Distance transformations in digital images
Computer Vision, Graphics, and Image Processing
Applied multivariate statistical analysis
Applied multivariate statistical analysis
Decomposition of gray-scale morphological structuring elements
Pattern Recognition
A fast algorithm for Euclidean distance maps of a 2-D binary image
Information Processing Letters
A unified linear-time algorithm for computing distance maps
Information Processing Letters
Sequential Operations in Digital Picture Processing
Journal of the ACM (JACM)
Digital Picture Processing
A Euclidean Distance Transform Using Grayscale Morphology Decomposition
IEEE Transactions on Pattern Analysis and Machine Intelligence
Linear Time Euclidean Distance Algorithms
IEEE Transactions on Pattern Analysis and Machine Intelligence
Image Analysis and Mathematical Morphology
Image Analysis and Mathematical Morphology
Fast distance transformation on irregular two-dimensional grids
Pattern Recognition
Anti-aliased Euclidean distance transform
Pattern Recognition Letters
Linear Time Algorithms for Exact Distance Transform
Journal of Mathematical Imaging and Vision
Elliptical distance transforms and applications
DGCI'06 Proceedings of the 13th international conference on Discrete Geometry for Computer Imagery
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In this paper we prove an equivalence relation between the distancetransform of a binary image, where the underlying distance is based ona positive definite quadratic form, and the erosion of itscharacteristic function by an elliptic poweroid structuringelement. The algorithms devised by Shih and Mitchell [18] and Huang and Mitchell [7], for calculating the exact Euclidean distance transform (EDT) of abinary digital image manifested on a square grid, are particular cases ofthis result. The former algorithm uses erosion by a circular cone tocalculate the EDT whilst the latter uses erosion by an ellipticparaboloid (which allows for pixel aspect ratio correction) tocalculate the square of the EDT. Huang and Mitchell‘s algorithm[7] is arguably the better of the two because: (i)the structuring element can be decomposed into a sequence of dilationsby 3 × 3 structuring elements (a similar decomposition is notpossible for the circular cone) thus reducing the complexity of theerosion, and (ii) the algorithm only requires integer arithmetic (itproduces squared distance). The algorithm is amenable to both hardwareimplementation using a pipeline architecture and efficientimplementation on serial machines. Unfortunately the algorithm doesnot directly transpose to, nor has a corresponding analogue on, thehexagonal grid (the same is also true for Shih and Mitchell‘salgorithm [7]). In this paper, however, we show thatif the hexagonal grid image is embedded in a rectangular grid thenHuang and Mitchell‘s algorithm [7] can be applied,with aspect ratio correction, to obtain the exact EDT on the hexagonalgrid.