Current Trends in Mathematical Image Analysis

A Survey

Igor Gurevich and Vera Yashina

Mathematical and Applied Problems of Image Analysis, Dorodnicyn Computing Center of the Russian Academy of

Sciences, Moscow, Russian Federation

Keywords: Algebraic Approach, Descriptive Approach, Image Analysis.

Abstract: The main task of the survey is to explain and discuss the opportunities and limitations of algebraic

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. 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 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.

1 INTRODUCTION

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.

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.

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

analysed is transformed so that standard recognition

algorithms could process it. 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.

At present, automated image mining is the main

strategic goal of fundamental research in image

analysis, recognition and understanding and

development of the proper information technology

and algorithmic software systems.

To ensure such automatization, we need to

develop and evolve a new approach to analysing and

evaluating information represented in the form of

images. To do it, the “Algebraic Approach” of Yu. I.

Gurevich I. and Yashina V..

Current Trends in Mathematical Image Analysis - A Survey.

DOI: 10.5220/0005461800580070

In Proceedings of the 5th International Workshop on Image Mining. Theory and Applications (IMTA-5-2015), pages 58-70

ISBN: 978-989-758-094-9

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

Zhuravlev (Zhuravlev, 1998) 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 (Gurevich, 1989, 1991,

2005), (Gurevich, Jernova, 2003), (Gurevich,

Koryabkina, 2006), (Gurevich, Yashina, 2006, 2008,

2012), (Gurevich et al., 2008).

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, and 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 1970s, Yu. I. Zhuravlev proposed the so-

called “Algebraic Approach to Recognition and

Classification Problems” (Zhuravlev, 1998), 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” (Grenander, 1976, 1978, 1981,

1993, 1996), where he considered methods of data

representation and transformation in recognition

problems in terms of regular combinatorial

structures, leveraging algebraic and probabilistic

apparatus. Both approaches dealt with the

recognition problem in its classical statement and

did not touch upon representation of initial data in

the form of images.

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.

By the middle of 1990s, it became obvious that

for the development of image analysis and

recognition, it is critical to: understand the nature of

the initial information – images, find methods of

image representation and description that allow

constructing image models designed for recognition

problems, establish the mathematical language

designed for unified description of image models

and their transformations that allow constructing

image models and solving recognition problems, and

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 develop and implement these models and the

mathematical language (Gurevich, 1989, 1991). 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 analysed and evaluated.

In this work, we give a brief review of the main

algebraic methods and features. The work consists

of seven main sections (along with Introduction and

Conclusions).

“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.

The section “Steps of the algebraization”

presents leading approaches of mathematical theory

for image analysis oriented for automation of image

analysis and understanding.

The section “The algebraic approach to

recognition classification and forecasting problems

by Zhuravlev” contains main aspects of algebraic

theory of Yu.I.Zhuravlev.

The section “Image Algebras” consists of brief

description of different image algebras.

The section “Descriptive approach to image

analysis” presents a methodology, mathematical and

computational techniques for automation of image

mining on the base of Descriptive Approach to

Image Analysis.

In conclusion, there are some words about

opportunities of algebraic techniques via an example

of biomedical image analysis practical problem and

discussion the prospects of the mathematical image

analysis development.

2 STATE OF THE ART OF

MATHEMATICAL THEORY OF

IMAGE ANALYSIS

To automate image mining, we need an integrated

CurrentTrendsinMathematicalImageAnalysis-ASurvey

approach to leverage the potential of mathematical

apparatus of the main lines in transforming and

analysing 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 (Zhuravlev, 1998).

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, analyse 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, 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 the 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 (Zhuravlev, 1998)

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.

What makes image analysis and recognition

problems peculiar, complex and thus difficult and

catching is the necessity to find a compromise

between rather contradictory factors. These factors

are the requirements imposed on the analysis, the

nature of visual perception, the ways to obtain, form

and reproduce images and the existing mathematical

and technical ways to process them. The main

contradiction is between the nature of the image and

the analysis based on formal description (a model, in

essence) of the object, which lies in the fact that to

leverage the fact that information is represented in

the form of images, it is necessary to make this

information non-depictive since the corresponding

algorithms can only process certain symbolic

descriptions.

