On Image Representing in Image Analysis
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 presentation is devoted to the research of mathematical fundamentals for image analysis and
recognition procedures being conducted currently in the Dorodnicyn Computing Centre of the Russian
Academy of Sciences, Moscow, Russian Federation. The paper presents and discusses the main results
obtained using the Descriptive Approach to Analysing and Understanding of Images when solving
fundamental problems of the formalization and systematization of the methods and forms of representing
information in the problems of the analysis, recognition, and understanding of images. In particular, the
problems arise in connection with the automation of information extraction from images in order to make
intelligent decisions (diagnostics, prediction, detection, evaluation, and identification of patterns). The final
goal of this research is automated image mining: a) automated design, test and adaptation of techniques and
algorithms for image recognition, estimation and understanding; b) automated selection of techniques and
algorithms for image recognition, estimation and understanding; c) automated testing of the raw data quality
and suitability for solving the image recognition problem.
1 INTRODUCTION
The automation of processing, analysing, evaluating,
and understanding of information provided in the
form of images is one of the critical breakthrough
problems of theoretical computer science. The image
is one of the main means of representing and
transmitting information needed to automate
intelligent decision making in a variety of
application domains.
To date, in the analysis and evaluation of images,
there is extensive experience in the application of
mathematical methods from different branches 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, the variety of methods used
does not replace the need for some regular basis for
ordering and selecting appropriate methods of image
analysis, a uniform representation of the processed
data (images) that meet standard requirements of
pattern recognition algorithms to the source data, the
construction of mathematical models of images
focused on the identification problem, and the
general availability of a universal language for the
uniform description of images and their
transformations
This paper presents the main results on the
formalization and systematization of methods and
forms of information representation in problems of
analysis, recognition, and understanding of images.
We have summarized the development of a
descriptive approach (DA) to analysing and
understanding images formulated by I.Gurevich
(Gurevich, 1989, 1991, 2005). This is a direction of
research concerning the formalization and
representation of images. Recall that DA is a
specialization of the algebraic approach of
Yu.Zhuravlev (Zhuravlev, 1998) to the case of the
representation of information in the form of images.
Axiomatics and formal structures of the DA
provide methods and tools for presenting and
describing images for their subsequent analysis and
evaluation. The theoretical basis of the research is
the DA; general algebraic methods; and methods of
the mathematical theories of image processing,
image analysis, and pattern recognition.
It is established that the overall success and
effectiveness of the analysis and evaluation of
29
Gurevich I. and Yashina V..
On Image Representing in Image Analysis.
DOI: 10.5220/0005460300290037
In Proceedings of the 5th International Workshop on Image Mining. Theory and Applications (IMTA-5-2015), pages 29-37
ISBN: 978-989-758-094-9
Copyright
c
2015 SCITEPRESS (Science and Technology Publications, Lda.)
information provided in the form of images are
determined by the possibilities of reducing images to
a form suitable for recognition (RIFR).
RIFR processes are crucial for solving applied
problems of image analysis and, in particular, to
make intelligent decisions based on information
extraction from images. The DA provides the ability
to solve both problems associated with the
construction of formal descriptions of images as
objects of recognition and problems of synthesis of
procedures of pattern recognition and image
understanding. The operational approach to
characterizing images requires that processes of
analysing and evaluating information provided in the
form of images (the trajectory of problem solving)
as a whole could be viewed as a
sequence/combination of transformations and
computing of a set of interim and final (defining the
solution) evaluations. These transformations are
defined on the equivalence classes of images and
their representations. The latter are defined
descriptively, i.e., using a base set of prototypes and
corresponding generative transformations that are
functionally complete with respect to the
equivalence class of admissible transformations.
Now we outline the goals of theoretical
development in the DA framework (and image
analysis algebraization) (“What for”) and necessary
steps to finalize the DA (“What to Do or What to be
Done”) and the global problem of an image
reduction to a recognizable form.
2 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 RIFR proposed to
formalize and systematize the methods and forms of
representation of images.
