This Article 
 Bibliographic References 
 Add to: 
Fast Homotopy-Preserving Skeletons Using Mathematical Morphology
June 1992 (vol. 14 no. 6)
pp. 653-664

Two algorithms for skeletonization of 2-D binary images, each of which explicitly separates the two major aspects of skeletonization are described: the identification of points critical to shape representation, and the identification of further points necessary to preserve homotopy. Sets of points critical to shape representation are found by eroding the original image I with a nested sequence of structuring elements E/sub i/. By choosing appropriate (E/sub i/) and D, a structuring element, either algorithm is capable of producing a variety of skeletons corresponding to different distance functions. A sufficient condition is given for the original image to be reconstructed from the skeleton. In the case of the first algorithm, there are few restrictions on the set of structuring elements. It uses a simple search strategy to find points whose removal would alter homotopy. The second, faster, algorithm has a more constructional approach to finding points necessary for preserving homotopy, which limits it to a more restricted set of structuring elements than the first algorithm. However, it may still be used with a variety of distance functions.

[1] J. Hilditch, "Linear skeletons from square cupboards," inMachine Intelligence 4(B. Meltzer and D. Michie, Eds.). Edinburgh: Edinburgh University Press, 1969, pp. 404-420.
[2] A. Rosenfeld, "Connectivity in digital pictures,"J. ACM, Jan. 1970.
[3] S. Stefanelli and A. Rosenfeld, "Some parallel thinning algorithms for digital pictures,"J. ACM, vol. 18, pp. 255-264, 1971.
[4] F. L. Bookstein, "The line skeleton,"Comput. Graphics Image Processing, vol. 11, pp. 123-137, 1979.
[5] T. Pavlidis, "A thinning algorithm for discrete binary images,"Comput. Graphics Image Processing, vol. 13, pp. 142-157, 1980.
[6] P.-E. Danielsson, "Euclidean distance mapping,"Comput. Graphics Image Processing, vol. 14, pp. 227-248, 1980.
[7] E. R. Davies and A. P. N. Plummer, "Thinning algorithms: A critique and a new methodology,"Patt. Recog.vol. 14, pp. 53-63, 1981.
[8] C. J. Hilditch, "Comparison of thinning algorithms on a parallel processor,"Image Vision Comp., vol. 1, pp. 115-132, 1983.
[9] C. Arcelli and G. Sanniti de Baja, "A width-independent fast thinning algorithm,"IEEE Trans. Patt. Anal. Machine Intell., vol. PAMI-7, pp. 464-474, 1985.
[10] J. Piper, "Efficient implementation of skeletonisation using interval coding,"Patt. Recog. Lett., vol. 3, pp. 389-397, 1985.
[11] L. Dorst, "Pseudo-Euclidean skeletons," inProc. 8th lnt. Conf. Patt. Recog., 1986, pp. 286-288.
[12] R. W. Smith, "Computer processing of line images: A survey,"Patt. Recog., vol. 20, pp. 7-15, 1987.
[13] B. J. H. Verwer, "Improved metrics in image processing applied to the Hilditch skeleton," inProc. 9th Int. Conf. Patt. Recog., 1988, pp. 137-142.
[14] F. Meyer, "Skeletons in digital spaces, " inImage Analysis and Mathematical Morphology, vol. 2.(J. Serra, Ed.). New York: Academic, 1988, pp. 257-296.
[15] F. Meyer, "Skeletons and perceptual graphs,"Signal Processing, vol. 16. pp. 335-363, 1989.
[16] B.-K. Jang and R. T. Chin, "Analysis of thinning algorithms using mathematical morphology,"IEEE Trans. Patt. Anal. Machine Intell., vol. PAMI-12, pp. 541-551, 1990.
[17] A. Rosenfeld and J. Pfaltz, "Sequential operations in digital picture processing,"J. ACM, vol. 4, 1966.
[18] J. C. Mott-Smith, "Medial axis transformations," inPicture Processing and Psychopictorics(B. S. Lipkin and A. Rosenfeld, Eds.). New York: Academic, 1970, pp. 267-283.
[19] S. Yokoi, J-I. Toriwaki, and T. Fukumara, "On generalized distance transformation of digitised pictures,"IEEE Trans. Patt. Anal. Machine Intell. vol. PAMI-4, pp. 424-443, 1981.
[20] J. Serra,Image analysis and mathematical morphology. London: Academic, 1982.
