The Algebraic and Descriptive Approaches and

Techniques in Image Analysis

I. B. Gurevich, Yu. O. Trusova and V. V. Yashina

Dorodnicyn Computing Centre, Russian Academy of Sciences, Moscow, Russian Federation

Abstract. The main purpose of this review is to explain and discuss the

opportunities and limitations of algebraic, linguistic and descriptive approaches

in image analysis. 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 an universal algebraic language for representing

image analysis transforms and image models. So, the main goal of the

Algebraic Approach is designing of a unified scheme for representation of

objects under recognition and its transforms in the form of certain algebraic

structures. It makes possible to develop corresponding regular structures ready

for analysis by algebraic, geometrical and topological techniques. Development

of this line of image analysis and pattern recognition is of crucial importance

for automated image mining and application problems solving, in particular for

diversification classes and types of solvable problems and for essential

increasing of solution efficiency and quality.

1 Introduction

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 application problem types

and for increasing the efficiency of problem solving.

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 new mathematical means for IAE.

B. Gurevich I., O. Trusova Y. and V. Yashina V..

The Algebraic and Descriptive Approaches and Techniques in Image Analysis.

DOI: 10.5220/0004394300820093

In Proceedings of the 4th International Workshop on Image Mining. Theory and Applications (IMTA-4-2013), pages 82-93

ISBN: 978-989-8565-50-1

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

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.

Development of this line of image analysis and pattern recognition is of crucial

importance for automatic image-mining and application problems solving, in

particular for diversification classes and types of solvable problems and for essential

increasing of solution efficiency and quality.

It is one of the breakthrough challenges for theoretical computer science to find

automated ways to process, analyze, evaluate and understand information represented

in the form of images. It is critical for computer science to develop this branch in

terms of solving applied problems, in particular, increasing the diversity of classes of

problems that can be solved and the efficiency of the process significantly.

Images are one of the main tools to represent and transfer information needed to

automate the intellectual decision-making in many application areas. Increasing the

efficiency, including automatization, of gathering information from images can help

increase the efficiency of intellectual decision-making.

Recently, this part of image analysis called image mining in English publications

has been often set off into a separate line of research.

We list the functions of particular aspects of image handling. Image processing and

analysis provides for image mining, which is necessary for decision-making, while

the very decision-making is done by methods of mathematical theory of pattern

recognition. To link these two stages, the information gathered from the image after it

is analyzed is transformed so that standard recognition algorithms could process it

(we called this process “the reducing an image to a recognizable form” (RIRF)). Note

that although this stage seems to have an “intermediate” character, it is the

fundamental and necessary condition for the overall recognition to be feasible.

We need to develop and evolve a new approach to analyzing and evaluating

information represented in the form of images. To do it the “Algebraic Approach” of

Yu. I. Zhuravlev [43] was modified for the case when the initial information is

represented in the form of images. The result is the descriptive approach to image

analysis and understanding (DA) proposed and justified by I.B.Gurevich and

developed by his pupils [2, 6, 14-16].

In this work, we give a brief review of the main algebraic methods and features.

2 State of the Art of Mathematical Theory of Image Analysis

“State of the art of mathematical theory of image analysis” is the section that

describes modern trends in developing of mathematical tools for automation of image

analysis, in particular in image-mining.

To automate image mining, we need an integrated approach to leverage the

potential of mathematical apparatus of the main lines in transforming and analyzing

information represented in the form of images, viz. image processing, analysis,

recognition and understanding.

Done by pattern recognition methods, image mining now tends to multiplicity

(multialgorithmic and multimodel) and fusion of the results, i.e., several different

algorithms are applied in parallel to process the same model and several different

models of the same initial data to solve the problem and then the results are fused to

obtain the most accurate solution.

Multialgorithmic classifiers and multimodel and multiple-aspect image

representations are the common tools to implement this multiplicity and fusion. Note

that it was Yu. I. Zhuravlev who obtained the first and fundamental results in this area

in 1970s [43].

