New smoothing algorithms based on reversed-time lumped models

Dense nonrigid motion tracking from a sequence of velocity fields

“…The problem is then the following: Given the observations : m(O), ... , m(T), we wish to obtain an optimal estimate of the state vector s at any given time k within the interval, s(klk), that minimizes the mean-normsquared-error. There exists an efficient recursive algorithm to calculate this estimate: the fixed interval smoothing filter [11].…”

A Stochastic Realization Approach to the Smoothing Problem. Revised.

“…Our interest in the smoothing problem was caused by the Mayne-Fraser twofilter formula [5,6], on which topic a large number of papers have been written [7][8][9][12][13][14][15][16][17] . In some of these papers the authors have encountered difficulties in motivating this formula, and the many attempts to justify it stochastically have, in our opinion , been less than convincing~ In our stochastic realization setting the two filters have a natural interpretation :…”

“…Moreover , we have chosen to write the backward equation in terms of i rather than x as in [ 13 , 161 . (However , see the "ad joint" formulation in [ 16]. )…”

“…where x~ and x~ are given by Fraser two-filter f ormuZ-a [5,6], which has received considerable attention in the literature [7][8][9][13][14][15][16][17] . Although this algorithm is easy to derive formally [9], its probabilistic justification has caused considerable difficulty, partly due to the fact that Q*(t) as t -~ T. The system (3.20) has usually been interpreted as a backward filter , and in [14][15][16][17] [14 , 15 , 17], in which papers the back ward estima te…”

“…Although this algorithm is easy to derive formally [9], its probabilistic justification has caused considerable difficulty, partly due to the fact that Q*(t) as t -~ T. The system (3.20) has usually been interpreted as a backward filter , and in [14][15][16][17] [14 , 15 , 17], in which papers the back ward estima te…”