Ansgar Brunn 
  
with y; being a vector of observations. P(y;|s;) is called likelihood function and P(s;) prior distribution. Thus we are 
able to connect different probability distributions. Here we use the Bayes' Theorem by introducing background knowledge 
Js; about the random variables in some neighborhood of s; (markovinity) (Koch and Schmidt, 1994) 
P(si|y;, 0si) « P(y;|si, 0s;) P(si|0si). 
Further on we assume that the likelihood function only depends on the random variable s; itself, not on the neighboring 
random variables 
P(yi|si, 05i) = P(y;lsi). 
Therefore eq. 1 changes to 
P(si|y;. 0s;) « P(yi|si) P(si|0si) (D 
where the distribution P(s;|0s;) inherits the information about the building model. 
The theory of Markov-Random-Fields states that the local classification (cf. eq. 1) leads to a maximal probability of the 
complete random field. We calculate the probability of the classification of the complete random field by 
$ max P(s|y) — Il max P(sily;, Ôs:), 
2 
because in this application each subset of random variables is considered only once for local classification (Koch and 
Schmidt, 1994). Therefore, if the building type is unknown and from a set of building models M with M = {M;|i 
€ {1,..., nm }}, the maximization can be done for each building type. Then the maximum of all probabilities of the 
different building types gives the type of the complete building 
M = max P(s(M)|y). 
3.2 The building model 
We use simplicial complexes, which consist of 0, 1 and 2-simplices, to represent the topology of the surface of buildings. 
Each simplex is associated with some appearance attributes, which could be e. g. geometric or radiometric. The simplices 
should be classified. Therefore we define classes for each simplex type. 
We look at the classes of the simplices of the real building parts from different views (cf. fig. 4): at first we describe 
a semantic model which will be generalized to a geometric!. A semantic class scheme could consist of the following 
classes: 
e O-simplices: eaves corner point, ridge corner point, ground-plane corner point, eaves point (point on the eaves, not 
corner point), ridge point, ground-plane point, point on a wall, point on the roof and point outside the building in the 
ground-plane 
e 1-simplices: eaves edge, ridge edge, ground-plane edge, corner edge (mostly nearly vertical), edge in the roof, edge 
in a wall, edge outside the building in the ground-plane 
e 2-simplices: face in a wall, face in a roof, face in the ground-plane 
In this paper we generalize to the following geometric model: 
e O-simplices: corner point (CP) (on the border of a least three building planes), edge point (EP) (on the border of two 
building planes), face point (FP) (inside one plane of the building) 
e l-simplices: breakline (BL) (intersection of two building planes), face edge (FE) (inside one building plane) 
e 2-simplices: vertical (V), oblique (O) and horizontal (H) face 
The definition of local neighborhoods enables us to do local classification. We define different kind of neighborhoods for 
the three simplex types which are shown in fig. 5. Different building types are modeled by sets of conditional probabilities 
which coincide with the neighborhoods. For each building type the following chosen conditional probabilities have to 
known are priori or have to be learned (cf. sec. 33): 
  
! We use the notion "generalized" because the capability to distinguish between different objects in the geometric modeling is less than in the semantic 
modeling. Also the number of classes is reduced in the geometric model. —? Notation: 3€ means the "number of". 
  
120 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.