Most methods of image processing are purely

heuristic, with their quality essentially given by the

degree to which they are successful in coping with

the “depictive” nature of the image using the “non-

depictive” tools, i.e., in employing procedures that

do not depend on the fact that the information to be

processed is organized in the form of images.

When we solve an image recognition problem, it

is very important that we are able to choose the right

recognition algorithm in a great number of known

algorithms, i.e., we need to choose the best in some

sense algorithm in the particular situation. It is

obvious that both in image recognition and in

solving recognition problems with standard teaching

information (Zhuravlev, 1998), to make the choice

of the best algorithm systematic, we need to

introduce and formalize the corresponding objects of

mathematical theory of image recognition, in

particular, the concept of image recognition

algorithm.

It is known that the necessity to state and solve

the problem of choosing the algorithm with respect

to the recognition quality functional led to

introducing the concept of the model of recognizing

algorithm. To choose optimal or acceptable

procedure to solve the particular problem, one

needed to fix the class of algorithms somehow. This

is the first reason that led to the necessity to

synthesize models of recognition algorithms.

With the concept of the model of recognizing

algorithm, we can apply strict mathematical methods

to study the sets of incorrect recognition procedures

(i.e., heuristic procedures that are not justified

mathematically but were experimentally tested in

solving real recognition problems). Analysing the

totality of incorrect recognition algorithms as they

are accumulated, we can select and describe

IMTA-52015-5thInternationalWorkshoponImageMining.TheoryandApplications

particular algorithms as well as principles to form

them. Acting over subsets of algorithms and first

formed in a poorly formalized form, these principles

can then become accurate mathematical

descriptions. At this stage, principles are chosen on a

heuristic basis while algorithms generated according

to it can be constructed in a standard way. It is in

this sense that formalization of different principles

of constructing recognizing algorithms results in

models of recognizing algorithms.

To construct the model of recognizing algorithm,

we need to describe sets of incorrect procedures that

nevertheless are efficient in solving practical

problems in a uniform way. To give such set, we

specify variables, objects, functions, and parameters

and their exact variation area, thus introducing the

sought model of the algorithm. Given some set of

the corresponding variables, objects, parameters and

types of functions, we can single out some fixed

algorithm from the model we consider.

To construct the model of an image recognition

algorithm and determine the proper class of

recognition algorithms, it is not enough to transfer

the concept of the model of recognizing algorithm

developed in the mathematical recognition theory

automatically to the image domain and directly use

formal representations of a number of known

recognition models studied in classical recognition

theory (Zhuravlev, 1998). As noted above, the

nature and matter of image recognition problems

differ from that of the mathematical recognition

theory in its classical statement. When we move

from classical recognition problems to image

recognition problems, there arise mathematical

problems due to formal description of the image as

the object to be analysed.

To obtain formal descriptions of images as

objects to be analysed and form and choose

recognition procedures, we study the internal

structure and content of the image as the result of the

operations that can be performed to construct it of

sub-images and other objects of simpler nature, i.e.,

primitives and objects singled out on the image

during different stages of handling it (depending on

the aspect, morphological and/or scale level used to

form the image model). Since this way of

characterizing the image is operational, we can

consider the whole process of image processing and

recognition, including construction of formal

description –model of the image, as a system of

transformations implemented on the image and

given on the equivalence classes that represent

ensembles of admissible images (Gurevich, Yashina,

2006). Hence, we operate with the hierarchy of

formal descriptions of images, i.e., image models

used in recognition relate to different aspects and/or

morphological (scale) levels of image

representation. In essence, these are multiple-aspect

and/or multilevel models that allow choosing and

changing the necessary degree of detail of

description of the recognition object in the course of

solving the problem. This approach to formal

description of images forms the basis for the

multimodel representation of images in recognition

problems.

