Efficient Euclidean distance transform algorithm of binary images in arbitrary dimensions

Jun Wang, Ying Tan

Abstract: In this paper, we propose an efficient algorithm, i.e., PBEDT, for short, to compute the exact Euclidean distance transform (EDT) of a binary image in arbitrary dimensions. The PBEDT is based on independent scan and implemented in a recursive way, i.e., the EDT of a d-dimensional image is able to be computed from the EDTs of its (d−1)-dimensional<math><mo stretchy="false" is="true">(</mo><mi is="true">d</mi><mo is="true">−</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo><mi mathvariant="normal" is="true">-</mi><mi is="true">dimensional</mi></math> sub-images. In each recursion, all of the rows in the current dimensional direction are processed one by one. The points in the current processing row and their closest feature points in (d−1)-dimensional<math><mo stretchy="false" is="true">(</mo><mi is="true">d</mi><mo is="true">−</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo><mi mathvariant="normal" is="true">-</mi><mi is="true">dimensional</mi></math> sub-images can be shown in a Euclidean plane. By using the geometric properties of the perpendicular bisector, the closest feature points of (d−1)-dimensional<math><mo stretchy="false" is="true">(</mo><mi is="true">d</mi><mo is="true">−</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo><mi mathvariant="normal" is="true">-</mi><mi is="true">dimensional</mi></math> sub-images are easily verified so as to lead to the EDT of a d-dimensional image after eliminating the invalid points. The time complexity of the PBEDT algorithm is linear in the amount of both image points and dimensions with a small coefficient. Compared with the state-of-the-art EDT algorithms, the PBEDT algorithm is much faster and more stable in most cases.