=Paper=
{{Paper
|id=Vol-3609/paper17
|storemode=property
|title=Parallel Algorithms for Interpolation with Bezier Curves and B-Splines for Medical Data Recovery
|pdfUrl=https://ceur-ws.org/Vol-3609/short3.pdf
|volume=Vol-3609
|authors=Lesia Mochurad,Yulianna Mochurad
|dblpUrl=https://dblp.org/rec/conf/iddm/MochuradM23
}}
==Parallel Algorithms for Interpolation with Bezier Curves and B-Splines for Medical Data Recovery==
https://ceur-ws.org/Vol-3609/short3.pdf
Parallel Algorithms for Interpolation with Bezier Curves and B-
Splines for Medical Data Recovery
Lesia Mochurad a, Yulianna Mochurad b
a
Lviv Polytechnic National University, 12 Bandera street, Lviv, 79013, Ukraine
b
Danylo Halytsky Lviv National Medical University, 69 Pekarska street, Lviv, 79010, Ukraine
Abstract
Clinical data are increasingly being used to generate new medical knowledge to improve
diagnostic accuracy, better personalize therapeutic regimens, improve clinical outcomes, and
more efficiently use healthcare resources. However, clinical data often become available only
at irregular intervals, contingent upon patients and data types, with missing records or
unknown indicators. Consequently, missing data often constitute a primary impediment to
optimal knowledge extraction from clinical data. This study analyzes proposed parallel
interpolation algorithms using Bézier curves and B-splines for medical data restoration.
Parallelization is achieved through CUDA technology. The research reveals that the
considered algorithms do not compromise the accuracy of subsequent forecasting; in fact,
they marginally enhance it. Acceleration indices are obtained, indicating an acceleration
factor of 11 for B-splines and – 97 for Bézier curves. The latter allows for swift data
restoration, exerting minimal impact on the overall forecasting time – a crucial aspect in
cardiovascular disease scenarios. Further exploration of this research can involve variations
and selection of an optimal degree of interpolation curves, forecasting methodologies, and the
size of the studied dataset.
Keywords 1
Medical diagnosis, interpolation, Bézier curves, B-splines, CUDA parallel computing
technology.
1. Introduction
Medical diagnostics are advancing each year, reaching new heights of precision through emerging
technologies [1-3]. However, the challenge of medical data loss during collection or transmission
persists, leading to difficulties or even rendering subsequent analysis infeasible. This underscores the
need for exploring various approaches to restore lost medical data for further processing and
utilization. One such approach is interpolation, which facilitates the recovery of incomplete data
based on known values, thereby generating more accurate forecasts.
Particularly, interpolation finds application in various domains including Geodesy and
Cartography (for constructing maps and Earth surface models using geodetic data) [4]; Physics (for
approximating and constructing smooth functions describing experimental data) [5-8]; Computer
Graphics (for building smooth curves and surfaces utilized in animation and 3D modeling) [9];
Finance (for market prediction and data analysis) [10, 11]; and Medicine (interpolation is often used
for analyzing and processing medical images) [12].
Different types of interpolation methods exist: linear interpolation, nearest-neighbor method, cubic
spline interpolation method, Lanczos interpolation, shape-preserving method, thin plate spline
method, biharmonic interpolation method, Bézier curves, and B-splines. In [13], the author conducts
research on and compares various interpolation methods. Unlike this study, our research is conducted
on a medical dataset, whereas the examined article deals with image interpolation.
IDDM’2023: 6th International Conference on Informatics & Data-Driven Medicine, November 17 - 19, 2023, Bratislava, Slovakia
EMAIL: lesia.i.mochurad@lpnu.ua (L. Mochurad); yuliannamochurad@gmail.ua (Yu. Mochurad)
ORCID: 0000-0002-4957-1512 (L. Mochurad); 0009-0008-4486-7139 (Yu. Mochurad)
© 2023 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Wor
Pr
ks
hop
oceedi
ngs
ht
I
tp:
//
ceur
-
SSN1613-
ws
.or
0073
g
CEUR Workshop Proceedings (CEUR-WS.org)
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
Article [14] describes the Data Analytics Challenge on Missing data Imputation (DACMI), which
introduced a joint clinical dataset with known outcomes to assess and develop advanced methods for
filling missing data in clinical time series. The dataset involved thirteen commonly measured blood
laboratory indicators. Methods such as Recurrent Neural Network (RNN) combined with Multilayer
Perceptron (MLP), weighted averaging from K-Nearest Neighbors (KNN), linear interpolation,
piecewise-linear interpolation, nonlinear extension of Multivariate Imputation by Chained Equations
(MICE), among others, were employed to evaluate the effectiveness of missing data imputation.
