Statistically Optimal Averaging for Image Restoration and Optical Flow Estimation

In this paper we introduce a Bayesian best linear unbiased estimator (Bayesian BLUE) and apply it to generate optimal averaging filters. Linear filtering of signals is a basic operation frequently used in low level vision. In many applications, filter selection is ad hoc without proper theoretical justification. For example input signals are often convolved with Gaussian filter masks, i.e masks that are constructed from truncated and normalized Gaussian functions, in order to reduce the signal noise. In this contribution, statistical estimation theory is explored to derive statical optimal filter masks from first principles. Their shape and size are fully determined by the signal and noise characteristics. Adaption of the estimation theoretical point of view not only allows to learn optimal filter masks but also to estimate the variance of the estimate. The statistically learned filter masks are validated experimentally on image reconstruction and optical flow estimation. In these experiments our approach outperforms comparable approaches based on ad hoc assumptions on signal and noise or even do not relate their method at all to the signal at hand.