{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T23:27:08Z","timestamp":1770334028347,"version":"3.49.0"},"reference-count":44,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,8,27]],"date-time":"2024-08-27T00:00:00Z","timestamp":1724716800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Tchebichef polynomials (TPs) play a crucial role in various fields of mathematics and applied sciences, including numerical analysis, image and signal processing, and computer vision. This is due to the unique properties of the TPs and their remarkable performance. Nowadays, the demand for high-quality images (2D signals) is increasing and is expected to continue growing. The processing of these signals requires the generation of accurate and fast polynomials. The existing algorithms generate the TPs sequentially, and this is considered as computationally costly for high-order and larger-sized polynomials. To this end, we present a new efficient solution to overcome the limitation of sequential algorithms. The presented algorithm uses the parallel processing paradigm to leverage the computation cost. This is performed by utilizing the multicore and multithreading features of a CPU. The implementation of multithreaded algorithms for computing TP coefficients segments the computations into sub-tasks. These sub-tasks are executed concurrently on several threads across the available cores. The performance of the multithreaded algorithm is evaluated on various TP sizes, which demonstrates a significant improvement in computation time. Furthermore, a selection for the appropriate number of threads for the proposed algorithm is introduced. The results reveal that the proposed algorithm enhances the computation performance to provide a quick, steady, and accurate computation of the TP coefficients, making it a practical solution for different applications.<\/jats:p>","DOI":"10.3390\/a17090381","type":"journal-article","created":{"date-parts":[[2024,8,27]],"date-time":"2024-08-27T09:26:51Z","timestamp":1724750811000},"page":"381","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Multithreading-Based Algorithm for High-Performance Tchebichef Polynomials with Higher Orders"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6072-9926","authenticated-orcid":false,"given":"Ahlam Hanoon","family":"Al-sudani","sequence":"first","affiliation":[{"name":"Department of Computer Engineering, University of Baghdad, Al-Jadriya, Baghdad 10071, Iraq"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4121-0843","authenticated-orcid":false,"given":"Basheera M.","family":"Mahmmod","sequence":"additional","affiliation":[{"name":"Department of Computer Engineering, University of Baghdad, Al-Jadriya, Baghdad 10071, Iraq"}]},{"given":"Firas A.","family":"Sabir","sequence":"additional","affiliation":[{"name":"Department of Computer Engineering, University of Baghdad, Al-Jadriya, Baghdad 10071, Iraq"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6439-0082","authenticated-orcid":false,"given":"Sadiq H.","family":"Abdulhussain","sequence":"additional","affiliation":[{"name":"Department of Computer Engineering, University of Baghdad, Al-Jadriya, Baghdad 10071, Iraq"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7937-3093","authenticated-orcid":false,"given":"Muntadher","family":"Alsabah","sequence":"additional","affiliation":[{"name":"Medical Technical College, Al-Farahidi University, Baghdad 10071, Iraq"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5610-0586","authenticated-orcid":false,"given":"Wameedh Nazar","family":"Flayyih","sequence":"additional","affiliation":[{"name":"Department of Computer Engineering, University of Baghdad, Al-Jadriya, Baghdad 10071, Iraq"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Abd-Elhameed, W.M., and Al-Harbi, M.S. (2022). Some formulas and recurrences of certain orthogonal polynomials generalizing Chebyshev polynomials of the third-kind. Symmetry, 14.","DOI":"10.3390\/sym14112309"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"111372","DOI":"10.1016\/j.chaos.2021.111372","article-title":"Numerical analysis of variable fractional viscoelastic column based on two-dimensional Legendre wavelets algorithm","volume":"152","author":"Sun","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"AbdelFattah, H., Al-Johani, A., and El-Beltagy, M. (2020). Analysis of the stochastic quarter-five spot problem using polynomial chaos. Molecules, 25.","DOI":"10.3390\/molecules25153370"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1024","DOI":"10.1134\/S0040577923070097","article-title":"Energy spectrum design and potential function engineering","volume":"216","author":"Alhaidari","year":"2023","journal-title":"Theor. Math. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"3266","DOI":"10.1109\/TIP.2023.3279525","article-title":"Local Orthogonal Moments for Local Features","volume":"32","author":"Yang","year":"2023","journal-title":"IEEE Trans. Image Process."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1109\/TASSP.1975.1162724","article-title":"Roundoff noise characteristics of a class of orthogonal polynomial structures","volume":"23","author":"Markel","year":"1975","journal-title":"IEEE Trans. Acoust. Speech Signal Process."