Particle filtering is frequently used for visual tracking problems since it provides a general framework for estimating and propagating probability density functions for nonlinear and non-Gaussian dynamic systems. However, this algorithm is based on a Monte Carlo approach and the cost of sampling and measurement is a problematic issue, especially for high-dimensional problems. We describe an alternative to the classical particle filter in which the underlying density function has an analytic representation for better approximation and effective propagation. The techniques of density interpolation and density approximation are introduced to represent the likelihood and the posterior densities with Gaussian mixtures, where all relevant parameters are automatically determined. The proposed analytic approach is shown to perform more efficiently in sampling in high-dimensional space. We apply the algorithm to real-time tracking problems and demonstrate its performance on real video sequences as well as synthetic examples.