No Access

Descriptive Image Analysis. Foundations and Descriptive Image Algebras

Federal Research Center “Computer Sciences and Control” of the Russian, Academy of Sciences 44, Vavilov Str., Building 2, Moscow 119333, Russian Federation

E-mail Address: igourevi@ccas.ru

Search for more papers by this author

and

Vera Yashina

http://orcid.org/0000-0002-5120-3081

Federal Research Center “Computer Sciences and Control” of the Russian, Academy of Sciences 44, Vavilov Str., Building 2, Moscow 119333, Russian Federation

E-mail Address: werayashina@gmail.com

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0218001419400184Cited by:16 (Source: Crossref)

This article is part of the issue:

Special Issue on the International Conference on Pattern Recognition and Artificial Intelligence (ICPRAI 2018)
Guest Editors: Ching Yee Suen (Concordia University, Canada), Pong C. Yuen (Hong Kong Baptist University), Adam Krzyzak (Concordia University, Canada), Simone Marinai (Università degli Studi di Firenze, Italy) and Patrick S. P. Wang (Northeastern University, USA)

Abstract

The paper is devoted to Descriptive Image Analysis (DA) — a leading line of the modern mathematical theory of image analysis. DA is a logically organized set of descriptive methods, mathematical objects, and models and representations aimed at analyzing and evaluating the information represented in the form of images, as well as for automating the extraction from images of knowledge and data needed for intelligent decision-making.

The basic idea of DA consists of embedding all processes of analysis (processing, recognition, understanding) of images into an image formalization space and reducing it to (1) construction of models/representations/formalized descriptions of images; (2) construction of models/representations/formalized descriptions of transformations over models and representations of images.

We briefly discuss the basic ideas, methodological principles, mathematical methods, objects, and components of DA and the basic results determining the current state of the art in the field. Image algebras (IA) are considered in the context of a unified language for describing mathematical objects and operations used in image analysis (the standard IA by Ritter and the descriptive IA by Gurevich).

Keywords:

