The Method of Probabilistic Nodes Combination in Biomechanics
DARIUSZ JACEK JAKÓBCZAK
Department of Electronics and Computer Science,
Technical University of Koszalin,
Sniadeckich 2, 75-453 Koszalin,
POLAND
Abstract: - Proposed method, called Probabilistic Nodes Combination (PNC), is the method of 2D curve
modeling and handwriting identification by using the set of key points. Nodes are treated as characteristic
points of signature or handwriting for modeling and writer recognition. Identification of handwritten letters or
symbols need modeling and the model of each individual symbol or character is built by a choice of probability
distribution function and nodes combination. PNC modeling via nodes combination and parameter γ as
probability distribution function enables curve parameterization and interpolation for each specific letter or
symbol. Two-dimensional curve is modeled and interpolated via nodes combination and different functions as
continuous probability distribution functions: polynomial, sine, cosine, tangent, cotangent, logarithm, exponent,
arc sin, arc cos, arc tan, arc cot or power function.
Key-words: - Handwriting identification, shape modeling, curve interpolation, PNC method, nodes
combination, probabilistic modeling.
Received: September 22, 2021. Revised: September 12, 2022. Accepted: October 10, 2022. Published: November 10, 2022.
1 Introduction
Handwriting identification and writer verification
are still the open questions in artificial intelligence
and computer vision. Handwriting based author
recognition offers a huge number of significant
implementations which make it an important
research area in pattern recognition, [1]. There are
so many possibilities and applications of the
recognition algorithms that implemented methods
have to be concerned with a single problem.
Handwriting and signature identification represents
such a significant problem. In the case of writer
recognition, described in this paper, each person is
represented by the set of modeled letters or
symbols. The sketch of the proposed method
consists of three steps: first a handwritten letter or
symbol must be modeled by a curve, then
compared with an unknown letter and finally there
is a decision of identification. Author recognition
of handwriting and signature is based on the choice
of key points and curve modeling. Reconstructed
curve does not have to be smooth in the nodes
because a writer does not think about smoothing
during the handwriting. Curve interpolation in
handwriting identification is not only a pure
mathematical problem but an important task in
pattern recognition and artificial intelligence such
as: biometric recognition [2], [3], [4], personalized
handwriting recognition [5], automatic forensic
document examination [6], [7], classification of
ancient manuscripts [8]. Also writer recognition in
monolingual handwritten texts is an extensive area
of study and the methods independent from the
language are well-seen. Proposed method
represents a language-independent and text-
independent approach because it identifies the
author via a single letter or symbol from the
sample. This novel method is also applicable to
short handwritten text.
Writer recognition methods in the recent years
are going to various directions: writer recognition
using multi-script handwritten texts, [9],
introduction of new features, [10], combining
different types of features, [3], studying the
sensitivity of character size on writer identification,
[11], investigating writer identification in multi-
script environments, [9], impact of ruling lines on
writer identification, [12], model perturbed
handwriting [13], methods based on run-length
features [14,3], the edge-direction and edge-hinge
features [2], a combination of codebook and visual
features extracted from chain code and polygonized
representation of contours, [15], the autoregressive
coefficients [9], codebook and efficient code
extraction methods [16], texture analysis with
Gabor filters and extracting features, [17], using
Hidden Markov Model [18], [19]. [20] or Gaussian
Mixture Model, [1]. But no method is dealing with
writer identification via curve modeling or
interpolation and points comparison as it is
presented in this paper.
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The author wants to approach a problem of
curve interpolation, [21], [22], [23], and shape
modeling, [24] by characteristic points in
handwriting identification. Proposed method relies
on node combination and functional modeling of
curve points situated between the basic set of key
points. The functions that are used in calculations
represent a whole family of elementary functions
with inverse functions: polynomials, trigonometric,
cyclometric, logarithmic, exponential and power
functions. These functions are treated as probability
distribution functions in the range [0;1]. Nowadays
methods apply mainly polynomial functions, for
example Bernstein polynomials in Bezier curves,
splines and NURBS, [25]. But Bezier curves do not
represent the interpolation method and cannot be
used for example in signature and handwriting
modeling with characteristic points (nodes).
Numerical methods for data interpolation are based
on polynomial or trigonometric functions, for
example Lagrange, Newton, Aitken and Hermite
methods. These methods have some weak sides,
[26] and are not sufficient for curve interpolation in
the situations when the curve cannot be built by
polynomials or trigonometric functions. Proposed
2D curve interpolation is the functional modeling
via any elementary functions and it helps us to fit
the curve during handwriting identification.