From 1970s, the most part of image recognition applications and considerable part

of research in artificial intelligence deal with images. As a result, new technical tools

emerged to obtain information that allow representing recorded and accumulated data

in the form of images and the image recognition itself became more popular as the

powerful and efficient methodology to process and analyze data mathematically and

detect hidden regularities. Various scientific and technical, economic and social

factors make the application domain of image recognition experience grow

constantly.

There are internal scientific problems that have arisen within image recognition.

First of all, these imply algebraizing the image recognition theory, arranging image

recognition algorithms, estimating the algorithmic complexity of the image

recognition problem, automating the synthesis of the corresponding efficient

procedures, formalizing the description of the image as the recognition object, making

the choice of the system of representations of the image in the recognition process

regular, and some others. It is these problems that form the basis of the mathematical

agenda of the descriptive theory of image recognition developed using the ideas of the

algebraic approach to recognition [43] to create a systematized set of methods and

tools of data processing in image recognition and analysis problems.

There are three main issues one need to solve when dealing with images–describe

(simulate) images; develop, study and optimize the selection of mathematical methods

and tools of data processing in image recognition; and implement mathematical

methods of image analysis on a software and hardware basis.

3 Algebraization of Pattern Recognition and Image Analysis

(1970 – Till Now)

This section contains steps of the algebraization in image analysis, fundamentals and

the basic theories of pattern recognition, different image algebras. Some words are

concerned with contribution of the Russian mathematical school.

3.1 Fundamentals and the Basic Theories in Pattern Recognition

By now, image analysis and evaluation have a wide experience gained in applying

mathematical methods from different sections of mathematics, computer science and

physics, in particular algebra, geometry, discrete mathematics, mathematical logic,

probability theory, mathematical statistics, mathematical analysis, mathematical

theory of pattern recognition, digital signal processing, and optics.

On the other hand, with all this diversity of applied methods, we still need to have

a regular basis to arrange and choose suitable methods of image analysis, represent, in

an unified way, the processed data (images), meeting the requirements standard

recognition algorithms impose on initial information, construct mathematical models

of images designed for recognition problems, and, on the whole, establish the

universal language for unified description of images and transformations over them.

In applied mathematics and computer science, constructing and applying

mathematical and simulation models of objects and procedures used to transform

them is the conventional method of standardization. It was largely the necessity to

solve complex recognition problems and develop structural recognition methods and

specialized image languages that generated the interest in formal descriptions–models

of initial data and formalization of descriptions of procedures of their transformation

in the area of pattern recognition (and especially in image recognition in 1960s).

As for the substantial achievements in this “descriptive” line of study, we mention

publications by A. Rosenfeld [34], T. Evans [12], R. Narasimhan [29], R. Kirsh [21],

A. Shaw [37], H. Barrow, A. Ambler, and R. Burstall[1], S. Kaneff [20].

In 1970s Yu. I. Zhuravlev proposed the so called “Algebraic Approach to

Recognition and Classification Problems” [42], where he defined formalization

methods for describing heuristic algorithms of pattern recognition and proposed the

universal structure of recognition algorithms. In the same years, U. Grenander stated

his “Pattern Theory” [18], where he considered methods of data representation and

transformation in recognition problems in terms of regular combinatorial structures,

leveraging algebraic and probabilistic apparatus. M.Pavel [31] was introduced

“Theory of Categories Techniques in Pattern Recognition”, that is formal describing

of pattern recognition algorithms via transforms of initial data preserving its class

membership.

Then, up to the middle of 1990s, there was a slight drop in the interest in

descriptive and algebraic aspects in pattern recognition and image analysis.

Also we present a very brief description of the most important original results on

algebraic tools for pattern recognition and image analysis in Russian mathematical

school. There are algebras on algorithms, algebraic multiple classifiers, algebraic

committees of algorithms, combinatorial algorithms for recognition of 2-D data[1],

descriptive image models, 2-D formal grammars [34].

