=Paper=
{{Paper
|id=Vol-3723/paper26
|storemode=property
|title=Peculiarities of a symbol classification and recognition of expression structures system development
|pdfUrl=https://ceur-ws.org/Vol-3723/paper26.pdf
|volume=Vol-3723
|authors=Taras Basyuk,Andrii Vasyliuk
|dblpUrl=https://dblp.org/rec/conf/modast/BasyukV24a
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==Peculiarities of a symbol classification and recognition of expression structures system development==
https://ceur-ws.org/Vol-3723/paper26.pdf
Peculiarities of a symbol classification and recognition
of expression structures system development
Taras Basyuk1,∗,†, Andrii Vasyliuk1,∗,†
1 Lviv Polytechnic National University, Bandera str.12, 79013, Lviv, Ukraine
Abstract
The article analyzes the existing methods and approaches used in the pattern recognition process,
which made it possible to identify the peculiarities of their application and outlined the range of
problems that arise. A symbol and relation classifiers were designed, which allowed to categorize
symbols into one of four classes: 'small', 'descending', 'increasing', 'variable range', and to determine
what is the relationship between them. A one-pass algorithm was implemented, including search
with returns, which provided fast structure recognition of the expression using the algebra of
algorithms. The software system was designed using the object approach and displaying the created
diagrams in accordance with the UML language. Sequence and class diagrams are given, and their
detailed description is carried out. The study presents the structure of basic actions, which allowed
to display in more detail the process of classification and expression structures recognition. An
applied software system has been developed using the Java object-oriented programming language,
which implements the process of symbol classification and recognition of image structures. With the
help of the developed system, it is possible to recognize the structure of expressions and classify
symbols. As for now, the software solution works in the form of a prototype.
Further research will be directed to testing and improving the system, eliminating conflicts, and
expanding functionality in accordance with the specified requirements.
Keywords 1
classification system, expression recognition, functional model, object-oriented design
1. Introduction
Computer vision is an activity in which statistical methods are used to obtain data and models,
built using geometry, physics, and learning theory [1,2]. Computer vision is used quite widely,
both in old fields (for example, mobile robot control, military applications, industrial
surveillance) and in new ones (human/computer interaction, image retrieval in libraries,
medical image analysis and realistic rendering of simulated scenes in computer graphics). A
distinctive feature of computer vision is the extraction of descriptions from images or sequences
of images. For example, such an aspect of computer vision as the detection of structure by
motion allows you to get an idea from a series of images about how the camera moves and what
is depicted in the picture. In the entertainment industry, similar methods are used to filter out
MoDaST-2024: 6th International Workshop on Modern Data Science Technologies, May, 31 - June, 1, 2024, Lviv-Shatsk,
Ukraine
∗ Corresponding author.
† These authors contributed equally.
Taras.M.Basyuk@lpnu.ua (T. Basyuk); Andrii.S.Vasyliuk@lpnu.ua (A. Vasyliuk)
0000-0003-0813-0785 (T. Basyuk); 0000-0002-3666-7232 (A. Vasyliuk)
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
motion and build three-dimensional computer models of a building with structural integrity.
With the help of a few photos, you can get good, simple, accurate and convenient models.
This technology is now at a special point of its development. Only relatively recently
appeared the opportunity to create practical programs using the ideas of computer vision. The
speed of modern digital devices and the possibility of parallel calculations enable the
implementation of many algorithms for working with libraries of digital images. Thus,
conducting important research and solving many everyday tasks (for example, organizing a
collection of photographs, creating a three-dimensional model of the surrounding world,
managing, and making changes to a collection of video recordings) is now possible with the
help of computer vision methods.
Character recognition is performed by classic OCR methods, for example, using support
vector methods, pattern matching. Analysis of structure is carried out by means of geometric
reasoning based on implicit rules or grammatical rules.
These rules take character identity into account, and only a few studies have attempted to
separate structure analysis from character recognition. Uncertainty in mathematical
expressions, especially in handwritten ones, is accepted. This may be uncertainty about the
structure symbol meaning, and only a few studies address this issue, maintaining multiple
interpretations about the symbol meaning until the structure resolves the ambiguity [3].
Therefore, considering the mentioned features, an urgent task is analyzing symbol classification
in known systems and recognition of expression structures with further development of a
software tool for solving the given task.
1.1. Analysis of recent researches and publications
1.1.1. Analysis of known approaches to problem-solving
The problem created is not new, and therefore there are many studies related to it. We will
analyze the most famous scientific works.