This paper presents novel Probabilistic Nodes
Combination (PNC) method of curve interpolation
and takes up PNC method of two-dimensional
curve modeling via the examples using the family
of Hurwitz-Radon matrices (MHR method), [27],
but not only (other nodes combinations). The
method of PNC requires minimal assumptions: the
only information about a curve is the set of at least
two nodes. Proposed PNC method is applied in
handwriting identification via different coefficients:
polynomial, sinusoidal, cosinusoidal, tangent,
cotangent, logarithmic, exponential, arc sin, arc
cos, arc tan, arc cot or power. Function for PNC
calculations is chosen individually at each
modeling and it represents the probability
every point situated between two successive
interpolation knots. PNC method uses nodes of the
curve pi = (xi,yiR2, i = 1,2,…n:
1. PNC needs 2 knots or more (n ≥ 2);
2. If first node and last node are the same (p1
= pn), then curve is closed (contour);
3. For more precise modeling knots ought to
be settled at key points of the curve, for
example local minimum or maximum and
at least one node between two successive
local extrema.
Condition 3 means for example the highest point of
the curve in a particular orientation, convexity
changing or curvature extrema. The goal of this
paper is to answer the question: how to model a
handwritten letter or symbol by a set of knots [28]?
2 Probabilistic Interpolation
The method of PNC is computing points between
two successive nodes of the curve: calculated
points are interpolated and parameterized for real
number [0;1] in the range of two successive
nodes. PNC method uses the combinations of nodes
p1=(x1,y1), p2=(x2,y2),…, pn=(xn,yn) as h(p1,p2,…,pm)
and m = 1,2,…n to interpolate second coordinate y
for first coordinate c = xi+ (1-)xi+1, i = 1,2,…n-
1:
),...,,()1()1()( 211 mii ppphyycy
, (1)
α [0;1], γ = F(α) [0;1].
Here are the examples of h computed for MHR
method [29]:
2
1
1
21 ),( x
x
y
pph
1
2
2x
x
y
(2)
or
)(
1432243441221
2
4
2
2
yxxyxxyxxyxx
xx
.
The examples of other nodes combinations:
12
12
21
21
21 ),( yx
xy
yx
xy
pph
or
1
12
2
21
21 ),( y
xy
y
xy
pph
or
221121 ),( yxyxpph
or
212121 ),( yyxxpph
or
0),...,,( 21
m
ppph
or
111)( yxph
or others. Nodes combination is chosen
individually for each curve. Formula (1) represents
curve parameterization as α [0;1]:
x() = xi + (1-)xi+1
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and
),...,,())(1)(())(1()()( 211 mii ppphFFyFyFy
,
1211 )),...,,())(1(()()( imii yppphFyyFy
.
Proposed parameterization gives us the infinite
number of possibilities for curve calculations
(determined by choice of F and h) as there is the
infinite number of human signatures, handwritten
letters and symbols. Nodes combination is the
individual feature of each modeled curve (for
example a handwritten letter or signature).
Coefficient γ = F(α) and nodes combination h are
key factors in PNC curve interpolation and shape
modeling.
2.1 Interpolating Functions in PNC
Modeling
Points settled between the nodes are computed
using the PNC method. Each real number c [a;b]
is calculated by a convex combination c = a +
(1 - ) b for
ab
cb
[0;1].
Key question is dealing with coefficient γ in (1).
The simplest way of PNC calculation means h = 0
and γ = α (basic probability distribution). Then
PNC represents a linear interpolation. MHR
method, [30], is not a linear interpolation. MHR,
[31], is the example of PNC modeling. Each
interpolation requires specific distribution of
parameter α and γ (1) depends on parameter α
[0;1]:
γ = F(α), F:[0;1]→[0;1], F(0) = 0, F(1) = 1
and F is strictly monotonic. Coefficient γ is
calculated using different functions (polynomials,
power functions, sine, cosine, tangent, cotangent,
logarithm, exponent, arc sin, arc cos, arc tan or arc
cot, also inverse functions) and choice of function
is connected with initial requirements and curve
specifications. Different values of coefficient γ are
connected with applied functions F(α). These
functions γ = F(α) represent the examples of
probability distribution functions for random
variable α[0;1] and real number s > 0:
γ=αs, γ=sin(αs·π/2), γ=sins(α·π/2), γ=1-
cos(αs·π/2), γ=1-coss(α·π/2), γ=tan(αs·π/4),
γ=tans(α·π/4), γ=log2(αs+1), γ=log2s(α+1),
γ=(2α–1)s, γ=2/π·arcsin(αs),
γ=(2/π·arcsinα)s, γ=1-2/π·arccos(αs), γ=1-
(2/π·arccosα)s, γ=4/π·arctan(αs),
γ=(4/π·arctanα)s, γ=ctg(π/2–αs·π/4),
γ=ctgs(π/2-α·π/4), γ=2-4/π·arcctg(αs), γ=(2-
4/π·arcctgα)s.
