The algorithm presented in this paper is an application of a general framework for morphological processing of signals on weighted graphs. Here we apply it to images by defining what we call a co-circularity graph. In this graph, the vertices are the pixels and the weighted edges depend on a consistency criterion (co-circularity) between local orientations estimated from the structure tensors. This graph induces anisotropic adaptive morphological operators which are related both to anisotropic diffusion in images and path optimality in graphs. We present several applications such as the enhancement of fibre-like structures, completion of interrupted edges and the regularisation of gray-scale images. We also discuss the parameters setting depending on the application.