The automatic edge detection of cracks on concrete structures plays an important role in the damage assessment process for cracked structures. In this paper, we proposed an automatic method for accurate edge detection of concrete cracks from real 2D images of concrete surfaces containing noisy and unintended objects. In the 2D image of a damaged concrete surface, cracks are usually observed as tree-like topology dark objects of which the branches are line-like and have local symmetry across their center axes. We utilize these two geometric properties of cracks to detect crack edges and discriminate them with edges of other unintended objects. The novel automatic crack edge detection is composed of two sequential stages. In the first stage, cracks are enhanced by a novel phase symmetry-based crack enhancement filter (PSCEF) based on their symmetric and line-like properties while non-crack objects are removed. Estimated crack center-lines are then obtained by thresholding the filtered images and applying morphological thinning algorithm to the binary image. In the second stage, the estimated center lines of the detected cracks are fitted by cubic splines and the pixel intensity profiles in the directions perpendicular to the splines are used to determine the edge points. The edge points are linked together to form the desired continuous crack edges. Various experiments of real concrete crack images are used to demonstrate the excellent performance of the proposed method.