Functions above, used in γ calculations, are strictly
monotonic for the random variable α[0;1] as γ =
F(α) is probability distribution function. Also
inverse functions F-1(α) are appropriate for γ
calculations. Choice of function and value s
depends on curve specifications and individual
requirements. Considering nowadays used
probability distribution functions for random
variable α[0;1] - one distribution is dealing with
the range [0;1]: beta distribution. Probability
density function f for random variable α[0;1] is:
rs
cf )1()(
, s ≥ 0, r ≥ 0. (3)
When r = 0 probability density function (3)
represents
s
cf
)(
and then probability
distribution function F is like
2
3)(
f
and γ =
α3. If s and r are positive integer numbers then γ is
the polynomial, for example
)1(6)(
f
and γ
= 3α2-2α3. Beta distribution gives us coefficient γ in
(1) as polynomial because of interdependence
between probability density f and distribution F
functions:
)(')(
Ff
,
dttfF )()(
0
. (4)
For example (4):
ef )(
and
1)1()(
eF
.
What is very important in PNC method: two curves
(for example a handwritten letter or signature) may
have the same set of nodes but different h or γ
results in different interpolations (Fig.6-14).
Algorithm of the PNC interpolation and
modeling (1) looks as follows:
Step 1: Choice of knots pi at key points.
Step 2: Choice of nodes combination
h(p1,p2,…,pm).
Step 3: Choice of distribution γ = F(α).
Step 4: Determining values of α: α = 0.1, 0.2…0.9
(nine points) or 0.01, 0.02…0.99 (99 points) or
others.
Step 5: The computations (1).
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These five steps can be treated as the algorithm of
the PNC method of curve modeling and
interpolation (1).
Curve interpolation has to implement the
coefficients γ. Each strictly monotonic function F
between points (0;0) and (1;1) can be used in PNC
interpolation.
3 Handwriting Modeling and
Recognition
The PNC method enables signature and
handwriting recognition. This process of
recognition consists of three parts:
1. Modeling – choice of nodes combination and
probabilistic distribution function (1) for
known signature or handwritten letters;
2. Unknown writer - choice of characteristic
points (nodes) for unknown signature or
handwritten word and the coefficients of
points between nodes;
3. Decision of recognition - comparing the
results of PNC interpolation for known
models with coordinates of unknown text.
3.1 Modeling – The Basis of Patterns
Known letters or symbols ought to be modeled by
the choice of nodes, determining specific nodes
combination and characteristic probabilistic
distribution function. For example a handwritten
word or signature “rw” may look different for
persons A, B or others. How to model “rw” for
some persons via the PNC method? Each model
has to be described by the set of nodes for letters
“r” and “w”, nodes combination h and a function
γ=F(α) for each letter. Less complicated models
can take h(p1,p2,…,pm) = 0 and then the formula of
interpolation (1) looks as follows:
1
)1()(
ii yycy
.
It is linear interpolation for basic probability
distribution (γ = α). How does the first letter “r” be
modeled in three versions for nodes combination h
= 0 and α=0.1,0.2…0.9? Of course α is a random
variable and α[0;1].
Person A
Nodes (1;3), (3;1), (5;3), (7;3) and γ = F(α) = α2:
Fig. 1: PNC modeling for nine reconstructed points
between nodes.
Person B
Nodes (1;3), (3;1), (5;3), (7;2) and γ = F(α) = α2:
Fig. 2: PNC modeling of letter “r” with four nodes.
Person C
Nodes (1;3), (3;1), (5;3), (7;4) and γ = F(α) = α3:
Fig. 3: PNC modeling of handwritten letter “r”.
These three versions of letter “r” (Fig.1-3) with
nodes combination h = 0 differ at fourth node and
probability distribution functions γ = F(α). Much
more possibilities of modeling are connected with a
choice of nodes combination h(p1,p2,…,pm). MHR
method, [32], uses the combination (2) with good
features because of orthogonal rows and columns at
Hurwitz-Radon family of matrices:
11),( i
i
i
ii x
x
y
pph
i
i
ix
x
y
1
1
and then (1)
),,()1()1()( 11 iiii pphyycy
.
