values the match between initial and current position and orienta-
tion respectively. For orientation the angle between initial normal
direction 1;4;; and moved normal n(p) contributes to the cost
function whereas for position the projection of (c(p) — cinit)
onto the plane normal is used. Thus a position shift perpendic-
ular to the plane normal does not have any influence on the cost
function.
All values w(i, j) of the part's area in the certainty map (fig. 2
right) are integrated into the constraint function weight we ac-
cording to the following formula:
um M uj.
(3,7)ER
Larger walls (regarding their area in the image plane) with more
reliable certainty values lead to model parts that are less moveable
by the optimization than smaller parts. Together with internal
constraints (e.g. the parallel constraint) smaller extensions are
aligned to larger parts whose spatial position can be estimated
more accurately from image data.
4.6 Edge constraint
As shown in figure 7 the goal of an edge constraint is to make the
projection of a model's 3-D edge congruent to the correspond-
ing image contour. To have the 3-D edge's projection lie on the
image edge is particularly important for texturing, which is the
backprojection of image information onto the 3-D model.
auxiliary planes
image plane
camera focus Pz
Figure 7: Using edge information for 3-D modelling
The image edge is represented by the points p1 and p», which are
used to span two auxiliary planes. These planes are perpendicular
to the so called plane of sight which is determined by the focal
point of the camera, p; and p». The 3-D model edge intersects
with the auxiliary planes in S4 and Sy. The orthogonal distances
6, and 62 between the intersections and the plane of sight have
influence on the cost function:
Cedge (P) = Wedge * (61 (p) = 62 (p)), (5)
with the weight Wwedge that controls the relative influence of the
contraint during optimization.
All above constraints are designed to meet the needs of human
spectators regarding geometric properties.
5 Incremental Surface Reconstruction
The goal of surface reconstruction is to find the object's shape and
perform the accompanying viewpoint registration. The latter is
used to integrate new information if further viewpoints are added
to the scene description.
As described in section 3 the model is represented a planes in
space which can be moved through the parameter vector p. To-
gether with the model parts a number of geometric constraints,
as described in the previous section, are derived from the generic
knowledge base during interpretation. Due to faulty image data
the resulting set of constraints generally lacks consistency. Nu-
merical optimization is applied and leads to a model which best
meets all constraints. Therefore the optimizing algorithm finds
the parameter vector p such that the global sum of all N cost
functions c? is minimized:
N
Cgiod(P) = > Uic c (p) — Min, (6)
iz
where w; denotes the respective weight for the cost function. For
details on how to choose the weights refer to[9].
In the approach presented here the method of conjugated gradi-
ents is employed for minimizing the global sum of cost functions.
The use of a general minimization technique like conjugated gra-
dients makes it easy to extend the existing set of constraints. Only
a new cost function is needed to integrate the judgement of new
features into the system.
To avoid local minima and to reduce the parameter space needed
for optimization, the search for the best fit is done hierarchically,
by use of the system's control structure. In a the first step the
shape is only approximated by a few planes and then refined in
following minimizations.
In addition the cost functions are used for verification of hypothe-
ses generated during interpretation.If for instance an angle be-
tween two parts has been wrongly predicted to 90 degrees in-
stead of, say, 135 degrees, the affiliated cost function will return
an unusually high value. This indicates that the hypothesis has
probably to be rejected by the interpretation.
3rd Kamera-
position
first CME e
position
2nd cameraposition
Figure 8: Registration of new camera viewpoints
Internal constraints apply only to objects. The measurement con-
straints like edge constraint or position constraint, on the other
hand, are used for integrating camera related image features like
edges or depth information into the model. Thus two ways of
minimizing the latter constraints are possible: Either the model
parts or the camera model can be moved in space by numerical
optimization. This allows not only to further optimize the model
but also to incrementally add new camera viewpoints to the scene
description. In this mode the constraints are minimized by mov-
ing the new camera to the position best matching the constraints.
As depicted in figure 8 the first camera viewpoint is used as a
reference position. Together with a first shape approximation the
second viewpoint is determined. Using this shape approximation
the following camera positions are estimated. In final steps the
model is completed with smaller extensions.
596
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996
Constr
structio
and clo:
This se
achieve
tal reco
part de:
with th«
61 H
Figure ©
for the 1
mep of
a manu:
Figure ‘
map (b)
In a firs
only of |
edge ba:
first vie
are then
model's
In the s:
