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# Reduced-Order Kalman Filtering for Processing Relative Measurements

- Wednesday, 28 May 2008

### A Kalman filter can be propagated using fewer computations.

A study in Kalman-filter theory has led to a method of processing relative measurements to estimate the current state of a physical system, using less computation than has previously been thought necessary. As used here, “relative measurements” signifies measurements that yield information on the relationship between a later and an earlier state of the system. An important example of relative measurements arises in computer vision: Information on relative motion is extracted by comparing images taken at two different times.

Relative measurements do not directly fit into standard Kalman filter theory, in which measurements are restricted to those indicative of only the current state of the system. One approach heretofore followed in utilizing relative measurements in Kalman filtering, denoted state augmentation, involves augmenting the state of the system at the earlier of two time instants and then propagating the state to the later time instant. While state augmentation is conceptually simple, it can also be computationally prohibitive because it doubles the number of states in the Kalman filter.

In many practical applications, relative measurements are not functions of entire earlier states but rather may be a function of only a subset of elements of the earlier state. A relative measurement that can be thus characterized is denoted a partial relative measurement. For example, in computer vision, relative-measurement information is usually a function of position rather than velocity, acceleration, or other elements of the state.

When processing a relative measurement, if one were to follow the state-augmentation approach as practiced heretofore, one would find it necessary to propagate the full augmented state Kalman filter from the earlier time to the later time and then select out the reduced-order components. The main result of the study reported here is proof of a property called reduced-order equivalence (ROE). The main consequence of ROE is that it is not necessary to augment with the full state, but, rather, only the portion of the state that is explicitly used in the partial relative measurement. In other words, it suffices to select the reduced-order components first and then propagate the partial augmented state Kalman filter from the earlier time to the later time; the amount of computation needed to do this can be substantially less than that needed for propagating the full augmented Kalman state filter.

*This work was done by David S. Bayard of Caltech for NASA’s Jet Propulsion Laboratory. For more information, contact This email address is being protected from spambots. You need JavaScript enabled to view it.
. NPO-44427*

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