References

1. H. G. Barrow, A. P. Ambler and R. M. Burstall , Some techniques for recognizing structures in pictures, in Proc. Int. Conf. Frontiers of Pattern Recognition, ed. S. Watanabe (Academic Press, New York, London, 1972), pp. 1–30. Google Scholar
2. G. Birkhoff and J. D. Lipson , Heterogeneous algebras, J. Combinat. Theory 8 (1970) 115–133. Google Scholar
3. V. M. Chernov , On defining equations for the elements of associative and commutative algebras, in Space-Time Structure. Algebra and Geometry, eds. D. Pavlov, Gh. Atanasiu and V. Balan (Lilia Print, 2007), pp. 182–188. Google Scholar
4. J. L. Davidson , Classification of lattice transformations in image processing, Comput. Vis. Graph. Image Process. Image Understand. 57(3) (1993) 283–306. Web of Science, Google Scholar
5. E. R. Dougherty , A homogeneous unification of image algebra. Part I: The homogenous algebra, Part II: Unification of image algebra, Imaging Sci. 33(4) (1989) 136–149. Web of Science, Google Scholar
6. E. R. Dougherty and D. Sinha , Computational Gray-Scale Mathematical Morphology on Lattices (A Comparator-Based Image Algebra), Part 1: Architecture. Real-Time Imaging, Part 2: Image Operators, Real-Time Imaging, 1 (1995) 69–85; 283–295. Google Scholar
7. M. J. B. Duff, D. M. Watson, T. J. Fountain and G. K. Shaw , A cellular logic array for image processing, Pattern Recognit. 5(3) (1973) 229–247. Web of Science, Google Scholar
8. T. G. Evans , Descriptive pattern analysis techniques: Potentialities and problems, in Proc. Int. Conf. Methodologies of Pattern Recognition (Academic Press, New York, London, 1969), pp. 149–157. Google Scholar
9. Ya. A. Furman , Parallel recognition of different classes of patterns, Pattern Recognit. Image Anal. 19(3) (2009) 380–393. Google Scholar
10. Ya. A. Furman, R. V. Eruslanov and I. L. Egoshina , Recognition of images and recognition of polyhedral objects, Pattern Recogn. Image Anal. 22(1) (2012) 196–209. Google Scholar
11. P. D. Gader, M. A. Khabou and A. Koldobsky , Morphological regularization neural networks, Pattern Recognit. 33 (2000) 935–944. Web of Science, Google Scholar
12. U. Grenander , General Pattern Theory. A Mathematical Study of Regular Structure (Clarendon Press, Oxford, 1993). Google Scholar
13. I. B. Gurevich , The descriptive framework for an image recognition problem, in Proc. 6th Scandinavian Conf. Image Analysis (Pattern Recognition Society of Finland, 1989), Vol. 1, pp. 220–227. Google Scholar
14. I. B. Gurevich , Descriptive technique for image description, representation and recognition, Pattern Recognit. Image Anal. Adv. Math. Theory Appl. USSR 1 (1991) 50–53. Google Scholar
15. I. B. Gurevich , The descriptive approach to image analysis. Current state and prospects, in Proc. 14th Scandinavian Conf. Image Analysis (Springer-Verlag, Berlin, Heidelberg, 2005), pp. 214–223. Google Scholar
16. I. B. Gurevich and I. A. Jernova , The joint use of image equivalents and image invariants in image recognition, Pattern Recognit. Image Anal. Adv. Math. Theory Appl. 13(4) (2003) 570–578. Google Scholar
17. I. B. Gurevich and I. V. Koryabkina , Comparative analysis and classification of features for image models, Pattern Recognit. Image Anal. Adv. Math. Theory Appl. 16(3) (2006) 265–297. Google Scholar
18. I. B. Gurevich and A. V. Nefyodov , Block diagram representation of a 2D-AEC algorithm with rectangular support sets// pattern recognition and image analysis: Advances in mathematical theory and applications, MAIK, “ Nauka/Interperiodica ” (Russia), 15(1) (2005) 187–191. Google Scholar
19. I. B. Gurevich and V. V. Yashina , Descriptive image algebras with one ring, Pattern Recognit. Image Anal. Adv. Math. Theory Appl. 13(4) (2003) 579–599. Google Scholar
20. I. B. Gurevich and V. V. Yashina , Algorithmic scheme based on a descriptive image algebra with one ring: Image analysis example, Pattern Recognit. Image Anal. Adv. Math. Theory Appl. 15(1) (2005) 192–194. Google Scholar
21. I. B. Gurevich and V. V. Yashina , Generating descriptive trees, in Proc. 10th Int. Fall Workshop on Vision, Modeling, and Visualization (2005), pp. 367–374. Google Scholar