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1242","DOI":"10.1109\/TEMC.2014.2312003","article-title":"EMP signal reconstruction using associated-Hermite orthogonal functions","volume":"56","author":"Ma","year":"2014","journal-title":"IEEE Trans. Electromagn. Compat."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Abdulhussain, S.H., Mahmmod, B.M., Flusser, J., AL-Utaibi, K.A., and Sait, S.M. (2022). Fast overlapping block processing algorithm for feature extraction. Symmetry, 14.","DOI":"10.3390\/sym14040715"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"114455","DOI":"10.1109\/ACCESS.2022.3217225","article-title":"Plain, edge, and texture detection based on orthogonal moment","volume":"10","author":"Abdulqader","year":"2022","journal-title":"IEEE Access"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1303","DOI":"10.1109\/18.412678","article-title":"Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces","volume":"41","author":"Levenshtein","year":"1995","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1349","DOI":"10.1137\/0521075","article-title":"Linearization of Meixner, Krawtchouk, and Charlier polynomial products","volume":"21","author":"Zeng","year":"1990","journal-title":"SIAM J. Math. Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1998","DOI":"10.1016\/j.patcog.2013.01.008","article-title":"Generalized dual Hahn moment invariants","volume":"46","author":"Karakasis","year":"2013","journal-title":"Pattern Recognit."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Mahmmod, B.M., Flayyih, W.N., Fakhri, Z.H., Abdulhussain, S.H., Khan, W., and Hussain, A. (2023). Performance enhancement of high order Hahn polynomials using multithreading. PLoS ONE, 18.","DOI":"10.1371\/journal.pone.0286878"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1016\/j.sigpro.2006.07.007","article-title":"Image analysis by discrete orthogonal Racah moments","volume":"87","author":"Zhu","year":"2007","journal-title":"Signal Process."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"110502","DOI":"10.1109\/ACCESS.2023.3321969","article-title":"Accelerated and improved stabilization for high order moments of Racah polynomials","volume":"11","author":"Mahmmod","year":"2023","journal-title":"IEEE Access"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Veerasamy, M., Jaganathan, S.C.B., Dhasarathan, C., Mubarakali, A., Ramasamy, V., Kalpana, R., and Marina, N. (2023). Legendre Neural Network Method for Solving Nonlinear Singular Systems. Intelligent Technologies for Sensors, Apple Academic Press.","DOI":"10.1201\/9781003314851-4"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"035202","DOI":"10.1088\/1751-8113\/44\/3\/035202","article-title":"Recurrence coefficients of generalized Meixner polynomials and Painlev\u00e9 equations","volume":"44","author":"Boelen","year":"2010","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_18","first-page":"1","article-title":"Associated Hermite Polynomials. Some Applications","volume":"22","author":"Vasileva","year":"2023","journal-title":"Int. J. Differ. Equ. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Schweizer, W., and Schweizer, W. (2021). Laguerre Polynomials. Special Functions in Physics with MATLAB, Springer.","DOI":"10.1007\/978-3-030-64232-7"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1357","DOI":"10.1109\/83.941859","article-title":"Image analysis by Tchebichef moments","volume":"10","author":"Mukundan","year":"2001","journal-title":"IEEE Trans. Image Process."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Li, J., Wang, P., Ni, C., Zhang, D., and Hao, W. (2022). Loop Closure Detection for Mobile Robot based on Multidimensional Image Feature Fusion. Machines, 11.","DOI":"10.3390\/machines11010016"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"920","DOI":"10.1364\/JOSA.70.000920","article-title":"Image analysis via the general theory of moments","volume":"70","author":"Teague","year":"1980","journal-title":"J. Opt. Soc. Am."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"12183","DOI":"10.1007\/s11042-018-6757-z","article-title":"Fast computation of Charlier moments and its inverses using Clenshaw\u2019s recurrence formula for image analysis","volume":"78","author":"Jahid","year":"2019","journal-title":"Multimed. Tools Appl."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"den Brinker, A.C. (2021). Stable calculation of Krawtchouk functions from triplet relations. Mathematics, 9.","DOI":"10.3390\/math9161972"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"den Brinker, A.C. (2022). Stable Calculation of Discrete Hahn Functions. Symmetry, 14.","DOI":"10.3390\/sym14030437"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Aldakheel, E.A., Khafaga, D.S., Fathi, I.S., Hosny, K.M., and Hassan, G. (2023). Efficient Analysis of Large-Size Bio-Signals Based on Orthogonal Generalized Laguerre Moments of Fractional Orders and Schwarz\u2013Rutishauser Algorithm. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7110826"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Costas-Santos, R.S., Soria-Lorente, A., and Vilaire, J.M. (2022). On Polynomials Orthogonal with Respect to an Inner Product Involving Higher-Order Differences: The Meixner Case. Mathematics, 10.","DOI":"10.3390\/math10111952"},{"key":"ref_28","unstructured":"Fern\u00e1ndez-Irisarri, I., and Ma\u00f1as, M. (2021). Pearson equations for discrete orthogonal polynomials: II. Generalized Charlier, Meixner and Hahn of type I cases. arXiv."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Bourzik, A., Bouikhalen, B., El-Mekkaoui, J., and Hjouji, A. (2023, January 24\u201326). A comparative study and performance evaluation of discrete Tchebichef moments for image analysis. Proceedings of the 6th International Conference on Networking, Intelligent Systems & Security, Larache, Morocco.","DOI":"10.1145\/3607720.3607772"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1055","DOI":"10.1109\/TIP.2004.828430","article-title":"Some Computational Aspects of Discrete Orthonormal Moments","volume":"13","author":"Mukundan","year":"2004","journal-title":"IEEE Trans. Image Process."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2470","DOI":"10.1109\/ACCESS.2017.2669218","article-title":"On Computational Aspects of Tchebichef Polynomials for Higher Polynomial Order","volume":"5","author":"Abdulhussain","year":"2017","journal-title":"IEEE Access"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"332","DOI":"10.1016\/j.patrec.2018.08.020","article-title":"Some computational aspects of Tchebichef moments for higher orders","volume":"112","year":"2018","journal-title":"Pattern Recognit. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"e7311","DOI":"10.1002\/cpe.7311","article-title":"Fast and accurate computation of high-order Tchebichef polynomials","volume":"34","author":"Abdulhussain","year":"2022","journal-title":"Concurr. Comput. Pract. Exp."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Kumar, R., Tullsen, D.M., and Jouppi, N.P. (2006, January 16\u201320). Core architecture optimization for heterogeneous chip multiprocessors. Proceedings of the 15th International Conference on Parallel Architectures and Compilation Techniques, Seattle, WA, USA.","DOI":"10.1145\/1152154.1152162"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2557","DOI":"10.1109\/TPDS.2021.3064709","article-title":"Virtualization overhead of multithreading in X86 state-of-the-art & remaining challenges","volume":"32","author":"Schildermans","year":"2021","journal-title":"IEEE Trans. Parallel Distrib. Syst."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"4675","DOI":"10.1007\/s00366-022-01713-7","article-title":"Multithreaded runtime framework for parallel and adaptive applications","volume":"38","author":"Thomadakis","year":"2022","journal-title":"Eng. Comput."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Kim, E., Choi, S., Kim, C.G., and Park, W.C. (2023). Multi-Threaded Sound Propagation Algorithm to Improve Performance on Mobile Devices. Sensors, 23.","DOI":"10.3390\/s23020973"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1016\/j.aej.2020.06.055","article-title":"Multi-core-, multi-thread-based optimization algorithm for large-scale traveling salesman problem","volume":"60","author":"Wei","year":"2021","journal-title":"Alex. Eng. J."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"3746","DOI":"10.1109\/TPDS.2022.3169410","article-title":"Online thread auto-tuning for performance improvement and resource saving","volume":"33","author":"Luan","year":"2022","journal-title":"IEEE Trans. Parallel Distrib. Syst."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1109\/TIT.2006.871582","article-title":"Compressed sensing","volume":"52","author":"Donoho","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_41","unstructured":"Abramowitz, M., and Stegun, I.A. (1964). Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Dover Publications."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/BF02986888","article-title":"Irresistible integrals: Symbolics, analysis and experiments in the evaluation of integrals","volume":"28","author":"Foncannon","year":"2006","journal-title":"Math. Intell."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1137\/1009002","article-title":"Computational aspects of three-term recurrence relations","volume":"9","author":"Gautschi","year":"1967","journal-title":"SIAM Rev."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1090\/S0025-5718-1985-0804941-2","article-title":"Recurrence relations for hypergeometric functions of unit argument","volume":"45","author":"Lewanowicz","year":"1985","journal-title":"Math. Comput."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/17\/9\/381\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:43:33Z","timestamp":1760111013000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/17\/9\/381"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,27]]},"references-count":44,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2024,9]]}},"alternative-id":["a17090381"],"URL":"https:\/\/doi.org\/10.3390\/a17090381","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,8,27]]}}}