Richard Zanibbi's works analyze the baselines present in expressions [4,5]. Consider the
dominant baseline, the line on which the expression will be written, and the nested baselines
corresponding to the indices. In the first step, a tree is built based on these baselines. Then,
using knowledge of mathematical notation properties for some tree transformations, one
recognizes that the sequence 'sin' corresponds to the function of the same name, or that in lim
n→1, n→1 is the argument of the limit. Further, with their help, a tree is obtained, which
represents the content of the equation. To build a tree of baseline structures, you need to
recognize individual symbols. To do this, character classes such as upper (e.g., 'd', 'b'), lower
(e.g., 'y', 'p'), variable range (e.g., Σ, ∪), etc. are defined. Using these classes, areas around the
symbol are defined, where subscripts, superscripts, etc. should be found, if any. The specified
analysis of baselines was also used in other works [6,7].
In subsequent works, the ability to recognize subscripts and superscripts was improved
using fuzzy regions [8,9]. At the same time, they were motivated by the fact that most
ambiguities in handwritten mathematical expressions refer to the options' index/line and
line/superscript. Additionally, using fuzzy logic allowed them to return to a ranked list of
interpretations.
The online recognition method is performed in four stages: stroke recognition, structure
recognition, character recognition, and subscript/superscript recognition [10]. After analyzing
the baseline, they first recognize dominant symbols such as ∑ or fractional lines, which they
call parent symbols, and identify the child blocks in which the arguments must be found. The
rest of the structure is then recognized using bounding boxes and vertical symbol positions.
Richard A. Tapia and Raúl Rojas González first obtain baselines like Richard Zanibbi and
recursively build a minimal tree in which each node is a symbol. Next, the distance between the
two symbols is determined depending on whether it is in the range of the other. In this method,
they use different symbol classes (top, bottom, centered) and the regions around the symbol are
built accordingly with clear boundaries. Studies also consider points of attraction according to
the class of operators where the argument usually occurs. For example, the ∑ operator usually
has top and bottom arguments, so they place the attraction points in the middle of the upper
and lower bounds of that symbol.
Qi Xiangwei also uses symbol dominance and minimal spanning tree but performs further
analysis of symbol locations to construct groups of symbols more accurately [11,12]. In
particular, the main character on the dominant baseline or group of characters representing the
function name (for example, 'sin', 'lim') is determined.
Erik G. Miller and Emanuele Viola maintain ambiguity during the character recognition
stage. They then calculate the probability of each character being of a certain class (lowercase
letter, number, binary operator, etc.) and the probability of being an index, superscript, or linear
expression according to character recognition and some location properties. Next, stochastic
context-free grammar is used for syntactic analysis of the expression. The system uses a convex
hull instead of bounding rectangles, models character positions using a Gaussian distribution,
and uses an A-star algorithm to search the space of interpretations.
After character recognition, a set of rules is applied. Three sets of rules are applied, which
are used consecutively. Mathematical rules are grammatical rules for parsing and
understanding an expression. Sentiment-based rules are used to disambiguate the layout.
Experience-based rules handle uncertainty in the semantics of an expression. The last two
classes of rules are learnt and modified using feedback. At the end, they use the results to build
a location tree and a semantic tree, and then turn them into a Latex tape.
In his writings, Walaa Aly conducts research, the essence of which is the correct recognition
of indexes and superstructures [13]. They use normalized bounding rectangles as the main
feature of the symbol. It has been proven that with normalized bounding rectangles along with
special handling of irregular symbols, it is possible to effectively recognize the relationship
using a Bayesian classifier. His work uses the InftyCDB dataset, which contains expressions
extracted from more than 70 articles with different settings.
This article [14,15] explains the normalization of bounding rectangles in more detail. In
addition, they define more character classes. While there are usually three main classes (top,
bottom, small), they define six classes to handle variations in the positioning of some symbols.
At the same time, the Bayesian classifier is also used, which allows obtaining positive results
on a large sample.
Summarizing the research, it can be concluded that some recognition methods are based
only on spatial considerations, such as baselines. Other methods use rule-based systems, such
as grammar, and analyze an expression to interpret it [16]. Part of the algorithms considers
knowledge of mathematical symbols and operators and their spatial properties. At the same
time, not all studies use machine learning methods, so the classification of symbols and
recognition of expression structures with the subsequent development of a software tool for
solving the given task is an urgent issue.
