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This paper addresses the problem of motion analysis performed in a signal sampled on an irregular grid spread in 3-dimensional space and time (3D+T). Nanosensors can be randomly scattered in the field to form a “sensor network”. Once released, each nanosensor transmits at its own fixed pace information which corresponds to some physical variable measured in the field. Each nanosensor is supposed to have a limited lifetime given by a Poisson-exponential distribution after release. The motion analysis is supported by a model based on a Lie group called the Galilei group that refers to the actual mechanics that takes place on some given geometry. The Galilei group has representations in the Hilbert space of the captured signals. Those representations have the properties to be unitary, irreducible and square-integrable and to enable the existence of admissible continuous wavelets fit for motion analysis. The motion analysis can be considered as a so-called “inverse problem” where the physical model is inferred to estimate the kinematical parameters of interest. The estimation of the kinematical parameters is performed by a gradient algorithm. The gradient algorithm extends in the trajectory determination. Trajectory computation is related to a Lagrangian-Hamiltonian formulation and fits into a neuro-dynamic programming approach that can be implemented in the form of a Q-learning algorithm. Applications relevant for this problem can be found in medical imaging, Earth science, military, and neurophysiology.
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