Table 1.

A list of mathematical symbols used in the Kalman filter, and what they represent in the proposed kV-based 3D tumor tracking application.

Symbol	Kalman filter	kV-based 3D tumor tracking
k	Current time frame	Current imaging frame
x̂k|k−1,x̂k|k	Predicted/updated state	Predicted/updated 3D tumor position
Pk|k−1,Pk|k	Predicted/updated state covariance matrix	Predicted/updated covariance matrix of the distribution of 3D tumor positions
zk	Measurement	2D tumor position on the kV image measured using template matching
hk(․)	Measurement response function	3D-to-2D forward-projection function
Qk	Prediction covariance matrix	Uncertainty attributed to the respiratory-correlated prediction model
Rk	Measurement covariance matrix	Uncertainty attributed to template matching
Kk	Kalman gain	The weightings of how much the template matching result is used to correct the prediction
Fk	Jacobian of the prediction model	Jacobian of the respiratory-correlated prediction (reduced to I)
Hk	Jacobian of the measurement response function	Jacobian of the 3D-to-2D forward-projection function