laplacian pyramid was developed by Burt and Adelson (1983) to compress
images. The term
Laplacian
was used by these authors for the difference between
two successive levels in a pyramid, defined itself in turn by repeatedly applying a
low-pass (smoothing) filtering operation. After the filtering, only one sample out
of two is kept. The number of samples decreases by a factor 2 at each scale. The
difference between images is obtained by expanding (or interpolating) one of the
pair of images in the sequence associated with the pyramid.
The 1-D Laplacian pyramid transform and its usual inverse are depicted in
Fig. 3.14. First, the scaling coefficients are computed by discrete convolution and
down-sampling (see Fig. 3.14 (top)):
[
h
c
j
+
1
[
l
]
=
h
[
k
−
2
l
]
c
j
[
k
]
=
c
j
]
↓
2
[
l
]
.
(3.26)
k
Then the detail coefficients
w
1
are computed as the difference
j
+
w
j
+
1
[
l
]
=
c
j
[
l
]
−
c
j
[
l
]
,
(3.27)
where
c
j
is the prediction obtained by zero-interpolation and discrete convolution:
2
k
c
j
[
l
]
=
h
[
l
−
2
k
]
c
j
+
1
[
k
]
=
2(
h
c
j
+
1
)[
l
]
.
(3.28)
The scaling coefficients
c
j
are reconstructed from
c
j
+
1
and
w
j
+
1
, as illustrated in
Fig. 3.14 (bottom), that is,
c
j
[
l
]
=
2(
h
c
j
+
1
)[
l
]
+
w
j
+
1
[
l
]
.
(3.29)
