In this paper we present a method for representing a continu ous curve in the plane by a function defined on a unit interval, which we call the angular characteristic function of the curve. This representation is invariant under translations, rotations and universal scaling and is well suited for curve shape recognition. We have used this function for feature extraction in an experimental system for optical recognition of handwritten characters and the approach, briefly described in this paper, has shown good results. For graphic primitives - circular arcs and line segments - the angular characteristic function representing them has a particularly simple form. This circumstance allows using the angular characteristic function in motion planning problems and in robotics. As an example of such applications, we present a n algorithm that can be used in a system for automatic program generation for a robotic manipulator used for a cutting process. This is an algorithm for approximating a smooth curve in the plane by a sequence of such graphic primitives forming a continuous smooth curve (in which each segment is conjugate to its neighbors).
Over the past few years, as machine learning and computer vision have experienced rapid advancements, image classification and recognition as well as increasing efficiency of the robot’s sensor and control information processing have gained widespread application across diverse domains [1-3]. For instance, it is employed in security for facial recognition technology, in transportation for vehicle recognition, and in aerospace for aviation remote sensing, among other areas. The process of identifying graphical patterns stands out from other methods of pattern recognition due to its distinct approach. It involves several preliminary stages of preprocessing the input information. In the initial phases, the image undergoes transformations to make it suitable for subsequent analysis. Following this, a critical step involves selecting the type and quantity of informational attributes that can facilitate pattern recognition, whether with or without a reference sample [4]. Among these methods, the most crucial stage is the selection of informative attributes. These attributes play a pivotal role in distinguishing a pattern from a group of surrounding objects.
A method for representing a continuous plane curve is proposed in this paper and applications of this method to image recognition problems and robotics are discussed. The essence of the method is that a curve on a plane is assigned a function, defined on a unit interval, which we call the function of the angular characteristic of the curve. This representation is invariant under translations, rotations, and uniform scaling, which makes it very convenient for image recognition problems. The method was designed primarily for optical handwritten symbol recognition [5]. There are known studies on image recognition, in which algorithms for recognizing curves in an image based on curve segmentation were proposed [6-8]. An algorithm for automatic segmentation of a curve on a plane (such segmentation in which each segment is well approximated by either a straight-line segment or a circular arc) can be used to control a robot manipulator [9, 10], and is also promising for trajectory control of mobile robots [11, 12].
Strokes on paper that a person perceives as 1D curved lines correspond to curvilinear structures in a digital image. Those curvilinear structures possess a width and hence are not one-dimensional manifold. But their so called medial-axis, or simply center line is one dimensional manifold. Some detectors of curvilinear structures search for exactly these central lines. As an example, the algorithm described by Steger in the paper [13] can be considered. Usually, the detected central line is represented by an ordered sequence of vertices. Of course, some central lines may be closed, and then such a sequence of vertices defines a polygon. It is possible that the central lines may intersect, and in general, the output of such a detector can be considered as a planar graph. More precisely, an algorithm has been developed that transforms a black-and-white digital image into a planar graph in such a way that the points in the image correspon ding to the graph vertices lie on imaginary central lines.
To avoid misunderstandings, two remarks should be made here. First, the term "extract a planar graph from an image" usually has a different meaning, namely, the extraction procedure is usually associated with image segmentation, and its result does not apply to curvilinear structures [14]. Second, to the best of the authors' knowledge, there are no known algorithms for detecting curvilinear structures that would directly transform an image into a planar graph. On the other hand, detecting curvilinear structures is a fairly broad area in which completely different problems are often set. The current state of affairs in this area can be found in the thesis [15].
The purpose of transforming an image in to a planar graph of the type described above was to extract features for recognition from the planar graph. To analyze the obtained graph, procedures were carried out to identify connected subgraphs and construct spanning trees for connected subgraphs. This type of planar graph analysis can be called topological analysis. The chains of a planar graph correspond to broken lines with two ends. It is for recognizing their shape that we use the angular characteristic function.
Thus, we used the angular charac teristic function for representation of continuous but not smooth plane curves. The angular characteristic function of a non-smooth curve (in particular, a broken line) has discontinuities. It is easy to see that if the graph of the angular characteristic function has the form of a broken line, that is, if the angular characteristic function is a piecewise linear function, then such an angular characteristic function represents a continuous smooth curve of a special type, which we call a polysegment line. A polysegment line is a sequence of conjugate arcs of circles and line segments. As shown in Section 2, using the angular characteristic function, one can construct an efficient algorithm for approximating a plane curve with a polysegment line.
