A physics based approach to nonlinear filters
Fred Daum


We have invented a new particle filter with vastly superior performance compared with the classic particle filter and the extended Kalman filter. In particular, we achieve estimation errors that are many orders of magnitude smaller than the classic particle filter and several orders of magnitude better than the EKF. Performance of the new filter is also excellent for problems with multimodal densities. We do not use any proposal density, and we do not resample! This is a radical departure from any other particle filter. We evaluate performance for high dimensional fully-coupled (i.e., non-sparse) but smooth problems (d = 1 to 24). We have tested five classes of examples, with various nonlinearities, including quadratic and cubic.

The key idea is to compute Bayes’ rule using a flow of particles rather than as a pointwise multiplication. This is analogous to the flow of particles used to model the dynamics of the system in standard particle filters. We do not have to use any proposal density or resampling, because we move the particles to the correct distribution in state space using our particle flow. We completely avoid particle collapse or so-called degeneracy. We avoid the curse of dimensionality for certain problems that enjoy concentration of measure (e.g., log-concave probability densities). We never compute the density itself, but rather we represent the unnormalized log-density. Our algorithm is extremely robust, and in contrast with the EKF, it requires exactly no tuning of parameters. We show the flow of particles using very interesting movies. In particular, for an important radar application, we show the correct non-Gaussian density using our particles, which has been called the “contact lens” for obvious reasons. For this radar problem, our filter has much better velocity estimation accuracy compared with the extended Kalman filter.

The differential equation for particle flow is derived using Liouville’s criterion, borrowed from physics. Other ingredients include: the chain rule, the Moore-Penrose inverse and a log-homotopy. It turns out that a homotopy does not work at all, owing to the singularity of the resulting ODE, but a log-homotopy removes the singularity and works extremely well. The most interesting and challenging part of this filter is the approximation of the gradient of the log-homotopy; we studied 17 distinct methods for this, and we now use a simple but effective approach borrowed from geology, combined with a fast approximate k-NN algorithm. This talk is for normal engineers, who do not have log-homotopy for breakfast.