Mathematical Morphology is a common strategy for non-linear filtering of image data. In its traditional form the filters used, known as structuring elements, have constant shape once set. Such rigid structuring elements are excellent for detecting patterns of a specific shape, but risk destroying valuable information in the data as they do not adapt in any way to its structure.We present a novel method for adaptive morphological filtering where the local structure tensor, a well-known method for estimation of structure within image data, is used to construct adaptive elliptical structuring elements which vary from pixel to pixel depending on the local image structure. More specifically, their shape varies from lines in regions of strong single-directional characteristics to disks at locations where the data has no prevalent direction.