Notably, KNN-based methods outperformed 3D-MICE, highlighting limitations in relying solely on a
restricted search for similar patients when imputing missing values.
In [15], a new method for data imputation using artificial neural networks is proposed. Despite its
high accuracy, it demands significant time for training, particularly in the analysis of high-
dimensional datasets.
Based on the analysis of available literature, the use of B-splines and Bézier curves in medical data
interpolation is relatively rare. This indicates the necessity for further in-depth investigation and
performance comparison of these methods in the context of medical data. Additionally, it's essential
for the data restoration phase to be swift and minimally affect the overall analysis time in more
complex tasks involving medium and large-sized data. Thus, the study of parallelization processes to
optimize interpolation methods becomes pertinent.
Article [16] proposes a parallel version of the cross-interpolation algorithm for calculating high-
dimensional integrals, particularly those arising in the Ising model in quantum physics. This algorithm
differs from traditional methods like Monte Carlo and quasi-Monte Carlo, using computation results
on individual dimensions instead of random points. The authors demonstrate the algorithm's high
efficiency, especially with strongly superlinear convergence. Due to its flexibility and potential for
parallel execution across processes, the algorithm is deemed beneficial for high-dimensional
integration tasks in various fields including statistics, probability, and quantum physics.
In [17], the authors explore the application of B-spline interpolation in the medical field. This
method enables the reconstruction of three-dimensional objects derived from medical images and is
used for image-guided surgery (IGS) navigation and medical image registration tasks. Given the
complexity and data-intensive nature of IGS tasks, the authors propose utilizing graphical processing
units (GPUs) for parallelization. A novel B-spline interpolation method on GPUs is introduced,
minimizing data transfer between memory and computational cores during input mesh loading.
Trilinear interpolation is also employed to reduce computational complexity and enhance method
accuracy. The authors integrate the method into a liver deformation compensation workflow,
demonstrating a 6.5-fold interpolation acceleration on their dataset using GPU parallelization. The
goal of our work is to develop, analyze, and compare parallel interpolation algorithms using Bézier
curves and B-splines for the restoration of medical data. To achieve this objective, we will conduct a
study of interpolation algorithms, perform experiments to compare the effectiveness of Bézier curves
and B-splines in medical data restoration, and explore the possibility of optimizing these methods
through parallelization to enhance computational speed and result accuracy.
The research task can be outlined as follows: Let us consider an input dataset, denoted as D –a
matrix comprising n rows and s columns. This dataset contains missing values. The aim is to
determine the values of the missing elements within the matrix D using Bézier curve and B-spline
interpolation. For each missing value to be restored, an interpolation function is employed, based on
known values of neighboring points.
It is of paramount importance to rapidly restore missing data, as the resulting dataset D' will
subsequently be utilized for the analysis and prediction of heart attacks.
2. Methodic and materials
2.1. Dataset description
For this study, the 'Heart Failure Prediction Dataset' [18] was utilized, consisting of 918 rows and
12 columns. Cardiovascular diseases (CVD) are the number one cause of death worldwide, claiming
approximately 17.9 million lives each year, representing 31% of all global deaths. Four out of five
CVD-related deaths are caused by heart attacks and strokes, with one-third of these deaths occurring
prematurely in individuals under 70 years of age. Heart failure is a prevalent event resulting from
cardiovascular diseases, and this dataset comprises 11 features that can be employed to predict
possible heart conditions.
