International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 7-4-3 W6, Valladolid, Spain, 3-4 June, 1999 
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(a) (b) 
Fig. 2. GGP (a) and GLP (b), both with scale ratio p/q — 3/2 
and layers 0 to 3, of a 513 x 513 (513 divides by 3, 512 does 
not)) detail from Landsat TM Band 5 of Elba Island. 
Frequency response 
(a) 
Frequency response 
while (3) becomes expansion by p followed by reduction by q: 
expand p / q [Gk] = reduce q {expand p [Gk]}- (7) 
The Generalized Laplacian Pyramid (GLP) (Aiazzi, 1997a) with 
p/q scale factor between two adjacent layers, Lk, can thus be 
defined, for k = 0,..., K — 1, as 
Lk (m, n) = Gk(m,n) — expand p/q [Gk+i](rn,n) (8) 
which may also be written, by replacing (6) in (8), as 
Lk(m,n) = Gk(m,n) — expand p / q {reduce p / q [Gk]}{m,n) 
(9) 
Again, the base-band layer is added to the GLP: L/c(m,n) = 
Gk (m, n), to yield a complete multi-resolution description, suit 
able for merging data imaged with a p/q scale ratio, p and q being 
two integers which are assumed to be prime to each other to yield 
a unique representation of all possible cases, and thus to avoid 
designing different and unnecessarily cumbersome filters. 
Figure 2 portrays the GGP and the GLP (p/q — 3/2) of the 
sample TM-5 image. Notice the non-octave low-pass and band 
pass behaviours of the former and of the latter. 
(b) 
Frequency response 
(C) 
Fig. 3. Frequency responses of pyramid-generating filters: (a) 
polynomial half-band, (b) with cut-off at one-third of band 
width, (c) with cut-off at one-fifth of bandwidth (29 coeffi 
cients or taps. Reduction filters are used for expansion, with 
a DC (or zero-frequency) gain set equal to their expansion 
factors 2, 3 and 5. 
2.3 Pyramid-generating filters 
Filters with different frequency responses are to be designed 
to сорю with the scaling requirements of the pyramid algorithm. 
In particular, for a p/q scale ratio, p > q, if normalized fre 
quency is considered, only one filter with 1 /p cut-off is needed. 
In fact, when (4) is cascaded to (5), the low-pass filtering step 
can be omitted after up-sampling by q (t q) in (6), as well as 
before down-sampling by q (| q) in (7), by assuming that fil 
ters are close to exhibit ideal frequency cut-offs, i.e rectangular 
responses. Although this assumption would require an infinite 
number of coefficients (the samples of a sine function), it may be 
approximated with a reasonable number. 
In the case p = 2, polynomial kernels with 3 (linear), 7 (cu-