In the framework of scientific school of Yu.I.Zhuravlev several essential results

were obtained in algebraic direction by V.L.Matrosov [26], by K.V.Rudakov [35] and

V.D.Mazurov [27].

Algebraic method of analysis and estimation of information represented as signals.

Apart from basic researches of Yu.I.Zhuravlev scientific school there are significant

number of papers concerned with algebraic methods of analysis and estimation of

information represented as signals, in partially V.G.Labunec [22], Ya.A.Furman [13],

V.M.Chernov [4].

3.2 Steps of the Algebraization

The section presents leading approaches of mathematical theory for image analysis

oriented tor automation of image analysis and understanding. First of all there is the

history of developing algebraic construction for image analysis and processing –

formal grammars, cellular automata, mathematical morphology, image algebras,

multiple algorithms, descriptive approach.

Algebraization of pattern recognition and image analysis has attracted and

continues to attract the attention of many researchers. Appreciable attempts to create a

formal apparatus ensuring a unified and compact representation for procedures of

image processing and image analysis were inspired by practical requirements for

effective implementation of algorithmic tools to process and analyze images on

computers with specialized architectures, in particular, cellular and parallel.

The idea of constructing a unified language for concepts and operations used in

image processing appeared for the first time in works by Unger [42], who suggested

to parallelize algorithms for processing and image analysis on computers with cellular

architecture.

Mathematical morphology, developed by G. Matheron [25] and Z. Serra [36],

became a starting point for a new mathematical wave in handling and image analysis.

Serra and Sternberg [39] were the first to succeed in constructing an integrated

algebraic theory of processing and image analysis on the basis of mathematical

morphology. It is believed [28] that it was precisely Sternberg who introduced the

term “image algebra” in the current standard sense. (We note that U. Grenander used

this concept in the 1970s; however, he was talking about another algebraic

construction [18]). Within the limits of this direction, an array of works continues to

be written, devoted to the development of specialized algebraic constructions

implementing or improving upon methods of mathematical morphology.

From that time till 1990’s the interest to descriptive and algebraic aspects of image

analysis is failing. The final view of idea of IA has become Standard Image Algebra

by G.Ritter [32] (algebraic presentation of image analysis and processing operations).

DIA is created as a new IA provided possibility to operate with main image models

and with basic models of procedure of transforms, which lead to effective synthesis

and realization of basic procedures of formal image description, processing, analysis

and recognition. DIA is introduced by I.B.Gurevich and developed by him and his

pupils [14-16].

The history of algebraization: J.von Neumann [30], S.Unger [42] (studies of

interactive image transformations in cellular space); M. Duff, D. Watson, T. Fountain,

and G. Shaw [10] (a cellular logic array for image Processing); A. Rosenfeld [33]

(digital topology); H. Minkowski and H.Hadwiger (pixel neighborhood arithmetic and

mathematical morphology); G.Matheron, J.Serra, S.Sternberg [25, 36, 39] (a coherent

algebraic theory specifically designed for image processing and image analysis -

mathematical morphology); S. Sternberg [39] (the first to use the term “image

algebra”); P. Maragos [24] (introduced a new theory unifying a large class of linear

and nonlinear systems under the theory of mathematical morphology); L. Davidson

[9] (completed the mathematical foundation of mathematical morphology by

formulating its embedding into the lattice algebra known as Mini-Max algebra);

G.Ritter [32] (Image Algebra); I.B.Gurebich [15] (Descriptive Image Algebra); T.R.

Crimmins and W.M. Brown, R.M. Haralick, L. Shapiro, R.W. Schafer, J. Goutsias,

L. Koskinen and Jaako Astola, E.R. Dougherty, P.D. Gader, M.A. Khabou, A.

Koldobsky, B. Radunacu, M.Grana, F.X. Albizuri, P. Sussner [8, 7, 10, 11, 19, 40]

(recent papers on mathematical morphology and image algebras).

4 Descriptive Approach to Image Analysis and Understanding

This section contains a brief description of the principal features of the DA needed to

understand the meaning of the introduction of the conceptual apparatus and schemes

of synthesis of image models proposed to formalize and systematize the methods and

forms of image representation.