Features:
• Age: patient's age [years];
• Sex: patient's gender [M: male, F: female];
• ChestPainType: type of chest pain [TA: typical angina, ATA: atypical angina, NAP: non-anginal
pain, ASY: asymptomatic];
• BloodPressure: resting blood pressure [mm Hg];
• Cholesterol: serum cholesterol [mg/dl];
• FastingBS: fasting blood sugar level [1: if FastingBS > 120 mg/dl, 0: otherwise];
• RestingECG: resting electrocardiogram results [Normal: normal, ST: ST-T wave abnormality (T
wave inversion and/or ST elevation or depression > 0.05 mV), LVH: possible or definite left
ventricular hypertrophy by Estes' criteria];
• MaxHR: maximum heart rate achieved [numerical value from 60 to 202];
• ExerciseAngina: exercise-induced angina [Y: Yes, N: No];
• Oldpeak: oldpeak = ST [numerical value measured in depression];
• ST_Slope: the slope of the peak exercise ST segment [Up: upsloping, Flat: flat, Down:
downsloping];
• HeartDisease: output class [1: heart disease, 0: normal].
This dataset does not contain missing data, so we create missing values as follows:
Replacing random values with NaN
for col in cols:
mask = np.random.rand(len(origin)) < 0.2
origin.loc[mask, col] = np.nan
As a result, we will have two datasets, one loaded without missing values, and the other one
generated by us for interpolation purposes. Having a dataset without missing values will help us
analyze the reliability and advantages of using the proposed interpolation algorithms more effectively.
2.2 B-Splines
B-Splines are integral components of many algorithms used in computer graphics, signal
processing, image processing, and various other fields of science and technology [19]. B-Splines can
be constructed using a recursive formula, dependent on their degree and the number of control points.
Let's consider a vector of control points 𝒕𝒊 , 𝒊 = 𝟎, '''''''
𝒎, and a function 𝒚 = 𝒇(𝒙) that we wish to
interpolate. The process of B-Spline interpolation involves constructing B-Spline basis functions
𝑩𝒊,𝒌 (𝒙), where 𝒌 is the degree, satisfying the conditions:
• each basis function 𝐵$,% (𝑥) is defined over the interval [𝑡$ , 𝑡$&%&' ];
• 𝑩𝒊,𝒌 (𝒙) is a polynomial of degree k within each interval [𝒕𝒊 , 𝒕𝒊&𝒌&𝟏 ];
• each point 𝒙 lies within at most 𝒌 + 𝟏 neighboring basis functions.
B-Spline basis functions can be computed recursively using formulas (1)-(2), which define 𝒌 basis
functions in terms of degree 𝒌 − 𝟏 basis functions:
1, 𝑡$ ≤ 𝑥 ≤ 𝑡$&' (1)
𝐵$,) (𝑥) = 8 ,
0, 𝑒𝑙𝑠𝑒.
𝑥 − 𝑥$ 𝑡$&%&' − 𝑥 (2)
𝐵$,% (𝑥) = 𝐵$,%*' (𝑥) + 𝐵$&',%*' (𝑥).
𝑡$&% − 𝑡$ 𝑡$&%&' − 𝑡$&'
Further, we will formulate a general algorithm for finding missing values in the dataset:
1. Input: 𝑫 (dataset with missing values).
2. For each of the 𝒔 columns in the dataset:
• Find those n rows (index 𝒊) for which there are missing values in the column.
• For each of the 𝒏 rows with missing values:
o Find the nearest preceding row with a value (index 𝒊 − 𝟏) and the nearest succeeding
row with a value (index 𝒊 + 𝟏).
o Record the values from these rows as 𝒙_𝟏 and 𝒙_𝟐, respectively.
o Record the corresponding y values, 𝒚_𝟏 and 𝒚_𝟐.
o Calculate B-Spline coefficients.
o Compute the interpolated value using B-Spline coefficients and 𝒙 values.
o Replace the missing value in the dataset with the interpolated value.
• Output the updated dataset 𝑫′ with filled missing values.
Description of the parallelization algorithm using CUDA [20] and Python:
1. Set up and configure the CUDA-compatible environment on the respective computing device
(GPU).
2. Load the dataset into the device's memory (GPU).
3. Create a CUDA kernel for B-Spline computation.
4. Divide the dataset into blocks for parallel processing.
5. Launch the CUDA kernel concurrently for each data block, computing B-Spline for each
missing value.
6. Store the interpolated values on the device (GPU).
7. Load the updated values from the device (GPU) back to the host memory (CPU).
8. Replace missing values in the dataset with updated values.
9. Output the updated dataset with filled missing values.
In this pseudocode, functions from the PyCUDA library are used for interaction with CUDA. The
parallelization process involves computing B-Spline for each missing value on the GPU. Each
missing value is processed individually in its own computation thread.
