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<p>A fast and exact Euclidean distance transformation using decomposed grayscale morphological operators is presented. Applied on a binary image, a distance transformation assigns each object pixel a value that corresponds to the shortest distance between the object pixel and the background pixels. It is shown that the large structuring element required for the Euclidean distance transformation can be easily decomposed into 3/spl times/3 windows. This is possible because the square of the Euclidean distance matrix changes uniformly both in the vertical and horizontal directions. A simple extension for a 3D Euclidean distance transformation is discussed. A fast distance transform for serial computers is also presented. Acting like thinning algorithms, the version for serial computers focuses operations only on the potential changing pixels and propagates from the boundary of objects, significantly reducing execution time. Nonsquare pixels can also be used in this algorithm. An example application, shape filtering using arbitrary sized circular dilation and erosion, is discussed. Rotation-invariant basic morphological operations can be done using this example application.</p>
mathematical morphology; transforms; filtering and prediction theory; image processing; Euclidean distance transform; gray scale morphology decomposition; binary image; object pixel; background pixels; 3D Euclidean distance; shape filtering; circular dilation; erosion; structuring element decomposition
O.R. Mitchell, C.T. Huang, "A Euclidean Distance Transform Using Grayscale Morphology Decomposition", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 16, no. , pp. 443-448, April 1994, doi:10.1109/34.277600
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