International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 7-4-3 W6, Valladolid, Spain, 3-4 June, 1999
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2. MULTIRESOLUTION IMAGE ANALYSIS
2.1 Gaussian and Laplacian pyramids
The Laplacian pyramid is derived from the Gaussian pyramid
(GP) which is a multi-resolution image representation obtained
through a recursive reduction (low-pass filtering and decimation)
of the image data set.
Let Go(m, n), m = 0,..., M — 1, and n = 0,..., N — 1,
M = u x 2 h , N — v x 2 h , be a grey-scale image. The classic
Burt’s GP (Burt, 1983, 1984) is defined with a decimation factor
of 2 (| 2) as
L r L r
Gk(m,n) = reduce2[Gk-i](rn, n) = EE
i=-L r j = -L r
T2(i) X r 2 (j)Gk-1 (2m + i, 2n + ;') (1)
for A; = m = 0, ...,M/2 k — 1, and n =
0,..., N/2 k — 1; in which k identifies the level of the pyra
mid, K being the top, or root, or base-band, of size u x v. The
2D reduction low-pass filter is given as the outer product of a
linear symmetric kernel, generally odd-sized, i.e. {r 2 (i), i =
—L r , ■ ■ •, L r } which should have the —3 dB cut-off at one half
of the bandwidth of the signal, to minimize the effects of aliasing
(Crochiere, 83), although this requirement was not always strictly
observed (Burt, 1983, 1984).
From the GP, the enhanced LP (ELP) (Baronti, 1994; Aiazzi,
1997b) is defined, for k = 0,..., K — 1, as
Lk(m,n) = Gk{rn,n) - expand2[Gk+i\(m,n) (2)
in which expand2[Gk+1] denotes the (k + l)st GP level ex
panded by 2 to match the size of the underlying Ac-th level:
L e L e
expand2[Gk+i](rn,n) — EE e 2 (i) x e 2 (j)
i=-L e j — — L e
(j+n) mod 2=0
(i+m) mod 2=0
X (3)
for m = 0,..., M/2 k - 1, n = 0,..., N/2 k - 1, and k =
0,..., K — 1. The 2D low-pass filter for expansion is given as the
outer product of a linear symmetric odd-sized kernel {e 2 (z), i =
—L e , • • •, L e }, which must cut-off at one half of the bandwidth of
the signal to reject the spectral images introduced by up-sampling
by 2 (j' 2) (Crochiere, 1983). Summation terms are taken to be
null for noninteger values of (i + m)/2 and (j + n)/2, corre
sponding to interleaving zeroes. The base-band is usually added
to the band-pass ELP, that is Lk (m, n) = Gk (m, n), to yield a
complete multi-resolution image description.
The attribute enhanced (Baronti, 1994) depends on the expan
sion filter being forced to be half-band, i.e. an interpolator by 2
(Crochiere, 1983), and chosen independently of the reduction fil
ter, which may be half-band as well, or not. The ELP outperforms
Burt’s LP for image compression (Aiazzi, 1997b), incidentally by
using different filters (Burt’s Gaussian-like kernel for reduction),
thanks to its layers being almost completely uticorrelated with
one another.
(a) (b)
Fig. 1. GP (a) and ELP (b) layers 0 to 3 of 512x512 detail from
Landsat TM Band 5 portraying Elba Island and Tyrrhenian
Sea, in Italy.
Figure 1 shows the GP (1) and ELP (2) of a typical optic re
motely sensed image. Notice the low-pass octave structure of GP
layers, as well as the band-pass octave structure of ELP layers.
2.2 Generalized Laplacian pyramid
When the desired scale ratio is not a power of 2, but a ratio
nal number, (1) and (3) need be generalized to deal with rational
factors for reduction and expansion (Kim, 1993).
Reduction by an integer factor p is defined as
L r L r
reduce p [Gk](rn,n) = EE r P (i) x r p (j)
i=-L r j=-L r
x G k (pm -\-i,pn + j). (4)
The reduction filter {r p (i), i = —L r , ■ ■ ■, L r } must cut-off at
one p-th of bandwidth, to prevent from introduction of aliasing.
Analogously, an expansion by p is defined as
Le L e
expand p [Gk](m,n) = EE e p (i) x e p (j)
Le j = Le
(j+n) mod p=0
(i+m) mod p=0
x Gl fi±E,i±»y (5)
V p p J
The low-pass filter for expansion {e p (i), i = -L e ,- ■ ■ ,L e }
must cut-off at one p-th of bandwidth. Summation terms are
taken to be null for noninteger values of (i + m)/p and (j + n)/p,
corresponding to interleaved zero samples.
If p/q > 1 is the desired scale ratio, (1) modifies into the
cascade of an expansion by q and a reduction by p, to yield a
generalized GP (GGP) (Kim 1993):
Gk+i = reduce p / q [Gk] = reduce p {expand q [Gk]} (6)