International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 7-4-3 W6, Valladolid, Spain, 3^1 June, 1999 
(a) 
(b) 
Fig. 1. Flowchart of the fuzzy DPCM with context-coding: (a) encoder; (b) decoder. 
3. SPATIAL AND SPECTRAL DE-CORRELATION 
The multi-spectral compression scheme (Aiazzi, 1999c) is 
highlighted in Figure 1. Prediction of a pixel to be encoded is 
based on a linear combination of L surrounding pixels lying in 
its causal neighbourhood, i.e. pixels that have been already en 
coded, as shown in Figure 2. In the 3D case, pixels both on the 
current band and on previously encoded bands may be used. Size 
and shape of the causal neighbourhood determine the trade-off 
between coding performance and computational cost. 
(b) 
Fig. 2. 3D causal neighbourhood of current pixel g(n): (a) 13-th 
order, i.e. 13-pixels, on current band and one previous 
band; (b) 22-th order on current and two previous bands. 
Alternatively, once the (k — l) st is available, first the k th band 
is skipped and the (k + l) sf band is predicted from the (k — l) st 
one; then, both these two bands are used to predict the k th band 
in a spatially causal but spectrally non-causal fashion (Aiazzi, 
1999d; Rao, 1996), as shown in Figure 3. 
As patterns of pixel values occurring within the causal neigh 
bourhood are related to local spatial/spectral features, a number 
M of representative prototypes are identified by a preliminary 
classification of the occurrences of the causal basis, obtained by 
means of fuzzy-clustering. Once the M prototypes have been 
identified and stored in a centroid matrix C, each occurrence is 
given a degree of membership to each prototype, based on its Eu 
clidean distance. The membership array U is used in the two next 
stages. 
For each prototype, the L coefficients of a linear regression 
predictor are calculated by an LS algorithm fed by the occur 
rences whose membership to that prototype exceeds a threshold 
•q. Predictors are arranged into an L x M regression matrix 4>. 
During the de-correlation, each pixel is predicted as a weighted 
sum of all the predictors: each one is given a weight according 
to the membership associated to its occurrence. Differences be 
tween the pixel value and its prediction are considered for coding. 
When the algorithm utilizes a unique cluster, a minimum mean 
square error (MMSE) linear regression predictor is obtained. 
j-W j j+W 
Fig. 3. Spectrally noncausal 3D neighbourhood for bidirectional 
prediction, (a) 13-th order prediction with middle band 
skipped, (b) 22-th order bidirectional prediction of k- 
th band sandwiched between previously encoded bands. 
The neighbourhoods must always be spatially causal. 
The coding performances of the algorithm also rely on the use 
of a context-based entropy-coding strategy (Wu, 1997) which 
takes advantage both of the nonstationarity and of the small 
residual correlation of prediction errors, which are partitioned 
into equi-populated, statistically homogeneous classes, based on 
RMS measured in the causal neighbourhood of the current pixel. 
Once prediction errors have been classified, adaptive arithmetic 
coding of each class (16 in this work) is executed. 
The decoder is much simpler, as no training is needed. Over 
head information at the decoder is: the values of the prototypes 
(centroids), stored in an L x M matrix C, the prediction matrix 
3>, and thresholds 0fc, k = 1, • • •, 15, for context decoding. 
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