Ermes, Pierre
constraint equations in (5) provides a formulation that is free of singularities. Note that the formulation of the direction
constraint is very similar to the rotation representation of a cylinder in (1).
The constraints discussed above only give a general description; Table 1 relates these constraints to the parameters of a
cylinder and a torus. The constraint equations are linearized and included as weighted observations together with the
measurements in the adjustment for the model parameters. The constraint equations are weighted relative to the
measurements by the operator. The weights of the measurements can be computed from the expected measurement
accuracy in the images (weight — 1/variance).
Cylinder Cylinder Torus Torus
bottom top bottom top
Direction 0 0 0 0
R cyl 0 R cyl 0 R torus 0 Ru sin ex
-1 1 —l cosaz
Position 0 0
T T * R,, 0 T ru: T is +R torus R(1 COS ©
L Rsinc
Table 1. The directions and positions of possible connections for a cylinder and a torus, expressed in
terms of their parameters.
The weights of the constraints can be related to physical properties such as the manufacturing accuracy of the part.
Especially for standardized components this data is available. Connections between piping elements come in different
types, such as welded connections or flanged connections. These types of connections are also standardized and,
therefore, should meet accuracy requirements.
When the accuracy data is not available, the weights can be approximated by assuming a realistic accuracy. Where the
position constraint is expressed in object coordinates, the direction constraint is expressed using dimensionless
numbers. The weight of a direction constraint can be related to the weight of the position constraint when the effect of a
change in direction is estimated in object space. For example, the maximum effect of change in direction of a cylinder is
its length or its radius, whichever is biggest. Multiplying the weight of the position constraint by the squared size of a
model for the weight of the direction constraint produces 'resistance' in object space the same order of magnitude for
both constraints. Note that the expressions in Table 1 for the positions contain rotation parameters that are also
multiplied by shape parameters thus producing the same result in the adjustment as the weighting described above.
A third method for the computation of the weights is Variance Components Estimation (VCE) (Luhmann, 2000). VCE
is a statistical technique and is applicable when a large number of redundant measurements and constraints are
available. By dividing the measurements and constraints into two groups (components) the residuals of both groups can
be analyzed and the variance of both groups can be estimated. The group of constraints can even be further divided into
groups of different constraint types to estimate the weights of the different types.
5 BUNDLE ADJUSTMENT
Piper also contains functionality for estimating the exterior orientations of the images, this estimation was used in the
experiment described in the next section. So far, we used the measurement method described in Ermes et al. (1999) for
estimating the pose and shape of CSG models assuming known camera orientations. However, the camera orientations
can also be estimated using the same measurements and this avoids a separate triangulation stage for obtaining the
camera orientations. A point on an edge of a model p, is backprojected into the image at point p, according to:
P; ETTTP, (6)
Where T, denotes the pose of the model in object coordinates, T, denotes the pose of the object space in camera
coordinates (the inverse of the exterior orientation), and T, is the interior orientation of the camera. The linearized
observation equations as discussed in Ermes et al. (1999), are extended with the partial derivatives of the parameters of
T, , which is parameterized with a translation and a quaternion.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 219