The main function 'interpolate_missing_values' performs the following steps:
1. Identifies the indices of rows with missing values in the dataset.
2. Creates and copies data to the device (GPU).
3. Defines block sizes and grids for launching the CUDA kernel.
4. Launches the CUDA kernel 'compute_spline' in parallel for each data block.
5. Loads interpolated values from the device back to the host (CPU).
6. Replaces missing values in the dataset with the updated values.
7. Returns the filled dataset.
Evaluation of the computational complexity of each algorithm stage:
1. Identification of rows with missing values: This stage requires iterating through all rows in
the dataset, thus having a complexity of 𝑶(𝒏).
2. Finding nearest preceding and succeeding rows with values: This stage also requires iterating
through all rows in the dataset, resulting in a complexity of 𝑶(𝒏).
3. Calculation of B-Spline coefficients: This stage doesn't require iterations over all rows, but
for each missing value, coefficients need to be calculated. Considering this, the complexity of this
stage can be estimated as 𝑶(𝟏).
4. Computation of interpolated values: This stage also has a complexity of 𝑶(𝟏) since it
involves basic mathematical operations on B-Spline coefficients and x values.
5. Replacement of missing values with interpolated values: This stage also has a complexity of
𝑶(𝒏), as it requires updating corresponding values in the dataset.
Therefore, the overall computational complexity of the algorithm for one iteration of B-Spline
interpolation for one dataset column is estimated to be 𝑶(𝒏).
For the entire dataset containing n rows, the total computational complexity of the algorithm will
depend on the number of missing values in the columns where B-Spline interpolation is performed. If
we denote p as the count of missing values in the dataset, the total computational complexity of the
algorithm for the entire dataset will be 𝑶(𝒏 ∙ 𝒑). This is due to the fact that each missing value
requires performing an iteration of B-Spline interpolation, resulting in a complexity of 𝑶(𝒏).
2.3 Bezier curve
A Bézier curve of degree k is defined by the formula:
𝑅(𝑡) = ∑%$+) 𝐵$% (𝑡)𝑃$ , 𝑡 ∈ [0, 1], (3)
where 𝐵$% (𝑡) = C%$ 𝑡 $ (1 − 𝑡)%*$ – Bézier curve basis functions, also known as Bernstein polynomials,
%!
𝐶%$ = $!(%*$)!, 𝑃$ = O0/! P.
!
Some of the properties of Bernstein polynomials significantly affect the behavior of Bézier curves.
Let's mention the main ones: Bernstein polynomials take non-negative values; the sum of Bernstein
polynomials yields one, satisfying condition (2); Bernstein polynomials do not depend on the vertices
of the array 𝑃 but only on the number of points in it.
%
Q 𝐵$% (𝑡) = 1 (4)
$+)
In scalar form, equation (3) is written as follows:
𝑥(𝑡) = ∑%$+) 𝐶%$ 𝑡 $ (1 − 𝑡)%*$ 𝑥$ ; 𝑦(𝑡) = ∑%$+) 𝐶%$ 𝑡 $ (1 − 𝑡)%*$ 𝑦$ (5)
The general algorithm for finding missing values in the dataset based on Bézier curves and its
parallelization using CUDA is analogous to the use of B-splines. As a result, the computational
complexity estimation of the proposed algorithm is as follows:
1. Reading the dataset has a complexity of 𝑂(𝑛 ∙ 𝑠), where 𝑛 is the number of rows, and 𝑠 is the
number of columns.
2. Iterating over each column to find missing values has a complexity of 𝑂(𝑠).
3. For each missing value, searching for neighboring rows, computing intermediate points, and
replacing the missing value have a complexity of 𝑂(1).
4. The overall computational complexity of the algorithm for a single CUDA kernel is 𝑂(𝑠).
5. The overall computational complexity of the algorithm with parallelization depends on the
1
number of CUDA kernels. If we have p CUDA kernels, the complexity becomes 𝑂( + 𝑝).
2
Therefore, the total computational complexity of the algorithm for the entire dataset with
1
parallelization using p processes or CUDA kernels is 𝑂(𝑛 ∙ 𝑠 + 2 + 𝑝).
