Quantification and modelling of curvilinear structures in 2D and 3D images is a common challenge in a wide range of biomedical applications. Image enhancement is a crucial pre-processing step for curvilinear structure quantification. Many of the existing state-of-the-art enhancement approaches still suffer from contrast variations and noise. In this paper, we propose to address such problems via the use of a multiscale image processing approach, called Multiscale Top-Hat Tensor (MTHT). MTHT produces a better quality enhancement of curvilinear structures in low contrast and noisy images compared with other approaches in a range of 2D and 3D biomedical images. The proposed approach combines multiscale morphological filtering with a local tensor representation of curvilinear structure. The MTHT approach is validated on 2D and 3D synthetic and real images, and is also compared to the state-of-the-art curvilinear structure enhancement approaches. The obtained results demonstrate that the proposed approach provides high-quality curvilinear structure enhancement, allowing high accuracy segmentation and quantification in a wide range of 2D and 3D image datasets.