Here are two examples of PNC modeling with
MHR combination (2).
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Person D
Nodes (1;3), (3;1), (5;3) and γ = F(α) = α2:
Fig. 4: PNC modeling of letter “r” with three
nodes.
Person E
Nodes (1;3), (3;1), (5;3) and γ = F(α) = α1.5:
Fig. 5: PNC modeling of handwritten letter “r”.
Fig.1-5 shows modeling of the letter “r”. Now let
us consider a letter “w” with nodes combination h =
0.
Person A
Nodes (2;2), (3;1), (4;2), (5;1), (6;2) and γ = F(α) =
(5α - 1)/4:
Fig. 6: PNC modeling for nine reconstructed points
between nodes.
Person B
Nodes (2;2), (3;1), (4;2), (5;1), (6;2) and γ = F(α) =
sin(α·π/2):
Fig. 7: PNC modeling of letter “w” with five nodes.
Person C
Nodes (2;2), (3;1), (4;2), (5;1), (6;2) and γ = F(α) =
sin3.5(α·π/2):
Fig. 8: PNC modeling of handwritten letter “w”.
These three versions of letter “w” (Fig.6-8) with
nodes combination h = 0 and the same nodes differ
only at probability distribution functions γ = F(α).
Fig.9 is the example of nodes combination h (2)
from MHR method:
Person D
Nodes (2;2), (3;1), (4;1), (5;1), (6;2) and γ = F(α) =
2α - 1:
Fig. 9: PNC modeling for nine reconstructed points
between nodes.
Examples above have one function γ = F(α) and
one combination h for all ranges between nodes.
But it is possible to create a model with functions γi
= Fi(α) and combinations hi individually for a range
of nodes (pi;pi+1). It enables very precise modeling
of handwritten symbols between each successive
pair of nodes.
Each person has its own characteristic and
individual handwritten letters, numbers or other
marks. The range of coefficients x has to be the
same for all models because of comparing
appropriate coordinates y. Every letter is modeled
by PNC via three factors: the set of nodes,
probability distribution function γ = F(α) and nodes
combination h. These three factors are chosen
individually for each letter, therefore this
information about modeled letters seems to be
enough for specific PNC curve interpolation,
comparing and handwriting identification. Function
γ is selected via the analysis of points between
nodes and we may assume h = 0 at the beginning.
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What is very important - PNC modeling is
independent of the language or a kind of symbol
(letters, numbers or others). One person may have
several patterns for one handwritten letter.
Summarize: every person has the basis of patterns
for each handwritten letter or symbol, described by
the set of nodes, probability distribution function γ
= F(α) and nodes combination h. Whole basis of
patterns consists of models Sj for j = 0,1,2,3…K.
3.2 Unknown Author – Points of
Handwritten Character
Choice of characteristic points (nodes) for
unknown letters or handwritten symbols is a crucial
factor in object recognition. The range of
coefficients x has to be the same like the x range on
the basis of patterns. Knots of the curve (opened or
closed) ought to be settled at key points, for
example local minimum or maximum (the highest
point of the curve in a particular orientation),
convexity changing or curvature maximum and at
least one node between two successive key points.
When the nodes are fixed, each coordinate of every
chosen point on the curve (x0c,y0c), (x1c,y1c),…,
(xMc,yMc) is accessible to be used for comparison
with the models. Then probability distribution
function γ = F(α) and nodes combination h have to
be taken from the basis of modeled letters to
calculate appropriate second coordinates yi(j) of the
pattern Sj for first coordinates xic, i = 0,1,…,M.
After interpolation it is possible to compare given
handwritten symbol with a letter in the basis of
patterns.
3.3 Recognition – The Writer
Comparing the results of PNC interpolation for
required second coordinates of a model in the basis
of patterns with points on the curve (x0c,y0c),
(x1c,y1c),…, (xMc,yMc), we can say if the letter or
symbol is written by person A, B or another. The
comparison and decision of recognition, [33], is
done via minimal distance criterion. Curve points
of unknown handwritten symbol are: (x0c,y0c),
(x1c,y1c),…, (xMc,yMc). The criterion of recognition
for models Sj = {(x0c,y0(j)), (x1c,y1(j)),…, (xMc,yM(j))},
j=0,1,2,3…K is given as:
min
0
)(
M
i
j
i
c
iyy
.
