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Adaptive Shape Diagrams for Multiscale Morphometrical Image Analysis

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Abstract

Shape diagrams are integral geometric representations in the Euclidean plane introduced to study 2D connected compact sets. Such a set is represented by a point within a shape diagram whose coordinates are morphometrical functionals defined as normalized ratios of geometrical functionals. In addition, the General Adaptive Neighborhoods (GANs) are spatial neighborhoods defined around each point of the spatial support of a gray-tone image, that fit with the image local structures. The aim of this paper is to introduce and study the GAN-based shape diagrams, which allow a gray-tone image morphometrical analysis to be realized in a local, adaptive and multiscale way. The GAN-based shape diagrams will be illustrated on standard images and also applied in the biomedical and materials areas.

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References

  1. Arce, G.R., Foster, R.E.: Detail-preserving ranked-order based filters for image processing. IEEE Trans. Acoust. Speech Signal Process. 37(1), 83–98 (1989)

    Article  Google Scholar 

  2. Brailean, J.C., Sullivan, B.J., Chen, C.T., Giger, M.L.: Evaluating the EM algorithm for image processing using a human visual fidelity criterion. In: Proceedings of the International Conference on Acoustics, Speech and Signal Processing, pp. 2957–2960 (1991)

    Google Scholar 

  3. Breen, E., Jones, R.: Attribute openings, thinnings, and granulometries. Comput. Vis. Image Underst. 64(3), 377–389 (1996)

    Article  Google Scholar 

  4. Buzuloiu, V., Ciuc, M., Rangayyan, R.M., Vertan, C.: Adaptive-neighborhood histogram equalization of color images. J. Electron. Imaging 10(2), 445–459 (2001)

    Article  Google Scholar 

  5. Cifre, M.A.H., Salinas, G., Gomis, S.S.: Complete systems of inequalities. JIPAM. J. Inequal. Pure Appl. Math. 2(1–10), 1–12 (2001)

    MathSciNet  Google Scholar 

  6. Ciuc, M., Rangayyan, R.M., Zaharia, T., Buzuloiu, V.: Filtering noise in color images using adaptive-neighborhood statistics. J. Electron. Imaging 9(4), 484–494 (2000)

    Article  Google Scholar 

  7. Coster, M., Chermant, J.L.: Précis D’analyse D’image. Hermès, Paris (1985)

    Google Scholar 

  8. Deb-Joardar, N., Thuret, G., Zhao, M., Acquart, S., Peoc’h, M., Garraud, O., Gain, P.: Comparison of two semiautomated methods for evaluating endothelial cells of eye bank corneas. Investig. Ophthalmol. Vis. Sci. 48(7), 3077–3082 (2007)

    Article  Google Scholar 

  9. Debayle, J., Pinoli, J.C.: Spatially adaptive morphological image filtering using intrinsic structuring elements. Image Anal. Stereol. 24, 145–158 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Debayle, J., Pinoli, J.C.: General Adaptive Neighborhood Image Processing—Part I: Introduction and Theorical Aspects. J. Math. Imaging Vis. 25(2), 245–266 (2006)

    Article  MathSciNet  Google Scholar 

  11. Debayle, J., Pinoli, J.C.: General Adaptive Neighborhood Image Processing—Part II: Practical Application Examples. J. Math. Imaging Vis. 25(2), 267–284 (2006)

    Article  MathSciNet  Google Scholar 

  12. Debayle, J., Pinoli, J.C.: General adaptive neighborhood Choquet image filtering. J. Math. Imaging Vis. 35(3), 173–185 (2009)

    Article  MathSciNet  Google Scholar 

  13. Debayle, J., Pinoli, J.C.: Advances in imaging and electron physics. In: Theory and Applications of General Adaptive Neighborhood Image Processing, vol. 167, pp. 121–183. Elsevier, Amsterdam (2011)

    Google Scholar 

  14. Debayle, J., Pinoli, J.C.: General adaptive neighborhood-based pretopological image filtering. J. Math. Imaging Vis. 41(3), 210–221 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  15. Deng, D.: An entropy interpretation of the logarithmic image processing model with application to contrast enhancement. IEEE Trans. Image Process. 18, 1135–1140 (2011)

