Prolegomena toward Algebraic Image Analysis
Igor Gurevich and Vera Yashina
Dorodnicyn Computing Center, Russian Academy of Sciences, Vavilov str. 40
Moscow, 119333 Russian Federation
Abstract. The paper is an extended abstract of analytical tutorial devoted to
algebraization of image analysis.
1 Summary
Automation of image processing, analysis, estimating and understanding is one of the
crucial points of theoretical computer science having decisive importance for
applications, in particular, for diversification of solvable problem types and for
increasing the efficiency of problem solving. The main subgoals are developing and
applying of mathematical theory for constructing image models accepted by efficient
pattern recognition algorithms and for standardized representation and selection of
image analysis transforms. Automation of image-mining is possible by combined
application techniques for image analysis, understanding and recognition.
The specificity, complexity and difficulties of image analysis and estimation (IAE)
problems stem from necessity to achieve some balance between such highly
contradictory factors as goals and tasks of a problem solving, the nature of visual
perception, ways and means of an image acquisition, formation, reproduction and
rendering, and mathematical, computational and technological means allowable for
the IAE.
The mathematical theory of image analysis is not finished and is passing through a
developing stage. It is only recently came understanding of the fact that only
intensive creating of comprehensive mathematical theory of image analysis and
recognition (in addition to the mathematical theory of pattern recognition) could bring
a real opportunity to solve efficiently application problems via extracting from
images the information necessary for intellectual decision making. The transition to
practical, reliable and efficient automation of image-mining is directly dependent on
introducing and developing of mathematical means for IAE.
This work was partially supported by the Russian Foundation for Basic Research Grant 08-
01-00469, by the project “Algorithmic schemes of descriptive image analysis” of the
Program of Basic Research “Algebraic and Combinatorial Techniques of Mathematical
Cybernetics” of the Department of Mathematical Sciences of the RAS and by the project of
the Program of the Presidium of the Russian Academy of Sciences “Fundamental Problems
of Computer Science and Information Technologies” (the project no. 2.14)
Gurevich I. and Yashina V. (2009).
Prolegomena toward Algebraic Image Analysis.
In Proceedings of the 2nd International Workshop on Image Mining Theory and Applications, pages 3-8
DOI: 10.5220/0001963000030008
During recent years there was accepted that algebraic techniques, in particular
different kinds of image algebras, is the most prospective direction of construction of
the mathematical theory of image analysis and of development of an universal
algebraic language for representing image analysis transforms and image models.
The purposes of this tutorial are:
to set forth the state of the art of mathematical theory of image analysis;
to consider the algebraic approaches and techniques acceptable for image
to present a methodology, mathematical and computational techniques for
automation of image mining on the base of Descriptive Approach to Image Analysis
(DAIA) and to consider an example (automated diagnosis of hematological deceases).
The program of the tutorial is following:
Introduction «On a way to a unified theory»
The introduction presents a history of image analysis algebraization (M.Duff,
G.Matheron, J.Serra, J.von Neumann, S.Sternberg, S.Unger, and others).
Chapter 1 “State of the art of mathematical theory of image analysis”.
In the chapter a current state of the mathematical theory of image analysis is
Chapter 2 “The Algebraic Approaches and Techniques in Image Analysis.
Algebraization of Pattern Recognition (1970 – till now)”.
The chapter considers leading approaches in the mathematical theory of image
analysis oriented for automation of image analysis and understanding.
Chapter 2 consists of the two following sections: 2.1 «The Basic Theories»; 2.2
«Image Algebras».
Section 2.1 «The Basic Theories» describes three basic theoretical approaches in
the field of pattern recognition: 1) “Pattern Theory” (U.Grenander) – techniques for
data representation and transformation on the base of regular combinatorial structures
and algebraic and probabilistic means in pattern recognition; 2) “Theory of
Categories Techniques in Pattern Recognition” (M.Pavel) – formal descriptions of
pattern recognition algorithms via transforms of initial data preserving its class
membership; 3) “The Algebraic Approach to Recognition, Classification and
Forecasting Problems” (Yu.Zhuravlev) – mathematical set-up of a pattern recognition
problem, correctness and regularity conditions, multiple classifiers.
Section 2.2 «Image Algebras» contains a description of two known image
algebras: 1) Standard Image Algebra by G.Ritter – a unified algebraic representation
of image processing and analysis operations; 2) Descriptive Image Algebra by
I.Gurevich – a unified algebraic language for describing, performance estimating and
standardizing algorithms for image analysis, recognition and understanding.
Chapter 3 “Descriptive Approach to Image Analysis and Understanding
(DAIA) and its main tools”
The chapter describes basic concepts and mathematical tools of DAIA of
I.B.Gurevich and his school (conceptualization of notions for characterizing images
in pattern recognition problems; basic model of image recognition process;
descriptive image models).
Chapter consists of the three following sections: 3.1 “Descriptive Approaches – basic
steps”; 3.2 “DAIA”; 3.3 “Descriptive Image Models”.
Section 3.1 «Descriptive Approaches – basic steps» contains a description of
papers in the field of pattern and image recognition in 1960’s, which gives main
attention to a formal description of initial data, and to a formalization of description
procedures of their transforms (F.Ambler, G.Barrow, R.Burstall, T.Evans, S.Kaneff,
R.Kirsh, R.Narasimhan, A.Rosenfeld, A.Shaw).
Section 3.2 «DAIA» contains a description of basic concepts of DAIA. The main
intention of DAIA is to structure different techniques, operations and representations
being applied in image analysis and recognition. The axiomatic and formal
constructions of DAIA establishes conceptual and mathematical base for representing
and describing images and its analysis and estimation. The DAIA provides an
opportunity to solve the problems connected with the development of formal
descriptions for an image as a recognition object as well as the synthesis of
procedures for an image recognition and understanding. The analysis of the problems
is based on the investigation of inner structure and content of an image as a result of
the procedures “constructing” it from its primitives, objects, descriptors, features and
Section 3.3 «Descriptive Image Models (DIM)» contains a description of
mathematical objects providing representation of information carried by an image and
by an image legend (context) in a form acceptable for a recognition algorithm. This
section places high emphasis on multiple DIM and multi-aspect image
Chapter 4 “Example”
Chapter “Example” demonstrates application of the descriptive techniques in an
application problem - automating of morphologic analysis of cytological specimens
(lymphatic system tumors).
This chapter consists of the three following sections: 4.1 “Problem set-up”, 4.2
“Mathematical means used for formal representation of a descriptive model of an
information technology for early diagnostic analysis of cytological specimens”; 4.3
“Discussion of the results’.
• Conclusions
The last chapter “Conclusions” discusses open questions of the mathematical theory
of image analysis. Future researches for development of this field are outlined.
The program of the tutorial is briefly presented at Table 1.
Table 1. Tutorial summary.
Name of a Part Content
State of the art of mathematical theory of
image analysis.
2.1 The Basic Theories The Algebraic Approaches and Techniques
in Image Analysis. Algebraization of Pattern
Recognition (1970 – till now).
2.2 Image Algebras
Descriptive Approaches
Descriptive Approach to Image Analysis and
Understanding (DAIA) and its main tools
Descriptive Image Models
Problem set-up
Mathematical means used for formal
representation of a descriptive model of an
information technology for early diagnostic
analysis of cytological specimens
Discussion of the results
Figure 1 presents a classification reflecting the authors’ point of view on the
contemporary hierarchy of algebras and the place of DIA in this hierarchy.
Fig. 1. Algebraic links.
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