Publication 2003 Issue No. 10 - October Abstract - An O(1)Time Algorithm for the 3D Euclidean Distance Transform on the CRCW PRAM Model
An O(1)Time Algorithm for the 3D Euclidean Distance Transform on the CRCW PRAM Model
October 2003 (vol. 14 no. 10)
pp. 973-982
 ASCII Text x Yuh-Rau Wang, Shi-Jinn Horng, "An O(1)Time Algorithm for the 3D Euclidean Distance Transform on the CRCW PRAM Model," IEEE Transactions on Parallel and Distributed Systems, vol. 14, no. 10, pp. 973-982, October, 2003.
 BibTex x @article{ 10.1109/TPDS.2003.1239866,author = {Yuh-Rau Wang and Shi-Jinn Horng},title = {An O(1)Time Algorithm for the 3D Euclidean Distance Transform on the CRCW PRAM Model},journal ={IEEE Transactions on Parallel and Distributed Systems},volume = {14},number = {10},issn = {1045-9219},year = {2003},pages = {973-982},doi = {http://doi.ieeecomputersociety.org/10.1109/TPDS.2003.1239866},publisher = {IEEE Computer Society},address = {Los Alamitos, CA, USA},}
 RefWorks Procite/RefMan/Endnote x TY - JOURJO - IEEE Transactions on Parallel and Distributed SystemsTI - An O(1)Time Algorithm for the 3D Euclidean Distance Transform on the CRCW PRAM ModelIS - 10SN - 1045-9219SP973EP982EPD - 973-982A1 - Yuh-Rau Wang, A1 - Shi-Jinn Horng, PY - 2003KW - Computer visionKW - Euclidean distance transformKW - image processingKW - parallel algorithmKW - Voronoi diagramKW - CRCW PRAM model.VL - 14JA - IEEE Transactions on Parallel and Distributed SystemsER -

Abstract—In this paper, we develop a parallel algorithm for the 2D Euclidean distance transform (2D_EDT, for short) of a binary image of size N x N in O(1)time using N^{2+\delta+\epsilon} CRCW processors and a parallel algorithm for the 3D Euclidean distance transform (3D_EDT, for short) of a binary image of size N x N x N in O(1)time using N^{3+\delta+\epsilon} CRCW processors, where \delta= {1\over k}, \epsilon= {1\over 2^{c+1}-1}, k, and c are constants and positive integers. Our 2D_EDT (3D_EDT) parallel algorithm can be used to build up Voronoi diagram and Voronoi polygons (polyhedra) in a 2D (3D) binary image also. All of these parallel algorithms can be performed in O(1) time using N^{2+\delta+\epsilon} (N^{3+\delta+\epsilon}) CRCW processors. To the best of our knowledge, all results derived above are the best O(1) time algorithms known.