3. Numerous experiments
For the analysis and comparison of interpolation results in the study, the following metrics were
utilized:
• Mean Absolute Error (MAE): This metric measures the average absolute difference between
interpolated values and original values. A lower MAE indicates better performance.
• Coefficient of Determination (R-squared, R^2): This metric measures the proportion of the
variance in the original values that can be explained by the interpolated model. R^2 values range
from 0 to 1, where a value of 1 indicates a perfect fit. A higher R^2 value indicates a better fit.
These metrics provide insights into the quality and accuracy of the interpolation methods used. A
lower MAE and a higher R^2 value are indicative of more accurate interpolation and better model fit.
Table 1
The values of the metrics for the utilized interpolation methods are as follows
metrics B-Spline interpolation Bezier curve interpolation
МАЕ 0.19019 0.09976
R^2 0.8664 0.9259
As evident from Table 1, Bezier curve interpolation yields better results than B-spline
interpolation. Based on these values, one can conclude that the Bezier curve interpolation model can
explain approximately 92.6% of the variation between interpolated and original values. This suggests
that the model fits the data quite well and performs interpolation effectively.
Since the dataset considered in the study is created for predicting heart attacks, it is important to
conduct post-interpolation forecasting to ensure that interpolation doesn't compromise prediction
accuracy.
For the purpose of comparison and visualization of prediction results, the following metrics are
used:
• Accuracy: The ratio of correctly classified samples to the total number of samples in the test
set. This metric indicates how accurately the model predicts classes.
• Precision: The ratio of correctly classified positive samples to the total number of predicted
positive samples. This metric measures how well the model identifies positive classes.
• Recall: The ratio of correctly classified positive samples to the total number of actual positive
samples in the test set. This metric shows how well the model detects positive classes.
• F1-score: The harmonic mean of precision and recall. This metric is used to assess model
accuracy in cases of class imbalance.
• ROC AUC: A metric that measures the effectiveness of the model at different binarization
thresholds. The ROC curve illustrates the relationship between the true positive rate and false
positive rate. A higher AUC indicates a better model.
Without loss of generality, logistic regression is used for prediction. It is relatively simple to
implement and performs well for binary classification tasks.
Table 2
Metrics Values Before and After Interpolation Prediction
Based on Interpolation
Without
Metrics Using B-spline Using Bezier
Interpolation
Interpolation Curves
Accuracy 0.8432 0.8745 0.8691
Precision 0.8571 0.8712 0.8664
Recall 0.8532 0.8695 0.8641
F1-score 0.8539 0.8699 0.8646
AUC-ROC 0.9257 0.9392 0.9283
Based on the results from Table 2, the following conclusions can be drawn:
1. Prediction without prior interpolation has the lowest values for all metrics, indicating that a
model not accounting for missing values is not accurate enough.
2. Prediction using Bezier curves yields slightly lower AUC-ROC values than the model with B-
spline interpolation. Bezier curves are interpolation curves that use control points to model curve
dependencies in the data.
Therefore, interpolation of missing values can enhance the quality of prediction models, and it's
preferable to use more complex interpolation methods like B-splines and Bezier curves to achieve
better results.
For a more objective assessment, additional research should be conducted on different datasets
using alternative prediction methods. However, it's important to note that the use of the proposed
algorithms does not worsen the accuracy of predicted data; in some cases, it even improves it.
To determine acceleration factors, the following formula will be utilized:
𝑆 = 𝑇_𝑠𝑒𝑞⁄𝑇_𝑝𝑎𝑟, (6)
where 𝑻_𝒔𝒆𝒒 is the execution time of the sequential version of the algorithm, and 𝑻_𝒑𝒂𝒓 is the
execution time of the parallel version.
Table 3
Execution Time of Interpolation with and without Parallelization and Corresponding Acceleration
Metrics of the Proposed Algorithm
CPU GPU
Metrics B-spline Bezier Curve B-spline Bezier Curve
itnerpolation Interpolation itnerpolation Interpolation
Time,seconds 0.76152 7.1316 0.028451 0.073027
𝑆 11.45265 97.65703
Based on the results presented in Table 3, the obtained findings indicate significant acceleration of
computations using CUDA compared to the conventional sequential algorithm execution. The
utilization of parallel computations on GPU enables the simultaneous utilization of multiple cores,
resulting in accelerated calculations.