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 analysing 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 complex use of 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 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, RIFR techniques, conceptual system of
analysis and recognition image, descriptive image
models (DIM) classes, the descriptive image algebra
(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 (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, 2006), (4)
the mathematical formulation of the problem of
IMTA-52015-5thInternationalWorkshoponImageMining.TheoryandApplications
30
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)
(Gurevich, 2005).
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 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 (Zhuravlev, 1998) and
their algorithmic algebra and (2) the theory of
images of Grenander (Grenander, 1993, 1996), in
particular algebraic methods for the representation
of source data in image recognition problems
developed in it.
As was 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 (Zhuravlev, 1998) and the research of
Sternberg (Sternberg, 1985) and Ritter (Ritter,
2001), (Ritter, Wilson, 2001), 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 (Gurevich, 1989, 1991, 2005),
(Gurevich, Zhernova, 2003), (Gurevich, Koryabkina,
2006), (Gurevich, Yashina, 2006, Computer-Aided),
(Gurevich, Yashina, 2006, Operations), (Gurevich,
Yashina, 2008), (Gurevich et al., 2008), (Gurevich,
Yashina, 2012).
3 WHAT TO DO OR WHAT TO BE
DONE. BASIC STEPS
The critical points of an image analysis problem
solution are: 1) precise setting of a problem; 2)
correct and “computable” representation of raw and
processed data for each algorithm at each stage of
processing; 3) automated selection of an algorithm:
a) decomposition of the solution process for main
stages; b) indication points of potential improvement
of the solution (“branching points”); c) collection
and application of problem solving experience; d)
selection for each problem solution stage of basic
algorithms, basic operations and basic models
(operands); e) classification of the basic elements; 4)
performance evaluation at each step of processing
and of the solution: a) analysis, estimation and
utilization of the raw data specificity;
b).diversification of mathematical tools used for
performance evaluation; c) reduction of raw data to
the real requirements of the selected algorithms.
Basic steps of the development of image analysis
theory are: a) mathematical settings of an image
recognition problem (step 1); b) image formalization
space and descriptive image models (step 2); c)
generating descriptive trees and multimodel
representation of images (step 3); d) image
equivalence (step 4); e) image metrics (step 5); f)
descriptive image algebras (step 6).
3.1 Mathematical Settings of an Image
Recognition Problem
DONE:
1) Descriptive Model of Image Recognition
Procedures
2) Mathematical Setting of an Image Recognition
Problem. Image Equivalence Case.
OnImageRepresentinginImageAnalysis
31
Image analysis and recognition deal with properties
of the object (scene) shown and deformations
associated with the way and procedure of obtaining
the image. In this case, to formalize image
processing, we need to specify three sets (models) of
images, on which we postulate the existence of
classes of equivalence and sets of admissible
transformations given on the classes of equivalence
(Gurevich, Yashina, 2006). Introducing classes of
equivalence on the sets of image models, we accept
that any image possesses some regularity or a mix of
regularities of different types. Under this
assumption, analysis and recognition problem is
reduced to making a difference between images that
preserve their own regularity and images, the
regularity of which can be broken.
Figure 1 shows the descriptive model of the
image recognition problem.
Figure 1: Descriptive model of the image recognition
problem.
Here, {J} is the set of ideal images, {J*} is the
set of observable images, {J
R
} is the set of images
obtained as a result of solving the recognition
problem, {T
F
} is the set of admissible
transformations to form the image, {T
R
} is the set of
admissible transformations to recognize the image,
and {K
i
} are classes of equivalence.
Let J be some true image of the object involved.
