Review: Prediction of Respiratory Motion - Prediction and Classification of Respiratory Motion - page 14

x ( t ) a 0 x ˆ t )

Σ

z - 1

a 1

z - 1

a 2

z - 1

a n

Fig. 2.5 Linear predictor with tapped-delay line. The predicted value is a linear combination of
previous observations x(t - n) and predictor coefficients a n that are not changing over time

User defined

u ( t+ 1) z ( t+ 1)

u ( t ) z ( t ) Hidden state

Visible

Hidden

H

B V , R B H V , R

F F

x ( t -1) x ( t ) x ( t +1)

W , Q t W , Q t +1

t

−

1

Fig. 2.6 Roles of the variables in the Kalman filter. u(t)isann-dimensional known vector, and
z(t) is a measurement vector. The next state is calculated based on the dynamic equation, such as
xt þ ð Þ¼ Fx ðÞþ Bu ðÞþ V . Here, V and W are process noise and measurement noise with
covariance R and Q

2.3.1.2 Kalman Filter

The Kalman filter (KF) is one of the most commonly used prediction methods in
real-time filtering technologies [ 72 , 73 , 75 , 81 - 83 ]. KF provides a recursive
solution to minimize mean square error within the class of linear estimators, where
linear process and measurement equations to predict a tumor motion can be
expressed as follows [ 84 ]:

x ð t Þ¼ Fx ð t 1 Þþ Bu ð t 1 Þþ W ; z ð t Þ¼ Hx ð t Þþ V ; ð 2 : 2 Þ

where we denote the state transition matrix as F, the control-input matrix as B, and
the measurement matrix as H. u(t)isann-dimensional known vector, and z(t)isa
measurement vector. The random variables W and V represent the process and
measurement noise with the property of the zero-mean white Gaussian noise with
covariance,

¼ Q ðÞ , respectively. The
matrices F, B, W, H, and V are assumed known and possibly time-varying (Fig. 2.6 ).

EW ðÞ W ðÞ T ¼ R ðÞ and E V ðÞ V ðÞ T