Ansgar Brunn
[E m
semantic classes
Figure 4: Three different views on the real objects result in three different sets of object groups.
& gh | NS
Figure 5: Neighborhood system for vertices, edges and triangles in the Markov-Random-Field.
e Conditional probability for O-simplex classification: the classification of a O-simplex depends on the number of
breaklines and the number of corner points in the neighborhood
Py (l,| 34 BL(v), 4: C P(v)).
With the assumption that the conditional probabilities can be separated,
m (I | #BL(v), #C P(v)) = Pu(lbl#BL(v)) + Pu(lu|#CP(0)) Q)
follows.
e Conditional probability for edge classification: the classification of a 1-simplex depends on the classifications of the
neighboring faces, neighboring vertices and the number of other breaklines bounding the neighboring faces
Pmlle |. Mn (e) tg (e) Up (0) Ip. (6) EBL(e))
We assume separability of the conditional probability:
Pmlle | {y (e),ly,(e)}, tp, (€) su #BL(e))
= Purely (e), Ly. (€ ia (e),lp,(e)}) - Pm(le|#HBL(e)) (3)
e Conditional probability for triangle (face) classification: The classification of 2-simplices depends on the number of
breaklines bounding the 2-simplex.
Pm(l;|#BTL(f)) (4)
We assume the likelihood function as independent of the building type. Therefore, we just have to define one set of
likelihood function for each combination of each simplex with each data type or, of course, a joint likelihood function for
all data types, which is mostly not practicable. These functions P(y|s) transfer the observations y into probabilities for
all classes of each random variable. Assuming a normal distribution for a observation vector y, a likelihood may have the
following form
P(y|s) = (27)-""? (det X)? exp (zw -X8,)! x (y - x8,))
The vector A is a class specific parameter vector, X a matrix of linear coefficients and 3 the matrix of covariances of the
observations.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 121
