How many Kalman filters?

Consider a multitarget tracking problem with M=3 measurements per time instant and K=2 tracks. Suppose that at some time instant we have 1 Kalman filter per measurement and are determining the next states using the K-path Viterbi algorithm [1]. The N=3 states used by the Viterbi algorithm, representing combinations of the measurement indices (1,2,3) taken two at a time, are denoted by the pairs (1,2), (1,3), and (2,3). The best path to any state must also note which of the K! permutations of the measurement indices is being used. This is represented in the figure below by having two labels for each state:

The measurement tracks corresponding to the state transitions shown in the figure are:

1 -> 2
2 -> 1
1 -> 1
3 -> 3
2 -> 3
3 -> 2

In this example each destination measurement index has two possible predecessors. For example, index 1 must be represented using two Kalman filters, one with predecessor 2 and the other with predecessor 1.

The conclusion is, in the general case, N*K Kalman filters must be stored for each time instant.

[1]
	J. K. Wolf, A. M. Viterbi, and G. S. Dixon, Finding the Best Set of K Paths Through a Trellis with Application to Multitarget Tracking, IEEE AES-25, No. 2, March 1989, pp. 287-296