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This work presents new methods and algorithms for tracking the shape andtrajectory of moving reflecting obstacles with broken rays, or rays reflectingat an obstacle. While in tomography the focus of the reconstruction method isto recover the velocity structure of the domain, the shape and trajectoryreconstruction procedure directly finds the shape and trajectory of theobstacle. The physical signal carrier for this innovative method are ultrasonicbeams. When the speed of sound is constant, the rays are straight line segmentsand the shape and trajectory of moving objects will be reconstructed withmethods based on the travel time equation and ellipsoid geometry. For variablespeed of sound, we start with the eikonal equation and a system of differentialequations that has its origins in acoustics and seismology. In this case, therays are curves that are not necessarily straight line segments and we developalgorithms for shape and trajectory tracking based on the numerical solution ofthese equations. We present methods and algorithms for shape and trajectorytracking of moving obstacles with reflected rays when the location of thereceiver of the reflected ray is not known in advance. The shape and trajectorytracking method is very efficient because it is not necessary for the reflectedsignal to traverse the whole domain or the same path back to the transmitter.It could be received close to the point of reflection or far away from thetransmitter. This optimizes the energy spent by transmitters for tracking theobject, reduces signal attenuation and improves image resolution. It is a safeand secure method. We also present algorithms for tracking the shape andtrajectory of absorbing obstacles. The new methods and algorithms for shape andtrajectory tracking enable new applications and an application to one-hopInternet routing is presented.

Shape and Trajectory Tracking of Moving Obstacles