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Issue No.05 - May (2008 vol.30)
pp: 823-836
ABSTRACT
We develop a general theory of spatially-variant (SV) mathematical morphology for binary images in the Euclidean space. The basic SV morphological operators (i.e., SV erosion, SV dilation, SV opening and SV closing) are defined. We demonstrate the ubiquity of SV morphological operators by providing a SV kernel representation of increasing operators. The latter representation is a generalization of Matheron's representation theorem of increasing and translation-invariant operators. The SV kernel representation is redundant, in the sense that a smaller subset of the SV kernel is sufficient for the representation of increasing operators. We provide sufficient conditions for the existence of the minimal basis representation in terms of upper-semi-continuity in the hit-or-miss topology. The latter minimal basis representation is a generalization of Maragos' minimal basis representation for increasing and translation-invariant operators. Moreover, we investigate the upper-semi-continuity property of the basic SV morphological operators. Several examples are used to demonstrate that the theory of spatially-variant mathematical morphology provides a general framework for the unification of various morphological schemes based on spatiallyvariant geometrical structuring elements (e.g., circular, affine and motion morphology). Simulation results illustrate the theory of the proposed spatially-variant morphological framework and show its potential power in various image processing applications.
INDEX TERMS
Morphological, Filtering
CITATION
Mohammed Charif-Chefchaouni, Nidhal Bouaynaya, "Theoretical Foundations of Spatially-Variant Mathematical Morphology Part I: Binary Images", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.30, no. 5, pp. 823-836, May 2008, doi:10.1109/TPAMI.2007.70754
REFERENCES
[1] J. Angulo and J. Serra, “Automatic Analysis of DNA Microarray Images Using Mathematical Morphology,” Bioinformatics, vol. 19, no. 5, pp. 553-562, 2003.
[2] G.J.F. Banon and J. Barrera, “Minimal Representation for Translation-Invariant Set Mappings by Mathematical Morphology,” SIAM J. Applied Math., vol. 51, pp. 1782-1798, Dec. 1991.
[3] G.J.F. Banon and J. Barrera, “Decomposition of Mappings between Complete Lattices by Mathematical Morphology—Part I: General Lattices,” Signal Processing, vol. 30, no. 3, pp. 299-327, Feb. 1993.
[4] R.G. Bartle, The Elements of Real Analysis. John Wiley & Sons, 1976.
[5] S. Beucher, J.M. Blosseville, and F. Lenoir, “Traffic Spatial Measurements Using Video Image Processing,” Proc. SPIE Intelligent Robots and Computer Vision, vol. 848, pp. 648-655, Nov. 1987.
[6] G. Birkhoff, Lattice Theory. Am. Math. Soc., 1984.
[7] N. Bouaynaya, M. Charif-Chefchaouni, and D. Schonfeld, “Spatially-Variant Morphological Restoration and Skeleton Representation,” IEEE Trans. Image Processing, vol. 15, no. 11, pp. 3579-3591, Nov. 2006.
[8] M. Charif-Chefchaouni, “Morphological Representation of Non-Linear Operators: Theory and Applications,” PhD dissertation, Univ. of Illinois at Chicago, 1993.
[9] M. Charif-Chefchaouni and D. Schonfeld, “Spatially-Variant Mathematical Morphology,” Proc. IEEE Int'l Conf. Image Processing, vol. 2, pp. 555-559, Nov. 1994.
[10] C.-S. Chen, J.-L. Wu, and Y.-P. Hung, “Statistical Analysis of Space-Varying Morphological Openings with Flat Structuring Elements,” IEEE Trans. Signal Processing, vol. 44, no. 4, pp. 1010-1014, Apr. 1996.
[11] C.-S. Chen, J.-L. Wu, and Y.-P. Hung, “Theoretical Aspects of Vertically Invariant Gray-Level Morphological Operators and Their Application on Adaptive Signal and Image Filtering,” IEEE Trans. Signal Processing, vol. 47, no. 4, pp. 1049-1060, Apr. 1999.
[12] F. Cheng and A.N. Venetsanopoulos, “An Adaptive Morphological Filter for Image Processing,” IEEE Trans. Image Processing, vol. 1, no. 4, pp. 533-539, Oct. 1992.
[13] J. Crespo, J. Serra, and R.W. Schafer, “Theoretical Aspects of Morphological Filters by Reconstruction,” Signal Processing, vol. 47, no. 2, pp. 201-225, Nov. 1995.
[14] O. Cuisenaire, “Locally Adaptable Mathematical Morphology,” Proc. IEEE Int'l Conf. Image Processing, vol. 2, pp. 125-128, Sept. 2005.
[15] J. Debayle and J.C. Pinoli, “General Adaptive Neighborhood Image Processing—Part I: Introduction and Theoretical Aspects,” J. Math. Imaging and Vision, vol. 25, no. 2, pp. 245-266, Sept. 2006.