A smooth plane curve given parametrically as ( ) ∈ ℝ2 can be considered as a spatial trajectory of
a point moving along the plane, with the velocity vector of the point being defined as ̇ . and the
acceleration vector as ̈ .It is easy to prove that for the angular velocity o f rotation of the vector ̇ ,
the formula (
̇ ∧̇ ̈ ,
= | ̇ |2
(
Let us now proceed to the parameterization of the by a natural parameter , imagining that an
imaginary point moves along the plane so that at any moment | ̇ | = 1. So, if we denote the length
of by , then equality ∀ ∈ [0. ] | ̇ ( )| = 1 holds true, and will denote ̇ ( ) as ⃗( ) to emphasize
this fact. For an arbitrary point ( 0) ∈ , we define ( 0) as (
∀ ∈ [0. ] ( ) = ̇ ( ) ∧ ̈ ( ).
We define the function ( ) by the formula (
( )
true.
where ∧ denotes the external product of vectors , ∈ ℝ2. If we denote as ( , ) the oriented
area of the parallelogram, spanned by vectors , , then in any Cartesian basis the equalities (
Assuming some uncertainty, we can give the following clear geometric definition of the function
( ): Let ⃗0 = ⃗(0) be a tangent vector to at endpoint . Then ( ) is the angle between the
on the interval = [0,1] the function ( ) by expression (
The function ( ) is defined on the interval [0, ], where is the length of the curve . We define ( 0) = lim ∠ ⃗( 0+∆ ), ⃗( 0) .
∆ →0
∆ 1 It should be noted that | ( 0)| =
, where is the radius of the tangent circle, is the curvature
of the curve at point ( 0). Note that as ∀ ∈ [0. ] | ̇ ( )| = 1
∆ →0
∠ ⃗( 0+| ̇ ( )|∆ ), ⃗( 0)
| ̇ ( )|∆
= , therefore, is analytically determined by the formula (
= ∆ ( ) = ( ∙ ). used in symbol recognition tasks.
We will call ( ) the function of the angular characteristic of the curve (or simply the angular characteristic of ). As will be shown below, this function has properties that allow it to be effectively
1) Invariance to similarity transformations of the plane.
transformations means that equality (
The most important property of the function of angular characteristics is its invariance to
similarity transformations of the plane. Below we provide the necessary explanations: A similarity
is a transformation of Euclidean plane which maps lines to lines and preserves the sizes of angels.
The set of all similarities is the similarity group (
∀ ∈ [0,1] 1( ) = 2( ). ∀ ∈ [0,1] ( ) = 0.
( ) = ∙ .
defined by (
If the curve γ is a line segment, then its function of the angular characteristic of the γ is It should be noted that the curves forming the symbol in the image can be approximated by curves made up of segments and arcs of a circle. The last two properties of the angular characteristics function make it easy to imagine the angular characteristics of these curves. In addition, the idea of approximating curves by joints of segments and arcs suggests an idea of using segmentation methods to recognize curves.
{ ⃗1, ⃗2, … , ⃗ } given by equalities (
We denote a broken line by a sequence of vertices { 0, 1
, … , }. Let ( , ) denote Cartesian
coordinates of the -th vertex. Then the broken line corresponds to a sequence of vectors
⃗ = ( − −1, − −1)
We define the sequences { 1, 2, … , } and { 1, 2, … , }. by equalities (
=1
, ≠ 0 ( ) = ∑ =1 ∙ ( ),
all points 0, 1, … ,
. Now we define on the interval [0, ] the step function by (
The function of the angular characteristic of a broken line will be determined (as for a
regular smooth curve, see (
Some problems in motion planning and robotics require that the plane curve representing a spatial trajectory be approximated by a special type of line, which we will call polysegment lines, defined as follows:
Definition 1. We define a polysegment line as a pair ( , ), where = { 1, 2, … , } is a sequence of vertices ∀ ∈ {1, . . , } ∈ ℝ 2 defining the polyline , and = { 1, 2, … , −1} is a sequence of − 1 segments. Each segment where ∈ {1, . . , − 1} is a plane curve with two ends at vertices and −1 which is either a line segment or an arc of a circle, connecting vertices and −1. A polysegment line in which any two adjacent segments and +1are either conjugate circular arcs or a straight-line segment, conjugate with an arc of a circle, we will call a conjugate polysegment line.
