Improved smoothness priors using bilinear transform

Smoothness priors is a well-known and most commonly used method in the analysis of stochastic processes making it very useful in the field of stochastic signal processing. It is particularly suited for smoothing the noisy data and detrending the time-series signals. The method is based on an optimization problem where the n-th order derivative of the signal enters as a constraint. When the method is designed in discrete time domain, the backward difference rule is used to perform differential-to-difference conversion. Moreover, the solution depends on a smoothness trade-off parameter. An efficient algorithm for the trade-off parameter selection remains an important and challenging issue. In this paper, first, we propose a closed-form expression for the trade-off parameter. The closed-form expression resulted from a frequency domain interpretation of the smoothness priors procedure. The trade-off parameter determines the amount of frequency components that the procedure allows to pass. We show that the trade-off parameter is related to the arbitrary choice of cutoff frequency. Second, we introduce a new way to the design and implementation of smoothness priors using bilinear transformation method. Frequency analysis and experiments on both synthetic and real world signals with different levels of noise demonstrate that bilinear transform is indeed more effective for smoothness priors implementation when compared with the traditional ones, i.e., the backward difference rule.