Abstract
Mathematical Morphology (MM) is a general method for image processing based on set theory. The two basic morphological operators are dilation and erosion. From these, several non linear filters have been developed usually with polynomial complexity, and this because the two basic operators depend strongly on the definition of the structural element. Most efforts to improve the algorithm's speed for each operator are based on structural element decomposition and/or efficient codification.
A new framework and a theoretical basis toward the construction of fast morphological operators (of zero complexity) for an infinite (countable) family of regular metric spaces are presented in work. The framework is completely defined by the three axioms of metric. The theoretical basis here developed points out properties of some metric spaces and relationships between metric spaces in the same family, just in terms of the properties of the four basic metrics stated in this work. Concepts such as bounds, neighborhoods and contours are also related by the same framework.
The presented results, are general in the sense that they cover the most commonly used metrics such as the chamfer, the city block and the chess board metrics. Generalizations and new results related with distances and distance transforms, which in turn are used to develop the morphologic operations in constant time, in contrast with the polynomial time algorithms are also given.
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References
G. Agam, H. Luo, and I. Dinstain, “Morphological approach for dashed lines detection,” in Proc. of the First International Workshop on Graphics Recognition, 1995, pp. 92-105.
B. Albert and H. Li, “Digital compression and morphological image coding,” in Proc. of the ICAASP, 1989, pp. 1898-1900.
M.A. Armstrong, Topologia Básica, UNAM, México, 1990.
A.L.D. Beckers and A.W.M Smeulders, “A comment on: A note on distance transformations in digital images,” Computer, Vision, Graphics and Image Processing, Vol. 47, pp. 89-91, 1989.
S.B. Berestein, “Algebraic analysis of the generating functional for discrete randon sets and statistical inference for intensity in the discrete boolean random set model,” Journal of Mathematical Imaging and Vision, Vol. 4, pp. 273-290, 1994.
B. Bettoli and E.R. Dougherty, “Linear granulometric moments of noisy images,” Journal of Mathematical Imaging and Vision, Vol. 3, pp. 299-319, 1993.
P. Bhattacharya, K. Quian, and X. Lu, “An algebraic approach for morphological operations on 2D and 3D images,” Pattern Recognition, Vol. 26, No. 12, pp. 1785-1796, 1993.
A. Bleau, J. Guise, and A.R. LeBlanc, “A new set of fast algorithms for mathematical morphology I,” Computer, Vision, Graphics and Image Processing, Vol. 56, No. 2, pp. 178-209, 1992.
A. Bleau, J. Guise, and A.R. LeBlanc, “A new set of fast algorithms for mathematical morphology II,” Computer, Vision, Graphics and Image Processing, Vol. 56, No. 2, pp. 210-229, 1992.
G. Borgefors, “Distance transformations in digital images,” Computer Graphics and Image Processing, Vol. 34, pp. 344-371, 1986.
G. Borgefors, “Distance transformations on hexagonal grids,” Pattern Recognition Letters, Vol. 9, pp. 97-105, 1989.
G. Borgefors, “Another comment on: A note on distance transformations in digital images,” Computer, Vision, Graphics and Image Processing, Vol. 54, pp. 301-306, 1991.
G. Borgefors, “Applications using distance transforms,” in Aspects of Visual Form Processing, 1994, pp. 83-108.
J. Bronskill and A. Venetsanopoulus, “Multidimensional shape description and recognition using mathematical morphology,” Intelligent and Robotic Systems I: Theory and Applications, Vol. 1, No. 2, 1988.
B. Chanda, “Application of binary mathematical morphology to separate overlapped objects,” Pattern Recognition Letters, Vol. 13, pp. 639-645, 1992.
S. Chen and R.M. Haralick, “Recursive erosion, dilation, opening and closing transforms,” IEEE Transaction on Image Processing, Vol. 4, No. 3, pp. 335-345, 1995.
F.Y. Chyang and O.R. Mitchell, “Decomposition of gray scale morphological structuring elements,” Pattern Recognition, Vol. 24, No. 3, pp. 195-203, 1991.
