Dr Graeme A. Jones tools from the vision tool box Kalman Tracker - noise and filter design

Reviewing the Kalman Equations Predict state pˆ, and state uncertainty Pˆ Predict observation zˆ, and observation uncertainty Zˆ Update state p, P from actual observation z, Z:

Reducing the uncertainty of the state Starting from an initial uncertain state, the state uncertainty should steadily reduce with the number of observations. Observed state and observed state uncertainty x y Constant Acceleration True Observation Noise Initial State

Observation Noise What is the relationship between the computed state uncertainty X t and the estimated uncertainty Z t of each observation z t ? (Not necessarily the real underlying uncertainty Z t * ) –the asymptotic value of the state covariance X t is directly related to the estimated uncertainty Z t rather than the real uncertainty Z t *

True Observation Noise

Observation Noise What is the relationship between the computed state uncertainty X t and the estimated uncertainty Z t of each observation z t ? (Not necessarily the real underlying uncertainty Z t * ) –the asymptotic value of the state covariance X t is directly related to the estimated uncertainty Z t rather than the real uncertainty Z t * –when the estimated uncertainty Z t is too small, the necessary data association stage starts to reject even those true observations with modest amounts of noise.

True Observation Noise  2 Threshold = 5.0 (92%)

The role of system noise Q is to enable the filter to adapt to deviations from the assumed trajectory model. –expands the state uncertainty (and hence the uncertainty of the predicted position) System Noise

Linear Model / Quadratic Trajectory

No Data Association Quadratic Model / Linear Trajectory

Noisy Observation Stream Since Gaussian PDF is infinite, the thresholded gate i.e. χ thres would miss a predictable number of true observations present in the stream. True observations are typically accompanied by noisy observations (uniformly distributed?)

Noisy Observation Stream Linear Model / Quadratic Trajectory (10 Uniformly distributed noise samples)

Noisy Observation Stream Linear Model / Quadratic Trajectory (10 Uniformly distributed noise samples)

Noisy Observation Stream Linear Model / Quadratic Trajectory (10 Uniformly distributed noise samples)

Noisy Observation Stream Quadratic Model / Quadratic Trajectory (10 Uniformly distributed noise samples) Higher dimensional model particularly vulnerable near initiation where state covariance high. Recommended solution is to constrain model to linear trajectory using tight initial state covariance and allow added system noise to enable acceleration term. Higher dimensional model particularly vulnerable near initiation where state covariance high. Recommended solution is to constrain model to linear trajectory using tight initial state covariance and allow added system noise to enable acceleration term.

Summary The estimated observation noise should be at least as large as the underlying observation noise. System noise should reflect deviation from trajectory model Both observation and system noise significantly increase the size of the predicted position uncertainty, and, hence the size of the data association gate. The data association gate is itself a significant source of noise into the system (typically tackled by including appearance matching). (No satisfactory practical method of handling update stage when dealing with missing data.)