The achieved outcomes reveal substantial potential for parallelization using CUDA to enhance the
computational efficiency of B-spline and Bezier curve interpolations. Leveraging GPUs can markedly
amplify computation speed and improve interpolation accuracy. These results underscore the
significance of further investigations into the efficiency of parallel computation algorithms and their
practical applicability, especially within the realm of medical data processing [21-23].
4. Conclusions and future research
A majority of medical datasets suffer from missing information, which can lead to significant
errors in data processing and forecasting analyses. This challenge is particularly relevant for
cardiology systems aiming to predict cardiovascular diseases. These datasets encompass a plethora of
physiological indicators derived from periodic examinations and studies, including arterial pressure,
serum cholesterol, blood glucose levels, and more. The adequacy of interpreting such data in the
presence of missing values remains questionable.
This work explored one approach to addressing the problem of missing data recovery, specifically
investigating interpolation methods based on B-splines and Bezier curves. Ensuring that the
preprocessing of data does not compromise the accuracy of subsequent predictions is of paramount
importance. This assertion is convincingly supported by a series of experiments employing the
proposed algorithms, where not only is the accuracy maintained but sometimes even improved.
Another requirement when dealing with missing data is to maximize the acceleration of this process,
so that the data preprocessing does not unduly impact the overall forecasting time. The proposed
solution in this study, employing parallel computation technology CUDA, has significantly
accelerated the process of data imputation.
B-spline interpolation presents a viable option for medical data, as it ensures smoother
approximation and reduced data distortion. B-splines yield more accurate results, especially when
substantial gaps exist between neighboring data points. However, it should be noted that B-splines
might lead to excessive smoothness or overfitting, which can impact the precision of reproduction.
Bezier curve interpolation demonstrated superior results. It provides flexibility and yields smooth
outcomes. Nevertheless, employing Bezier curves might require greater computational power and
time compared to the previous method.
For further research in this domain, it is recommended to conduct additional investigations and
validations to determine the optimal interpolation method for specific medical data, accounting for the
peculiarities of the particular application.
5. Acknowledgements
The authors would like to thank the Armed Forces of Ukraine for providing security to perform
this work. This work has become possible only because of the resilience and courage of the Ukrainian
Army.
6. References
[1] L. Mochurad, R, Panto. A Parallel Algorithm for the Detection of Eye Disease, In: Hu, Z., Wang,
Y., He, M. (eds) Advances in Intelligent Systems, Computer Science and Digital Economics IV.
CSDEIS (2022). Lecture Notes on Data Engineering and Communications Technologies, vol
158. Springer, Cham. doi:10.1007/978-3-031-24475-9_10.
[2] L. Mochurad, Ya.Hladun, Modeling of Psychomotor Reactions of a Person Based on
Modification of the Tapping Test, International Journal of Computing, (2021), 20(2), 190-200.
doi: 10.47839/ijc.20.2.2166.
[3] Ya. Tolstyak, M. Havryliuk, An Assessment of the Transplant's Survival Level for Recipients
after Kidney Transplantations using Cox Proportional-Hazards Model. Proceedings of the 5th
International Conference on Informatics & Data-Driven Medicine, Lyon, France, November 18 -
20, CEUR-WS.org, (2022), 260-265.
[4] X. Zhu, K.C. Thompson, T.J. Martínez, Geodesic interpolation for reaction pathways, J. Chem.
Phys. 2019 Apr 28;150(16):164103. doi: 10.1063/1.5090303.
[5] L. Zhou, Sh. Ren, G. Meng, Zh. Ma, Node-based smoothed radial point interpolation method for
electromagnetic-thermal coupled analysis, Applied Mathematical Modelling, Volume 78 (2020)
841-862. doi: 10.1016/j.apm.2019.09.047.
[6] S. Ljaskovska, Y. Martyn, I. Malets, O. Velyka.Optimization of parameters of technological
processes means of the flexsim simulation program. Proceedings of the 2020 IEEE 3rd
International Conference on Data Stream Mining and Processing, DSMP 2020, (2020): 391–397,
9204029.
[7] S. Liaskovska, Y. Martyn. Investigation of Anomalous Situations in the Machine-Building
Industry Using Phase Trajectories Method. Lecture Notes in Networks and Systemsthis link is
disabled, 463, (2022): 49–59.