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. Ritter, J. Serra, S. Sternberg and

others (Under, 1958), (Duff et al., 1973), (Ritter,

2001), (Serra, 1982), (Sternberg, 1985) 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 ring (Gurevich, Yashina [2001

to date]).

3 ZHURAVLEV ALGEBRAIC

APPROACH

“The Algebraic Approach to Recognition,

Classification and Forecasting Problems”

(Yu.Zhuravlev) (Zhuravlev, 1998) is mathematical

set-up of a pattern recognition problem, correctness

and regularity conditions, multiple classifiers.

One of the topical problems in image recognition

is searching for an algorithm that would provide a

correct classification of an image by its description

(i.e. the algorithms that produce zero errors on a

control set of objects). The approach to image

recognition that is developed by the present authors

is a specialization of the algebraic approach to

recognition and classification problems originally

relational for this approach is the fact that there are

CurrentTrendsinMathematicalImageAnalysis-ASurvey

no accurate mathematical models for weakly

formalized fields such as geology, biology,

medicine, and sociology. However, in many cases,

inexact methods based on heuristic considerations

are practically effective. Therefore, it is sufficient to

construct a family of such heuristic algorithms for

solving appropriate problems and then construct the

algebraic closure of this family. The existence

theorem has been proved, which states that any

problem among the set of problems associated with

the study of poorly formalized situations is solvable

in this closure.

Suppose we give a certain set of admissible

patterns described by n-dimensional vectors of

features. The set of admissible patterns is covered by

a finite number of subsets, called classes. Let there

exist l classes K

, …, K

. There is a recognition

algorithm A that constructs an l-dimensional

information vector by an n-dimensional description

vector. Recall that an information vector is the

vector of membership of an object in the classes in

which the values of elements of the information

vector 0, 1, Δ are interpreted, according to

(Zhuravlev, 1998), as “an object does not belong to

the class,” “an object belongs to the class,” and “the

algorithm cannot determine whether or not an object

belongs to the class.” We will assume that each

recognition algorithm A ∈ {A} can be represented as

a sequential execution of algorithms B and C, where

B is the recognition operator that transforms learning

information and the description of an admissible

object into a numerical vector, called the estimate

vector, and C is the decision rule that transforms an

arbitrary numerical vector into an information

vector.

The operation of the recognition algorithm can

be schematically represented as follows.

Feature description of an object α = (α

, α

, …,

)

↓ Recognition algorithm B

Vector of estimates for a class β = (β

, β

, …, β

)

↓ Decision rule C

Information vector γ = (γ

, γ

, …, γ

Thus, during the solution of a recognition

problem, the object of recognition, i.e., an image, is

described by three different vectors: the n-

dimensional vector of features, the l-dimensional

vector of estimates for a class, and the l-dimensional

information vector.

Let us briefly recall the pattern recognition

problem in the standard statement that was

formulated by Zhuravlev.

Z(I

, S

, …, S

, P

, …, P

) is a recognition

problem, where I

is admissible initial information;

, …, S

is the set of admissible objects described

by feature vectors; K

, …, K

is a set of classes; and

, …, P

is a set of predicates on the admissible

objects, P

= P

(S), i =1, 2, …, l. Problem Z consists

in finding the values of the predicates P

, …, P

Definition. An algorithm is said to be correct for

problem Z if the following equality holds:

()

,,, ,, ,

qlij

AIS S P P

=

, where α

One of the main tasks of pattern recognition is

searching for an algorithm that correctly solves the

image recognition problem. Zhuravlev proves the

existence theorem for such an algorithm stating that

the algebraic closure of AECs for the image

recognition problem is correct. AECs are based on

the formalization of the concepts of precedence or

partial precedence: an algorithm analyzes the

proximity between the parts of descriptions of earlier

classified objects and the object to be recognized.