T. Crimmons and W. Brown, “Image algebra and automatic shape representation,” IEEE Transaction on Aerospace and Electronic Systems, Vol. 21, No. 1, pp. 60-69, 1985.
P.P. Das and B.N. Chatterji, “A note on distance transformations in arbitrary dimensions,” Computer, Vision, Graphics and Image Processing, Vol. 43, pp. 368-385, 1988.
J.L. Díaz de León, “Esqueletonizing algorithms for binary images,” Master's thesis, PCINVESTAV-IPN, Electrical Engineering Deparment, 1993. In Spanish.
J.L. Díaz de León, “Mathematical morphology based on linear combined metric spaces in Z2,” PhD thesis, CINVESTAV-IPN, Electrical Engineering Deparment, 1996. In Spanish.
J.L. Díaz de León and H. Sossa, “Automatic path planning for an autoguided vehicle based on a topological approach,” in Proc. of the Int. Conf. on Systems, Man and Cybernetics, San Antonio Texas, USA, 1994, pp. 2786-2791.
J.L. Díaz de León and H. Sossa, “Automatic path planning for a mobile robot among obstacles of arbitrary shape,” IEEE Transactions on Systems, Man and Cybernetics, Part B: Cybernetics, Vol. 28, No. 3, pp. 467-472, 1998.
E. Dougherty, “Morphological image segmentation by local granulometric size distributions,” Electronic Imaging, Vol. 1, No. 1, 1992.
E. Dougherty, “Optimal mean-square n-observation digital morphological filters-part I: Optimal binary filters,” Computer, Vision, Graphics and Image Processing, Vol. 55, No. 1, pp. 36-54, 1992.
E. Dougherty, “Optimal mean-square n-observation digital morphological filters-part II: Optimal gray-scale filters,” Computer, Vision, Graphics and Image Processing, Vol. 55, No. 1, pp. 55-72, 1992.
Dujundji, Topology, Allin and Bacon, Bosoton, 1976.
R. Feehs and G. Arce, “Multidimensional morphological edge detection,” in Proc. of the SPIE, 1987, Vol. 845.
R. Foltyniewics, “Automatic face recognition via wavelets and mathematical morphology,” in Proc. of the ICPR, 1996, pp. 13-17.
P.D. Gader, “Separable decompositions and approximations of grayscale morphological templates,” Computer, Vision, Graphics and Image Processing, Vol. 53, No. 3, pp. 288-296, 1991.
P. Ghosh, “A mathematical model for shape description using Minkowski operators,” Computer, Vision, Graphics and Image Processing, Vol. 44, pp. 239-269, 1989.
G. Gordon and L. Vincent, “Application of morphology to feature extraction for face recognition,” in Proc. of the SPIE, 1992, Vol. 1658.
R.M. Haralick, S.R. Sternberg, and X. Zhuang, “Image analysis using mathematical morphology,” IEEE Transactions on Pattern Mathematical Morphology (Part I) 153 Analysis and Machine Intelligence, Vol. 9, No. 4, pp. 532-550, 1987.
H. Heijmans and C. Ronse, “The algebraic basis of mathematical morphology, I. dilations and erosions,” Computer, Vision, Graphics and Image Processing, Vol. 50, No. 3, pp. 245-295, 1990.
H. Heijmans and C. Ronse, “The algebraic basis of mathematical morphology, II. Openeings and closings,” Computer, Vision, Graphics and Image Processing,Vol. 54, No. 1, pp. 74-97, 1991.
L. Ji, J. Piper, and J.Y. Tang, “Erosion and dilation of binary images by arbitrary structuring elements using interval coding,” Pattern Recognition Letters, Vol. 9, pp. 201-209, 1989.
T. Kanungo and R.M. Haralick, “Vector-space solution for a morphological shape decomposition problem,” Journal of Mathematical Imaging and Vision, Vol. 2, pp. 51-82, 1992.
J.C. Klein and J. Serra, “The texture analyser,” Journal of Microscopy, Vol. 95, pp. 349-356, 1972.
