Interpolation of signals with missing data using Principal Component Analysis

Abstract

A non-iterative methodology for the interpolation and regularization of multidimensional sampled signals with missing data resorting to Principal Component Analysis (PCA) is introduced. Based on unbiased sub-optimal estimators for the mean and covariance of signals corrupted by zero-mean noise, the PCA is performed and the signals are interpolated and regularized. The optimal solution is obtained from a weighted least mean square minimization problem, and upper and lower bounds are provided for the mean square interpolation error. This solution is a refinement to a previously introduced method proposed by the author Oliveira (Proceedings of the IEEE international conference on acoustics, speech, and signal processing—ICASSP06, Toulouse, France, 2006), where three extensions are exploited: (i) mean substitution for covariance estimation, (ii) Tikhonov regularization method and, (iii) dynamic principal components selection. Performance assessment benchmarks relative to averaging, Papoulis-Gerchberg, and Power Factorization methods are included, given the results obtained from a series of Monte Carlo experiments with 1-D audio and 2-D image signals. Tight upper and lower bounds were observed, and improved performance was attained for the refined method. The generalization to multidimensional signals is immediate.