We can consider processes of obtaining, forming,
discretization, etc. (all procedures that make it
possible to work with the image) as if the true image
were transferred via the noisy channel. As a result,
we analyze some real (observable) image J* rather
than the true image. This real image is to be
classified in the course of analysis, i.e., we should
determine the prototype in the true class of
equivalence or find the regularity (regularities) of
the given type JR on the observable image J*. Thus,
we can specify the sets {J}, {J*}, and {J
R
} and
transformations to form (T
F
) and recognize (T
A
) the
image
T
F
: J-> J*, (1)
T
A
: J*->J
R
. (2)
To perform image recognition, we need to give
algebraic systems of transformations {T
F
} and {T
A
}
on classes of equivalence of the set {J} and apply
them to observable images J* to perform the
backward analysis, i.e., classify images according to
the nature of their regularity (restore true images,
i.e., indicate classes of equivalence they belong to),
and the forward analysis, i.e., search the image J*
for regularities of the certain type J
R
and localize
them.
Stating the analysis problem in such a way, we
can give the class of image processing procedures,
analysis process of which is of fixed structure, with
interpretation (particular implementation) depending
on the purposes and type of analysis. There are
following main stages of analysis.
Now three mathematical statements of image
recognition problems are considered. The first one Z
is introduced by Yu. Zhuravlev (Zhuravlev, 1998).
When solving real image recognition problems,
we deal with the images of objects rather than with
the objects themselves. Therefore, we will assume
that the whole set of images is somehow divided into
equivalence classes. We also assume that there is
correspondence between the equivalence classes of
images and the objects; however, in the future, we
will not mention objects in the statement of the
recognition problem. Taking into account the
concept of equivalence of images introduced above,
we can formulate the image recognition problem as
follows.
The difference between problems Z
2
and Z
1
is
that each equivalence class in problem Z
2
is replaced
by a unique image—a representative of the class—
with the number n
i
, 1≤ n
i
≤ p
i
, where i is the number
of the equivalence class. This replacement is
performed by introducing the concept of an
admissible transformation.
Problem Z
1
differs from Z by the fact that it
explicitly uses equivalence classes of images. To
reduce the image recognition problem Z
1
to the
standard recognition problem Z, one should pass
from the classification of a group of objects to the
classification of a single object. Under certain
constraints on admissible transformations, problem
Z
2
, which differs from Z
1
in that it contains
admissible transformations that do not take the
image beyond the equivalence class, allows one to
handle a single image for each equivalence class—a
representative of this equivalence class.
TO BE DONE:
1) establishing of interrelations and mutual
correspondence between image recognition
problem classes and image equivalence classes;
IMTA-52015-5thInternationalWorkshoponImageMining.TheoryandApplications
32
2) new mathematical settings of an image
recognition problem connected with image
equivalency;
3) new mathematical settings of an image
recognition problem connected with an image
multimodel representation and image image data
fusion.
3.2 Image Formalization Space and
Descriptive Image Models
DONE:
1) the conceptualization of a system of concepts
that describe the initial information (images) in
recognition problems has been carried out;
2) descriptive models of images focused on the
recognition problem have been defined;
3) the image formalization space has been
introduced, the elements of which include
different forms (states, phases) of representing
the image transformed from the original form
into the recognizable one, i.e., into the image
model;
4) the basic axioms of the descriptive approach
were introduced.
Image formalization space (IFS) is the space
including sets of an image “states” and sets of image
transforming schema for formalization and
systematization of techniques and forms of
information representations in image analysis,
recognition and understanding problems. More
detailed description of IFS (Gurevich, Yashina,
2012) includes: a) construction of algorithmic
schema generating phase trajectories for solving
image analysis and recognition problems; b) DIM -
mathematical objects providing representation in a
form acceptable for a recognition algorithm of
information carried by an image and by an image
legend (context); c) multiple DIM and multi-aspect
image representations; d) topological properties of
the Image Formalization Space.
Recall that, in the DA, the processes of analyzing
and evaluating the information presented in the form
of images are considered as sequences of the
transformations and calculations of a set of
intermediate and final (defining the solution)
evaluations. These estimates are essential
characteristics of representations of the source image
obtained at each stage of RISR. Final estimates are
used in the final stage of solving the problem of
recognition/classification of the source image in the
application of algorithms for the
recognition/classification of the image model created
by RIFR.
