International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004
extended Kalman filter (EKF) can be formulated. It is a
standard tool in engineering that is frequently used when either
the system model or the observation model are non-linear. In an
interesting paper Julier and Uhlmann (1996) have demonstrated
that even in a seemingly innocuous situation - a road vehicle
moving along a circle - the EKF does not handle the non-
linearity in an acceptable manner. After a quarter circle, the
covariance propagation results in error ellipses that do not
represent the actual situation. The authors show quite
convincingly that this is due to the linearized covariance
propagation. To rectify the situation, the authors propose to
change this part of the EKF. They approximate the Gaussian
distribution at a carefully chosen set of points and propagate
this information through the non-linear equations. In this way
the transformed Gaussian distribution will reflect the non-
linearities of the system better. The new filter, called the
Unscented Kalman Filter (UKF) by its authors, has received
some attention during the past few years, see for instance Julier
and Uhlmann (2002) and Crassidis and Markley (2003). A
paper that combines in-motion alignment, large azimuth
modeling and UKF for MEMS /DGPS integration is Shin and
El-Sheimy (2004). Although the UKF does not often show a
substantial increase in accuracy, it seems to be more robust than
the EKF in critical situations.
Another approach based on Neural Networks (NN) has been
proposed by Chiang and El-Sheimy (2002). They suggested an
INS/GPS integration algorithm utilizing multi-layer neural
networks for fusing data from DGPS and either navigation
grade IMUs or tactical grade IMUs. Artificial Neural Networks
(ANNs) have been quite promising in offering alternative
solutions to many engineering problems, where traditional
models have failed or were too complicated to build. Due to the
nonlinear nature of ANNs, they are able to express much more
complex phenomena than some linear modeling techniques.
They extract the essential characteristics from the numerical
data as opposed to memorizing all of it. ANNs, therefore, offer
a convenient way to form an implicit model without necessarily
establishing a traditional, physical mathematical model of the
underlying phenomenon (see Figure 6). In contrast to traditional
Kalman Filtering models, ANNs require only a little or no a
priori knowledge of the underlying mathematical process. For
GPS/INS integration, this simply means that integration
architecture is platform and system independent, as long as the
implicit functional relationship between the input and output is
fixed.
Observed
d Unknown model Output
» 11) »
d
Figure 6: Supervised learning as model identification or
function approximation
S. FILTERING AND SMOOTHING
The discussion in the last chapter indicated already that
modeling and estimation are closely connected in the geo-
referencing problem. In terms of interesting recent contributions
in filtering and smoothing, three topics will be discussed:
Denoising, AR modeling, and simplified smoothing. All three
are post-mission methods and are well suited to mobile
mapping. Denoising is an important aspect in post-mission IMU
modeling because the noise level of inertial sensors is very
high, typically 20 000 — 30 000 times higher than the minimum
signal to be resolved. In real-time applications the standard way
of treating this problem is to trust integration to work as a filter
and to carefully select the white noise components in the
Kalman filter. In mobile mapping where most applications are
processed in post mission, denoising often allows a more
refined analysis because the spectral band of interest can be
defined and the high-noise band can be eliminated. This is of
importance when one tries to model the bias terms in the
Kalman filter, as in case of autoregressive modeling (AR).
Without denoising the results of an AR analysis become
meaningless. Post-mission processing has also the advantage
that trajectory constraints can be applied in both forward and
backward direction, while in real-time Kalman filtering this is
only possible in forward direction. Since an optimal smoother is
time consuming and requires considerable storage capacity, a
simplified model for backward smoothing will also be briefly
discussed.
Band limiting and denoising describes a variety of techniques
that can be used to eliminate white or colored noise from
observations. Skaloud et al (1999) were the first to apply
wavelet denoising to the raw data from inertial sensors. They
were able to show that the accuracy of the estimated orientation
parameters improved by a factor of five, resulting in standard
deviations of about 10 arcseconds for pitch and roll, and of 20
arcseconds for azimuth for a medium accuracy IMU. In Figure
7, the rather dramatic noise reduction is shown when applying a
wavelet filter to a set of accelerometer measurements. The noise
drops from a standard deviation of about 2 000 mGal (a) to
about 10 mGal (b). Further work in this area was done by
Noureldin et al (2002) who used forward linear prediction to
design a tap delay line filter to improve the performance of a
FOG gyro by eliminating the short-term angle random walk.
iere À à i zi ee leri =
60 120 180 240 300 360 420 480
Time (minutes)
Figure 7: Y-Accelerometer Specific Force Measurements
(a) Before Wavelet De-noising
(b) After Applying the Wavelet 6" LOD