J.W. Klinger, C.L. Vaughan, T.D. Fraker, and L.T. Andrews, “Segmentation of echocardiographic images using mathematical morphology,” IEEE Transactions on Biomedical Engineering, Vol. 35, No. 11, pp. 925-934, 1988.
B. Lay, “Recursive algorithms in mathematical morphology,” in Acta Stereologica, Vol. 6/III Proc. 7th Int. Congress for Stereology, Caen, France, 1987, pp. 691-699.
J.S.J. Lee, R.M. Haralick, and L.G. Shapiro, “Morphological edge detection,” IEEE Transactions on Robotics and Automation, Vol. 3, No. 2, pp. 142-255, 1987.
D. Li, “Morphological template decomposition with maxpolinomials,” Journal of Mathematical Imaging and Vision, Vol. 1, pp. 215-221, 1992.
E.H. Liang and E.K. Wong, “Hierarchical algorithms for morphological image processing,” Pattern Recognition, Vol. 26, No. 4, pp. 511-529, 1993.
P.L. Lin and S. Chang, “A shortest path algorithm for a nonrotaing object among obstacles of arbitrary shape,” IEEE Transactions on Systems, Man and Cybernetics, Vol. 23, pp. 825-833, 1996.
H. Luo and I. Dinstain, “Using directional mathematical morphology for separation of character strings from text/graphics image,” in Proc. of the Fifth International Workshop on Structural and Syntactic Pattern Recognition, 1994, pp. 372-381.
P. Maragos, “Tutorial on advances in morphological image processing and analysis,” Optical Engineering, Vol. 26, No. 7, pp. 623-632, 1987.
P. Maragos and R. Schafer, “Morphological filters-part i: Their set-theoretic analysis and relations to linear shift-invariant filters,” IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. 35, No. 8, pp. 1153-1169, 1987.
P. Maragos and R. Schafer, “Morphological filters-part ii: Their relations to median, order-statistic, and stack filters,” IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. 35, No. 8, pp. 1170-1184, 1987.
G. Matheron, Random Sets and Integral Geometry, Willey, New York, 1975.
F.Meyer, “Automatic screening of cytological specimens,” Computer Graphics and Image Processing, Vol. 35, pp. 356-369, 1986.
H. Minkowski, “Volumen und oberflache, Math. Ann., Vol. 57, pp. 447-495, 1903.
P.F.M. Nacken, “Chamfer metrics in mathematical morphology,” Journal of Mathematical Imaging and Vision, Vol. 4, pp. 233-253, 1994.
P.F.M. Nacken, “Chamfer metrics the medial axis and mathematical morphology,” Journal of Mathematical Imaging and Vision, Vol. 6, pp. 235-248, 1996.
D.W. Paglieroni, “Distance transforms: Properties and machine vision applications,” Computer Graphics and Image Processing, Vol. 54, pp. 56-74, 1992.
I. Pitas and A. Maglara, “Range analysis by using morphological signal decomposition,” Pattern Recognition, Vol. 24, No. 2, pp. 165-181, 1991.
I. Pitas and N.D. Sidiropoulus, “Pattern recognition of binary image objects using morphological shape decomposition,” Computer, Vision, Graphics and Image Processing, Vol. 54, pp. 279-305, 1992.
I. Pitas and A.N. Venetsanopoulus, “Morphological shape representation,” Pattern Recognition, Vol. 25, No. 6, pp. 555-565, 1992.
I. Ragnemalm, “Fast erosion and dilation by contour processing and thresholding of distance maps,” Pattern Recognition Letters, Vol. 13, pp. 161-166, 1992.
J. Rigaut, “Automated image segmentation by morphological morphology and fractal geometry,” Journal of Microscopy, Vol. 150, 1988.
A. Rosenfeld and J.L. Pfaltz, “Distance functions on digital pictures,” Pattern Recognition, Vol. 1, pp. 33-61, 1968.
J. Serra, Image Analysis and Mathematical Morphology, Vol. 1, Academic Press, San Diego, 1982.
J. Serra, “Introduction to mathematical morphology,” Computer, Vision, Graphics and Image Processing, Vol. 35, No. 3, 1986.