The descriptive algebraic scheme of the image
representation (DASIR), which is a formal scheme
designed to produce a standardized formal
description of surfaces, point configurations, shapes
that form the image, and the relations between them,
is recorded using the DIA.
DASIR reflect sequential and/or parallel use of
transformations from the set of transformations to
the initial information from the space of initial data.
In (Gurevich, et al., 2008), there is an example of
constructing DASIR to solve the problem of the
morphological analysis of blood cells. All steps of
the algorithmic scheme for training recognition
algorithms for problems of analyzing cytological
preparations and classifying the new image using
three diagnoses based on a recognition algorithm
with adjusted parameters were defined and described
using the DIA with one ring.
In the DA, three classes of admissible
transformations of images are considered (Gurevich,
Yashina, 2006), i.e., procedural transformations,
parametric transformations, and generative
transformations. The basic classes of
transformations of images (procedural, parametric,
and generating) are defined, as well as related
concepts of the structuring element, which generates
rules and correct generative transformation.
All transformations for processing and analyzing
images are conducted using the DIA record on the
transformations of images. It makes it possible to
vary the methods for solving the subproblem using
different operations of the image analysis of fixed
DIA and keeping the whole scheme of the
technology of RISF and the extraction of
information from images unchanged.
In order to apply of pattern-recognition
algorithms to the created formal descriptions of
images, it is necessary to implement the created
schemes (to set specific transformations from the
fixed DIA and parameters of transformations
selected in the schemes) and apply them the initial
information, i.e., to create models of images.
An example of the DASIR implementation can
be found in (Gurevich, Yashina, 2006), i.e., at each
step of the algorithmic scheme the created DASIR
was concretized by the selection of a transformation
that belongs to a specific DIA that describes the
step.
In the general case, we can say that the use of the
convolution of structuring elements and admissible
transformations of the image to the initial
information about the image leads to the
OnImageRepresentinginImageAnalysis
33
transformation of the initial information in the image
model. Specific allowable image transformations
and specific methods of applying them to the initial
information are selected based on the set problem of
the analysis and recognition of images.
Axiomatization of algebraic image analysis
constitutes a base for unification of image analysis
algorithms representations and image models
representations. The axioms define properties and
structure of the Image Formalization Space (IFS).
It was shown that all image representations and
procedures of RFSR form a topological space (IFS).
The main properties of this space, as well as the
conceptual basis of the synthesis of image models,
are defined by the following axioms that constitute
basic provisions of the DA.
Image models are the results of RIFR (taking
into account all the information about the image).
On the set of image models, basic DIA are
introduced on image models of three classes in
accordance with operations used for their
construction. Note that these models are descriptive
image models (DIM).
TO BE DONE:
1) Creation of image models catalogue
2) Selection and study of basic operations on image
models for different types of image models
(including construction of bases of operations)
3) Use of information properties of images in image
models
4) Study of multimodel representations of images.
3.3 Generating Descriptive Trees and
Multimodel Representation of
Images
DONE:
Generating Descriptive Tree (GDT) - a new data
structure for generation plural models of an image is
introduced.
The introduction of axiomatics of DA and
definition of three classes of DIM has led to the
introduction of a new mathematical object for
structuring representations of images and generation
of image models.
Three types of appropriate conversions
generating rules and a source image are necessary
for constructing the three classes of representations
of images (procedural, parametrical, and generating
representations of images). The source image is
described by means of a set of its implementations
and by means of context-sensitive and semantic
information.
According to the introduced axiomatics and
definitions of various classes of representations of
images, in this way, for merging and combination of
various properties of image models, it is necessary to
introduce the following hierarchies: the hierarchy of
possible implementations of images; the hierarchies
of semantic and context-sensitive information in
images; the hierarchies of parametrical, procedural,
and generating conversions; and hierarchies of
generating rules. It is suggested to implement such
structures in the form of special trees.
Specialization of the concept of a tree on the
whole is related to specialization of nodes of a tree.
