Nonlinear filtering of systems governed by Ito differential equations with jump parameters

Hypoelliptic Diffusion and Human Vision: A Semidiscrete New Twist

“…• The lifting procedure to SE(2, N ) that we implement is the most simple that can be thought: after smoothing the image (Gaussian smoothing), the gradient at the point is well defined (both the smoothing and the gradient computation are performed at the level of the FFT of the image), and we just lift to the closest value of the angle. 4 To simplify, even when we know where the image is corrupted we always compute the gradient in this rough way. This point could certainly be improved on.…”

“…Firstly, we show that a certain semi-discrete (discretization with respect to the angle) model of the diffusion is compatible with the limit continuous model. For the considerations in this section, one may refer to the paper [4]. The developments in this section may be considered as elementary by probabilistic readers, however they are very instructive with respect to the nature of our semi-discrete model.…”

Linear Filtering for a Class of Jump Processes Arising in Navigation Systems

“…Here we consider the filtering problem for the case where the process r) is allowed to have multiple jumps. It is clear from equation (5) that, if, for some t eJ, r](t) = e, (1 ^ i =s M), then, for sufficiently small At>0, t](t + At) = e/_i or e t or e /+1 . This fact is indicated in the state transition diagram (Fig.…”

“…where t k is the discrete time point at which the observation y{t k ) is available (see equation (8)). Further, during any interval of time [t k , t k+l ) (1 «s k «s N), the process r) can only make one transition from one state to another and that the infinitesimal rates {k t j : i,j = 1, 2,..., M} satisfy the property (5). Under these assumptions, one can determine, by observing y(t k ) and y(t k+1 ), whether or not a jump has occurred in x during the time interval (t k , t k+i ].…”

“…Hence the process y carries information about the jump process TJ, and therefore x is conditionally Gaussian relative to y. Using this fact, the filtering problem reduces to solving a finite set of differential or difference equations instead of solving linear or nonlinear partial differential equations, as indicated in [5][6][7][8]13].…”