This paper proposes a robust method of detecting edge points by thresholding edge reliabilities with fixed threshold values. Traditionally edge detection is done by thresholding edge height (or edge intensity). One crucial problem is that the threshold is determined ad hoc for the given input image. We use edge reliability instead of edge height, and show that two fixed thresholds work well. In a previous paper, we proposed an edge vector extractor and stated necessity of a measure, edge reliability, which can discriminate a low but steady edge from a noisy edge of large edge height. Our idea of extracting edge points is to detect points, which are neither in a uniform region nor in a gradational region. The two detection thresholds, square root of 2/pi and square root of 3/4, are derived from the analysis of edge reliability in uniform regions and gradational regions contaminated by white Gaussian noises, respectively. Experiments show that our method can detect edge points stably, discriminating them from the points in uniform or gradational regions influenced by noises.