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We propose a new edge enhancer based on a modified multilayer neural network, which is called a neural edge enhancer (NEE), for enhancing the desired edges clearly from noisy images. The NEE is a supervised edge enhancer: Through training with a set of input noisy images and teaching edges, the NEE acquires the function of a desired edge enhancer. The input images are synthesized from noiseless images by addition of noise. The teaching edges are made from the noiseless images by performing the desired edge enhancer. To investigate the performance, we carried out experiments to enhance edges from noisy artificial and natural images. By comparison with conventional edge enhancers, the following was demonstrated: The NEE was robust against noise, was able to enhance continuous edges from noisy images, and was superior to the conventional edge enhancers in similarity to the desired edges. To gain insight into the nonlinear kernel of the NEE, we performed analyses on the trained NEE. The results suggested that the trained NEE acquired directional gradient operators with smoothing. Furthermore, we propose a method for edge localization for the NEE. We compared the NEE, together with the proposed edge localization method, with a leading edge detector. The NEE was proven to be useful for enhancing edges from noisy images.