1.2. The main tasks of the research and their significance
The purpose of this research is to develop a pattern recognition system, or rather a
mathematical expression recognition system based on machine learning methods, where
symbol classification and structure analysis are separate tasks. The conducted research will
provide means for classifying symbols and determining the structure of expressions, including
mathematical ones. To achieve the goal, the following tasks must be solved: to analyze the
existing approaches and methods used in the process of symbol classification and recognition
of the structure of expressions; to determine the main tasks that arise at the same time; to
develop the structure and determine the main components of the recognized expression; to
formulate an algorithm for symbol classification and recognition of an expression structure,
and to carry out its mathematical description using the algebra of algorithms; to design a
software system using the object approach; to construct an applied software system that
implements the process of symbol classification and expression structure recognition.
The results of the study solve the actual scientific and practical task of analyzing the
structure of an expression and classifying symbols, using a small amount of knowledge about
the syntax of a mathematical expression.
2. Major research results
Mathematical expression recognition is a task in which an image representing a mathematical
expression is interpreted by a computer so that it can be stored, processed, and reused.
Expression recognition consists of two stages [17]:
• Symbol recognition: each foreground pixel in the image belongs to a symbol, and each
symbol has a value in the expression and conveys some information.
• Pattern recognition: the two-dimensional layout of an expression obeys some rules, and
each pattern corresponds to a specific value.
Character recognition. A character recognition task is a procedure by which each character
is recognized and classified [18]. This is not an easy task due to the enormous number of
characters, given that there is no dictionary of words like for text recognition. The same symbol
can appear in different contexts, and it is sometimes important to distinguish between, for
example, the summation symbol Σ and the Greek letter Σ. Some different symbols have the
same shape, for example, p and P. Recognizing them correctly is not an easy task. Even more
problems arise when dealing with handwritten expressions, such as the q-9 problem.
Figure 1: “q-9” problem
This task is usually performed by neural networks or support vector methods.
Structure recognition. Although the former task is easy to understand, pattern recognition
can be difficult, depending on what level of interpretation needs to be achieved. In the research
process, several goals were defined, along with the format in which the structure should be
analyzed.
For example, for simple digitization in Latex, recognizing spatial relationships regardless of
their meaning is usually sufficient. For example, in the example presented earlier: xi/x(i), it is
not necessary to know that ‘i’ is a power in the first case and an index in the other. In Latex,
x^i and x^{(i)} will simply be written.
It is necessary to understand the meaning of the expression. It is important to know that 1
+ 2 is not only a sequence of characters in one line, but also an addition. In this case, the value
of operators, both explicit (for example, +) and implicit (for example, multiplication by xy), must
be known.
Sometimes a deeper analysis is required. In mathematical expressions, the meaning derives
not from the symbols, but from what they represent. As mentioned earlier, the same function
(or variable) can have a different name in different contexts. It is possible that f(x) and g(y) are
the same in two different documents.
Thus, recognizing the structure of an expression is often a combination of location analysis
and interpretation of what symbols and relations represent.
Complex mathematical symbols. As stated earlier, the range of symbols and rules used to
write mathematical expressions is not fixed. Common symbols and structural rules are only a
subset of an infinite number, since symbols and their new meanings can be invented at any
time. Indeed, new fields of science continue to appear, bringing with them the need for new
mathematical notations.
Quantum mechanics, for example, defined new uses for < and | to denote quantum states.
The symbol <, which represents a comparison of numbers in 1 < 2, can be used as a kind of
bracket in ⟨x, y⟩, or in the definition of some mathematical objects in (C, <). These are examples
where existing symbols are reused for a different purpose. It is also possible to find completely
new symbols. Therefore, it is necessary to consider this fact when recognizing mathematical
expressions. A convenient way to solve this problem is to reduce the impact of the symbol. In
most systems, character recognition determines how the structure will be recognized. As a
result of the study, it turned out that character identification is not necessary for structure
recognition. In addition, we believe that symbols can be classified using only their bounding
box and context (Fig.2).
Figure 2: Hypotheses. Left: Context helps classify symbols. Right: character identification is
not required for structure recognition
Figure 3: Variable superscript position
Figure 4: Arguments are on one line
When we analyze different symbols separately, we can see different spatial properties. For
example, the relative position of the superscript, as seen in Fig. 3, will not be locally the same,
with the symbol ‘b’ or ‘q’. However, at the global level, all superscripts must be written in one
line (Fig. 4). At the same time, the input format is an image of a handwritten mathematical
expression. Tables 1 and 2 show the character classes and relations used, respectively.