[16] J. Debayle and J.C. Pinoli, “General Adaptive Neighborhood Image Processing—Part II: Practical Application Examples,” J.Math. Imaging and Vision, vol. 25, no. 2, pp. 267-284, Sept. 2006.
[17] E.R. Dougherty and C.R. Giardina, “A Digital Version of the Matheron Representation Theorem for Increasing $\tau$ -Mappings in Terms of a Basis for the Kernel,” Proc. IEEE Computer Vision and Pattern Recognition, pp. 534-536, June 1986.
[18] E.R. Dougherty and C.R. Giardina, Morphological Methods in Image Processing. Prentice Hall, 1988.
[19] T. Fang, M.A. Jafari, S.C. Danforth, and A. Safari, “Signature Analysis and Defect Detection in Layered Manufacturing of Ceramic Sensors and Actuators,” Machine Vision and Applications, vol. 15, no. 2, pp. 63-75, Dec. 2003.
[20] R. Gordon and R.M. Rangayyan, “Feature Enhancement of Mammograms Using Fixed and Adaptive Neighborhoods,” Applied Optics, vol. 23, no. 4, pp. 560-564, Feb. 1984.
[21] J. Goutsias and S. Batman, “Morphological Methods for Biomedical Image Analysis,” Handbook of Medical Imaging, Progress in Medical Image Processing and Analysis, vol. 3, M. Sonka and J.M.Fitzpatrick, eds., pp. 175-272. SPIE Optical Eng. Press, 2000.
[22] A.G. Hanbury and J. Serra, “Analysis of Oriented Textures Using Mathematical Morphology,” Vision with Non-Traditional Sensors, Sept. 2002.
[23] J.C. Handley, “Minimal-Memory Bit-Vector Architecture for Computational Mathematical Morphology Using Subspace Projections,” IEEE Trans. Image Processing, vol. 14, no. 8, pp. 1088-1095, Aug. 2005.
[24] H.J.A.M. Heijmans, “Mathematical Morphology: An Algebraic Approach,” CWI Newsletter, vol. 14, pp. 7-27, 1987.
[25] H.J.A.M. Heijmans, Morphological Image Operators. Academic Press, 1994.
[26] H.J.A.M. Heijmans and J. Goutsias, “Nonlinear Multiresolution Signal Decomposition Schemes—I: Morphological Pyramids,” IEEE Trans. Image Processing, vol. 9, no. 11, pp. 1862-1876, Nov. 2000.
[27] H.J.A.M. Heijmans and J. Goutsias, “Nonlinear Multiresolution Signal Decomposition Schemes—II: Morphological Wavelets,” IEEE Trans. Image Processing, vol. 9, no. 11, pp. 1897-1913, Nov. 2000.
[28] H.J.A.M. Heijmans and C. Ronse, “The Algebraic Basis of Mathematical Morphology—I: Dilations and Erosions,” Computer Vision, Graphics and Image Processing: Image Understanding, vol. 50, no. 3, pp. 245-295, June 1990.
[29] H.J.A.M. Heijmans and C. Ronse, “The Algebraic Basis of Mathematical Morphology—II: Openings and Closings,” Computer Vision, Graphics and Image Processing: Image Understanding, vol. 54, no. 1, pp. 74-97, July 1991.
[30] S. Hou, S. Wu, and S. Ma, “Doppler Frequency Extraction of Radar Echo Signal Based on Time-Frequency Analysis and Morphological Operation,” Proc. Int'l Conf. Signal Processing, vol. 3, pp. 2017-2020, 2004.
[31] A.C. Jalba, M.H.F. Wilkinson, and J.B.T.M. Roerdink, “Shape Representation and Recognition through Morphological Curvature Scale Spaces,” IEEE Trans. Image Processing, vol. 15, no. 2, pp.331-341, Feb. 2006.
[32] A.N. Kolmogoroff and S.V. Fomin, Introductory Real Analysis. Dover, 1975.
[33] R. Kresch and D. Malah, “Morphological Reduction of Skeleton Redundancy,” Signal Processing, vol. 38, no. 1, pp. 143-151, July 1994.
[34] R. Kresch and D. Malah, “Skeleton-Based Morphological Coding of Binary Images,” IEEE Trans. Image Processing, vol. 7, no. 10, pp.1387-1399, Oct. 1998.
[35] C. Lantuejoul, “La Squelettization et son Application aux Mesures Topologuiques des Mosaiques Polycristallines,” PhD dissertation, School of Mines, 1978.
[36] R. Lerallut, M. Goehm, E. Decenciere, and F. Meyer, “Noise Reduction in 3D Images Using Morphological Amoebas,” Proc. IEEE Int'l Conf. Image Processing, vol. 1, pp. 109-112, Sept. 2005.
[37] P.A. Maragos, A Unified Theory of Translation-Invariant Systems with Applications to Morphological Analysis and Coding of Images, PhD dissertation, Georgia Inst. of Tech nology, July 1985.
