Tensor median filtering and M-smoothing

Abstract

Median filters for scalar-valued data are well-known tools for image denoising and analysis. They preserve discontinuities and are robust under noise. We generalise median filtering to matrix-valued data using a minimisation approach. Experiments on DT-MRI and fluid dynamics tensor data demonstrate that tensorvalued median filtering shares important properties of its scalar-valued counterpart, including the robustness as well as the existence of non-trivial steady states (root signals). A straightforward extension of the definition allows the introduction of matrixvalued mid-range filters and, more general, M-smoothers. Mid-range filters can also serve as a building block in constructing further (e.g. supremum-based) tensor image filters.