This paper presents a nonlinear derivative approach to addressing the problem of discrete edge detection. This edge detection scheme is based on the nonlinear combination of two polarized derivatives. Its main property is a favorable signal-to-noise ratio (SNR) at a very low computation cost and without any regularization. A 2D extension of the method is presented and the benefits of the 2D localization are discussed. The performance of the localization and SNR are compared to that obtained using classical edge detection schemes. Tests of the regularized versions and a theoretical estimation of the SNR improvement complete this work.