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# Kalman Filtering Technique

Nota: The first four subsections are extracted from the book [25]: 3D Dynamic Scene Analysis: A Stereo Based Approach, by Z. Zhang & O. Faugeras (Springer Berlin 1992).

Kalman filtering, as pointed out by Lowe [12], is likely to have applications throughout Computer Vision as a general method for integrating noisy measurements.

The behavior of a dynamic system can be described by the evolution of a set of variables, called state variables.  In practice, the individual state variables of a dynamic system cannot be determined exactly by direct measurements; instead, we usually find that the measurements that we make are functions of the state variables and that these measurements are corrupted by random noise. The system itself may also be subjected to random disturbances. It is then required to estimate the state variables from the noisy observations.

If we denote the state vector by and denote the measurement vector by , a dynamic system  (in discrete-time form) can be described by

where is the vector of random disturbance of the dynamic system and is usually modeled as white noise:

In practice, the system noise covariance is usually determined on the basis of experience and intuition (i.e., it is guessed). In (18), the vector is called the measurement vector. In practice, the measurements that can be made contain random errors. We assume the measurement system  is disturbed by additive white noise, i.e., the real observed measurement is expressed as

where

The measurement noise covariance is either provided by some signal processing algorithm or guessed in the same manner as the system noise. In general, these noise levels are determined independently. We assume then there is no correlation between the noise process of the system and that of the observation, that is

Next: Standard Kalman Filter Up: Parameter Estimation Techniques: A Previous: Bias-Corrected Renormalization Fitting

Zhengyou Zhang
Thu Feb 8 11:42:20 MET 1996