By the middle of 1990s, it became obvious that for the development of image

analysis and recognition, it is critical to: 1) understand the nature of the initial

information – images, 2) find methods of image representation and description that

allow constructing image models designed for recognition problems, 3) establish the

mathematical language designed for unified description of image models and their

transformations that allow constructing image models and solving recognition

problems, and 4) construct models to solve recognition problems in the form of

standard algorithmic schemes that allow, in the general case, moving from the initial

image to its model and from the model to the sought solution.

The DA gives a single conceptual structure that helps to develop and implement

these models and the mathematical language [14-16]. The main DA purpose is to

structure and standardize different methods, operations and representations used in

image recognition and analysis. The DA provides the conceptual and mathematical

basis for image mining, with its axiomatic and formal configurations giving the ways

and tools to represent and describe images to be analyzed and evaluated.

The automated extraction of information from images includes (1) automating the

development, testing, and adaptation of methods and algorithms for the analysis and

evaluation of images; (2) the automation of the selection of methods and algorithms

for analyzing and evaluating images; (3) the automation of the evaluation of quality

and adequacy of the initial data for solving the problem of image recognition; and (4)

the development of standard technological schemes for detecting, assessing,

understanding, and retrieving images.

The automation of information extraction from images requires combined use of

all features of the mathematical apparatus used or potentially suitable for use in

determining transformations of information provided in the form of images, namely in

problems of processing, analysis, recognition, and understanding of images.

Experience in the development of the mathematical theory of image analysis and

its use to solve applied problems shows that, when working with images, it is

necessary to solve problems that arise in connection with the three basic issues of

image analysis, i.e., (1) the description (modeling) of images; (2) the development,

exploration, and optimization of the selection of mathematical methods and tools for

information processing in the analysis of images; and (3) the hardware and software

implementation of the mathematical methods of image analysis.

The main purpose of the DA is to structure and standardize a variety of methods,

processes, and concepts used in the analysis and recognition of images. The DA is

proposed and developed as a conceptual and logical basis of the extraction of

information from images. This includes the following basic tools of analysis and

recognition of images: a set of methods of analysis and recognition of images, BFSR

techniques, conceptual system of analysis and recognition image, DIM classes, the

DIA language, statement of problems of analysis and recognition of images, and the

basic model of image recognition.

The main areas of research within the DA are (1) the creation of axiomatics of

analysis and recognition of images, (2) the development and implementation of a

common language to describe the processes of analysis and recognition of images (the

study of DIA), and (3) the introduction of formal systems based on some regular

structures to determine the processes of analysis and recognition of images (see [14-

17]).

Mathematical foundations of the DA are as follows: (1) the algebraization of the

extraction of information from images, (2) the specialization of the Zhuravlev algebra

to the case of representation of recognition source data in the form of images, (3) a

standard language for describing the procedures of the analysis and recognition of

images (DIA) [14-16], (4) the mathematical formulation of the problem of image

recognition, (5) mathematical theories of image analysis and pattern recognition, and

(6) a model of the process for solving a standard problem of image recognition. The

main objects and means of the DA are as follows: (1) images; (2) a universal language

(DIA); (3) two types of descriptive models, i.e., (a) an image model and (b) a model

for solving procedures of problems of image recognition and their implementation;

(4) descriptive algebraic schemes of image representation (DASIR); and (5)

multimodel and multiaspect representations of images, which are based on generating

descriptive trees (GDT) [14-16].

The basic methodological principles of the DA are as follows: (1) the

algebraization of the image analysis, (2) the standardization of the representation of

problems of analysis and recognition of images, (3) the conceptualization and

formalization of phases through which the image passes during transformation while

the recognition problem is solved, (4) the classification and specification of

admissible models of images (descriptive image model - DIM), (5) RIRF, (6) the use

of the standard algebraic language of DIA for describing models of images and

procedures for their construction and transformation, (7) the combination of

algorithms in the multialgorithmic schemes, (8) the use of multimodel and

multiaspect representations of images, (9) the construction and use of a basic model

of the solution process for the standard problem of image recognition, and (10) the

definition and use of nonclassical mathematical theory for the recognition of new

formulations of problems of analyzing and recognizing images.