Suppose we are given standard descriptions of

objects

{

}

SSK∈



and

{

}

SSK

′′

∉

, and a

method for determining the degree of proximity

between certain parts of the description of S and the

corresponding parts of the descriptions

()

{

}

()

{

}

,IS IS

′



; S, j=1,2,…,l, is the object of

recognition. Calculating estimates for the proximity

between the parts of the descriptions

()

{

}



and

()

{

}

′

and, respectively, between

()

and

()

′

, one can construct a generalized estimate for

the proximity between S and the sets of objects

{

}

{}

,SS

′



(in the simplest case, the generalized

estimate is equal to the sum of estimates for the

proximity between the parts of descriptions). Then,

using the set of estimates, one forms a general

estimate of an object over a class, which is precisely

the value of the membership function of the object in

the class.

For the algebraic closure of the AECs, the

following existence theorem for an AEC is proved,

which correctly solves recognition problem Z.

Theorem. Suppose that natural assumptions on

the difference between the descriptions of classes

and recognition objects hold for the vectors of

features in recognition problem Z. Then the

algebraic closure of the class of AECs is correct for

problem Z.

The image recognition problem is one of the

classical examples of problems with incompletely

IMTA-52015-5thInternationalWorkshoponImageMining.TheoryandApplications

formalized and partially contradictory data. This

suggests that the application of an algebraic

approach to image recognition may lead to important

results; hence, the “algebraization” of this field is the

most promising approach for development of the

required mathematical apparatus for the analysis and

estimation of information represented by images.

When the recognition objects are images, this

theorem cannot be directly applied. There are

several reasons for this. First, representation of an

image by a vector of features (as in the case of a

standard recognition object) often leads to the loss of

a considerable part of the information about the

image and, consequently, to an incorrect

classification. Second, the existence of equivalence

classes is an essential difference between the image

recognition problem and the recognition problem in

the classical formulation.

Passage from the algebra of pattern recognition

algorithms to an algebra of image recognition

algorithms requires a choice, first, of algorithms

used as elements of algebra, and second, of algebraic

representations of images that make it possible to

formalize the task of choosing descriptors. It is

expedient to select representations taking into

account the possibility of combining the initial

information and algorithms of different types. For

the first time, the idea of a combination of qualifiers

with optimization of their operation by algebraic

correction was suggested and justified by Yu.I.

Zhuravlev. The complex of mathematical methods

related to synthesis and research of such qualifiers is

known under the common title “Algebraic Approach

to Tasks of Recognition, Classification, and

Prediction.” In the English-Language literature for

the designation of qualifiers, the term Multiple

Classifiers (Winbridge, Kittler, 2001) is used.

Recently, quite interesting results have been

achieved in the field of theoretical-informational

analysis of combined qualifiers (Grin et al., 2001),

developments of specific strategies for merging

algorithms (Kittler, Alkoot, 2001), and usage of

methods of code theory in tomography (Tax, Duin,

2001).

Image analysis and understanding have a certain

peculiarity, due to which the use of the Zhuravlev

algebraic approach in the general form is

inconvenient. The reasons are the following:

• the character of the considered problem is not

taken into account if algebraic methods are applied

to the information represented in the form of images;

• the results of application of the theory cannot

always be simply interpreted;

• there are many natural transformations of

images which are easily interpreted from the user’s

point of view (for instance, rotation, contraction,

stretching, colour inversion, etc.) but are hardly

representable by standard algebraic operations.

The necessity arises of using algebraic tools to

record natural transformations of images. Moreover,

the algebraization of image analysis and

understanding must include the construction of

algebraic descriptions of both the images themselves

and algorithms for their processing, analysis, and

recognition.

Analysing the publications related to

applications of algebraic methods to image analysis

and understanding, we distinguish the following

advantages of unified representation of images and

algorithms for their processing and analysis:

• construction of unified representations for

descriptions of images;

• efficiency of transition from input data in the

form of images to different formal models of the

images;

• naturalness of uniting the algebraic

representation of the information with the developed

algebraic tools for pattern recognition, which has

been successfully

• employed;

• the possibility of using the methods of

mathematical modelling employed in applied

domains to which the processed images belong;

• the possibility of using the image descriptions

in the form of group-theoretic representations;

• naturalness of uniting the methods of

structural analysis of images with tools of

probabilistic analysis;

• the possibility of a formalized description for

problems of parallelizing with due regard for the

specifics of particular computational architectures.

