Recent clinical research has highlighted important links between a number of diseases and the tortuosity of curvilinear anatomical structures like corneal nerve fibres, suggesting that tortuosity changes might detect early stages of specific conditions. Currently, clinical studies are mainly based on subjective, visual assessment, with limited repeatability and inter-observer agreement. To address these problems, we propose a fully automated framework for image-level tortuosity estimation, consisting of a hybrid segmentation method and a highly adaptable, definition-free tortuosity estimation algorithm. The former combines an appearance model, based on a Scale and Curvature-Invariant Ridge Detector (SCIRD), with a context model, including multi-range learned context filters. The latter is based on a novel tortuosity estimation paradigm in which discriminative, multi-scale features can be automatically learned for specific anatomical objects and diseases. Experimental results on 140 in vivo confocal microscopy images of corneal nerve fibres from healthy and unhealthy subjects demonstrate the excellent performance of our method compared to state-of-the-art approaches and ground truth annotations from 3 expert observers.