Kalman Filter

TL;DR

Combines noisy sensor measurements with model-based predictions to estimate a system’s state over time.
Produces smooth, accurate estimates despite measurement uncertainty or noise.
Commonly applied to problems such as tracking position/velocity and estimating temperature in a building.

Definition

The Kalman filter is a mathematical algorithm that uses a series of data measurements and predictions to estimate the state of a system or process.

Explanation

The Kalman filter operates by combining noisy, uncertain measurements with predictions derived from a mathematical model of the underlying process. At each step it uses the model (based on the system’s previous state, such as prior position and velocity) to predict the current state, and then refines that prediction by incorporating the latest measurements. This iterative combination of prediction and measurement yields a smooth and accurate estimate of the system state over time, even when measurements are noisy or the process is uncertain.

The filter is applicable wherever sensors produce noisy readings or where the process dynamics are not perfectly known; for example, sensor errors, drift, camera shake, and variations in lighting are sources of measurement uncertainty that the Kalman filter is designed to accommodate.

Examples

Tracking a moving object

A camera provides a series of position measurements of a car moving along a straight road. Those measurements are noisy and uncertain due to factors such as camera shake and variations in lighting. A Kalman filter combines the noisy camera measurements with predictions of the car’s movement based on its previous position and velocity, using a mathematical model of the car’s motion to produce a smooth and accurate estimate of position and velocity over time.

Estimating temperature in a building

Multiple sensors measure temperature in different parts of a building, but the sensors may exhibit errors or drift and the building temperature can vary due to weather changes or opening and closing windows. A Kalman filter combines the noisy sensor measurements with predictions of the building’s temperature based on its previous state and a mathematical model of heat transfer, producing a smooth and accurate estimate of temperature over time.

Use cases

Tracking the position and velocity of a moving object (e.g., camera-based tracking of a car).
Estimating overall temperature in a building from multiple, noisy sensors.

Notes or pitfalls

Measurements used by the filter can be inherently noisy or uncertain (examples in the source: camera shake, variations in lighting, sensor errors, and sensor drift).
Reliable performance depends on having a mathematical model of the underlying process (e.g., motion model for tracking, heat transfer model for temperature estimation).

State
Process
Measurements
Prediction
Mathematical model
Noise
Uncertainty
Heat transfer
Position
Velocity
Sensors