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working environment of the harvester, any algorithm that attempts to correct the functioning of some
system in the harvester must be robust. Currently the harvester measures the diameter online while the
stem is processed without using future measurements to estimate the current diameter (filtering case).
The estimate is used whenever a diameter measurement is needed, for instance to predict the tapering of
the stem and to compute the volume of a log. In this paper three approaches for processing the meas-
urements are discussed: online filtering, online smoothing and offline smoothing.
A dataset consisting of 479 logs was used to evaluate the performance of the methods. The diameter
profile of each log in the dataset was measured manually. The measurements were taken with a measur-
ing interval of 0.3 meters at an accuracy of 1 mm. The diameter was measured along two perpendicular
lines, and the mean of the measurements was taken to be the final value. The manual measurements
were regarded to be correct and were compared to the diameter profile measured by the harvester. The
sum of square errors at each measurement point and at the cutting points divided with the number of the
measurement points were used as metrics for the performance of the methods.

ONLINE FILTERING

First a simple linear approximation was used. The diameter measurements on the level sections of the
stem profile are rejected and estimated by fitting a linear function in least squares sense to the valid
measurements. Estimated diameters are obtained by evaluating the function at the measurement points.

A second approach was to use a Kalman filter. Kalman filters are estimators that are used for deducing
the true value of a variable in a dynamical system. If the measurements given to a Kalman filter contain
normally distributed uncorrelated noise, then the estimates are optimal with respect to all quadratic func-
tions of the estimation error. (Grewal et al., 1993) The Kalman filter needs a model of the system to
work. Tree tapering curves have been studied previously to some extent, for example polynomial models
were used by Laasasenaho (1982) and in mixed linear regression models by Lappi (1986). These ap-
proaches were not used in this study because of the relatively high complexity of the models. It can be
concluded from the abovementioned studies that the parameters of the models vary by geographic region
and by tree species, so an adaptive model is needed. The current solution is to use a tapering matrix
which is updated during harvesting, and this is the approach that was used also in this study. In this ap-
plication a first order time-variant filter was used. The tapering matrix is a model of the change of the
stem diameter between two consecutive measurements at each relative height. If the measured diameter
stays constant, the magnitude of the noise covariance is increased to account for the increased uncer-
tainty in the measurements. This results in reducing the weight that is given to the difference between
the measured and estimated diameter values when the next diameter estimate is computed. The meas-
urement noise in this application is neither normally distributed nor uncorrelated, which degrades the
optimality of the estimates. However, a Kalman filter is still worth studying because of its ability to han-
dle measurements with varying uncertainty. Kalman filtering was also applied such that the output of the
filter is used at the level sections of the stem profile. When a level section is found, the tapering rate of
the filtered profile is set to be the same as the current tapering rate obtained from the Kalman filter.

Finally, a backward approximation method was used. The method tries to fix previous measurements by
recalculating them. Each time the algorithm detects a large drop in the diameter profile it checks if there
is a level section before the drop. If this is the case, the algorithm connects the beginning of the level
section with the measurement at the bottom of the drop.
The two first filters improve the accuracy of the diameter measurement for the whole stem as well as in
the cutting points. Since the Kalman filter is based on a model of the stem profile, the results are de-
pendant of the quality of the model. Even if the model was good for the majority of the trees on a stand,
most likely some of the trees would have a very different profile due to environmental conditions, and