[1] S.G. Akl, Parallel Computation: Models and Methods. Prentice-Hall, 1997.
[2] C. Arcelli and G. Sanniti di Baja, Computing Voronoi Diagrams in Digital Pictures Pattern Recognition Letters, vol. 4, pp. 383-389, 1986.
[3] H. Blum, A Transformation for Extracting New Descriptors of Shape Models for the Perception of Speech and Visual Form, W. Wathen Dunn, ed., pp. 362-380, 1967.
[4] G. Borgefors, Distance Transformations in Arbitrary Dimensions Computer Vision, Graphics, and Image Processing, vol. 27, pp. 321-345, 1984.
[5] G. Borgefors, Applications of Distance Transforms Aspects of Visual Form Processing, pp. 83-108, 1994.
[6] G. Borgefors, On Digital Distance Transforms in Three Dimensions Computer Vision, Graphics, and Image Processing, vol. 64, pp. 368-376, 1996.
[7] L. Boxer and R. Miller, Efficient Computing of the Euclidean Distance Transform Computer Vision and Image Understanding, vol. 80, pp. 379-383, 2000.
[8] H. Breu, J. Gil, D. Kirkpatrick, and M. Werman, “Linear time Euclidean Distance Transform Algorithms,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 17, pp. 529-533, 1995.
[9] L. Chen, Optimal Algorithm for Complete Euclidean Distance Transform Chinese J. Computers, vol. 18, no. 8, pp. 611-616, 1995.
[10] L. Chen and H.Y.H. Chuang, A Fast Algorithm for Euclidean Distance Transforms of a 2D Binary Image Information Processing Letters, vol. 51, pp. 25-29, 1994.
[11] P.E. Danielsson, Euclidean Distance Mapping Computer Vision, Graphics, and Image Processing, vol. 14, pp. 227-248, 1980.
[12] A. Datta and S. Soundaralakshmi, Constant-Time Algorithm for the Euclidean Distance Transform on Reconfigurable Meshes J. Parallel and Distributed Computing, vol. 61, pp. 1439-1455, 2001.
[13] A. Datta and S. Soundaralakshmi, Fast and Scalable Algorithms for the Euclidean Distance Transform on the LARPBS Proc. 15th Int'l Parallel and Distributed Processing Symp., pp. 1393-1400, 2001.
[14] H.N. Djidjev and A. Lingas, On Computing Voronoi Diagrams for Sorted Point Sets Int'l J. Computational Geometry and Applications, 1995.
[15] A. Fujiwara, T. Masuzawa, and H. Fujiwara, An Optimal Parallel Algorithm for the Euclidean Distance Maps of 2D Binary Images Information Processing Letters, vol. 54, pp. 295-300, 1995.
[16] T. Hayashi, K. Nakano, and S. Olariu, Optimal Parallel Algorithms for Finding Proximate Points, with Applications IEEE Trans. Parallel and Distributed Systems, vol. 9, no. 3, pp. 1153-1166, Mar. 1998.
[17] T. Hirata, A Unified Linear-Time Algorithm for Computing Distant Maps Information Processing Letters, vol. 58, pp. 129-133, 1996.
[18] J. JáJá, An Introduction to Parallel Algorithms. Addison-Wesley, 1992.
[19] M.N. Kolountzakis and K.N. Kutulakos, Fast Computation of the Euclidean Distance Maps for Binary Images Information Processing Letters, vol. 43, pp. 181-184, 1992.
[20] Y.H. Lee, S.J. Horng, and J. Seitzer, Fast Computation of the 3D Euclidean Distance Transform on the EREW PRAM Model Proc. 2001 Int'l Conf. Parallel Processing, pp. 471-478, 2001.
[21] Y.H. Lee, S.J. Horng, and J. Seitzer, Parallel Computation of the Euclidean Distance Transform on a Three Dimensional Image Array IEEE Trans. Parallel and Distributed Systems, vol. 14, no. 3, pp. 203-212, Mar. 2003.
[22] Y.H. Lee, S.J. Horng, T.W. Kao, F.S. Jaung, Y.J. Chen, and H.R. Tsai, Parallel Computation of Exact Euclidean Distance Transform Parallel Computing, vol. 22, pp. 311-325, 1996.
[23] Y. Pan and K. Li, Constant-Time Algorithm for Computing the Euclidean Distance Maps of Binary Images on 2D Meshes with Reconfigurable Buses Information Science, vol. 120, pp. 209-221, 1999.
[24] Y. Pan, Y. Li, J. Li, K. Li, and S.-Q. Zheng, Computing Distance Maps Efficiently Using an Optical Bus Proc. IPDPS 2000 Workshop Parallel and Distributed Computing in Image Processing, Video Processing and Multimedia (PDIVM 2000), pp. 178-185, 2000.
[25] Y. Pan, Y. Li, J. Li, K. Li, and S.-Q. Zheng, Efficient Parallel Algorithms for Distance Maps of 2D Binary Images Using an Optical Bus IEEE Trans. Systems, Man, and Cybernetics-Part A: Systems and Humans, vol. 32, no. 2, pp. 228-236, 2002.
[26] S. Pavel and S.G. Akl, Efficient Algorithms for Euclidean Distance Transform Parallel Processing Letters, vol. 5, no. 2, pp. 205-212, 1995.
[27] A. Rosenfeld and J.L. Pfalz, Sequential Operations in Digital Picture Processing J. ACM, vol. 13, pp. 471-494, 1966.
[28] A. Rosenfeld and J.L. Pfalz, Distance Function on Digital Pictures Pattern Recognition, vol. 1, pp. 33-61, 1968.
[29] T. Saito and J.I. Toriwaki, New Algorithms for Euclidean Distance Transformation of an n-Dimensional Digitized Picture with Applications Pattern Recognition, vol. 27, no. 11, pp. 1551-1565, 1994.
[30] Y. Shiloach and U. Vishkin, Finding the Maximum, Merging and Sorting in a Parallel Computation Model J. Algorithms, vol. 2, no. 1, pp. 88-102, 1981.
[31] Y.R. Wang, S.J. Horng, Y.H. Lee, and P.Z. Lee, Parallel Algorithms for Higher-Dimensional Euclidean Distance Transforms with Applications Proc. Third Int'l Conf. Parallel and Distributed Computing, Applications and Technologies (PDCAT '02), pp. 159-166, 2002.
[32] Y.R. Wang and S.J. Horng, An$O(1)$Time Parallel Algorithm for the 3D Euclidean Distance Transform on the AROB Proc. Int'l Conf. Parallel and Distributed Processing Techniques and Applications (PDPTA '02), pp. 1120-1126, 2002.
[33] Q.Z. Ye, "The Signed Euclidean Distance Transform and Its Applications," Int'l Conf. Pattern Recognition, pp. 495-499, 1988.

Index Terms:
Computer vision, Euclidean distance transform, image processing, parallel algorithm, Voronoi diagram, CRCW PRAM model.
Citation:
Yuh-Rau Wang, Shi-Jinn Horng, "An O(1)Time Algorithm for the 3D Euclidean Distance Transform on the CRCW PRAM Model," IEEE Transactions on Parallel and Distributed Systems, vol. 14, no. 10, pp. 973-982, Oct. 2003, doi:10.1109/TPDS.2003.1239866