# vision.KalmanFilter

Correction of measurement, state, and state estimation error covariance

## Description

The Kalman filter object is designed for tracking. You can use it to predict a physical object's future location, to reduce noise in the detected location, or to help associate multiple physical objects with their corresponding tracks. A Kalman filter object can be configured for each physical object for multiple object tracking. To use the Kalman filter, the object must be moving at constant velocity or constant acceleration.

## Creation

The Kalman filter algorithm involves two steps, prediction and correction (also known as the update step). The first step uses previous states to predict the current state. The second step uses the current measurement, such as object location, to correct the state. The Kalman filter implements a discrete time, linear State-Space System.

Note

To make configuring a Kalman filter easier, you can use the `configureKalmanFilter` object to configure a Kalman filter. It sets up the filter for tracking a physical object in a Cartesian coordinate system, moving with constant velocity or constant acceleration. The statistics are the same along all dimensions. If you need to configure a Kalman filter with different assumptions, do not use the function, use this object directly.

In the state space system, the state transition model, A, and the measurement model, H, are set as follows:

VariableValue
A`[1 1 0 0; 0 1 0 0; 0 0 1 1; 0 0 0 1]`
H`[1 0 0 0; 0 0 1 0]`

### Syntax

``kalmanFilter = vision.KalmanFilter``
``kalmanFilter = vision.KalmanFilter(StateTransitionModel,MeasurementModel)``
``kalmanFilter = vision.KalmanFilter(StateTransitionModel,MeasurementModel,ControlModel,Name,Value)``

### Description

example

````kalmanFilter = vision.KalmanFilter` returns a kalman filter for a discrete time, constant velocity system.```
````kalmanFilter = vision.KalmanFilter(StateTransitionModel,MeasurementModel)` additionally configures the control model, B.```
````kalmanFilter = vision.KalmanFilter(StateTransitionModel,MeasurementModel,ControlModel,Name,Value)` configures the Kalman filter object properties, specified as one or more `Name,Value` pair arguments. Unspecified properties have default values.```

## Properties

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Model describing state transition between time steps (A), specified as an M-by-M matrix. After the object is constructed, this property cannot be changed. This property relates to the A variable in the state-space model.

Model describing state to measurement transformation (H) , specified as an N-by-M matrix. After the object is constructed, this property cannot be changed. This property relates to the H variable in the state-space model.

Model describing control input to state transformation (B) , specified as an M-by-L matrix. After the object is constructed, this property cannot be changed. This property relates to the B variable in thestate-space model.

State (x), specified as a scalar or an M-element vector. If you specify `State` as a scalar, it will be extended to an M-element vector. This property relates to the x variable in the state-space model.

State estimation error covariance (P), specified as a scalar or an M-by-M matrix. If you specify `StateCovariance` as a scalar it will be extended to an M-by-M diagonal matrix. This property relates to the P variable in the state-space system.

Process noise covariance (Q) , specified as a scalar or an M-by-M matrix. If you specify `ProcessNoise` as a scalar it will be extended to an M-by-M diagonal matrix. This property relates to the Q variable in the state-space model.

Measurement noise covariance (R) , specified as a scalar or an N-by-N matrix. If you specify `MeasurementNoise` as a scalar it will be extended to an N-by-N diagonal matrix. This property relates to the R variable in the state-space model.

## Object Functions

Use the `predict` and `correct` functions based on detection results. Use the `distance` function to find the best matches.

• When the tracked object is detected, use the `predict` and `correct` functions with the Kalman filter object and the detection measurement. Call the functions in the following order:

```[...] = predict(`kalmanFilter`); [...] = correct(`kalmanFilter`,measurement);```

• When the tracked object is not detected, call the `predict` function, but not the `correct` function. When the tracked object is missing or occluded, no measurement is available. Set the functions up with the following logic:

```[...] = predict(`kalmanFilter`); If measurement exists [...] = correct(`kalmanFilter`,measurement); end```

• If the tracked object becomes available after missing for the past t-1 contiguous time steps, you can call the `predict` function t times. This syntax is particularly useful to process asynchronous video.. For example,

```for i = 1:k [...] = predict(kalmanFilter); end [...] = correct(kalmanFilter,measurement) ```

 `correct` Correction of measurement, state, and state estimation error covariance `predict` Prediction of measurement `distance` Confidence value of measurement

## Examples

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Track the location of a physical object moving in one direction.

Generate synthetic data which mimics the 1-D location of a physical object moving at a constant speed.

`detectedLocations = num2cell(2*randn(1,40) + (1:40));`

Simulate missing detections by setting some elements to empty.

```detectedLocations{1} = []; for idx = 16: 25 detectedLocations{idx} = []; end```

Create a figure to show the location of detections and the results of using the Kalman filter for tracking.

```figure; hold on; ylabel('Location'); ylim([0,50]); xlabel('Time'); xlim([0,length(detectedLocations)]);```

Create a 1-D, constant speed Kalman filter when the physical object is first detected. Predict the location of the object based on previous states. If the object is detected at the current time step, use its location to correct the states.

```kalman = []; for idx = 1: length(detectedLocations) location = detectedLocations{idx}; if isempty(kalman) if ~isempty(location) stateModel = [1 1;0 1]; measurementModel = [1 0]; kalman = vision.KalmanFilter(stateModel,measurementModel,'ProcessNoise',1e-4,'MeasurementNoise',4); kalman.State = [location, 0]; end else trackedLocation = predict(kalman); if ~isempty(location) plot(idx, location,'k+'); d = distance(kalman,location); title(sprintf('Distance:%f', d)); trackedLocation = correct(kalman,location); else title('Missing detection'); end pause(0.2); plot(idx,trackedLocation,'ro'); end end legend('Detected locations','Predicted/corrected locations');```

Use Kalman filter to remove noise from a random signal corrupted by a zero-mean Gaussian noise.

Synthesize a random signal that has value of 1 and is corrupted by a zero-mean Gaussian noise with standard deviation of 0.1.

```x = 1; len = 100; z = x + 0.1 * randn(1,len);```

Remove noise from the signal by using a Kalman filter. The state is expected to be constant, and the measurement is the same as state.

```stateTransitionModel = 1; measurementModel = 1; obj = vision.KalmanFilter(stateTransitionModel,measurementModel,'StateCovariance',1,'ProcessNoise',1e-5,'MeasurementNoise',1e-2); z_corr = zeros(1,len); for idx = 1: len predict(obj); z_corr(idx) = correct(obj,z(idx)); end```

Plot results.

```figure, plot(x * ones(1,len),'g-'); hold on; plot(1:len,z,'b+',1:len,z_corr,'r-'); legend('Original signal','Noisy signal','Filtered signal');```

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## References

[1] Welch, Greg, and Gary Bishop, An Introduction to the Kalman Filter, TR 95–041. University of North Carolina at Chapel Hill, Department of Computer Science.

[2] Blackman, S. Multiple-Target Tracking with Radar Applications. Artech House, Inc., pp. 93, 1986.

## Version History

Introduced in R2012b