[21] P. A. Maragos, R. W. Schafer, "Morphological skeleton representation and coding of binary images,"IEEE Trans. Acoustics Speech Signal Processing, vol. ASSP-34, pp. 1228-1244, 1986.
[22] D. Schonfeld and J. Goutsias, "A fast algorithm for the morphological coding of binary images," inProc. SPIE Visual Commun. Image Processing'88, pp. 138-145.
[23] Z. Zhou and A. N. Venetsanopoulos, "Pseudo-Euclidean morphological skeleton transform for machine vision," inProc. Int. Conf. Acoustics Speech Signal Processing, 1989, pp. 1695-1698.
[24] R. M. Haralick and L. Shapiro, "Glossary of computer vision terms,"Patt. Recog., vol. 24, pp. 69-93, 1991.
[25] A. Rosenfeld and A. Kak,Digital Picture Processing, New York: Academic, 1976.
[26] G. Matheron, "Examples of topological properties of skeletons," inImage Analysis and Mathematical Morphology, vol. 2. (J. Serra, Ed.). New York: Academic, 1988, pp. 217-238.
[27] D. Rutovitz, "Data structures for operations on digital images," inPictorial Pattern Recognition(G. C. Cheng, R. S. Ledley, D. K. Pollock, and A. Rosenfeld, Eds.). Washington, DC: Thompson, 1968, pp. 105-133.
[28] J. Piper and D. Rutovitz, "Data structures for image processing in a C language and Unix environment,"Patt. Recog. Lett. vol. 3, pp. 119-129, 1985.
[29] I. Ragnemalm, "Generation of Euclidean distance maps," Thesis no. 206, Linkoping Univ., Linkoping, Sweden, 1990.
[30] G. Borgefors, "Distance transformations in digital images,"Comput. Vision, Graphics, Image Processing, vol. 34, pp. 334-371, 1986.
[31] F. C. A. Groen and N. J. Foster, "A fast algorithm for cellular logic operations on sequential machines,"Patt. Recog. Lett., vol. 2, pp. 333-338, 1984.
[32] L. J. Van Vilet and B.J.H. Verwer, "A contour processing method for fast binary neighborhood operations,"Pattern Recog. Lett., vol. 7, pp. 27-36, Jan. 1988.
[33] X. Wang and G. Bertrand, "An algorithm for a generalized distance transformation based on Minkowski operations," inProc. 9th Int. Conf. Patt. Recog., 1988, pp. 1164-1168.
[34] L. Ji, J. Piper, and J. -Y. Tang, "Erosion and dilation of binary images by arbitrary structuring elements using interval coding,"Patt. Recog. Lett., vol. 9, pp. 201-209, 1989.
[35] R.M. Haralick, S.R. Sternberg, and X. Zhuang, "Image analysis using mathematical morphology,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-9, no. 4, pp. 532-550, 1987.
[36] C. Cherry, M. H. Kubba, D. E. Pearson, and M. P. Barton, "An experimental study of the possible bandwidth compression of visual image signals,"Proc. IEEE, vol. 51, pp. 1507-1517, 1963.
[37] J. W. Butler, M. K. Butler, and A. Stroud, "Automatic classification of chromosomes," inData Acquisition and Processing in Biology and Medicine(K. Enslein, Ed.). Oxford, UK: Pergamon, 1963, pp. 261-275.
[38] I. T. Young, R. L. Peverini, P. W. Verbeek, and P. J. van Otterloo, "A new implementation for the binary and Minkowski operators,"Comput. Graphics Image Processing, vol. 17, pp. 198-210, 1981.
[39] C. Ronse and P.A. Devijver,Connected Components in Binary Images: The Detection Problem. Letchworth, England: Research Studies Press, 1984.
[40] J. Piper and D. Rutovitz, "An investigation of object-oriented programming as the basis for an image processing and analysis system," inProc. 9th Int. Conf. Patt. Recog., 1988, pp. 1015-1019.
[41] L. Ji, J. Piper, and D. Rutovitz, "Topology proofs for the skeleton algorithm," Int. Rep. MRC, Human Genetics Unit, 1991.

Index Terms:
pattern recognition; picture processing; image reconstruction; combinatorial mathematics; homotopy-preserving skeletons; mathematical morphology; skeletonization; 2-D binary images; shape representation; sufficient condition; distance functions; combinatorial mathematics; pattern recognition; picture processing
L. Ji, J. Piper, "Fast Homotopy-Preserving Skeletons Using Mathematical Morphology," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 14, no. 6, pp. 653-664, June 1992, doi:10.1109/34.141555
Usage of this product signifies your acceptance of the Terms of Use.