As an example of the problems that arise in the field of robotic automation of manufacturing, we will consider the problem of automatic generation of a program in the AS language for a robot manipulator cutting from a flat sheet of metal. In simple terms, the robot manipulator program in the AS language is a linear sequence of commands with the mnemonics "move" and "circle", determining the movement of the manipulator's working tool, respectively, along a straight line or along an arc of a circle. Accordingly, the spatial trajectory of the working tool (the cutter in this case) and the trace of the trajectory on the surface being processed are described by polysegment lines. An attempt to solve the problem by using one of the algorithms for approximating a smooth line with a broken line leads to the following issue: The operands of the move and circle commands determine the speed of movement of the working tool. If the manipulator program specifies the movement of the working tool along a broken line, then regardless of how exactly the speeds of movement on the trajectory segments were determined, the manipulator physically cannot execute such a program exactly as it is. From the point of view of physics, the reason is completely clear: the movement defined by such a program causes infinite radial acceleration of the working tool at the moments of passing the vertices of the broken line. In practice, if you load such a program into the manipulator controller, then during execution, the manipulator will stop moving every time the working tool passes over one of the vertices of an imaginary broken line on the surface of the metal sheet (the so-called trajectory trace on the workpiece surface). For some types of robotic cutting (e.g., plasma cutting), the accuracy of temporal parameters of the trajectory is critical. If the speed of the working tool (cutting tool) differs significantly from the prescribed one, this will lead to failure or even damage to the cutting tool. This problem is eliminated if we use the approximation of a smooth curve by a conjugate polysegment line instead of the approximation by a polyline. It is clear that it is desirable to have as few conjugation points as possible on a polysegment line approximating a given smooth plane curve. It is also clear that by reducing the number of segments, we reduce the achievable approximation accuracy. Taking into account the last two remarks, we can formulate the problem of optimal approximation of a smooth curve on a plane as follows: For a given maximum permissible approximation error sigma, find a polysegment line with a minimum number of segments that approximates the given plane curve with an approximation error less than or equal to sigma. As will be shown below, using the angular characteristic function allows us to construct a simple and effective algorithm for solving this problem.
The idea underlying the proposed algorithm arises from a simple observation: the graph of the angular characteristic function of an arbitrary conjugate polysegment line is a broken line. The corresponding statement is given below in expanded form as Lemma 1. The angular characteristic function of a smooth curve on a plane will, of course, be a continuous smooth function. However, a continuous smooth function can be approximated by a broken line with any given approximation accuracy. This leads to the assumption that a broken line approximating the angular characteristic function of an arbitrarily smooth gamma curve on a plane will be the angular characteristic function of a polysegment line approximating the gamma curve. Of course, any angular characteristic function corresponds to a set of lines on a plane, and we must somehow choose an approximating polysegment line from this set (which will obviously be a set of polysegment lines). For an arbitrary angular characteristic function phi, there is an analytical expression that defines a set of curves on a plane for which phi will be a function of the angular characteristic. This set is parameterizable, and the parameterization has a simple geometric meaning. We give the precise formulation of these assertions as Lemma 2. Based on the assertions of Lemma 1 and Lemma 2, it is easy to describe the set of polysegment lines defined by a piecewise linear continuous function of the angular characteristic. Finally, having at our disposal the two lemmas mentioned above, we formulate an algorithm for the optimal approximation of a smooth curve on a plane by a polysegment line.
Lemma 1: The angular characteristic function of an arbitrary polysegment line ({ 1, , . . , }, { 1, … , −1}) with the number of vertices > 2 will be a continuous piecewise linear function, analytically defined by formulas (19).
First, we formulate an easily proven statement. Let be an arbitrary smooth curve on a plane with two ends at points and . Denote by the function of the angular characteristic of this curve. Let the length of the be equal to . Let us choose an arbitrary point on the curve . Let us denote by 1 the length of the arc of the (in this case, of course, the length of the arc will be equal to − 1). Let us denote by 1 the function of the angular characteristic of the curve and by 2 the function of the angular characteristic of the curve . Then the function phi can be expressed through the functions 1 and 2, and analytically the relationship between the functions , functions 1 and 2 is expressed by formula (17).