Table 1
Character classes used
Character classes Example
Small symbols a, e, r, u, o, s, m, x, c, n
Descending symbols y, p, q, g
Rising symbols A-Z, 0-9, t, d, h, k, l
Variable range symbols å, Õ, ⋃ , ⋂
Table 2
The relations used
Character classes Example
Linear xy, tan, 42, 10x, ∑n
Superscripts p! , b" , x #
Subscripts p" , b! , x$ å, Õ, ⋃ , ⋂
The goal of classification and recognition is to find the relationship between characters and
determine their classes. This can be thought of as linking symbols together and adding
information to an existing structure. At the beginning, a tree is created from the initial list.
Figure 5: An abstract expression tree and an example of a typical expression tree
Therefore, in the process of research, it is necessary not only to recognize the structure, but
also to try to find character classes using this structure. High order means a lot of contexts,
which should make classification easier. However, as expressions become more complex,
pattern recognition also becomes more difficult. That said, there is a trade-off between the
challenges of complexity and necessity in context, so recognizing all kinds of symbols and
relationships remains a challenge [19].
The next stage of the research was the system's design according to the object approach and
the UML language [20]. A class diagram is a structural diagram type in the UML (Unified
Modelling Language) used to visualize the structure of classes and their relationships in a
software system [21-23].
It allows you to show classes, their attributes and methods, and relationships between
classes. An extended description of the application of the class diagram includes the following
aspects: requirements analysis (used to analyze system requirements, namely, understanding
the structure of data and components allows you to analyze which classes are needed to
implement the functionality of the system and how they are related to each other);
communication with project participants (is an important communication tool between
different project participants and allows developers, architects, managers and other interested
parties to understand the structure of the system and its components); identifying potential
problems (can help identify potential problems and flaws in the system design, such as circular
dependencies between classes, excessive complexity, or insufficient modularity);
documentation (serves as an important means of documenting the project, namely providing
clear and specific information about the structure of the system for future developers and
support personnel).
The class diagram of the designed system is presented in Fig. 6.
Figure 6: Class diagram
Expressions are represented by the symbols of which they are composed, considered in their
context.
Symbol class. Objects of this class are representations of symbols. They contain information
about the symbol itself, out of context. A Symbol object is created from a bounding rectangle.
It provides geometric information such as the height, width, position, or center of a rectangle.
The character identifier is particularly useful when testing the system.
The SymbolClass class. A SymbolClass object is a symbol classification. It contains the results
of all the classifiers involved in the classification and the methods of providing the final
probability values.
Relationship class. A Relationship object represents a classification of relationships that exist
between a parent and subsidiary element. It stores the output of different classifiers and the
final probability values.
The Context class. An object in this class is referred to by the word 'symbol' throughout the
work. The described approach recognizes the structure and uses it to determine the character
class. Therefore, the context of the symbol is as important as the symbol itself. A Context object
is a node in a finite expression tree because it can be linked to other Context objects. A node
also contains a SymbolClass object and a Relationship object. Its key role is to present the
recognition result.
Expression class. An Expression object defines a list of all characters, sorted from left to right.
Provides a straightforward way to get information about the entire expression, such as its
interpretation, dimensions, leftmost character (the root of the tree after recognition), or to apply
any action to all characters (for example, to classify characters).
The main stages of the specified process include:
• Image capture. An initial handwritten image of a mathematical expression is fed to the
input of the recognition system.
• Image processing. The image is subject to processing such as downsizing, smoothing
and noise removal to improve quality and facilitate further analysis.
• Structure analysis. The resulting text is divided into individual characters and groups of
characters corresponding to subexpressions. This stage determines the structure of the
mathematical expression and establishes relationships between symbols.
• Classification of symbols. Provides the classification of the symbols of the handwritten
formula to the specified class.
• Context recognition. Given the context of the mathematical expression, such as the
order of operations, actions with parentheses, and operator priority, the received
symbols are analyzed to correctly recognize the expression.
• Generation of printed expressions. Based on the created expression tree, a printed
mathematical expression is built in the appropriate format.
The next stage was the creation of a functioning model using the algebra of algorithms [24].
The first stage of the implementation of the algebra of algorithms is the description of unit
terms and the synthesis of sequences, which is given below.
