Morphological gradient

In mathematical morphology and , a morphological gradient is the difference between the dilation and the erosion of a given image. It is an image where each pixel value (typically non-negative) indicates the contrast intensity in the close neighborhood of that pixel. It is useful for edge detection and applications. Let be a grayscale image, mapping points from a Euclidean space or discrete grid E (such as R2 or Z2) into the real line. Let be a grayscale structuring element. Usually, b is symmetric and has short-support, e.g., Then, the morphological gradient of f is given by: where and denote the dilation and the erosion, respectively. An internal gradient is given by: and an external gradient is given by: The internal and external gradients are "thinner" than the gradient, but the gradient peaks are located on the edges, whereas the internal and external ones are located at each side of the edges. Notice that . If , then all the three gradients have non-negative values at all pixels.