The “algebraic approach” to solve the tasks of

classification and/or pattern recognition was

developed in the school of Yu. Zhuravlev starting

from 1960s as means to build the correct algorithms

(i.e. the algorithms that produce zero errors on a

control set of objects) over specified sets of features.

Within the framework of the algebraic approach, the

algorithms are built as compositions of type where

A is the entire algorithm, B is an operator “base

classifier” that maps the feature space into a matrix

of estimates of the assignments of the objects’

classes, C is the “decision rule” operator that maps

the matrix of estimates into binary matrix of the

answers of the entire algorithm A.

In the framework of scientific school of

Yu.I.Zhuravlev several essential results were

obtained in algebraic direction by V.L.Matrosov

CurrentTrendsinMathematicalImageAnalysis-ASurvey

(Matrosov, 1985), (Khachai, 2010), by

K.V.Rudakov (Rudakov, 1987, 1988), (Rudakov,

Vorontsov, 1999), (Rudakov, Chekhovich, 2005,

2007) and V.D.Mazurov (Mazurov, 1971),

(Mazurov, Khachai, 2007).

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 (Labunec, 1984),

Yu.P.Pityev (Pityev, 2004), I.N.Sinicyn (Sinicyn,

2007), Ya.A.Furman (Furman, 2009), (Furman et al.,

2012), V.M.Chernov (Chernov, 2001, 2007),

(Felsberg et al., 2000).

4 IMAGE ALGEBRA

Mathematical morphology (Serra, 1982), (Soille,

1996, 2003, 2004), (Sternberg, 1980, 1985, 1986),

proposed by Minkowski and Hadwiger and

developed by Matheron and Serra, seems to be the

first attempt to create a theoretical apparatus that

allows one to describe many widespread operations

of image processing in the composition of a rather

small set of standard simple local operations. Such

representations allow one to formalize the choice of

procedures for image processing and are convenient

for implementation on parallel architectures. It might

have been the success of mathematical morphology

that initiated numerous attempts of algebraization

both in the domain of algorithm representations and

in closed domains. Mathematical morphology is an

efficient tool for uniform representation of local

operations of image processing, analysis, and

understanding in terms of algebras over sets. It

makes it possible to describe algorithms for image

transformations in terms of four basic local

operations, namely, those of erosion, dilatation,

opening, and closing; moreover, any two of these

operations form a basis, in terms of which the other

two operations may easily be expressed. This is very

convenient for the development of software systems,

in which the user can quickly design particular

algorithms from basic blocks.

On the basis of mathematical morphology,

Sternberg (Sternberg, 1980, 1985, 1986) introduced

the concept of an image algebra.

The image algebra made it possible to represent

algorithms for image processing in the form of

algebraic expressions, where variables are images

and operations are geometrical and logical

transformations of the images. It is known that the

possibilities of mathematical morphology are very

limited. In particular, many important and widely

used operations of image processing (feature

extraction based on the convolution operation,

Fourier transforms, use of the chain code,

equalization of a histogram, rotations, recording, and

nose elimination), except for the simplest cases, can

hardly (if ever) be realized in the class of

morphological operations.

The impossibility of constructing a universal

algebra for tasks of image processing on the basis of

the morphological algebra may be explained by the

limitation of the basis consisting of the set-

theoretical operations of addition and subtraction in

Minkowski’s sense.

It is known that this basis has the following

drawbacks (Miller, 1983): complicated realization of

widely used operations of image processing;

impossibility of establishing a correspondence

between the operations of mathematical morphology

and linear algebra; impossibility of using

mathematical morphology for transformations

between different algebraic structures, in particular,

sets including real and complex numbers and vector

quantities.

These problems have been solved in the standard

image algebra (IA) by G. Ritter (Ritter, 2001),

(Ritter, Wilson, 2001) on the basis of a more general

algebraic representation (Birkhoff, Lipson, 1970) of

operations of image processing and analysis.