Formed uniterms: S(i) – uniterm of capturing the input image; P(s) - is the uniterm of image
pre-processing; A(s) - uniterm of structure analysis; C(s) - is the uniterm of classification of
symbols; R(f) - uniterm of context recognition; G(f) - uniterm of generation of printed
expression; u1 – check whether the symbol is classified. As a result of the use of the apparatus
of the algebra of algorithms, the following sequences and eliminations were synthesized:
S1 – the sequence of system operation when the symbol is classified:
S2 – the sequence of system operation when the symbol is not classified:
L1 – elimination of the symbol classification check:
The next stage is the substitution of the corresponding sequences in the elimination.
As a result of using the properties of the algebra of algorithms, we subtract the common unit
terms by the sign of the elimination operation and obtain the following formula of the algebra
of algorithms:
The next stage was the development of the system, using modern software tools. The
developed system is presented as a desktop application. The application was created using Java
programming in the macOS operating system environment [25,26]. The power and flexibility
of the Java language make it an optimal choice for programmers in various fields. From support
for object-oriented programming to platform-independent capabilities through JVM to
automatic memory management, Java provides developers with powerful tools that enable them
to efficiently build complex applications [27]. With the developed system's help, it is possible
to classify symbols and determine the expression's structure. Structure recognition is performed
by the parentAndRelate() method of the RelationshipFinder object. It uses classifiers as
described earlier and two data structures. To easily retrieve the last characters on each baseline,
we defined the BaselineStructure class. The most recently viewed symbols are available via the
stack.
A BaselineStructure object. The purpose of the BaselineStructure object is to access the
symbols on each baseline without having to go through the entire expression tree. It consists
of a symbol and a list of nested baselines.
The classes that implement the graphical interface are in the 'GUI' package and are built
using web technologies. They can be divided into two components [28-30]:
• An input component that facilitates quick input of mathematical expressions.
• An output component that displays the recognition result in the form of a two-
dimensional image and a Latex expression.
DrawView class. An object of the DrawView class allows the user to draw (handwrite) a
mathematical expression.
ResultImageView class. An object of the ResultImageView class is used to display a
mathematical expression as a two-dimensional image.
ResultTextView class. An object of the ResultView class is a block containing a text area that
allows displaying results in a Latex message format [31,32].
At the same time, the symbol can only be a descendant of the last symbol of the baseline.
For example, in 𝑎𝑏 𝑐 𝑑 , c and d cannot be a subsidiary element of a, but they can be a subsidiary
element of b, which is the last character on its base. When a nested subsidiary element is found,
a new base structure is created containing an empty list of nested baselines and replaces the
parent's base structure. The entire basic structure is simplified and ensures that the complexity
does not grow exponentially with each symbol. When an unbuilt subsidiary element is found,
a BaselineStructure is created and added to its parent's nested baselines.
Figure 7: Baselines structure
𝑐
In Fig. 8 it is shown the evolution of the main base structure 𝑎𝑏𝑑 𝑒 𝑔ℎ𝑘 . The stack of last
characters. Each time a symbol is a parent, the created base structure is added to the last symbol
stack. This stack contains BaselineStructure objects rather than context objects to simplify
updating the entire structure during the recognition process.
Figure 8: Graphical user interface
The system was tested with 1, 2, 5, 7 and 10 iterations. The result was that 440 out of 570
characters (77.19%) were successfully classified. This means that, on average, the symbol is well
classified, or the actual class is considered the second most likely. Accuracy is given in Table 3.
Table 3
Evaluation of symbol recognition
Symbol classes Number of symbols Correctly classified Evaluation
Small symbols 225 212 94.22%
Descending symbols 68 57 83.82%
Rising symbols 218 137 62.84%
Variable range symbols 59 34 57.63%
There were 32 parentage recognition errors (94.39% success rate) and 52 relationship
recognition errors (9.12% errors). At the same time, the recognition of connections is
determined at the level of (0.797).
Conclusion
As a result of the research, the existing methods and approaches used in the process of
classifying symbols and determining the structure of expressions were analyzed, which made
it possible to identify the peculiarities of their application and outlined the range of problems
that arise. An iterative algorithm was developed to use the mutual constraints between the
structure and the type of symbols. A one-pass algorithm was implemented, which includes
search with returns, which ensured fast recognition of the expression structure, and its
representation was carried out using the algebra of algorithms. The next stage was the design
of the software system using the object approach and the display of the created diagrams in
accordance with the UML language. An applied software system was developed that
implements symbol classification and expression structure recognition. With the help of the
developed system, it is possible to recognize mathematical expressions by classifying symbols
and analyzing the structure of expressions. At the current moment, the software solution works
in the form of a prototype.
Further research will be directed to testing and improving the system, eliminating conflicts,
and expanding functionality in accordance with the specified requirements.
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