Note that the construction and use of mathematical and simulation models of

studied objects and procedures used for their transformation is the accepted method of

standardization in the applied mathematics and computer science.

The creation of the DA was significantly influenced by the following basic theories

of pattern recognition: (1) the algebraic approach to pattern recognition of Zhuravlev

and their algorithmic algebra [43] and (2) the theory of images of Grenander [18], in

particular algebraic methods for the representation of source data in image recognition

problems developed in it.

As already noted, in the DA, it is proposed to carry out the algebraization of the

analysis and recognition of images using DIA. DIA was developed from studies in the

field of the algebraization of pattern recognition and image analysis carried out since

the 1970s. The creation of a new algebra was directly influenced by algorithms of

Zhuravlev [43] and the research of Sternberg [39] and Ritter [32], which identified

classic versions of image algebras.

A more detailed description of methods and tools of the DA obtained in the

development of its results can be found in [14-16].

5 Ontology-based Approach to Image Analysis

This section briefly describes the use of ontologies for representation of knowledge

needed to support intellectual decision making in image analysis and understanding

tasks.

The automation of image analysis assumes that researchers and users of different

qualifications have at their disposal not only a standardized technology of automation,

but also a system supporting this technology, which accumulates and uses knowledge

on image processing, analysis and evaluation and provides adequate structural and

functional possibilities for supporting the more intelligent choice and synthesis of

methods and algorithms. The automated system (AS) for image analysis must combine

the possibilities of the instrumental environment for image processing and analysis

and a knowledge-based system. Therefore, one of its main components is a

knowledge base. Knowledge bases usually contain modules of universal knowledge,

which are not related to any subject domain (knowledge necessary for scheduling and

control of the processing, result mappings, estimation of the processing quality, object

recognition, and conflict resolution, as well as knowledge about methods of image

processing and analysis) and knowledge modules related to a certain subject domain

(segmentation strategies, object descriptions, and specialized strategies for feature

extraction and object identification). The AS must provide software implementation

of the hierarchies of classes of the main objects used in image analysis, have a

specialized user interface, contain a library of algorithms that allow one to solve the

main problems of image analysis and understanding with the help of efficient

computational procedures, and provide accumulation and structuring of knowledge

and experience in the domain of image analysis and understanding.

The need of efficient knowledge representation facilities can be fulfilled by using a

suite of ontologies. Ontologies as an effective way for knowledge representation

became very popular last years. Ontological knowledge representation has the

following advantages: (1) it provides the opportunity to establish a common

understanding of the considered field of knowledge, (2) it enables us to represent

knowledge in a convenient form for processing by automated information processing

and analysis systems, and (3) it provides the opportunity for acquisition and

accumulation of new knowledge and for multiple use of knowledge.

Different works related to usage of ontologies for solving image-based tasks have

been reported. For example, in [23], an approach devoted to semantic image

interpretation for complex object classification purposes is proposed. The described

framework is focused on the mapping between domain knowledge and image

processing knowledge. In the proposed knowledge acquisition methodology, the

mapping between the domain and the image is based on a visual concept ontology

composed of three different types of visual concepts - texture concepts, color concepts

and spatial concepts - associated with numerical descriptors useful for performing

object recognition. In [3], an integrated knowledge infrastructure and annotation

environment for multimedia description, analysis and reasoning is described. The

framework comprises ontologies for the description of low-level audio-visual features

and for linking these descriptions to concepts in domain ontologies based on a

prototype approach. The key idea is to associate domain concepts with instances that

serve as prototypes for these concepts. The proposed infrastructure includes a set of

ontologies including the core ontology, Visual Descriptor Ontology (VDO),

Multimedia Structure Ontology (MSO) and specific domain ontologies. The core

ontology serves as a basis for all components of the knowledge infrastructure. The