[8] Y. Martyn, S. Liaskovska, M. Gregus, O. Velyka. Optimization of Technological's Processes
Industry 4.0 Parameters for Details Manufacturing via Stamping: Rules of Queuing Systems.
Procedia Computer Science, 191, (2021): 290–295.
[9] G. Legnani, I. Fassi, A. Tasora, D. Fusai, A practical algorithm for smooth interpolation between
different angular positions, Mechanism and Machine Theory, Volume 162 (2021):104341.
doi:10.1016/j.mechmachtheory.2021.104341.
[10] S. Fujimoto, T. Mizuno, A. Ishikawa, Interpolation of non-random missing values in financial
statements’ big data using CatBoost, Journal of Computational Social Science, 5 (2022):1281–
1301. doi: 10.1007/s42001-022-00165-9.
[11] M. Havryliuk, I. Dumyn, O. Vovk, Extraction of Structural Elements of the Text Using
Pragmatic Features for the Nomenclature of Cases Verification. In: Hu, Z., Wang, Y., He, M.
(eds) Advances in Intelligent Systems, Computer Science and Digital Economics. Lecture Notes
on Data Engineering and Communications Technologies, 158(2023):703–711. doi: 10.1007/978-
3-031-24475-9_57.
[12] Yuan Luo, Evaluating the state of the art in missing data imputation for clinical data, Briefings in
Bioinformatics, Volume 23, Issue 1, (2022), bbab489. doi: 10.1093/bib/bbab489.
[13] Shreyas Fadnavis. Image Interpolation Techniques in Digital Image Processing: An Overview,
International Journal of Engineering Research and Applications, (2014), 4(10):2248-962270.
[14] Yuan Luo, Evaluating the state of the art in missing data imputation for clinical data, Briefings in
Bioinformatics, Volume 23, Issue 1, January 2022, bbab489. doi:10.1093/bib/bbab489.
[15] I. Izonin, R. Tkachenko, V. Verhun, Kh. Zub, An approach towards missing data management
using improved GRNN-SGTM ensemble method, Engineering Science and Technology, an
International Journal, (2021), 24(3):749-759. doi:10.1016/j.jestch.2020.10.005.
[16] S. Dolgov, D. Savostyanov, Parallel cross interpolation for high-precision calculation of high-
dimensional integrals, Computer Physics Communications, Volume 246, 2020, 106869.
doi:10.1016/j.cpc.2019.106869.
[17] O. Zachariadis, A. Teatini, . et al., Accelerating B-spline interpolation on GPUs: Application to
medical image registration, Computer Methods and Programs in Biomedicine, Volume 193,
(2020), 105431. doi:10.1016/j.cmpb.2020.105431.
[18] Heart Failure Prediction Dataset. Available online:
https://www.kaggle.com/datasets/fedesoriano/heart-failure-prediction (accessed on 13August
2023).
[19] J. Wang, J. Kosinka, A. Telea, Spline-Based Dense Medial Descriptors for Lossy Image
Compression, J. Imaging, (2021), 7, 153. doi:10.3390/jimaging7080153.
[20] R.S. Dehal, C. Munjal, A.A. Ansari and A.S. Kushwaha, GPU Computing Revolution: CUDA,
2018 International Conference on Advances in Computing, Communication Control and
Networking (ICACCCN) , Greater Noida, India, (2018), 197-201. doi:
10.1109/ICACCCN.2018.8748495.
[21] S. Dash, S.K. Shakyawar, M. Sharma et al. Big data in healthcare: management, analysis and
future prospects, J. Big Data 6, 54 (2019). doi:10.1186/s40537-019-0217-0.
[22] R PraveenKumar, Medical Data Processing and Prediction of Future Health Condition Using
Sensors Data Mining Techniques and R Programming, International Journal of Scientific
Research and Engineering Development, (2020), 3(4), 9 p. Available at
SSRN: https://ssrn.com/abstract=3686347.
[23] Ya. Tolstyak, V. Chopyak, M. Havryliuk, An investigation of the primary immunosuppressive
therapy's influence on kidney transplant survival at one month after transplantation, Transplant
Immunology, (2023). doi:10.1016/j.trim.2023.101832.