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Vessel segmentation is crucial in many medical image applications, such as detecting coronary stenoses, retinal vessel diseases and brain aneurysms. However, achieving high pixel-wise accuracy, complete topology structure and robustness to various contrast variations are critical and challenging, and most existing methods focus only on achieving one or two of these aspects. In this paper, we present a novel approach, the affinity feature strengthening network (AFN), which jointly models geometry and refines pixel-wise segmentation features using a contrast-insensitive, multiscale affinity approach. Specifically, we compute a multiscale affinity field for each pixel, capturing its semantic relationships with neighboring pixels in the predicted mask image. This field represents the local geometry of vessel segments of different sizes, allowing us to learn spatial- and scale-aware adaptive weights to strengthen vessel features. We evaluate our AFN on four different types of vascular datasets: X-ray angiography coronary vessel dataset (XCAD), portal vein dataset (PV), digital subtraction angiography cerebrovascular vessel dataset (DSA) and retinal vessel dataset (DRIVE). Extensive experimental results demonstrate that our AFN outperforms the state-of-the-art methods in terms of both higher accuracy and topological metrics, while also being more robust to various contrast changes. The source code of this work is available at https://github.com/TY-Shi/AFN.