[38] P.A. Maragos, “Pattern Spectrum and Multiscale Shape Representation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, no. 7, pp. 701-716, July 1989.
[39] P.A. Maragos, “A Representation Theory for Morphological Image and Signal Processing,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, no. 6, pp. 586-599, June 1989.
[40] P.A. Maragos, “Affine Morphology and Affine Signal Models,” Proc. SPIE Image Algebra and Morphological Image Processing, vol. 1350, pp. 31-43, 1990.
[41] P.A. Maragos, Morphological Filtering for Image Enhancement and Feature Detection, Chapter 3, pp. 135-156. Elsevier Academic Press, 2005.
[42] P.A. Maragos and R.W. Schafer, “Morphological Skeleton Representation and Coding of Binary Images,” IEEE Trans. Acoustics, Speech, and Signal Processing, vol. 34, no. 5, pp. 1228-1244, Oct. 1986.
[43] P.A. Maragos and R.W. Schafer, “Morphological Filters—Part I: Their Set-Theoric Analysis and Relations to Linear Shift-Invariant Filters,” IEEE Trans. Acoustics, Speech, and Signal Processing, vol. 35, no. 8, pp. 1153-1169, Aug. 1987.
[44] P.A. Maragos and R.W. Schafer, “Morphological Filters—Part II: Their Relation to Median, Order-Statistics and Stack Filters,” IEEE Trans. Acoustics, Speech, and Signal Processing, vol. 35, 1987.
[45] I. Masayasu, T. Masayoshi, and N. Akira, “Morphological Operations by Locally Variable Structuring Elements and Their Applications to Region Extraction in Ultrasound Images,” Systems and Computers in Japan, vol. 34, no. 3, pp. 33-43, Feb. 2003.
[46] G. Matheron, Random Sets and Integral Geometry. John Wiley & Sons, 1975.
[47] A. Morales, “Adaptive Structuring Element for Noise and Artifact Removal,” Proc. Conf. Information Sciences and Systems, Mar. 1989.
[48] J.B.T.M. Roerdink, “Group Morphology,” Pattern Recognition, vol. 33, no. 6, pp. 877-895, 2000.
[49] J.B.T.M. Roerdink, “Multiresolution Maximum Intensity Volume Rendering by Morphological Adjunction Pyramids,” IEEE Trans. Image Processing, vol. 12, no. 6, pp. 653-660, June 2003.
[50] J.B.T.M. Roerdink and H.J.A.M. Heijmans, “Mathematical Morphology for Structuring Elements without Translation Symmetry,” Signal Processing, vol. 15, no. 3, pp. 271-277, 1988.
[51] D. Schonfeld and J. Goutsias, “Morphological Representation of Discrete and Binary Images,” IEEE Trans. Signal Processing, vol. 39, no. 6, pp. 1369-1379, June 1991.
[52] D. Schonfeld and J. Goutsias, “Optimal Morphological Pattern Restoration from Noisy Binary Images,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, no. 1, pp. 14-29, Jan. 1991.
[53] J. Serra, Image Analysis and Math. Morphology. Academic Press, 1982.
[54] J. Serra, Image Analysis and Math. Morphology Theoretical Advances, vol. 2. Academic Press, 1988.
[55] J. Serra and P. Salembier, “Connected Operators and Pyramids,” Proc. SPIE, vol. 2030, pp. 65-76, 1993.
[56] F.Y. Shih and S. Cheng, “Adaptive Mathematical Morphology for Edge Linking,” Int'l J. Information Sciences Informatics and Computer Science, vol. 167, pp. 9-21, Dec. 2004.
[57] K. Shoji, “Generalized Skeleton Representation and Adaptive Rectangular Decomposition of Binary Images,” Proc. SPIE in Image Algebra and Morphological Image Processing, J.C. Serra, ed., vol. 1769, pp. 404-415, June 1992.
[58] J.G. Verly and R.L. Delanoy, “Adaptive Mathematical Morphology for Range Imagery,” IEEE Trans. Image Processing, vol. 2, no. 2, pp. 272-275, Apr. 1993.
[59] L. Vincent, “Morphological Grayscale Reconstruction in Image Analysis: Applications and Efficient Algorithms,” IEEE Trans. Image Processing, vol. 2, no. 2, pp. 176-201, Apr. 1993.
[60] Y. Xia, D. Feng, and R. Zhao, “Morphology-Based Multifractal Estimation for Texture Segmentation,” IEEE Trans. Image Processing, vol. 15, no. 3, pp. 614-623, Mar. 2006.
[61] J. Xu, “A Generalized Discrete Morphological Skeleton Transform with Multiple Structuring Elements for the Extraction of Structural Shape Components,” IEEE Trans. Image Processing, vol. 12, no. 2, pp. 1677-1686, Dec. 2003.
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