Review: Prediction of Respiratory Motion - Prediction and Classification of Respiratory Motion - page 27

Interaction

μ ( t )

Interacting
(mixing)

Model
probability

μ i|j ( t )

μ ( t )

Filtering
(KF)

Λ 2

CV filter
(model 1)

CA filter
(model 2)

x ˆ (t) (t)

P

2 2

Λ 1

x ˆ

P ( t +1)

( t +1)

Combination

Combination of
estimates and covariance

x ˆ

(t) P (t)

1 1

Fig. 2.15 An interactive multiple model for respiratory motion prediction. In the interaction
step, model and mixing probabilities are initialized and updated. In the filtering step, the mixed
filtering prediction (x i ) of target position and the associated covariance (P i ) are updated within
each model. In the combination step, the actual prediction of target position is computed for
output purposes with the mixing probability

filtering, and combination—are repeated by each time instant t. In the interaction
step, model probability (l j (t)) and mixing probability (l i|j (t)) are initialized and
updated based on a 2 9 2 Markovian transition matrix (P) with its component p ij
that represents the transition probability from model i to model j, satisfied with
R j p ij ¼ 1 for i = 1, 2, as follows [ 81 , 83 ]:

l j ð t Þ¼ X

2

p ij l i ð t 1 Þ; l i j j ð t Þ¼ p ij l i ð t 1 Þ= l j ð t Þ; ð 2 : 16 Þ

i ¼ 1

where we denote l j (t) as the predicted probability for model j at time step t, and
l i|j (t) as the weight for the conditional transition probability from model i for the
previous time step t - 1 to model j for the current time step t. In the filtering step,
the mixed filtering prediction of target position (x j (t)) and the associated covari-
ance (P j (t)) are updated with Kalman gain, likelihood update (K j ) and model
probability (l j (t)), shown in Fig. 2.15 [ 81 ]. In combination step, the actual pre-
diction of target position, i.e., combination of estimates and covariance, is com-
puted for output purposes with the mixing probability, such as estimation

^xt þ 1 ð Þ¼ R j x j t þ 1 ð Þ l j ðÞ; and covariance P t þ 1 ð Þ¼ R j P j ðÞþ x j ðÞ ^x ðÞ

x j ðÞ ^x ðÞ T g l j ðÞ [ 83 ].
Putra et al. showed that the prediction of IMM filter was better than the prediction
of the Kalman filters with CV and CA model, and that the errors of the IMM filter
were less than 0.98 mm with 200 ms latency [ 81 ]. The limitation of this method is
that the above hybrid method was proposed for dynamic iteration in one dimensional