Using Kalman Filter to Estimate Motion of Object Along Line Segment

[Lindley]

I want to use a Kalman filter to estimate the motion of an object. However, the catch is, the measurements I have only tell me that the object is somewhere along a particular line segment.

Typically Kalman filters require normally distributed measurements. I'm trying to work out how best to represent these line segments to the filter. Obviously, a 2D normal with extremely high covariance in the direction of the line would work to give relatively high likelihood to any point on the line segment; however, I also want likelihood to fall off quickly beyond the line, which won't occur in this case.

Any ideas?

 

[chingkui]

if u know the direction of the line, can you transform the 2D coordinates of your measurement to have one coordinate align with the line and the other orthogonal to that line, and then manipulate the measurement noise covariance?