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:

Variable	Value
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

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

StateTransitionModel — Model describing state transition between time steps (A)

[1 1 0 0; 0 1 0 0; 0 0 1 1; 0 0 0 1] (default) | M-by-M matrix

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.

MeasurementModel — Model describing state to measurement transformation (H)

[1 0 0 0; 0 0 1 0] (default) | N-by-M matrix

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.

ControlModel — Model describing control input to state transformation (B)

[] (default) | M-by-L matrix

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 the state-space model.

State — State (x)

[0] (default) | scalar | M-element vector.

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.

StateCovariance — State estimation error covariance (P)

[1] (default) | scalar | M-by-M matrix

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.

ProcessNoise — Process noise covariance (Q)

[1] (default) | scalar | M-by-M matrix

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.

MeasurementNoise — Measurement noise covariance (R)

[1] (default) | scalar | N-by-N matrix

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

Track Location of An Object

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');

Remove Noise From a Signal

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');

Algorithms

State Space Model

This object implements a discrete time, linear state-space system, described by the following equations.

State equation:	$$x(k)=Ax(k-1)+Bu(k-1)+w(k-1)$$
Measurement equation:	$$z(k)=Hx(k)+v(k)$$

Variable Definition

Variable	Description	Dimension
k	Time.	Scalar
x	State. Gaussian vector with covariance P. [$$x~N(\overline{x},P)$$]	M-element vector
P	State estimation error covariance.	M-by-M matrix
A	State transition model.	M-by-M matrix
B	Control model.	M-by-L matrix
u	Control input.	L-element vector
w	Process noise; Gaussian vector with zero mean and covariance Q. [$$w~N(0,Q)$$]	M-element vector
Q	Process noise covariance.	M-by-M matrix
z	Measurement. For example, location of detected object.	N-element vector
H	Measurement model.	N-by-M matrix
v	Measurement noise; Gaussian vector with zero mean and covariance R. [$$v~N(0,R)$$]	N-element vector
R	Measurement noise covariance.	N-by-N matrix

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.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• See System Objects in MATLAB Code Generation (MATLAB Coder).

Version History

Introduced in R2012b