e of
Ds
t of
w of
a. the relative positions of the cameras (camera
base, the exterior orientation position coordinates of
each camera pairs perspective centres); and
b. the relative rotations of the cameras (camera
convergence, the exterior orientation rotations of
each camera pairs coordinate axes).
The type of invariance is dependent upon the type of
stereo camera system used. Two stereo camera types
considered here are:
a. stereometric camera systems where the invariant
camera relationships are precisely known; and
b. non- or semi- metric camera systems where
cameras are placed in a fixed but not precisely
known relationship for specific projects as typified in
Fryer (1990).
By considering type b. camera systems as representing the
general case, the numerical values that describe the
invariant camera relationships must be solved for as part of
the bundle adjustment. Type a. cameras may then be
considered as a special case and the known values applied
as appropriate.
Two methods were considered in modelling the invariant
relationships:
a. by constraint equations between the camera
parameters of each stereopair; or
b. by modified collinearity equations relating the
camera parameters of each stereopair.
INVARIANCE MODELLED BY CONSTRAINT EQUATIONS
An introduction to the use of constraint equations in
analytical photogrammetry in given in Case (1961).
Constraint equations are used to express geometric or
physical relationships that exist between parameters of an
adjustment. In this case the parameters of interest are the
exterior orientation parameters of the cameras.
Constraint equations take the following form:
v, +CA = g (2)
v, = vector of constraint residuals;
C = matrix of constraint parameter
coefficients;
A = vector of parameters used in constraints;
g = vector of constraint constants,
One constraint equation is written for each physical or
geometric property that is required to be enforced.
The base constraint
For stereo photography the required constraint is that the
computed base distances for all stereopairs be the same.
As the camera base is unknown a priori the constraint
constant (the camera base distance) must be derived from
the adjustment. At each iteration the mean value of all the
bases is computed and used as the constraint constant.
The base constraint for stereopair i is then:
=
[X -XPP «v; -Y* «z^ -z^yr - B, (3
B, = mean camera base.
There is one base constraint equation for each stereopair.
The convergence constraint
The requirement here is that the relative orientations of the
camera axes are constant for all stereopairs. This may be
achieved by constraining the convergence angles of the
three pairs of camera axes (dot products of each pair of X,
Y & Z axes) to the mean of the corresponding convergence
angles for al! stereopairs. As the convergence angles are
independent for each axis pair there is one constraint
equation for each convergence angle. For stereopair i
these are:
Rit. Bit «RI2/. R12 «RISI. RIS - 4X,
R21/.R21P +R22/".R22/" +R23;.R23," = vY, (4)
R31,.R31* «R32/.R32^ «R33/.R33/ - yZ,,
R11 1. R33/° = rotation matrix elements of the
left and right cameras of
stereopair í,
1X, Y Y, YZn = Mean convergence angles for X,
Y and Z axes.
All constraint equations are non-linear in terms of the
camera parameters and must be linearised. Adding the
linearised constraint equations to the conventional
collinearity equation model changes equation (1) to:
V * BA = C (5)
V B B €
zulVi.- A = IC
V=|.,; B- Feb. uie. ur
V 0 -/ ^ C
V. C 0 g
C = matrix of partial derivatives of constraint
parameters;
g = vector of constraint discrepancies;
The corresponding normal equation matrix structure is
shown in figure 2.
Each stereopair is represented by a 12x12 symmetric sub-
matrix as a result of the constraint equations. The number
of camera parameters is unaltered from the conventional
model however there are 4s additional constraint equations
which will improve this model's degrees of freedom over
the conventional model.