J. Serra, Image Analysis and Mathematical Morphology, Vol. 2. Academic Press, San Diego, 1988.
J. Serra, “An overview of morphological filters,” Circuits, Systems and Signal Processing, Vol. 11, No. 1, 1992.
L.G. Shapiro, R.S. MacDonald, and S.T. Sternberg, “Ordered structural shape matching with primitive extraction by mathematical morphology,” Pattern Recognition, Vol. 20, No. 1, pp. 75-90, 1987.
Y.M. Sharahida and N. Christofides, “A graph-theoretic approach to distance transformations,” Pattern Recognition Letters, Vol. 15, pp. 1035-1041, 1994.
F.Y. Shi and C.C. Pu, “Morphological shape description using geometric spectrum on multidimensional binary images,” Pattern Recognition, Vol. 25, No. 9, pp. 921-927, 1992.
F.Y. Shih and H. Wu, “Decomposition of geometric-shaped structuring elements using morphological transformations of binary images,” Pattern Recognition, Vol. 25, No. 10, pp. 1097-1106, 1992.
M.M. Skolnick, “Application of morphological tranformations to the analysis of two-dimensional electrophoretic gels of biological materials,” Computer Graphics and Image Processing, Vol. 35, pp. 306-332, 1986.
R. Sternberg, “Grayscale morphology,” Computer, Vision, Graphics and Image Processing, Vol. 35, No. 3, pp. 333-355, 1986.
A. Tamariz, Curso de Topologia General, UNAM, Mexico, 1990.
A. Toet, “A morphological pyramidal image decomposition,” Pattern Recognition Letters, Vol. 9, pp. 255-261, 1989.
J. Toriwaki and S. Yokoi, Distance Transformations and Skeletons of Digitized Pictures with Applications, North-Holland Publishing Company, 1981, pp. 187-264.
S. Trakiti and P.D. Gader, “Local decomposition of grayscale morphological templates,” Journal of Mathematical Imaging and Vision, Vol. 2, pp. 39-50, 1992.
R. van der Boomgaard and R. van Balen, “Methods for fast morphological image transforms using bitmapped binary images,” Computer, Vision, Graphics and Image Processing, Vol. 54, No. 3, pp. 252-258, 1992.
L.J. van Vliet and B.J.H. Verwer, “A contour processing method for fast binary neighborhood perations,” Pattern Recognition Letters, Vol. 7, pp. 27-36, 1988.
B.J.H. Verwer, “Local distances for distance transformation in two and three dimensions,” Pattern Recognition Letters, Vol. 12, pp. 671-682, 1991.
A.M. Vossepoel, “A note on: Distance transformations in digital images,” Computer, Vision, Graphics and Image Processing, Vol. 43, pp. 88-97, 1988.
X. Wang and G. Bertrand, “Some sequential algorithms for a generalized distance transformation based on minkowski operations,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 14, No. 11, pp. 1114-1121, 1992.
D. Zhao and D. Daut, “Automated shape recognition in noisy environments based on morphological algorithms,” in Proc. of the SPIE, 1990, Vol. 1295.
D. Zhao and D. Daut, “An efficient approach to automatic shape recognition,” in Proc. of the ICASSP, 1990, pp. 2161-2164.
X. Zhuang, “Decomposition of morphological structuring elements,” Journal of Mathematical Imaging and Vision, Vol. 4, pp. 5-18, 1994.
X. Zhuang and R.M. Haralick, “Morphological structuring element descomposition,” Computer, Vision, Graphics and Image Processing, Vol. 35, pp. 370-382, 1986.
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Díaz De León S., J., Sossa-Azuela, J. Mathematical Morphology Based on Linear Combined Metric Spaces on Z2 (Part I): Fast Distance Transforms. Journal of Mathematical Imaging and Vision 12, 137–154 (2000). https://doi.org/10.1023/A:1008314406260
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DOI: https://doi.org/10.1023/A:1008314406260