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We propose a new signed or unsigned Euclidean distance transformation algorithm, based on the local corrections of the well-known 4SED algorithm of Danielsson. Those corrections are only applied to a small neighborhood of a small subset of pixels from the image, which keeps the cost of the operation low. In contrast with all fast algorithms previously published, our algorithm produces perfect Euclidean distance maps in a time linearly proportional to the number of pixels in the image. The computational cost is close to the cost of the 4SSED approximation.