    Article  Google Scholar 

  16. Deng, G., Cahill, L.W.: Multiscale image enhancement using the logarithmic image processing model. Electron. Lett. 29, 803–804 (1993)

    Article  Google Scholar 

  17. Deng, G., Cahill, L.W., Tobin, J.R.: A study of the logarithmic image processing model and its application to image enhancement. IEEE Trans. Image Process. 4, 506–512 (1995)

    Article  Google Scholar 

  18. Deng, G., Pinoli, J.C.: Differentiation-based detection using the logarithmic image processing model. J. Math. Imaging Vis. 8, 161–180 (1998)

    Article  MathSciNet  Google Scholar 

  19. Feret, L.R.: La grosseur des grains des matires pulvérulentes. In: Premières Communications de la Nouvelle Association Internationale pour l’Essai des Matériaux, Groupe D (1930)

    Google Scholar 

  20. Fernandes, M., Gavet, Y., Pinoli, J.C.: Improving focus measurements using logarithmic image processing. J. Microsc. 242(3), 228–241 (2011)

    Article  Google Scholar 

  21. Gordon, R., Rangayyan, R.M.: Feature enhancement of mammograms using fixed and adaptive neighborhoods. Appl. Opt. 23(4), 560–564 (1984)

    Article  Google Scholar 

  22. Gouinaud, H., Gavet, Y., Debayle, J., Pinoli, J.C.: Color correction in the framework of color logarithmic image processing. In: 7th International Symposium on Image and Signal Processing and Analysis, Dubrovnik, Croatia, pp. 129–133 (2011)

    Google Scholar 

  23. Grazzini, J., Soille, P.: Edge-preserving smoothing using a similarity measure in adaptive geodesic neighborhoods. Pattern Recognit. 42(10), 2306–2316 (2009)

    Article  MATH  Google Scholar 

  24. Gremillet, P., Jourlin, M., Pinoli, J.C.: LIP model-based three-dimensionnal reconstruction and visualisation of HIV infected entire cells. J. Microsc. 174, 31–38 (1994)

    Article  Google Scholar 

  25. Hafstrom, J.E.: Introduction to Analysis and Abstract Algebra. Saunders, Philadelphia (1967)

    MATH  Google Scholar 

  26. Jourlin, M., Pinoli, J.C.: A model for logarithmic image processing. J. Microsc. 149, 21–35 (1988)

    Article  Google Scholar 

  27. Jourlin, M., Pinoli, J.C.: Image dynamic range enhancement and stabilization in the context of the logarithmic image processing model. Signal Process. 41, 225–237 (1995)

    Article  MATH  Google Scholar 

  28. Jourlin, M., Pinoli, J.C.: Logarithmic image processing, the mathematical and physical framework for the representation and processing of transmitted images. In: Advances in Imaging and Electron Physics, vol. 115, pp. 129–196. Elsevier, Amsterdam (2001)

    Google Scholar 

  29. Jourlin, M., Pinoli, J.C., Zeboudj, R.: Contrast definition and contour detection for logarithmic images. J. Microsc. 156, 33–40 (1988)

    Article  Google Scholar 

  30. Lang, S.: Linear Algebra. Addison-Wesley, Reading (1966)

    MATH  Google Scholar 

  31. Lee, J.S.: Refined filtering of image using local statistics. Comput. Graph. Image Process. 15, 380–389 (1981)

    Article  Google Scholar 

  32. Maragos, P.: Pattern spectrum and multiscale shape representation. IEEE Trans. Pattern Anal. Mach. Intell. 11, 701–716 (1989)

    Article  MATH  Google Scholar 

  33. Maragos, P., Vachier, C.: Overview of adaptive morphology: trends and perspectives. In: IEEE International Conference on Image Processing, Cairo, Egypt, pp. 2241–2244 (2009)

    Google Scholar 

  34. Mayet, F., Pinoli, J.C., Jourlin, M.: Justifications physiques et applications du modèle LIP pour le traitement des images obtenues en lumière transmise. Trait. Signal 13, 251–262 (1996)