Let us denote as the length of the -th segment of the polysegment line ({ 1, , . . , }, { 1, … , −1}) and as the function of the angular characteristic of the -th segment, considered as a separate plane curve. Obviously, is determined by the formula ∀ ∈ [0,1] ( ) = , where = 0 if is straight line segment, otherwise, if is circular arc, then = , were is the radius of the circle, corresponded to . Let us denote as { } the sequence of values given by the formula ∀ ∈ {1, … , − 1} = =1 , where = ∑ =1 is a total −1 length of the polysegment line (). According to this definition ∀ ∈ {1, … , for the values ( ) of the angular characteristic function on the set of points { }. −1 = 1. Using formula (17) and taking into account previous notion, we obtain the formula (18) − 1} ∈ [0,1] and 1
∑ ( ) =
is defined on [0,1] by formula (19)
The points { } divide the
unit interval into − 1 intervals { } 1 = [0, 1) ∀ ∈ {2, … , − 1} = ( −1. ]. On each of these intervals the function will be a linear function, thus ∀ ∈ ( ) = −1 + − −1 .
Lemma 2: For a given triple ( 0, 0, ) where vector 0 = 0 , 0 defines a position of the characteristic function uniquely determines a plane curve , given by equation (20) point on the plane, 0 ∈ ℝ
2 is a unit vector, and > 0 is a scalar parameter, an arbitrary angular ∀ ∈ [0,1] ( ) =
, ⎨ ⎩ ( ) = 0 + ∫0
0 . ( ) = cos − sin sin . cos expression (21):
where ( ) is the orthogonal matrix of the two dimension rotation operator, defined by of .
Let us denote by an arbitrary continuous function defined on the interval [0,1] and satisfying
the condition (0) = 0. For this function and an arbitrary triple ( 0, 0, ), the formula (
will be a function of the angular characteristic. If we denote by the set of all triples ( 0, 0, ) and denote by the set of all plane curves for which will be an angular characteristic function, then the formula (20) defines the mapping : Obviously, two curves defined by the formula (20) for and two different triples 1, 2 ∈ be identical, so the mapping is injective. It is also obvious that any plane curve ∈ can be → . cannot represented by the formula (20) for the function and some triple ∈ , thus the image of the set under the mapping coincides with the set . This means that the mapping is bijective, that is, the sets and are equivalent and we can use the notation , where ∈ for the elements , that determines 0, is known
Now we can formulate an algorithm for constructing a polysegment line that approximates a smooth function plane curve with two ends and . length of the curve and calculate the tangent vector 0 = ̇ (0)(we assume that the position of point
Step 1: Calculate the triple of parameters ( 0, 0, ) for the given curve, i.e., calculate the 2. Step 2: Calculate the angular characteristic function of the plane curve . (17) (18) (19) (20) (21) Step 4. Build an approximating smooth function polysegment line ( ∗0, 0, ).
3. Step 3. Build an optimal approximation of the angular characteristic function by a piecewise linear function ∗ with a given accuracy > 0.
The spanning tree of a connected graph ( , )is the maximal acyclic connected subgraph (tree) of a given graph. All vertices of a graph are also vertices of its spanning tree (See Fig.3). Let be the set of edges of the spanning tree , and = − . If = ∅, then is a tree. Otherwise, the cycle generation procedure described below is applied to the connected cyclic graph. Let ( , ) ∈ be the arbitrary edge of , that is not an edge of spanning tree . Then there is only one path in , connecting the vertices and . If we add an edge ( , ) to the set of edges ( ), then we get a cycle of the graph corresponding to the edge ( , ). So, mapping : → ( ) , where ( ) is a set of cycles of the graph ,is defined. It is easy to show that a mapping is a one-to-one mapping. (Suppose the opposite: Let 1, 2 ∈ and ( 1) = ( 2). From this it follows that there is a path in the spanning tree containing one of the edges 1, 2, which is impossible). Let denote by ⊃ ( ) image of set under mapping . In the sequel, we will denote by ( ) the set of cycles defined in this way for an arbitrary spanning tree of a connected graph . In graph theory, it is proved that the set of cycles is a basis оf the linear space of the cycles over the field ℤ2.
