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Pr(w;|X, 24) (3)
It is also assumed that the effects of z, are external to the
original generation of the mean vector and covariance matrix of
w;. As a result the likelihood function Pr(w;|X) is unaltered by
the introduction of z,, but is simply modified by the conditional
probability
Pr(w,z, ) (4)
This is a process of identifying the association between spectral
class w; with census variable z,. For example, the spectral class
labelled as “low density residential” would be directly
associated by a conditional probability with the census variable,
“detached dwellings". In effect, w, is weighted by the
probability of z,, producing the prior probability of w,. In the
example its assumed that the prior probabilities of each of the
four dwelling types exist in inclusive m-dimensional feature
space, so that,
Pr(w;) + Pr(w,) + Pr(w;) + Pr(w,) = 10
The probability densities d;; — Pr(X]w;), dj; ^ Pr(X]w,), d, =
Pr(X]w;), dj, = Pr(X{w4), are known for each pixel. Let /,, be
the shorthand for the posterior probability Pr(w,|X;,z,) that pixel
i belongs to class w,, and p; as the shorthand for the prior
probabilities. The Bayesian modified ML is now represented as
JT dpi
(5)
dipi * dip, * dips * dap,
Likewise, /; — Pr(w,X;z,), 43 = Pr(wyX;z;) and [, =
Pr(w4|X,z,) may also be found, and of course, the sum of the
four posterior probabilities equals 1.0,
dip;
lj = HE LE (6)
k
2, dip,
j=1
Empirical Application
Let’s look at one application of the Bayesian-modified ML
classifier, the case of the settlement, Norwich in eastern
England (others may be found in Mesev 1995; Longley and
Mesev, 1996). The aim was to classify the four residential
dwelling types from a Landsat 5 (TM) image, taken on the 15th
July 1989, using census data from the April 1991 UK
Population Census (the 21 month discrepancy was unavoidable
and does not represent a period of high residential development
in eastern England). Norwich is a free-standing medium-sized
city located on land that is relatively flat and unaffected by
serious impediment to urban development.
The image was first geometrically corrected and enhanced
before classified into a binary distinction of “residential” and
"non-residential" using training sample selection and
559
postclassification sorting based on census probability pseudo-
surfaces within a GIS (see Mesev 1995: Mesev et al, 1995).
The "residential" stratum was then exposed to the modified ML
classifier, with a priori probabilities from the 1991 census
(figure 1). Before equation (6) could be implemented, a size
ratio between each relative dwellings type had to be calculated.
This would help to preserve the relative areal proportions of
each dwelling type, where for instance “detached” dwellings
occupy larger areas than “terraced” households. ^ Using
stereoscopic photographs, 20 samples of dwelling type sizes
were generated and average relative size ratios between
dwelling types were constructed. The ratios were 1 detached
dwelling to 1.5 semi-detached, 1 detached to 2.25 terraced, 1
semi-detached to 1.5 terraced, and 1 detached to 10 apartments.
Although these were approximations they are still more realistic
than assuming absolute linear relationships.
The results are thematic classifications of the four dwelling
density types based on maximum a posteriori probabilities.
together with area estimates. Table 1 quantifies how
classifications based on adjusted a priori probabilities produced
areal estimates that were in most cases closer to observed
census data than classifications assuming equal prior
probabilities. The Bayesian modified classifier performed best
for the detached category due perhaps to its larger size on the
ground and hence least spectrally mixed. The worst category
was apartments were the estimated size-ratio may not have truly
been representative.
MODELING OF URBAN REMOTELY-SENSED DATA
The second part of this paper will examine how ideas from
fractal geometry can be instigated within an urban modeling
approach. Specifically, the use of fractal geometry, and density
functions based on fractal properties, to describe and summarize
the spread of urban development in terms of size, form, and
density. For this, thematic urban categories generated by image
classification will be used as the source data. This represents a
departure from established work where census tract data and
derived residential data have been the usual baselines for urban
modeling (summary in Zielinski, 1979; Batty and Xie, 1994).
It will be argued here that urban data from classified images
represent the most appropriate source for the measurement of
fractal properties. More appropriate in the sense that the
inherent spatial irregularities associated with urban areas and
assumed as fractal, are represented by classified remotely-
sensed of varying spatial resolutions that exhibit a similar
amount of spatial irregularity (De Cola, 1989).
Estimating fractal dimensions and density functions
This paper will add support to the contention that cities exhibit
generalized fractal properties (Batty and Longley, 1994;
Frankhauser, 1994), and that fractal geometry provides a much
deeper insight into urban density functions than has so far been
recognized. In particular, emphasis is given to the ways the
form of urban development can be linked to its spread and
development (Batty and Kim, 1992). Measurement in this
sense is restricted to the manner and rate at which space is filled
with respect to distance from the CBD (Mesev et al, 1995).
The suggestion here is that the inverse power function,
HR) SER? (7)
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996