22. I. B. Gurevich and V. V. Yashina , Operations of descriptive image algebras with one ring, Pattern Recognit. Image Anal. Adv. Math. Theory Appl. 16(3) (2006) 298–328. Google Scholar
23. I. B. Gurevich and V. V. Yashina , Computer-aided image analysis based on the concepts of invariance and equivalence, Pattern Recognit. Image Anal. Adv. Math. Theory Appl. 16(4) (2006) 564–589. Google Scholar
24. I. B. Gurevich and V. V. Yashina , Descriptive approach to image analysis: Image models, Pattern Recognit. Image Anal. Adv. Math. Theory Appl. 18(4) (2008) 518–541. Google Scholar
25. I. Gurevich and V. Yashina , Image formalization via descriptive image algebras, in Proc. 3rd Int. Workshop on Image Mining Theory and Applications–IMTA 2010 (in Conjunction with VISIGRAPP 2010), eds. I. Gurevich, H. Niemann and O. Salvetti (INSTICC Press, Angers, France, 2010), pp. 19–28. Google Scholar
26. I. B. Gurevich and V. V. Yashina , Image formalization space: Formulation of tasks, structural properties, and elements, Pattern Recognit. Image Anal. Adv. Math. Theory Appl. 21(2) (2011) 134–139. Google Scholar
27. I. B. Gurevich and V. V. Yashina , Descriptive approach to image analysis: Image formalization space, Pattern Recognit. Image Anal. Adv. Math. Theory Appl. 22(4) (2012) 495–518. Google Scholar
28. I. Gurevich, Yu. Trusova and V. Yashina , Current trends in the algebraic image analysis. A survey, in Proc. 18th Iberoamerican Congress on Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications, CIARP 2013, Havana, Cuba, Nov. 20–23, 2013, eds. J. Ruiz-Shulcloper and G. Sanniti di Baja (Springer-Verlag, Berlin, Heidelberg, 2013), Vol. 1, pp. 423–430. Google Scholar
29. I. B. Gurevich and Yu. I. Zhuravlev , Computer science: Subject, fundamental research problems, methodology, structure, and applied problems, Pattern Recognit. Image Anal. Adv. Math. Theory Appl. 24(3) (2014) 333–346. Google Scholar
30. I. B. Gurevich, Yu. I. Zhuravlev, A. A. Myagkov, Yu. O. Trusova and V. V. Yashina , On basic problems of image recognition in neurosciences and heuristic methods for their solution, Pattern Recognit. Image Anal. Adv. Math. Theory Appl. 25(1) (2015) 132–160. Google Scholar
31. I. B. Gurevich and V. V. Yashina , Descriptive image analysis, genesis and current trends, Pattern Recognit. Image Anal. Adv. Math. Theory Appl. 27(4) (2017) 653–674. Google Scholar
32. H. Hadwiger , Uber Treffanzahlen bei translationsgleichen Eikorpern, Arch. Math. 8 (1957) 212–213. Google Scholar
33. R. Haralick and L. Shapiro , Image segmentation techniques, Comput. Vision, Graph. Image Process. 29 (1985) 100–132. Google Scholar
34. R. Haralick, L. Shapiro and J. Lee , Morphological edge detection, IEEE J. Robot. Automat. RA-3(1) (1987) 142–157. Google Scholar
35. J. Hyoman, R. N. Haralick and L. G. Shapiro , Toward the automatic generation of mathematical morphology procedures using predicate logic, in Proc. 3rd Int. Conf. Computer Vision, Osaka, 1990, pp. 156–165. Google Scholar
36. S. Kaneff , Pattern cognition and the organization of information, in Proc. Int. Conf. Frontiers of Pattern Recognition, ed. Satosi Watanabe (Academic Press, New York, London, 1972), pp. 193–222. Google Scholar
37. R. Kirsh , Computer interpretation of english text and picture patterns, IEEE-TEC EC-13(4) (1964). Google Scholar
38. U. Köthe, A. Montanvert and P. Soille (eds), Applications of Discrete Geometry and Mathematical Morphology, Lecture Notes in Computer Science, 7346 (Springer, 2012). Google Scholar
39. V. G. Labunetc , Algebraic Theory of Signals and Systems (Digital Signal Processing, Krasnoyarsk Univ., Krasnoyarsk, 1984). Google Scholar
40. J. S. J. Lee, R. M. Haralick and L. G. Shapiro , Morphologic edge detection, IEEE J. Robot. Automat. RA-3(2) (1987) 142–156. Google Scholar
41. D. A. Leites , Introduction to the theory of supermanifolds, Russ. Math. Survey 35(1) (1980) 1–64. Web of Science, Google Scholar
42. Yu. N. Maltsev and E. N. Kuzmin , A basis for identities of the algebra of second order matrices over a nite eld, Algebra Logic 17 (1978) 17–21. Google Scholar