As nodes we will select objects, operations, or rules
of image analysis tasks used to construct different
image models. Such nodes are called GDT
descriptors. The definitions of parent, calculated,
fixed, objective, and abstract GDT descriptors have
been introduced, but will not be dwelt on in this
work.
Definition 1 (Gurevich, 2005). The generating
descriptive tree (GDT) is the structure intended for
classification and automated generation of image
models and it possesses the following properties:(1)
GDT descriptors are GDT nodes; (2) Every GDT
combines the descriptors of one type; that is, GDTs
represent the same type of properties of an image;
(3) Each GDT element can be united with another
element to generate new partial multiaspect image
models; (4) Descriptors are linked among
themselves by parent–daughter relationships; (5)
Each descriptor has a relationship with a unique
parent descriptor and can have some links with
derived descriptors. If the descriptor has no parent, it
is called a radical GDT. If the descriptor has no
derived descriptors, it is called a leaf.
Note that parametrical GDTs are GDTs intended
for classification and automation of the generation of
parametrical image models. A parametrical GDT,
thus, contains GDT descriptors describing the
properties of parametrical conversions, leading to an
evaluation of features of images. A procedural GDT
is a GDT intended for classification and automation
of the generation of procedural image models. A
procedural GDT, thus, contains the GDT descriptors
describing the properties of procedural conversions.
TO BE DONE:
1) to define and to specify GDT;
2) to set up image recognition problem using GDT;
3) to define descriptive image algebra using GDT;
4) to construct a descriptive model of image
recognition procedures based on GDT using;
5) to select image feature sets for construction of P-
IMTA-52015-5thInternationalWorkshoponImageMining.TheoryandApplications
34
GDT;
6) to select image transform sets for construction of
T-GDT;
7) to define and study of criteria for selection of
GDT-primitives.
3.4 Image Equivalence
DONE:
There were introduced several types of image
equivalence: image equivalence based on the groups
of transformations; image equivalence directed at
the image recognition task; image equivalence with
respect to a metric.
We consider the problem of searching for a
correct algorithm for the image recognition problem.
We consider various methods for defining the
equivalence of images, namely, equivalence on the
basis of transformation groups, equivalence oriented
to a special statement of the image recognition
problem, and equivalence with respect to metric. In
the case of definition of equivalence based on
transformation groups, we construct examples of
equivalence classes. It is shown that the concept of
equivalence is one of the key concepts in image
recognition theory. We study the relationship
between equivalence and invariance of images.
Using the introduced concept of equivalence of
images, we modify the standard mathematical
statement of the image recognition problem and
formulate an image recognition problem in terms of
equivalence classes. We prove that, under certain
constraints on the image transformations, the
problem of image recognition in the standard
statement can be reduced to an abridged problem for
which there exists a correct algorithm within the
algebraic closure of the class of recognition
algorithms for calculating estimates (ACEs).
TO BE DONE:
1) to study image equivalence based on information
properties of the image;
2) to define and construct image equivalence
classes using template (generative) images and
transform groups;
3) to establish and to study links between image
equivalence and image invariance;
4) to establish and to study links between image
equivalence and appropriate types of image
models;
5) to establish and to study links between image
equivalence classes and sets of basic image
transforms.
3.5 Image Metrics
It is an open problem.
TO BE DONE:
1) to study, to classify, to define competence
domains of pattern recognition and image
analysis metrics;
2) to select workable pattern recognition and image
analysis metrics;
3) to construct and to study new image analysis-
oriented metrics;
4) to define an optimal image recognition-oriented
metric;
5) to construct new image recognition algorithms
on the base of metrics generating specific image
equivalence classes.
3.6 Descriptive Image Algebras
DONE:
1) Descriptive Image Algebras (DIA) with a single
ring were defined and studied (basic DIA);
2) it was shown which types of image models are
generated by main versions of DIA with a single
ring;
3) the technique for defining and testing of
necessary and sufficient conditions for
generating DIA with a single ring by a set of
image processing operations were suggested;
4) the necessary and sufficient conditions for
generating basic DIAs with a single ring were
formulated;
5) the hierarchical classification of image algebras
was suggested;
6) it was proved that the Ritter’s algebra could be
used for construction DIA’s without a “template
object”.
