Symmetrizing Smoothing Filters

Peyman Milanfar
2013 SIAM Journal of Imaging Sciences  
We study a general class of non-linear and shift-varying smoothing filters that operate based on averaging. This important class of filters includes many well-known examples such as the bilateral filter, non-local means, general adaptive moving average filters, and more 1 . They are frequently used in both signal and image processing as they are elegant, computationally simple, and high performing. The operators that implement such filters, however, are not symmetric in general. The main
more » ... al. The main contribution of this paper is to provide a provably stable method for symmetrizing the smoothing operators. Specifically, we propose a novel approximation of smoothing operators by symmetric doubly-stochastic matrices and show that this approximation is stable and accurate; even more so in higher dimensions. We demonstrate that there are several important advantages to this symmetrization. In particular, (1) they generally lead to improved performance of the baseline smoothing procedure, (2) when the smoothers are applied iteratively, only the symmetric ones can be guaranteed to lead to stable algorithms; and (2) symmetric smoothers allow an orthonormal eigen-decomposition which enables us to peer into the complex behavior of such non-linear and shift-varying filters in a locally-adapted basis using a familiar tool: principal components.
doi:10.1137/120875843 fatcat:g3ardp5r5vgtjlzfymb4pllfxq