Standard Image Algebra by G.Ritter is a unified

algebraic representation of image processing and

analysis operations. Image algebra generalizes the

known local methods for image analysis, in

particular, mathematical morphology, and provides

the following advantages as compared with

mathematical morphology: it makes it possible to

work with both real and complex quantities; it

allows one to include both scalar and vector data

into the input information; it makes image-algebra

structures consistent with linear structures; it

provides a more accurate and complete description

of its operations and operands; with the help of a

special structure “template,” composite operations of

image processing are divided into a number of

parallel simplest operations.

The bottleneck in applications of methods of

image algebra to image recognition is the choice of

the sequence of algebraic operations and templates

for representation of composite operations of image

processing.

At present, this choice is based, as a rule, on

general representations of the character of images

and tasks. Deficiencies of this approach are obvious:

first, it is subjective and its success depends to a

IMTA-52015-5thInternationalWorkshoponImageMining.TheoryandApplications

great extent on the user’s experience and, second, it

is intended to solve a specific narrow class of

problems. Image algebra generalizes the known

local methods for image analysis, in particular,

mathematical morphology.

Investigations in the area of algebraization and

image analysis of the 1970–1980s represent a source

of development of the descriptive image algebra

(DIA). DIA by I.Gurevich is a unified algebraic

language for describing, performance estimating and

standardizing representation of algorithms for image

analysis, recognition and understanding as well as

image models.

An object that lies most closely to the developed

mathematical object is the image algebra proposed

and developed by Ritter (Ritter, 2001). Ritter’s main

goal in developing the image algebra is the design of

a standardized language for description of

algorithms for image processing intended for

parallel execution of operations. A key difference in

the new image algebra from the standard Ritter

image algebra is that DIA is developed as a

descriptive tool, i.e., as a language for description of

algorithms and images rather than a language for

algorithm parallelizing.

The conceptual difference of the algebra under

development from the standard image algebra is that

objects of this algebra are (along with algorithms)

descriptions of input information. DIA generalizes

the standard image algebra and allows one to use (as

ring elements) basic models of images and

operations on images or the models and operations

simultaneously. In the general case, a DIA is the

direct sum of rings whose elements may be images,

image models, operations on images, and

morphisms. As operations, we may use both

standard algebraic operations and specialized

operations of image processing and transformations

represented in an algebraic form. To use DIA

actively, it is necessary to investigate its possibilities

and to attempt to unite all possible algebraic

approaches, for instance, to use the standard image

algebra as a convenient tool for recording certain

algorithms for image processing and understanding

or to use Grenander’s concepts for representation of

input information.

In the 1980s, Sternberg formalized the notion

“image algebra” and introduced the following

definition.

Definition 1. Image algebra is the representation

of algorithms for image processing on a cellular

computer in the form of algebraic expressions whose

variables are images and whose operations are

procedures for constructing logical and geometrical

combinations of images.

This image algebra is described on the basis of

mathematical morphology and is identified by the

author with mathematical morphology. In 1985,

Sternberg (Sternberg, 1985) noted that the languages

for image processing were being developed for each

processor architecture and none of them has been

created for one computer and run on another.

Ritter’s image algebra (Ritter, Wilson, 2001)

generalizes mathematical morphology, unites the

apparatus of local methods for image analysis with

linear algebra, and generates more complex

structures. The structure of the standard image

algebra may be extended by introducing new

operations. Hence, it may be successfully applied in

the cases where a satisfactory result cannot be

obtained with the help of morphology and linear

algebra.

Definition 2. A standard image algebra is a

heterogeneous (or multivalued) algebra with a

complex structure of operands and operations if the

basic operands are images (sets of points) and values

and characteristics related to these images (sets of

values related to these points).

Analysing 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: the new algebra must make possible

processing of images as objects of analysis and

recognition; 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;

the new algebra must make possible operations on

main models of procedures for image

transformations; 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.

Definition 3. An algebra is called a descriptive

image algebra if its operands are either image

models (for instance, as a model, we may take the

image itself or a collection of values and

characteristics related to the image) or operations on

images, or models and operations simultaneously.

