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CONTEXTUAL BAYESIAN CLASSIFIER
Miclial Haindl
Institute of Information Theory and Automation
Czechoslovak Academy of Sciences
Pod vodarenskou vezi 4, 182 08 Prague 8
Czechoslovakia
ISPRS Commission VII
Abstract
A new type of Bayesian classifier is introduced to recognize remote sensing image data. The classifier uses
contextual information about the classified pixel surrounding based on autoregression model prediction.
Key Words: Bayesian Classifier
1 INTRODUCTION
The conventional approach to remote sensing picture data
classification is to perform test on unknown pixel against
all classes using a spectral feature subset and then assign
the unknown pixel to one of these classes, not taking re
specting the large spatial correlation (Tubbs,2). The inter
pretation of picture element - pixel does not depend upon
any relationship with any other pixel, so we do not use all
available information. Not using context has fundamental
limitation influence on classification accuracy for machine
recognition. On the other hand the use of all context in
formation is paid in unsolvable increase of computational
demands. Possible solution is a compromise.
The developed decision rule is the optimal contextual
Bayesian classifier with several simplifications. We assume
local neighbour pixels to be conditionaly statisticaly inde
pendent, their class conditional density to be independent
on neighbour labels and we neglect the space arrangement
of local neighbour labels. In the first step the thematic map
as the output from per-point Bayesian classifier is created.
The second step consists of Bayesian classifier in which
formula the apriori class probabilities are replaced by non-
causal frequency predictor. This predictor is based on the
autoregressive model of class frequences estimated from the
first step classification.
2 CONTEXTUAL CLASSIFIER
Let us chose some direction of movement on the image
plane, for example row scanning from left to right and top
to bottom. According to this choice the following index is
used throughout this paper:
t = (i-l)N , + j (1)
where i,j = 1,2,..., N' is row and column index, respec
tively. N' x !\ rl is the size of classified image. Let us de
note X t (multidimensional pixel) in time t of scanning. u>,
i = 1,..., A class indicator,Y t the A—dimensional vector,
which i-th compound is the occurence frequency of u t in
a window D t .D t window is defined as the set of thematic
map entries within a given region centered around the class
indicator corresponding X t . The window is defined to be
perpendicular with odd number of class indicators in both
directions, so that each frequency vector Y can be assigned
to the center pixel of its respective window.
Let us denote the set of past thematic map windows
D (t) = {£> t , A-i, • • •, Di) (2)
and Lflb the set of all pixel indexes from D. We have
chosen the noncausal frequency predictor:
Y t = E[Y t \D (t ~ l) ] (3)
Let us denote
Mi(Xt) = (Xt - - ta) (4)
where /q, E, are mean value vectors and covariance ma
trices (in practise their estimators) of single classes. Now,
the modified Bayesian decision rule using the context in
formation abont the class membership of surrounding, can
be put in the form:
Assign feature vector X t into class uq if
Mi(X t ) + In |£i| - 2 In Y t (i)
= min {A/ J (A'«) + ln|E J |-21nV;(i)} (5)
The thematic map for prediction construction is as
sumed to be the output from per-point Bayesian classifier
. then the relation (3) can be put in the form (6)
Y t = E[Y t | min {Mj(X m ) + In |Ej|
- 2 In Pj} : m e £) (t-1) ] (6)
where P 3 are some prior class probabilities estimations.