In this paper, we investigate the computation of anisotropic metrics for geodesic distances using the heat equation and its application to image segmentation, particularly for tubular structures. Building upon the work of Bertrand et al. [2], we extend this approach to anisotropic media by incorporating spatially varying and direction-dependent diffusion tensors derived from the image’s structure tensor field. Our contributions are twofold: first, we formulate anisotropic geodesic distance computation using the heat equation and integrate it within deep learning models for image segmentation; second, we propose two methods that learn the anisotropic metric directly from image data—one requiring explicit seed point selection and another eliminating the need for seed points by predicting a probability map that serves as the initial condition for heat diffusion. Experiments on synthetic and medical image datasets demonstrate the effectiveness of our methods in accurately segmenting vascular tree structures by leveraging the anisotropic properties inherent in the images without relying on manual seed point selection.