Both the measurement of urban areas using modified image 
classifications, and the modelling of urban areas using fractal- 
based analysis, will be applied to Norwich, a medium-sized 
settlement in the United Kingdom, approximately at the time of 
the 1991 UK Population Census. 
MEASUREMENT OF URBAN REMOTELY-SENSED 
DATA 
In this paper, measurement will relate to image classification, in 
particular the conventional maximum-likelihood (ML) 
algorithm, its Bayesian modification, and links to GIS data. 
Conventional Maximum-Likelihood Classification 
As a parametric classifier, the ML algorithm relies on each 
training sample being represented by a Gaussian probability 
density function, completely described by the mean vector and 
variance-covariance matrix using all available spectral bands. 
Given these parameters, it is possible to compute the statistical 
probability of a pixel vector being a member of each spectral 
class (Thomas et al, 1987). The goal is to assign the most 
likely class w,, from a set of N classes, wj, . . . , Cy, to any 
feature vector X in the image. A feature vector X is the vector 
(X,, X4, ... , Xj, composed of pixel values in M features (in 
most cases, spectral bands). The most likely class w; for a 
given feature vector X is the one with the highest posterior 
probability Pr(w/X). Therefore, all Pr(w]X), i e [1 . . N] are 
calculated, and w; with the highest value is selected. The 
calculation of Pr(w,|X) is based on Bayes’ formula, 
Pr(X|w;) x Pr(w;) 
Pr(w, IX) = PX) 
(0) 
On the left hand side is the a posteriori probability that a pixel 
with feature vector X should be classified as belonging to class 
w, The right hand side is based on Bayes formula, where 
Pr(X|w;) is the conditional probability that some feature vector 
X occurs in a given class, in other words, the probability 
density of w; as a function of X. Supervised classifications, 
such as the ML, derive this information from training samples. 
Often, this is done parametrically by assuming normal class 
probability densities and estimating the mean vector and 
covariance matrix. Also on the numerator and coupled with the 
conditional probability is what is known in Bayes' formula as 
the prior probability of w; , shown as Pr(w;). This is the a priori 
probability of the occurrence of w; irrespective of its feature 
vector, and as such is open to estimation by prior knowledge 
external to the remotely-sensed image. External prior 
knowledge will typically include information on the 
distribution and relative areas covered by each class in the study 
scene and is most readily generated from GIS data. It follows 
that the accuracy of class priors is at best equal to the quality of 
GIS prior knowledge. In image classification terms, prior 
probabilities can be visualized as a means of shifting decision 
boundaries to produce larger volumes in M-dimensional feature 
space for classes that are expected to be large and smaller 
volumes for classes that are expected to be small (Mather, 
1985). The denominator in (1), Pr(X) is the unconditional 
probability density which is used to normalise the numerator 
such that 
Pr(X) = » Pr(X|w;) x Pr(w;) (2) 
i=l 
Normally, ML classifiers assume prior probabilities to be equal 
and assign each Pr(w;) a value of 1.0. However, it would seem 
intuitively more sensible to suggest that some classes are more 
likely to occur than others. By taking account extraneous 
information on the areal properties of each spectral class it will 
be possible to generate thematic per-pixel classifications that 
are more accurate than those produced from conventional ML 
techniques (Barnsley et a/, 1989; Maselli et al, 1992). The 
paper will now examine how prior probabilities may be 
modified to incorporate external GIS information on class area 
estimates. 
Modification of Prior Probabilities 
Before we examine precisely how prior probabilities may be 
modified, it is important to stress from the outset that our 
modifications can only be conducted within the more general 
framework of GIS/RS integration. This requires a systematic 
strategy which can co-ordinate the flow and coupling of GIS 
data within image classification procedures. In the worked 
example, prior probabilities will be modified using a 
hierarchical stratification strategy based upon data from the 
United Kingdom Population Census. The stratification will 
essentially allow census data to assist in the selection and 
hierarchical partition of spatial features from a satellite image. 
This hierarchical partition is critical to the statistical 
assumptions of ML prior probabilities, of which the most 
important being that all multi-dimensional feature space is 
subdivided between weighted classes. In other words, for prior 
probabilities to function most efficiently they need to be 
applied to inclusive feature space but mutually-exclusive 
classes. This essentially means that for the classification of 
mutually-exclusive residential dwelling classes, an image must 
only be composed of residential feature space. Census tract 
data have already been shown to be amenable to the generation 
of pseudo-surfaces of urban representations, especially 
residential surfaces (Martin and Bracken, 1991) from which 
such stratification is possible. These surfaces have been used 
by Mesev (1995) to enhance per-pixel classifications through 
training sample selection and post-classification sorting. The 
result is that satellite images have been routinely segmented 
into “urban” and “non-urban”, as well as “residential urban” 
and “non-residential urban” (Mesev et al, 1995). Using the 
“residential urban” category it will now be shown how prior 
probabilities of the surrogate residential density categories, 
“detached”, “semi-detached”, “terraced”, and “apartment” 
blocks, may be generated by census data and then inserted into 
the ML classifier. 
Consider z, as the census variable, “residential building type” 
(where k: 1 = “detached”, 2 = “semi-detached”, 3 = “terraced”, 
and 4 = “apartments”). When stratified into exclusively 
residential feature space, the four classes will have A pixels 
with feature values X,, where, X,, . . . , X, are not necessarily 
mutually-exclusive. The objective is to find the probability that 
a random pixel (within the “residential” stratum of the image) 
will be a member of a spectral class w; (where i: 1 = detached, 2 
= semi-detached, 3 = terraced, 4 = apartments), given its density 
vector of observed measurements X, in m-dimensional feature 
space and that it belongs to ancillary class z,, described as 
558 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996 
  
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