In view of the importance of the concept of a linear space of the cycles for understanding graph analysis procedures, we give a brief description of this mathematical construction below. The elements (vectors) of the linear space are some sets of edges of . Each element of can be represented as linear combinations of cycles of basis , considered as sets of edges. For arbitrary ∈ scalar multiplication is defined over {0,1} as follows: 1. 0 ∗ = {}, which shows that the zero-vector (the null set) is in Cycle Space 2. 1 ∗ = , which is identity
Let , ∈ then the addition operation = + be defined as a symmetric difference of the sets of edges: , ∈ 2 ( ), = ∆ . It is easy to show that 2 ( ) is an Abel group, relative to the addition operation defined in this way. It can be proved that for an arbitrary spanning tree of a connected graph , the corresponding set of cycles ( ), defined above, will be the basis of the linear space of cycles over the field ℤ2, with scalar product ∗: ℤ2 × → and addition operations +: × → defined above. The procedure of cycles generation for a given base consists in generating the set of cycles = { 1 + 2| 1, 2 ∈ ( 1) ∩ ( 2) ≠ ∅} and building the union = ∪ . The constructed set of cycles does not contain all the cycles of the graph, but it is enough for solving the problems of removing the artifacts of the planar graph and generating the chains and cycles for proper symbol recognition. The need to carry out the cycle generation operation can be explained with the example shown in Fig.4.
The fact is that it is impossible to predict in advance which of the spanning trees will be generated by the program. In this example, if the program generates the 1 spanning tree, then the corresponding cycle base 1 will not contain a “small” cycle (the top loop of the “8” symbol).
The function of the angular characteristics of the curve completely determines the geometric shape of this curve. More precisely, this function can be viewed as a mapping of the set of regular curves to the functional space ⊂ 2 of functions defined on the interval = [0,1]. As shown in the “Properties of angular characteristics function” section, this mapping is invariant with to plane similarity transformations. The concept of the function of the angular characteristics is easily transferred to broken lines. The functions of the angular characteristics of a broken line can be put in accordance with the feature vector (the procedure for extracting features is described in detail in the “feature extraction” section), which allows it to be classified. As mentioned above, the paths in the planar graph correspond to broken lines on the plane, which explains the key role of the concept of the function of the angular characteristics in the proposed method of recognizing handwritten characters.
An arbitrary function defined on an interval [ , ] can be considered as a periodic function ∀ ∈
ℝ ( + ) = ( ) with a period = − (See Fig. 6) By definition, the function of the angular
characteristics of arbitrary broken line is defined on the unit interval = [0,1] and (0) = 0, but
in the general case (
As is a piecewise continuous bounded function it is possible to define its absolute value ‖ ‖ by
= 2([−1,1]) be a 2 space of functions, defined on interval [−1,1]. Then,
∈ , due to the
fact that the norm ‖ ‖ < ∞ is defined by (
1 〈 1, 2〉 = ∫−1 1( ) ∙ 2( ) .
Each even function ∈ can be represented as a Fourier series in form (25) where functions is defined by equations (26) The system of functions (25) is orthonormal, which means that (27) holds true.
∀ , 〈 , 〉 = . vector ∈ onto the vectors of the orthonormal basis 1, 2 …: The coefficients 1, 2 … of Fourier series expansion (24) are defined as the projections (27) of the ∞ =1
1 −1 =1 cos( ∙ ∙ ) 1 < = 〈 , 〉 =
( ) ∙ ( ) . ( ) ≈
∙ ( ) For a given , the -th partial sum of the Fourier series (29) will approximate the original function. function ∈ and vector ⃗ ∈ ℝ 1, 2 ∈ approximations (30)-(31) hold true.
We will consider the real numbers 1, 2 … , as components of the vector ⃗ ∈ ℝ , so in some orthonormal basis of ℝ ⃗ = ( 1, 2 … ).Let designate by → ⃗ correspondence between even of Fourier series coefficients. Then, for arbitrary even functions
〈 1, 2〉 ≈ ⃗1 ∗ ⃗2. ‖ − ‖ ≈ ⃗ − ⃗ . (30) Both Hilbert space and Euclidean space are metric spaces. A metric in space (a set of points ) may be defined by the distance function , defined for all pairs of points , ∈ , which satisfies four ∀ , ∈ ( , ) = ( , ) ∀ , ∈ ( , ) ≥ 0
∀ ∈ ( , ) = 0 ∀ , , ∈ ( ) ≤ ( ) + ( ) As is known, in the Euclidean space ℝ , distance between ⃗, ⃗ ∈ ℝ is defined as ⃗, ⃗ = ⃗ − ⃗ . Similarly, for a Hilbert space 2 metric ∗ is defined by equation (32), where 1, 2 ∈ 2 ∗( 1, 2) = ‖ 1 − 2‖.
