Continuous-domain ARMA (Auto-Regressive Moving-Average) processes are widely used in control theory and in signal/image processing and analysis. Typical examples of applications are system identification and adaptive filtering, speech analysis and synthesis, stochastic differential equations, and image modeling. The linear estimation theory for ARMA processes is closely related to Sobolev spaces, since the reproducing kernel of a Sobolev space provides the generating function for the linear minimum-norm estimator of unknown sample values. Further, Sobolev norms provide the continuous-domain regularization term in numerous inverse problems. Their relation to ARMA modeling suggests a parameterized modeling of continuous-domain signals. In practice, the available data are discrete and one is usually required to estimate the underlying continuous-domain parameters from sample values. Potential examples are the continuous-domain structure modeling of physical phenomena, the identification of linear time-invariant (LTI) systems, as well as the numerical analysis of differential operators.
In this project, we consider the problem of estimating a continuous-time ARMA process from its sampled version. Our approach incorporates the sampling process into the problem formulation while introducing exponential models for both the continuous and the sampled processes. We derive an exact evaluation of the discrete-domain power spectrum using exponential B-splines and further suggest an estimation approach that is based on digitally filtering the available data. The proposed functional exhibits several local minima that originate from aliasing. The global minimum, however, corresponds to a maximum-likelihood estimator, regardless of the sampling step. Experimental results further indicate that the proposed approach closely follows the Cramér-Rao bound for various aliasing configurations.
We plan next to apply our 1D results to the ARMA modeling of 2D images. In particular, we shall propose a maximum-likelihood estimator for a 2D continuous-time ARMA process from its sampled version, providing an adaptive continuous-time modeling of an image. This, in turn, would allow us to introduce improved interpolation, deconvolution, and segmentation algorithms.
We also plan to investigate sampling-invariant properties of continuous-time ARMA models. For example, we want to characterize the uniqueness property of sampled ARMA processes. We also plan to investigate sampling-invariant characteristics of other, more-general, types of ARMA models that involve non-Gaussian innovation input as well as an additional deterministic component (Wold decomposition).