depth estimates and uncertainty values are computed at
each pixel of minified (256 x 256 pels) video frames and
that these estimates are refined incrementally over time.
The good performance of the Kalman filter is due to the
fact that only one parameter (depth value) constitutes the
state vector and is updated. Errors in orientation and cali-
bration of the video frames and thus correlations between
different elements of the depth map are not considered.
Also, only lateral motion of the CCD-camera is assumed.
Zhang and Faugeras (1990) use the Kalman update mech-
anism to track object motion in a sequence of stereo
frames. They track object features, called “tokens” (line
segments for example) in 3-D space from frame to frame
and estimate the motion parameters of these tokens in a
unified way (they also integrate a model of motion kine-
matics). Their state vector consists thus of three angular
velocity, three translational velocity and three translation-
al acceleration parameters of each object and in addition
six line segment parameters for each 3-D line representa-
tion. As before, the tokens are treated independently here,
which allows the state vector for estimation to remain
small in size and may result in straightforward paralleliza-
tion for any number of tokens.
Although we are aware that very fast (e.g. video rate 25
Hz) solutions to tracking problems with affordable com-
puter hardware still require substantial simplification of
the measurement problem at hand, we nevertheless
present in this paper the general solution for sequential
point positioning, based on the bundle solution. The solu-
tion is object point-bascd. Other features may be derived
from these 3-D point measurements. Any simplification in
measurement arrangement, as for instance non-moving
sensors, may readily be derived from the general concept.
Our solution may include self-calibration parameters for
systematic error modelling and statistical tests for blunder
detection. The sequential estimation procedure applies
Givens transformations for updating of the upper triangle
of the reduced normal equations (Blais, 1983, Gruen,
19852). We use Givens transformations as opposed to a
Kalman update because in robotics the covariance update
of the parameter vector is not required at every stage and
then, if at all, only at relatively sparse increments. Moreo-
ver, the varying size of the parameter vector (addition and
deletion of new object points, addition of frame exterior
orientation paramcters) leads to very poor computational
performance of the Kalman filter.
The presented approach treats only the 3-D point position-
ing problem in a sequential mode. A combination of sc-
quential algorithms in 2-D image measurement and 3-D
object point positioning within one unique system is feasi-
ble and meaningful, if for instance MPGC (Multiphoto
Geometrically Constrained) image matching, which de-
livers simultaneously object point coordinates, is execut-
ed for full frames in a pixel-by-pixel (iconic) mode. Also,
a complete bundle solution with integrated image match-
ing could be based on this concept. Another generaliza-
tion is possible if moving objects are included in the
system.
In section 2, a bricf description of the Givens transforma-
tions as applied to sequential bundle triangulation with
static object points is given. Section 3 presents a test ex-
ample using real image data produced with an arbitrarily
moving video camera over a 3-D testfield.
2. SEQUENTAL ESTIMATION IN BUNDLE
ADJUSTMENT WITH GIVENS
TRANSFORMATIONS
In this section we will present the formulae of Givens
transformations as applied to sequential estimation in
bundle systems in a concise fashion. For a more compre-
hensive treatment compare Blais (1983), Gruen (19852),
Runge (1987), and Holm (1989). In the following our
functional model for bundle adjustment will be set up
without the inclusion of parameters for self-calibration.
An extension by these parameters is straightforward and
does not alter the considerations and conclusions present-
ed here.
2.1. Least Squares approach for estimation
The Gauss-Markov model is the estimation model most
widely used in photogrammetric linear or linearized esti-
mation problems. An observation vector / of dimension n
x 1 is functionally related to a u x 1 parameter vector x
through
l-e- Ax. (1)
The design matrix A is an n x u matrix with n 2 u and
Rank (A) = u. There is no nced to work with rank-defi-
cient design matrices in on-line triangulation. Rank defi-
cient systems, caused by missing observations, generally
do not allow for a comprehensive model check. Observa-
tions should be accumulated until the system is regular
and can be solved using standard techniques. For rank de-
ficiency caused by incomplete datum, see Gruen (1985a).
Sequential least-squares estimation with pseudo-inverses
is very costly (compare Boullion, Odell, 1971, p. 50 ff).
The vector e represents the true errors. With the expecta-
tion E(e) = 0 and the dispersion operator D, we get
ED = Ax (2a)
D(l) = Cy » o2 P , and (2b)
D(e) C,, 5 Cy. (2c)
The estimation of x and 03 is usually attempted as unbi-
ased, minimum variance estimation performed by means
of least squares, and results in
zl
parameter vector & = (A"PA) A'PI, (3a)
residual vector v 2 AX - 1, (3b)
T
: 2
variance factor 6, = 7 5, r=n-u. (3c)
The architecture of A is determined by the type of triangu-
lation method used. As explained previously we chose the
bundle method for the purpose of generality and rigidity.
For bundle adjustment, Equation (1) can be written as
—e = Ag tA 4d SP (4a)
