The likelihood calculation of a vast number of particles forms the computational bottleneck for the particle filter in applications where the observation model is complicated, especially when map or image processing is involved. In this paper, a numerical fitting approach is proposed to speed up the particle filter in which the likelihood of particles is analytically inferred/fitted, explicitly or implicitly, based on that of a small number of so-called fulcrums. It is demonstrated to be of fairly good estimation accuracy when an appropriate fitting function and properly distributed fulcrums are used. The construction of the fitting function and fulcrums are addressed respectively in detail. To avoid intractable multivariate fitting in multi-dimensional models, a nonparametric kernel density estimator such as the nearest neighbor smoother or the uniform kernel average smoother can be employed for implicit likelihood fitting. Simulations based on a benchmark one-dimensional model and multi-dimensional mobile robot localization are provided
