International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 7-4-3 W6, Valladolid, Spain, 3-4 June, 1999 
194 
broad spectral reflective characteristics, edges are identical in 
most spectral bands and only vary in strength and polarity. 
(Tom, 1986; Schowengerdt, 1980). Therefore, a local 
correlation between different spectral bands can be assumed, 
even when no significant global correlation exists. The basic 
assumption to make this relation usable for image fusion is that 
a local correlation, once identified between a multispectral 
channel and the degraded panchromatic band should also apply 
to the higher resolution level. Consequently, the calculated local 
regression coefficients and residuals can be applied to the 
corresponding area of the high resolution panchromatic band. 
The required steps to implement this local correlation modelling 
approach (LCM) are the following (Figure 2): The 
geometrically co-registered panchromatic band is blurred to 
match the equivalent resolution of the multispectral image. The 
regression analysis within a small moving window (e.g. 5x5 
pixel) is applied to determine the optimal local modelling 
coefficients and the residual errors for the pixel neighbourhood 
using a single multispectral and the degraded panchromatic 
band. Thus, 
multi j 1 ™ = a j k>w +b j low * pan low +ej low 
(3) 
ej low = multi/™- (a j low +b j low * pan low ) 
(4) 
where [a j low ] and [b j tow ] are the coefficients and [e j*° w ] the 
residuals derived from the local regression analysis. The actual 
resolution enhancement is then computed by using the 
modelling coefficients with the original panchromatic band, 
where these are applied for a pixel neighbourhood the 
dimension of which is determined through the resolution 
difference between both images (the coefficient images are 
resampled to fit to the dimension of the high resolution 
panchromatic band). Thus, 
multi= aj low +bj low * pan 1 “ 8 * 1 + e/ ow (5) 
Substituting [e j low ] by Eq. 4, Eq. 5 can be rearranged to: 
multi ¡** = multi j tow + b j low * (pan 1 “ 8 * 1 - pan low ) (6) 
This implies that only the multiplicative component [b j low ] of 
the local regression analysis has to be calculated. Note that Eq. 
6 corresponds to the HFA (Eq. 1), with the important difference 
that [b j tow ] acts as an optimum local “scaling factor” for the 
high frequencies to be added. 
The net result of the restoration is an image which retains all the 
information contained in the multispectral image (i.e. blurring 
the restored image yields the original multispectral image), 
while the introduced high frequencies are scaled according to 
the local correlation properties between (degraded) 
panchromatic band and multispectral channel. Local contrast 
differences are adaptively modelled, even if the local relation 
between the datasets exhibits an inverse polarity. 
Further modifications of the LCM approach are mainly directed 
towards improving the algorithm in image sections where the 
local correlation between both image domains falls below a 
specified value (i.e. r < 0.66). This effect frequently occurs in 
more or less homogeneous image areas and may result in badly 
defined regression estimates. We are presently testing different 
options that appear suitable to successfully substitute the local 
correlation modelling (e.g. the HFM and LUT algorithm). 
Multi j low 
Pan low 
(a) Degrade (factor = resolution difference) 
< 
Multi j hi 9 h 
(b) Local regression 
analysis 
Pan hi s h 
(c) Local use of the 
local coefficients 
& residuals 
Fig. 2. Image fusion through local correlation modelling (flowchart).