se 
re 
divided. This division is dependent on the parcels size, form, 
and the number of partitions this parcel was subjected to 
already. With a statistical analysis an estimation of these 
dependencies and parameters, accompanied with an estima- 
tion of the probability of each rule is gained. 
Global parameters like maximal and minimal parcel size, 
maximal and minimal ratio of shorter side of a rectangle to 
larger side, and maximal hierarchy level can be estimated in 
a first step. These parameters determine when a subdivision 
stops. 
The dependency of the partition on the parcel attributes is 
modelled in terms of a Renewal Process. 
In this paper, only the outline of the estimation process is 
given. In order to reliably estimate the model parameters, a 
big sample of exemplary data has to be evaluated. Further- 
more, the functional dependencies of the different parameters 
have to be examined carefully. For simplicity in the follow- 
ing example dependencies on the size and form of the parcel 
and on the direction of the partition are neglected, and the 
following simplification is made: 
e Parcels are divided along their longer side. 
e The partition of a parcel only depends on the hierarchy 
level, namely the number of previous partitions. This is 
a fact that is easily verified with visual inspection of the 
data: the first cuts of a parcel try to generate a few big 
parcels, while the following split these parcels into the 
final individual fields, by higher number of cuts. The 
parameter A in equation 3 corresponds to the number 
of successors n. Different parameters A are estimated 
depending on the hierarchy level. 
The result of the estimation are both the functional depen- 
dencies and the corresponding probabilities. These values 
for the relation between hierarchy level level and number of 
cuts n for the above example (see Figure 3) are given in the 
following table: 
level 1 2 
n 3 5 2 
P(n|level) || 1.0 || 0.67 | 0.33 
  
  
  
  
  
  
  
  
  
These values are gained by simply counting the possible num- 
bers of successors of a level. On level 2 e.g. there are 3 par- 
cel to divide. Two of them are cut into 5, one is cut into 2 
parcels. Thus result the probabilities of 2/3 and 1/3 resp.. 
In order to rate an existing aggregation structure, the prob- 
abilities of the individual steps of its generation have to be 
evaluated in common. To this end the whole chain of depen- 
dencies has to be formulated. The probability of partition 
P(part) of a parcel N; with parcel sides wg and ho into n 
subparcels depends on: 
Wı Wo W3 
  
ho N; 
  
  
  
  
  
Wo 
e P(N,): the probability of ,father“-parcel NV; 
e P(n|level): the probability of n cuts given hierarchy 
level level 
861 
e the probability of dividing parcel N; into n cuts with 
widths w; 
P(w;,ws,::-,w4|Ni, n) 2 P(w|Ni,n)- 
P(w;|w;, Ni, n) es 
P(w,|wi, wa, --- ws 1, 1, n) 
Due to the ,lack of memory“-property of the Renewal 
Process, the probabilities of the individual cuts w; can 
be considered independent from each other, i.e. depend- 
ing only on the ,father“-parcel and the number of pre- 
vious partitions n, resulting in 
P(w;,ws,--- , wa |. N1, n) = P(wi|Ni,n) > 
P(w»|Ni,n) --- P(w,|Ni, n) 
e the probability of the width w; of a parcel given the 
number of partitions n is computed with equation 3: 
P(w;|N,,n) 2 P(X = 21 = n) 
Wi 
The probability of partition P(part) is then: 
P(part) = P(N,) - P(nilevel) - 
-P(w,|N,,n) - P(wa|N,,n)--- P(w,|N;,n) 
All the other dependencies have to be considered correspond- 
ingly. 
6.3 Representation in attributed 
stochastic grammar 
After estimating the model parameters, the stochastic gram- 
mar can be set up. 
e Vocabulary: 
Vv = PARCELA, 
Vr = parcel,,relation 
relation = |, — 
e Startsymbol S — PARCEL, 
e Production rules P: 
PARCEL, "€" PARCELarelationPARCEL, 
PARCEL, PUUP parcel, 
Nonterminal symbols (denoted in uppercase letters) stand 
for intermediate parcels; terminal symbols (lowercase letters) 
give the final parcels or the spatial relations between the 
parcels resp. (| — left-right; — — top-bottom connection). 
Each parcel has corresponding attributes A. 
This grammar represents a compact model description. It 
can be used in an algorithmic way to produce new objects, 
ie. new parcel aggregations. Given a startparcel, new par- 
cel are generated by applying the rules of the grammar. An 
example is given in the next subsection. On the other hand, 
the grammar can be used to object recognition: if an un- 
known object can be explained by the grammar, then it is 
recognized.