Modern medical imaging techniques enable the acquisition of in-vivo high resolution images of the vascular system. Most common methods for the detection of vessels in these images, such as multiscale Hessian-based operators and matched filters, rely on the assumption that, at each voxel there is a single cylinder. Such an assumption is clearly violated at the multitude of branching points that are easily observed in all but the most focused vascular image studies. In this paper, we propose a novel method for detecting vessels in medical images that relaxes this single cylinder constraint. Instead, we extract characteristics of the local intensity profile (in a spherical polar coordinate system) which we term as the polar neighborhood intensity profile enabling us to detect vessels even near branching points. Our method demonstrates improved performance over standard methods on both 2D synthetic images and MRA 3D animal vascular images, particularly close to vessel branching regions. This methodology is also applicable to the detection of other structures such as sheets with the appropriate choice of operators.