43. A. I. Maltcev , Algebraic Systems (Springer-Verlag, Berlin, 1973). Google Scholar
44. G. Matheron , Random Sets and Integral Geometry (Wiley, New York, 1975). Google Scholar
45. P. Maragos , Algebraic and PDE approaches for lattice scale-spaces with global constraints, Int. J. Comput. Vis. 52(2/3) (2003) 121–137. Web of Science, Google Scholar
46. P. Maragos and R. W. Schafer, Morphological filters. Part I: Their set-theoretic analysis and relations to linear shift-invariant filters, IEEE Trans. Acoustics, Speech, Signal Processing 35 (1987) 1153–1169; Part II: Their relations to median, order-statistic, and stack filters, IEEE Trans. Acoustics, Speech, Signal Processing 35 (1987) 1170–1184. Google Scholar
47. D. Marr , Vision (Freeman, New York, 1982). Google Scholar
48. X. X. Niu and C. Y. Suen , A novel hybrid CNN-SVM classifier for recognizing handwritten digits, Pattern Recognit. 45 (2012) 1318–1325. Web of Science, Google Scholar
49. R. Narasimhan , Picture languages, in Picture Language Machines, ed. by S. Kaneff (Academic Press, London, New York, 1970), pp. 1–30. Google Scholar
50. J. von Neumann , Theory of Self-Reproducing Automata (University of Illinois Press, Urbana, IL, 1966). Google Scholar
51. M. Pavel , Fundamentals of Pattern Recognition (Marcell Dekker, New York, 1989). Google Scholar
52. Yu. P. Pytiev , Method of Mathematical Modeling of Measuring and Computing Systems, 2nd edn. (MAIK Nauka, Moscow, 2004) [in Russian]. Google Scholar
53. Proc. 2nd Int. Workshop on Multiple Classifier Systems, MCS 2001, Cambridge, UK, July 2001 (Springer-Verlag, 2001). Google Scholar
54. B. Radunacu, M. Grana and F. X. Albizuri , Morphological scale spaces and associative morphological memories: Results on robustness and practical applications, J. Math. Imag. Vision 19 (2003) 113–131. Web of Science, Google Scholar
55. G. X. Ritter and J. N. Wilson , Handbook of Computer Vision Algorithms in Image Algebra, 2nd edn. (CRC Press, 2001). Google Scholar
56. G. X. Ritter , Image Algebra (Center for Computer Vision and Visualization, Dep. Computer and Information Science and Engineering, Univ. of Florida, Gainesville, FL, 2001). Google Scholar
57. A. Rosenfeld , Picture Languages. Formal Models for Picture Recognition (Academic Press, New York, San Francisco, London, 1979). Google Scholar
58. J. Serra , Image Analysis and Mathematical Morphology (Academic Press, London, 1982). Google Scholar
59. I. N. Sinitcyn , Calman and Pugachev Filters, 2nd edn. (Logos, Moscow, 2007) [in Russian]. Google Scholar
60. A. Shaw , A proposed language for the formal description of pictures, CGS Memo (Stanford University) 28 (1967). Google Scholar
61. M. Schlesinger and V. Hlavac , Ten Lectures on Statistical and Structural Pattern Recognition, Computational Imaging and Vision (Kluwer Academic Publishing, Dordrecht, Boston, London) Vol. 24 (2002). Google Scholar
62. P. Soille , Morphological Image Analysis. Principles and Applications, 2nd edn. (Springer-Verlag, Berlin, Heidelberg, New York, 2003, 2004). Google Scholar
63. S. R. Sternberg , Language and architecture for parallel image processing, in Proc. Conf. Pattern Recognition in Practice (North Holland, Amsterdam, 1980). Google Scholar
64. S. R. Sternberg , Grayscale morphology, Comput. Vision, Graphics Image Processing 35(3) (1986) 333–355. Google Scholar
65. P. Sussner , Observations on morphological associative memories and the kernel method, Neurocomput. 31 (2000) 167–183. Web of Science, Google Scholar
66. P. Sussner , Generalizing operations of binary autoassociative morphological memories using fuzzy set theory, J. Math. Imag. Vision 19 (2003) 81–93. Web of Science, Google Scholar
67. S. H. Unger , A computer oriented toward spatial problems, Proc. IRE 46 (1958) 1744–1750. Google Scholar
68. B. L. Van Der Waerden , Algebra I, Algebra II (Springer-Verlag, Berlin, Heidelberg, New York, 1971). Google Scholar
69. Yu. I. Zhuravlev , An algebraic approach to recognition and classification problems, Pattern Recognit. Image Anal. Adv. Math. Theory Appl. 8 (1998) 59–100. Google Scholar