Cross approximation-based quadrature filter

The Gauss-Hermite quadrature filter (GHQF) can achieve arbitrary degree of accuracy and high stability, but it suffers from heavy computational burden. Alternative high accurate filters, such as high-degree cubature Kalman filter (CKF) and high-degree sparse-grid quadrature filter (SGQF), can greatly reduce the computational cost but may have stability concerns. To give consideration to both filtering stability and efficiency, a cross approximation-based quadrature filter is proposed. The filter can achieve the same accuracy and stability as GHQF with much less computational burden. Firstly, tensors in GHQF are unfolded into matrices to incorporate cross approximation method, and low-rank representations of the unfolding matrices are obtained by only sampling a small subset of the sigma points. Secondly, taking advantage of the low-rank representations, the computational cost is further reduced using low-rank matrix operations. Simulation results show that the proposed filter only samples about 3% of the sigma points of 3-point GHQF in a 10-dimension target tracking problem, but achieves the same performance as GHQF.