International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
Using Bayes theorem, it comes:
pot [my = DEL) AES (3)
where P(D) does not depend on the model M, thus
M = argmax P(D|M)-P(M) (4)
MEM
We use images I, 3D segments S and focusing mask M, as
observations D although any additional observation could be
used (laser points clouds, manual selection, cadastral maps for
instance). Assuming independence between these observations
leads to
M = argmax P(S | M)-P(I | M)- P(Ma | M)-P(M) (5)
MeM
The model probability P(M) depends on the model complexity.
As recalled in section 1.1, one looks for the simplest model, given
the observations. The model probability is linked to the minimum
description length through:
— C(M)
P(M)--Cexp. ? (6)
where C is a normalization factor common to all models of M
that will be omitted in the following. 3 tunes the level of carica-
ture desired. A study on the influence of this parameter is done in
section 5.
4.2 Model Formulation
Constraints Inferring As stated above, constraints on prim-
itives represent the external knowledge brought in the decision
process to guide the choice of M. That way, we can also inte-
grate architectural knowledge to favor some types of architecture
when ambiguities remain in the choice of the model.
Constraints are first inferred on normals of the base primitives.
We adopt the principles of the system described in (Grossmann,
2002) since this system is used for enforcing constraints in the re-
construction. The algorithm uses one threshold o. Normals are
first clustered by angular proximity, thus grouping approxima-
tively parallel normals. For each cluster, a direction, average of
the clustered normals, is defined. This process ensures that mini-
mum angular distance between two directions is c4 and handles
parallelism constraints in the model.
A constraint graph is then deduced from these directions that are
nodes of the graph, whereas edges represent constraints between
directions: orthogonality, horizontal cross product (stating that
the intersection of both planes is horizontal, which is usual in ur-
ban environment) and vertical symmetry. An edge is created if
the relation is verified in the angular tolerance c4. Each edge q is
valued with a weight C'(q) related to the number of degree free-
dom that the constrain suppress on normal coordinates (Figure 6).
This will be of primary importance to integrate constraints in the
choice process (see next).
Notations Fm, Fm and Vm define respectively the number
of facets, edges and vertices for a given model M. Pm, Ry
represent the number of non-vertical directions and vertical di-
rections used in the models (after the clustering process) and
Du = Pm + Ry the total number of directions. Finally C;
is the number of constraints on directions in the model (the num-
ber of edges in the constraints graph related to M). The basic
idea for complexity computations follows the principles edicted
in (Kolbe, 1999) based on the transmission of information related
to the model: topological, geometrical and constraints.
Facades Planes
p
“py pij
- ; s Y Y
= >> x pet. clustering clustering
CP Sia
‘ ee +7 m *
LN
Fa; c Fa, T. pr N p:
|
: p 4 P4
—— horizontal edge Fa eL + E
—— orthogonality PE p
vertical symmetry
Figure 6: Constraints graph. The 4 facades have been grouped
into only 2 directions. Valuations on edges depend on the type of
constraints.
Topological description Topological description for a facet f
consists of the enumeration of its | f | points V'% and its direction
D ;, leading to:
Li(f) =| f | 1og(Vm) + log(Dm) (7)
When summed on all the facets, it simplifies to
L:(M) =. Fay log(Var) + Far - log(Dm) (8)
Geometrical description Assuming each coordinate can be
coded on 12 bits (which gives a lcm precision for point, given
a building of roughly 40 meters and a largely enough precision
on angular measures), we can enumerate directions and points
coordinates. Each facet brings also | f | —1 planarity equa-
tions such as D; - (V^ — V) — 0, thus reducing the number
of degrees of freedom and therefore coordinates that need being
coded. Finally, it comes:
Ly(M) = (2 + Pa + Rm +3Vm — Y f£ 170) «12
I (9)
= (2 * Part Hu-3Vu 2E PF) x12
Constraints Each admissible surface uses some directions and
thus defines a subgraph of the initial constraints graph. Each edge
in this subgraph represents a constraint inferred on the model.
This constraint is coded with its type (orthogonality, vertical sym-
metry for instance), directions it links and the value C'(q) related
to the type of the constraint q. Noting | c | the number of types
of constraints (3 in our case) and assuming constraints are inde-
pendent bring
LM) = > (log(| c]) - 21og(Du) - C(g) 12). (10)
q
Global Model Global model C(M) sums up the three previous
terms and thus takes into account topogical, geometrical com-
plexity as well as external information brought by constraints.
Complex models are penalized by this function whereas sim-
ple, symmetrical models and models embedding some usual con-
straints are conversely favored.
4.3 Observations
Images As for observations related to image, the model score
is given by correlating in all images the set H( M) of non vertical
facets that do not belong to z = z,. Let us note SP(M) the
surface of H(M) projected on z = z, and | SP(M ) | its area.
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