    MATH  Google Scholar 

  35. Meggido, N.: Linear-time algorithms for linear programming in R3 and related problems. SIAM J. Comput. 12(4), 759–776 (1983)

    Article  MathSciNet  Google Scholar 

  36. Michielsen, K., De Raedt, H.: Integral-geometry morphological image analysis. Phys. Rep. 347, 461–538 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  37. Ikeda, Y.: Region extraction and shape analysis in aerial photographs. Comput. Graph. Image Process. 10(3), 195–223 (1979)

    Article  Google Scholar 

  38. Nagao, N., Matsuyama, T.: Edge preserving smoothing. Comput. Graph. Image Process. 9, 394–407 (1979)

    Article  Google Scholar 

  39. Noyel, G., Angulo, J., Jeulin, D.: On distances, paths and connections for hyperspectral image segmentation. In: Banon, G., Barrera, J., Braga-Neto, U., Hirata, N. (eds.) Proceedings of International Symposium on Mathematical Morphology, Instituto Nacional de Pesquisas Espaciais (INPE), São José dos Campos, vol. 1, pp. 399–410 (2007)

    Google Scholar 

  40. Oppenheim, A.: Superposition in a class of nonlinear systems. Tech. Rep., Research Laboratory of Electronics, MIT, Cambridge, USA (1965)

  41. Oppenheim, A.: Generalized superposition. Inf. Control 11, 528–536 (1967)

    Article  MATH  Google Scholar 

  42. Osserman, R.: The isoperimetric inequality. Bull. Am. Math. Soc. 84, 1182–1238 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  43. Ouzounis, G., Soille, P.: Pattern spectra from partition pyramids and hierarchies. Lect. Notes Comput. Sci. 6671, 108–119 (2011)

    Article  MathSciNet  Google Scholar 

  44. Panetta, K.A., Agaian Zhou, S.Y., Wharton, E.J.: Parameterized logarithmic framework for image enhancement. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 41(2), 460–473 (2011)

    Article  Google Scholar 

  45. Paranjape, R.B., Rangayyan, R.M., Morrow, W.M.: Adaptive neighbourhood mean and median image filtering. J. Electron. Imaging 3(4), 360–367 (1994)

    Article  Google Scholar 

  46. Pinoli, J.C.: A general comparative study of the multiplicative homomorphic, log-ratio and logarithmic image processing approaches. Signal Process. 58, 11–45 (1997)

    Article  MATH  Google Scholar 

  47. Pinoli, J.C.: The logarithmic image processing model: connections with human brightness perception and contrast estimators. J. Math. Imaging Vis. 7, 341–358 (1997)

    Article  Google Scholar 

  48. Pinoli, J.C., Debayle, J.: Logarithmic adaptive neighborhood image processing (lanip): introduction, connections to human brightness perception and application issues. EURASIP J. Adv. Signal Process. 36105, 1–22 (2007)

    Google Scholar 

  49. Pinoli, J.C., Debayle, J.: Adaptive generalized metrics, distance maps and nearest neighbor transforms on gray tone images. Pattern Recognit. 45, 2758–2768 (2012)

    Article  MATH  Google Scholar 

  50. Presles, B., Debayle, J., Fevotte, G., Pinoli, J.C.: A novel image analysis method for in-situ monitoring the particle size distribution of batch crystallization processes. J. Electron. Imaging 19(3), 1–7 (2010)

    Google Scholar 

  51. Presles, B., Debayle, J., Pinoli, J.C.: Size and shape estimation of 3-d convex objects from their 2-d projections. Application to crystallization processes. J. Microsc. 248(2), 140–155 (2012)

    Article  Google Scholar 

  52. Rabie, T.F., Rangayyan, R.M., Paranjape, R.B.: Adaptive-neighborhood image deblurring. J. Electron. Imaging 3(4), 368–378 (1994)

    Article  Google Scholar 

  53. Rangayyan, R.M., Ciuc, M., Faghih, F.: Adaptive-neighborhood filtering of images corrupted by signal-dependent noise. Appl. Opt. 37(20), 4477–4487 (1998)

    Article  Google Scholar 

  54. Rangayyan, R.M., Das, A.: Filtering multiplicative noise in images using adaptive region-based statistics. J. Electron. Imaging 7(1), 222–230 (1998)