It is proved that conditions 1) –4) are satisfied for the metric ∗. Suppose that for a given function of the angular characteristic , the function is determined by the ratio (21) and → ⃗. If we consider the vector ⃗ as a feature vector of the Euclidean feature space, then based on the above considerations, it can be argued that such a method of feature extraction will be good suitable for curve classification tasks. But in practice we need to classify the step characteristic functions of broken lines, containing discontinuous jumps. It is highly desirable to smooth out such functions. To do this, we apply the operation described below to the vector ⃗ of the Fourier expansion coefficients of the function . Let denote by ( ) Gaussian function, defined as (33) ∞ −∞ ( ) ∙ ( ∙ ) = − 2∙ 2 Of course, the choice of the number n and the values of the parameter t strongly affect the quality of recognition. can be considered as the Fourier expansion coefficients of the function ∗, approximating the Gaussian function ( ). Let ⃗ = ( 1, 2, … , ) be the vector of the Fourier coefficients of the function , and ⃗∗ = ( 1∗, 2∗, … , ∗ ) the vector of the Fourier coefficients of the function ∗. We define the vector ⃗ = ( 1, 2, … , ) by equalities (37). of a filter with Gaussian kernel ∗ to the function (38) According to the convolution theorem for Fourier series, we can consider the components of the vector ⃗ as the Fourier expansion coefficients of the function ( ), which is a result of application
= ∙ ∗ . ( ) = ∫ −1 +1
( ) ∙ ∗( ) .
∞ where ∫−∞ arbitrary ∈ ℝ holds true.
∞ ∫−∞ ( ) ∙ cos( ∙ ) = ∫−∞ ( ) ∙ − .
∞ ( ) = 1, i.e., ( ) is normalized. As is an even function, equations (34)-(35), for For the values of the parameter , satisfying the non-equality < 2, we can assume that ( ) ≈ ∉ [−1,1]. Then the values ∗ defined by (36) 3
The method for identifying plane curves described in the article was practically tested. We developed software that implements this method and used it for experiments with character recognition of decimal digits on real photos. At the first stages of the recognition procedure, the raster image was transformed into a planar graph. As mentioned above, the planar graph obtained as a result of applying the conversion procedure contains all the necessary information for recognition, and after converting the image into a planar graph, the original image is not used in the recognition procedures. We have developed a simple text format for storing information about a planar graph in the form o f s - expressions. To visualize the resulting planar graphs, a special stand-alone application was developed (we called it “Graph Redactor”), which also allows the user to create artificial planar graphs using GUI tools (we used such artificially created planar graphs in experiments with character recognition software). All software was written in Python 2 language (Anaconda package). Since we used a small database to classify planar graphs corresponding to handwritten characters, the experimental results can only be considered as a proof of concept. However, we still consider it necessary to note that during the experiments, we observed high reliability of the implemented recognition method.
The suggested approach for recognizing plane curves was created with the intention of being utilized in tasks involving the recognition of handwritten characters. The core concept behind this method for identifying flat curves involves employing a unique function to discern the shape of a simple, continuous curve devoid of any self-intersections. Our primary focus was on character recognition techniques that involve the transformation of character images into a planar graph. This planar graph, generated through the conversion process, contains all the essential information required for recognition. As an outcome of our efforts, we designed software to put these algorithms into practice and applied it to conduct experiments involving the recognition of decimal digits in actual photographs.
Due to the radical difference between the proposed method and the existing ones, the software was created as proof of concept solely to confirm the operability of the method. For the experiments, images of the symbols of the digits 1, 2, 3, 5 were used (numbers for which the planar graphs corresponding to their symbols do not contain cycles were used) generated in a relatively small number of instances (from 10 to 20 for each symbol) by the participant of the experiment. Based on this sample, a small database was created that allowed character recognition. Then, an experiment was conducted to recognize characters generated by the same participant in the experiment. In the course of several similar experiments, 100% recognition accuracy was observed (no recognition errors). We emphasize that such experiments on small samples cannot be used for comparison with other methods and are only suitable as proof of concept.
This research was funded by the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP19679327).