A second-order polynomial kernel outperforms Gaussian kernels when smoothing Lagrangian particle trajectories

Accurate reconstruction of particle acceleration requires post-processing of Lagrangian particle trajectories to limit noise amplification by differentiation. Over the past two decades, many studies have used a convolution filter based on a truncated Gaussian kernel. The present work evaluates the performance of Gaussian kernels truncated at varying standard deviations. It is shown that, compared to the truncation typically used in Lagrangian particle tracking, a stronger truncation has a similar frequency response, but is superior in terms of overall noise reduction. For kernels of equal width, particle accelerations calculated using a kernel with stronger truncation have up to 20% lower noise. Alternatively, for a specified reduction in noise a shorter kernel can often be used compared to a Gaussian kernel at the commonly used truncation, resulting in less loss of data at trajectory endpoints. It is shown that at the optimal truncation, a Gaussian kernel is mathematically approximated by a second-order polynomial. In this limit, the use of a polynomial kernel has equal results at reduced computational expense compared to the Gaussian kernel.