It should be noted that, due to the variety of

“algebras”, we should indicate which algebra is

meant in definition of DIA. For the generality of the

results and extension of the domain of applications

of the new algebra, to define DIA with one ring, we

CurrentTrendsinMathematicalImageAnalysis-ASurvey

use the definition of the classical algebra of Van der

Waerden (Waerden, 1971).

Thus, a DIA with one ring must satisfy the

properties of classical algebras. A DIA with one ring

is a basic DIA, because it contains a ring of elements

of the same nature, i.e., either a ring of image

models or a ring of operations on images.

To design efficient algorithmic schemes for

image analysis and understanding, it is necessary to

investigate different types of operands and different

types of operations applicable to the chosen

operands, which generate the DIA.

5 DESCRIPTIVE APPROACH TO

IMAGE ANALYSIS

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 (Rosenfeld, 1979), T. Evans

(Evans, 1967, 1969), R. Narasimhan (Narasimhan,

1966, 1967, 1968), R. Kirsh (Kirsh, 1964), A. Shaw

(Shaw, 1967, 1968), H. Barrow, A. Ambler, and R.

Burstall (Barrow, et al., 1972), S. Kaneff (Kaneff,

1972), K.S.Fu (1972), Schlesinger (Schlesinger,

Hlavac, 2002). In 1970s, Yu. I. Zhuravlev proposed

the so called “Algebraic Approach to Recognition

and Classification Problems” (Zhuravlev, 1998),

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” (Grenander,

1976, 1978, 1981, 1993, 1996), where he considered

methods of data representation and transformation in

recognition problems in terms of regular

combinatorial structures, leveraging algebraic and

probabilistic apparatus. Both approaches dealt with

the recognition problem in its classical statement and

did not touch upon representation of initial data in

the form of images.

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.

The main intention of DA is to structure different

techniques, operations and representations being

applied in image analysis and recognition. The

axiomatics and formal constructions of DA establish

conceptual and mathematical base for representing

and describing images and its analysis and

estimation. The DA provides a methodology and a

theoretical base for solving the problems connected

with the development of formal descriptions for an

image as a recognition object as well as the synthesis

of transformation 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 tokens, and

relations between them.

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 representation of images.

The automation of information extraction from

images requires complex use all the 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 (modelling)

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 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, reducing images to a form suitable for

recognition (RIFR) 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.

IMTA-52015-5thInternationalWorkshoponImageMining.TheoryandApplications

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

(Gurevich, 1989, 1991)).

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 (Zhuravlev, 1998) 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) (Gurevich, Yashina,

2006), (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).

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 (DIM), (5) RIFR, (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 analysing 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.

A more detailed description of methods and tools

of the DA obtained in the development of its results

can be found in (Gurevich, 2005), (Gurevich,

Yashina, 2008, 2012).

6 CONCLUSIONS

Practical application of the algebraic instruments

DA was demonstrated: we have shown how to build,

by means of DIA, the model of a technology for

automating diagnostic analysis of cytological

preparations of patients with tumors of the lymphatic

system. This model has been used for the creation of

software for application of this technology, its

testing, and comparison of results.

The main contribution is construction of a model

for a method ensuring a unified representation of the

technology, instead of development of a method for

solving a medical task. This work, thus, solves a

dual task: first, it represents a technology in the form

of a well-structured mathematical model and,

second, shows how DIA can be used in an image

analysis task.

In the future, DA and its main instruments—

DIA, DIM and GDT—will be applied to

constructing models of an information technology

for automation of diagnostic analysis of medical

images in other areas of medicine.

ACKNOWLEDGEMENTS

This work was supported in part by the Russian

Foundation for Basic Research (projects no. 14–01–

00881), by the Presidium of the Russian Academy of

Sciences within the program of the Department of

Mathematical Sciences, Russian Academy of

Sciences “Algebraic and Combinatorial Methods of

Mathematical Cybernetics and Information Systems

of New Generation” (“Algorithmic schemes of

descriptive image analysis”) and “Information,

Control, and Intelligent Technologies and Systems”

(project no. 204).

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