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Our attention now turns to gray-scale morphology, which means that the morphological operators act on real-valued functions defined on n-dimensional Euclidean space or the n-dimensional Cartesian grid. For signals, n=1 , and for images, n=2 . It is useful to look at a gray-scale image as a surface. Figure 5.1 shows a gray-scale image made of three Gaussian-shape peaks of different heights and variances. The image is depicted in four different graphical representations: (a) the pixel values mapped in gray scale: low values are dark and high values are bright gray tones; (b) the pixel values also mapped in gray scale but in a reverse order: low values are bright and high values are dark gray tones; (c) the same image but as a top-view shading surface; and (d) a mesh plot of the same surface. Although our main concern is with image processing, we will develop the gray-scale theory for signals, our aim being to keep notation as simple as possible and to facilitate straightforward illustrative figures. Once the underlying gray-scale theory has been presented for signals, one need only recognize that by treating points on the line as spatial points in the plane the theory at once goes over into the imaging domain, the fundamental point being that the theory itself is independent of domain dimensionality.
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