Minimal distance criterion helps us to fix a
candidate for an unknown writer as a person from
the model Sj .
4 Conclusions
The method of Probabilistic Nodes Combination
(PNC) enables interpolation and modeling of two-
dimensional curves, [34], using nodes
combinations and different coefficients γ:
polynomial, sinusoidal, cosinusoidal, tangent,
cotangent, logarithmic, exponential, arc sin, arc
cos, arc tan, arc cot or power function, also inverse
functions. Function for γ calculations is chosen
individually at each curve modeling and it is treated
as probability distribution function: γ depends on
initial requirements and curve specifications. PNC
method leads to curve interpolation as handwriting
or signature identification via discrete set of fixed
knots. PNC makes possible the combination of two
important problems: interpolation and modeling in
a matter of writer identification. Main features of
PNC method are:
a) the smaller distance between knots the
better;
b) calculations for coordinates close to zero
and nearby extremum require more
attention because of importance of these
points;
c) PNC interpolation develops a linear
interpolation into other functions as
probability distribution functions;
d) PNC is a generalization of MHR method
via different nodes combinations;
e) interpolation of L points is connected with
the computational cost of rank O(L) as in
MHR method;
f) nodes combination and coefficient γ are
crucial in the process of curve probabilistic
parameterization and interpolation: they are
computed individually for a single curve.
Future works are going to: application of PNC
method in signature and handwriting recognition,
choice and features of nodes combinations and
coefficient γ, implementation of PNC in computer
vision and artificial intelligence: shape geometry,
contour modelling, object recognition and curve
parameterization.
References:
[1] Schlapbach, A., Bunke, H.: Off-line writer
identification using Gaussian mixture
models. In: International Conference on
Pattern Recognition, pp. 992–995 (2006)
[2] Bulacu, M., Schomaker, L.: Text-
independent writer identification and
verification using textural and allographic
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2022.17.50
Dariusz Jacek Jakóbczak
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features. IEEE Trans. Pattern Anal. Mach.
Intell. 29 (4), 701–717 (2007)
[3] Djeddi, C., Souici-Meslati, L.: A texture
based approach for Arabic writer
identification and verification. In:
International Conference on Machine and
Web Intelligence, pp. 115–120 (2010)
[4] Djeddi, C., Souici-Meslati, L.: Artificial
immune recognition system for Arabic writer
identification. In: International Symposium
on Innovation in Information and
Communication Technology, pp. 159–165
(2011)
[5] Nosary, A., Heutte, L., Paquet, T.:
Unsupervised writer adaption applied to
handwritten text recognition. Pattern Recogn.
Lett. 37 (2), 385–388 (2004)
[6] Van, E.M., Vuurpijl, L., Franke, K.,
Schomaker, L.: The WANDA measurement
tool for forensic document examination. J.
Forensic Doc. Exam. 16, 103–118 (2005)
[7] Schomaker, L., Franke, K., Bulacu, M.:
Using codebooks of fragmented connected-
component contours in forensic and historic
writer identification. Pattern Recogn. Lett. 28
(6), 719–727 (2007)
[8] Siddiqi, I., Cloppet, F., Vincent, N.: Contour
based features for the classification of
ancient manuscripts. In: Conference of the
International Graphonomics Society, pp.
226–229 (2009)
[9] Garain, U., Paquet, T.: Off-line multi-script
writer identification using AR coefficients.
In: International Conference on Document
Analysis and Recognition, pp. 991–995
(2009)
[10] Bulacu, M., Schomaker, L., Brink, A.: Text-
independent writer identification and
verification on off-line Arabic handwriting.
In: International Conference on Document
Analysis and Recognition, pp. 769–773
(2007)
[11] Ozaki, M., Adachi, Y., Ishii, N.: Examination
of effects of character size on accuracy of
writer recognition by new local arc method.
In: International Conference on Knowledge-
Based Intelligent Information and
Engineering Systems, pp.1170–1175 (2006)
[12] Chen, J., Lopresti, D., Kavallieratou, E.: The
impact of ruling lines on writer identification.
