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)