    Article  Google Scholar 

  55. Rivollier, S.: Analyse spectrale morphologique adaptative d’images. Master’s thesis, Ecole Nationale Supérieure des Mines de Saint-Etienne, France (2006)

  56. Rivollier, S., Debayle, J., Pinoli, J.C.: Shape diagrams for 2D compact sets—Part I: analytic convex sets. Aust. J. Math. Anal. Appl. 7(2–3), 1–27 (2010)

    Google Scholar 

  57. Rivollier, S., Debayle, J., Pinoli, J.C.: Shape diagrams for 2D compact sets—Part II: analytic simply connected sets. Aust. J. Math. Anal. Appl. 7(2–4), 1–21 (2010)

    Google Scholar 

  58. Rivollier, S., Debayle, J., Pinoli, J.C.: Shape diagrams for 2D compact sets—Part III: convexity discrimination for analytic and discretized simply connected sets. Aust. J. Math. Anal. Appl. 7(2–5), 1–18 (2010)

    Google Scholar 

  59. Ropert, M., Pelé, D.: Synthesis of adaptive weighted order statistic filters with gradient algorithms. In: Serra, J., Soille, P. (eds.) Mathematical Morphology and its Applications to Image Processing (1994)

    Google Scholar 

  60. Salembier, P.: Structuring element adaptation for morphological filters. J. Vis. Commun. Image Represent. 3(2), 115–136 (1992)

    Article  Google Scholar 

  61. Santalo, L.A.: Sobre los sistemas completos de desigualdades entre tres elementos de una figura plana. Math. Notae 17, 82–104 (1961)

    MATH  MathSciNet  Google Scholar 

  62. Scott, P.R., Awyong, P.W.: Inequalities for convex sets. JIPAM. J. Inequal. Pure Appl. Math. 1(1–6), 1–6 (2000)

    MathSciNet  Google Scholar 

  63. Shamos, M.I.: Computational geometry. Ph.D. Thesis, Yale University (1978)

  64. Siegel, A.: An isoperimetric theorem in plane geometry. Discrete Comput. Geom. 29(2), 239–255 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  65. Soille, P.: Morphological Image Analysis: Principles and Applications. Springer, New York (2003)

    Google Scholar 

  66. Soille, P.: Constrained connectivity for hierarchical image partitioning and simplification. IEEE Trans. Pattern Anal. Mach. Intell. 30(7), 1132–1145 (2008)

    Article  Google Scholar 

  67. Stockham, T.G.: The applications of generalized linearity to automatic gain control. IEEE Trans. Audio Electroacoust. 16(2), 267–270 (1968)

    Article  Google Scholar 

  68. Stockham, T.G.: Image processing in the context of a visual model. Proc. IEEE 60(7), 825–842 (1972)

    Article  Google Scholar 

  69. Strang, G.: Linear Algebra and Its Applications. Academic Press, New York (1976)

    MATH  Google Scholar 

  70. Urbach, E., Roerdink, J., Wilkinson, M.: Connected shape-size pattern spectra for rotation and scale-invariant classification of gray-scale images. IEEE Trans. Pattern Anal. Mach. Intell. 29(2), 272–285 (2007)

    Article  Google Scholar 

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Acknowledgements

The authors would like to thank the Professor P. Gain from the University Hospital Center of Saint-Etienne in France and the Professor G. Fevotte from the LGF UMR CNRS 5307 in France who have kindly supplied different original images used in this paper.

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  1. Ecole Nationale Supérieure des Mines, LGF UMR CNRS 5307, 158, cours Fauriel, 42023, Saint-Etienne cedex 2, France

    Séverine Rivollier, Johan Debayle & Jean-Charles Pinoli

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  1. Séverine Rivollier
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  2. Johan Debayle
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Correspondence to Johan Debayle.

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Rivollier, S., Debayle, J. & Pinoli, JC. Adaptive Shape Diagrams for Multiscale Morphometrical Image Analysis. J Math Imaging Vis 49, 51–68 (2014). https://doi.org/10.1007/s10851-013-0439-2

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