In: International Conference on Frontiers in
Handwriting Recognition, pp. 439–444
(2010)
[13] Chen, J., Cheng, W., Lopresti, D.: Using
perturbed handwriting to support writer
identification in the presence of severe data
constraints. In: Document Recognition and
Retrieval, pp. 1–10 (2011)
[14] Galloway, M.M.: Texture analysis using gray
level run lengths. Comput. Graphics Image
Process. 4 (2), 172–179 (1975)
[15] Siddiqi, I., Vincent, N.: Text independent
writer recognition using redundant writing
patterns with contour-based orientation and
curvature features. Pattern Recogn. Lett. 43
(11), 3853–3865 (2010)
[16] Ghiasi, G., Safabakhsh, R.: Offline text-
independent writer identification using
codebook and efficient code extraction
methods. Image and Vision Computing 31,
379–391 (2013)
[17] Shahabinejad, F., Rahmati, M.: A new
method for writer identification and
verification based on Farsi/Arabic
handwritten texts, Ninth International
Conference on Document Analysis and
Recognition (ICDAR 2007), pp. 829–833
(2007)
[18] Schlapbach, A., Bunke, H.: A writer
identification and verification system using
HMM based recognizers, Pattern Anal. Appl.
10, 33–43 (2007)
[19] Schlapbach, A., Bunke, H.: Using HMM
based recognizers for writer identification
and verification, 9th Int. Workshop on
Frontiers in Handwriting Recognition, pp.
167–172 (2004)
[20] Marti, U.-V., Bunke, H.: The IAM-database:
an English sentence database for offline
handwriting recognition, Int. J. Doc. Anal.
Recognit. 5, 39–46 (2002)
[21] Collins II, G.W.: Fundamental Numerical
Methods and Data Analysis. Case Western
Reserve University (2003)
[22] Chapra, S.C.: Applied Numerical Methods.
McGraw-Hill (2012)
[23] Ralston, A., Rabinowitz, P.: A First Course
in Numerical Analysis – Second Edition.
Dover Publications, New York (2001)
[24] Zhang, D., Lu, G.: Review of Shape
Representation and Description Techniques.
Pattern Recognition 1(37), 1-19 (2004)
[25] Schumaker, L.L.: Spline Functions: Basic
Theory. Cambridge Mathematical Library
(2007)
[26] Dahlquist, G., Bjoerck, A.: Numerical
Methods. Prentice Hall, New York (1974)
[27] Jakóbczak, D.: 2D and 3D Image Modeling
Using Hurwitz-Radon Matrices. Polish
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2022.17.50
Dariusz Jacek Jakóbczak
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Volume 17, 2022
Journal of Environmental Studies 4A(16),
104-107 (2007)
[28] Jakóbczak, D.: Shape Representation and
Shape Coefficients via Method of Hurwitz-
Radon Matrices. Lecture Notes in Computer
Science 6374 (Computer Vision and
Graphics: Proc. ICCVG 2010, Part I),
Springer-Verlag Berlin Heidelberg, 411-419
(2010)
[29] Jakóbczak, D.: Curve Interpolation Using
Hurwitz-Radon Matrices. Polish Journal of
Environmental Studies 3B(18), 126-130
(2009)
[30] Jakóbczak, D.: Application of Hurwitz-
Radon Matrices in Shape Representation. In:
Banaszak, Z., Świć, A. (eds.) Applied
Computer Science: Modelling of Production
Processes 1(6), pp. 63-74. Lublin University
of Technology Press, Lublin (2010)
[31] Jakóbczak, D.: Object Modeling Using
Method of Hurwitz-Radon Matrices of Rank
k. In: Wolski, W., Borawski, M. (eds.)
Computer Graphics: Selected Issues, pp. 79-
90. University of Szczecin Press, Szczecin
(2010)
[32] Jakóbczak, D.: Implementation of Hurwitz-
Radon Matrices in Shape Representation. In:
Choraś, R.S. (ed.) Advances in Intelligent
and Soft Computing 84, Image Processing
and Communications: Challenges 2, pp. 39-
50. Springer-Verlag, Berlin Heidelberg
(2010)
[33] Jakóbczak, D.: Object Recognition via
Contour Points Reconstruction Using
Hurwitz-Radon Matrices. In: Józefczyk, J.,
Orski, D. (eds.) Knowledge-Based Intelligent
System Advancements: Systemic and
Cybernetic Approaches, pp. 87-107. IGI
Global, Hershey PA, USA (2011)
[34] Jakóbczak, D.: Curve Parameterization and
Curvature via Method of Hurwitz-Radon
Matrices. Image Processing &
Communications- An International Journal 1-
2(16), 49-56 (2011).
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WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2022.17.50
Dariusz Jacek Jakóbczak
E-ISSN: 2224-2856
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Volume 17, 2022