This object is studied in developing a mathematical
apparatus for analysis and estimation of information
represented in the form of images. For a structural
description of possible algorithms for solving these
problems, we need a formal instrument that allows
us to describe and justify the chosen way of solution.
As formalization tools, we chose the algebraic
approach, which should provide a unique form of
procedures for describing the objects–images and
transformations of these objects–images.
The need to develop a mathematical language
that ensures that solutions of problems of image
processing, analysis, and understanding may be
uniformly described by structural algorithmic
schemes is justified by the following factors:
OnImageRepresentinginImageAnalysis
35
(1) there are many algorithms (designed and
introduced into practice) for analysis, estimation,
and understanding of information represented in
the form of images;
(2) the set of algorithms is neither structured nor
ordered;
(3) as a rule, methods for image analysis and
understanding are designed on the basis of
intuitive principles, because the information
represented in the form of images is hardly
formalized;
(4) the efficiency of these methods is estimated (as is
usual in experimental sciences) by the success in
solving actual problems—as a rule, the problem
of rigorous mathematical justification of an
algorithm is not considered.
“Algebraization” is one of the most topical and
promising directions of fundamental research in
image analysis and understanding. The main goal of
the algebraic approach is the development of a
theoretical basis for representations and
transformations of images in the form of algebraic
structures that enable one to use methods from
different areas of mathematics in image analysis and
understanding.
An object that lies most closely to the developed
DIA 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. In more detail, the
definition of the standard image algebra and that of
DIA are considered in (Gurevich, Yashina, 2006).
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.
The main attention was given to DIAs with one
ring, which form the main subclass of basic DIAs. In
future, we are going to consider DIAs based on
superalgebras and investigate other possibilities of
application of other algebraic concepts in the theory
being developed.
TO BE DONE:
1) to study DIA with a single ring, whose elements
are image models;
2) to study DIAs with several rings (super
algebras);
3) to define and study of DIA operation bases;
4) to construct standardized algebraic schemes for
solving image analysis and estimation problems
on the DIA base;
5) to generate DIA using equivalence and
invariance properties in an explicit form;
6) to demonstrate efficiency of using DIA in
applied problems;
7) to study alternative algebraic languages for
image analysis, recognition and understanding.
4 CONCLUSIONS
In principle, the success of image analysis and
recognition problem solution depends mainly on the
success of image reduction to a recognizable form,
which could be accepted by an appropriate image
analysis/recognition algorithm. All above mentioned
steps contribute to the development techniques for
this kind of image reduction/image modeling. It
appeared that an image reduction to a recognizable
form is a critical issue for image analysis
applications, in particular for qualified decision
making on the base of image mining. The main tasks
and problems of an image reduction to a
recognizable form are listed below:
1. Formal Description of Images:
1) Study and construction of image models (Step
2);
2) Study and construction of multimodel image
representations (Step 3);
3) Study and construction of metrics (Step 5).
IMTA-52015-5thInternationalWorkshoponImageMining.TheoryandApplications
36
2. Description of Image Classes Reducible to a
Recognizable Form:
1) Introduction of new mathematical settings of an
image recognition problem (Step 1);
2) Establishing and study of links between
multimodel representation of images and image
metrics (Steps 3, 5);
3) Study and use of image equivalencies (Step 4).
3. Development, Study and Application of an
Algebraic Language for Description of the
Procedures of an Image Reduction to a Recognizable
Form (Step 6).
We hope that after passing through the above
mentioned steps we’ll be able to formulate the
axiomatics of the descriptive (mathematical) theory
of image analysis.
ACKNOWLEDGEMENTS
This work was supported in part by the Russian
Foundation for Basic Research (projects no. 14–01–
00881), by the Russian Academy of Sciences within
the program of the Department of Mathematical
Sciences of the 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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