Hybrid Parametric-Nonparametric Modeling with Application to Natural Image Upsampling

Linear autoregressive (AR) model is widely used in signal processing. Usually the AR models are solved by classical least square (LS) method. An important issue with the LS solution of the AR model, which has been seemingly overlooked, is its numerical stability. The issue is related to the rank condition of the design matrix. We observed, in case of natural images, that the probability of numerical rank deficiency is rather high, roughly thirty-five per cent, due to discrete nature and structures of the digital images. Without care numerical rank deficiency can adversely affect the parameter estimation of the AR model. In this paper we use the rank revealing QR (RRQR) factorization to select optimal subset from the design matrix so as to effectively lower the condition number of the system. By removing the ill conditioned part of the right orthogonal matrix of the RRQR decomposition, we obtain a robust truncated solution to the linear system. On the other hand, for natural images, the unselected data tend to highly correlate with the pixel being modeled, and their exclusion from the modeling process waste valuable information. To avoid this loss we recycle the data including those discard by the parametric AR estimator into a nonparametrgic model of nonlocal type. Interestingly, the data that cause ill condition to the parametric AR model are of high quality for the non-local nonparametric modeling. Therefore, an approach of hybrid parametric-nonparametric modeling can make the best use of data and improve the model performance. The hybrid modeling approach is applied to image resolution upconversion, and it greatly improves the performance of the state-of-the-art image interpolator, achieving a gain of 3dB or more in PSNR in some cases.