International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 7-4-3 W6, Valladolid, Spain, 3-4 June, 1999
Fig. 4. Flowchart of GLP data fusion procedure for two multi-spectral images, whose scale ratio is p/q. | p indicates down-sampling
by P-1P indicates up-sampling by p, that is interleaving a couple of adjacent samples along a row or a column with p — 1 zeroes.
r p is the p—reduction low-pass filter with frequency cutoff at 1/p and unity DC gain. e p is the p-expansion filter with frequency
cutoff at 1/p and DC gain equal top. A vector of weights {wi, l = 1,..., L} is introduced to equalize spectral contributions.
bic), 11 (fifth-order), 15 (seventh-order), and 23 (eleventh-order)
coefficients have been assessed (Aiazzi, 1997a). The term poly
nomial stems from interpolation and denotes that filtering is equal
to fitting an n—th order polynomial to nonzero samples. The 7-
taps kernel is widespread to yield a bicubic interpolation. In the
case p = 3, two kernels with 17 and 23 taps have been designed
for pyramid generation. For p = 5, a kernels with 29 coeffi
cients has been designed. It is noteworthy that half-band filters
have the even order coefficients, except zeroth, all identically null
(Crochiere, 1983). Analogously for filters with frequency cut-off
at one p-th of bandwidth, coefficients whose order is a nonzero
multiple of p are null as well. Thus, the half-band 23-taps filter
has 13 nonzero coefficients, while the 23-taps filter with cut-off
at 1/3 has 17 nonzero coefficients. The frequency responses of
all the filters are plotted in Figure 3. Frequency is normalized to
the sampling frequency fs which is known to be twice the band
width available to the discrete signal. The filter design stems from
a trade-off between selectivity (sharp frequency cut-off) and com
putational cost (number of nonzero coefficients). In addition, the
assumption of negligible aliasing allows to define the equivalent
filters at the k-th pyramid layer of GP and LP (Ranganath 1991)
and, hence, of GGP and GLP, which turn out to be actually a
low-pass and band-pass structure, respectively.
3. GLP FUSION SCHEME
The flowchart reported in Figure 4 describes the data fusion al
gorithm for the general case of two image data sets, having differ
ent numbers of spectral bands, or equal number but wavelengths
not all the same, and ground scale ratio equal to p/q, that have
been preliminarily registered to each other, or better to the same
cartographic coordinate system. Note that fusion of two data sets
involves two levels of GLP, i.e. base-band at level K — 1.
Let Sj (1) , l = 1,..., L be the first data set made up of L multi-
spectral observations having lower resolution and size M x N.
Let S; (2) , / = 1,..., K be the second data set constituted by
another multi-spectral image having spatial resolution higher by
a factor p/q, but lower spectral resolution (K < L) and size
Mp/q x Np/q. The goal is to obtain a third set of L multi-
spectral images, S^\ each having the same spatial resolution as
S (2) . The upgrade of 5 (1) to the resolution of S (2) is the GLP of
S (2) calculated for k = 0. The images of the set S (1) have to be
expanded by p/q, i.e. interpolated by p and then reduced by q, to
match the scale of the data having finer spatial resolution. Then,
the high-pass component from S (2) is added to the expanded
l = 1,,L, which constitute the low-pass component,
in order to yield either a spatially enhanced set or a spectrally
enhanced set of multi-spectral observations, S\ z \ l = 1,..., L.
Equalization of high-pass features before merging is recom
mended for spatial enhancement, because the images to be fused
may exhibit different contrast. Thus, the high-frequency compo
nents of the data having the higher spatial resolution are weighted
by the ratio between the square root of the variances measured on
the base-bands.
Applicative cases of interest may be: 3 : 1, for spatial enhance
ment of Landsat TM (30 m) through SPOT-P (10 m) (Aiazzi,
1998), 3 : 2 for spectral enhancement of SPOT-XS (20 m)
through LANDSAT TM, 5 : 3 for spectral enhancement of multi-
spectral MOMS-2P (18 m) through Landsat TM. For spectral
enhancement, a decision based on physical congruence must be
embedded in the selection/combination of low-frequency coeffi
cients from the data having higher spectral resolution. Thus, the
weights wi may also be zero for certain couples of bands whose
fusion is a non-sense.