VDO represents the structure of the MPEG-7 visual part and models concepts and

properties that describe visual characteristics of objects. The MSO models basic

multimedia entities from the MPEG-7 Multimedia Description Scheme. Domain

ontologies model the content layer of multimedia content with respect to specific real-

world domains. The work described in [5] addresses the problem of explicit

representation of objectives when developing image processing applications. Authors

investigated kinds of information needed to design and evaluate image processing

software programs and proposed an ontology-based model for formulation of user

objectives. In [41], a cognitive architectural model for image and video interpretation

is discussed. The proposed framework demonstrates that ontology-based content

representation can be used as an effective way for hierarchical and goal-directed

inference in high-level visual analysis tasks.

In [6], a novel knowledge-oriented approach to image analysis based on the use of

thesauruses and ontologies as tools for representation of knowledge, which are

necessary for making intelligent decisions on the basis of information extracted from

images, is proposed. The main contribution of this work is the development of a

sufficiently detailed and well-structured Image Analysis Ontology (IAO) needed for

solving the following tasks: 1) construction of unified description and representation

of image-based tasks and methods for solving these tasks; 2) automation of image

analysis methods combination on the base of semantic integration; 3) automation of

navigation and retrieval in knowledge bases on image analysis. As a main source of

the information about concepts (including term definitions and basic relationships

between terms) the Image Analysis Thesaurus (IAT) [2] has been used. The important

feature of the IAT is a novel hierarchical classification of tasks and algorithms for

image processing, analysis and recognition. The following IAO classes were defined:

Task, Method, Data, Context and Requirements. The hierarchy of

subclasses is based on term relations fixed in the IAT. By integrating the ontology

with algorithms for image processing, analysis and understanding, high-level

semantic information can be extracted from images.

6 Conclusions

Note that the idea to create a single theory that embraces different approaches and

operations used in image and signal processing has a history of its own, with works of

von Neumann continued by S. Unger, M. Duff, G. Matheron, G. Ritter, J. Serra, S.

Sternberg and others [30, 10, 25, 32, 36, 39] playing an important role in it.

The main stages of algebraization are:

 Mathematical Morphology (G. Matheron, J. Serra [1970’s])

 Algorithm Algebra by Yu.I.Zhuravlev (Yu. Zhuravlev [1970’s]

 Pattern Theory (U. Grenander [1970’s]

 Theory of Categories Techniques in Pattern Recognition (M.Pavel [1970’s])

 Image Algebra (Serra, Sternberg [1980’s]

 Standard Image Algebra (Ritter [1990’s])

 Descriptive Image Algebra (DIA) (Gurevich [1990-2000])

 DIA with one king (Gurevich, Yashina [2001 to date])

Analyzing the existing algebraic apparatus, we came to the statement of the

following requirements on the language designed for recording algorithms for solving

problems of image processing and understanding: 1) the new algebra must make

possible processing of images as objects of analysis and recognition; 2) the new

algebra must make possible operations on image models, i.e., arbitrary formal

representations of images, which are objects and, sometimes, a result of analysis and

recognition; introduction of image models is a step in the formalization of the initial

data of the algorithms; 3) the new algebra must make possible operations on main

models of procedures for image transformations; 4) it is convenient to use the

procedures for image modifications both as operations of the new algebra and as its

operands for construction of compositions of basic models of procedures.

Acknowledgements

This work was supported in part by the Russian Foundation for Basic Research

(projects nos. 11-01-00990, 12-07-31123) and by the Presidium of the Russian

Academy of Sciences within the program “Fundamental Science to Medicine” as well

as within the program “Information, Control, and Intelligent Technologies and

Systems” (project no. 204) and the program of the Division of Computer Sciences,

Russian Academy of Sciences “Algebraic and Combinatorial Methods of

New_Generation Mathematical Cybernetics and Information Systems” (the project

“Algorithmic Schemes of Descriptive Image Analysis”).

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