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.