{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,5]],"date-time":"2026-05-05T12:13:16Z","timestamp":1777983196723,"version":"3.51.4"},"reference-count":39,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,6,14]],"date-time":"2024-06-14T00:00:00Z","timestamp":1718323200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"The National Natural Science Foundation of China","award":["62261055"],"award-info":[{"award-number":["62261055"]}]},{"name":"The National Natural Science Foundation of China","award":["61861044"],"award-info":[{"award-number":["61861044"]}]},{"name":"The National Natural Science Foundation of China","award":["2023-JC-YB-085"],"award-info":[{"award-number":["2023-JC-YB-085"]}]},{"name":"The National Natural Science Foundation of China","award":["2022JM-400"],"award-info":[{"award-number":["2022JM-400"]}]},{"name":"The project of Natural Science Foundation of Shaanxi Province","award":["62261055"],"award-info":[{"award-number":["62261055"]}]},{"name":"The project of Natural Science Foundation of Shaanxi Province","award":["61861044"],"award-info":[{"award-number":["61861044"]}]},{"name":"The project of Natural Science Foundation of Shaanxi Province","award":["2023-JC-YB-085"],"award-info":[{"award-number":["2023-JC-YB-085"]}]},{"name":"The project of Natural Science Foundation of Shaanxi Province","award":["2022JM-400"],"award-info":[{"award-number":["2022JM-400"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Convolution plays a pivotal role in the domains of signal processing and optics. This paper primarily focuses on studying the weighted convolution for quaternion linear canonical cosine transform (QLCcT) and its application in multiplicative filter analysis. Firstly, we propose QLCcT by combining quaternion algebra with linear canonical cosine transform (LCcT), which extends LCcT to Hamiltonian quaternion algebra. Secondly, we introduce weighted convolution and correlation operations for QLCcT, accompanied by their corresponding theorems. We also explore the properties of QLCcT. Thirdly, we utilize these proposed convolution structures to analyze multiplicative filter models that offer lower computational complexity compared to existing methods based on quaternion linear canonical transform (QLCT). Additionally, we discuss the rationale behind studying such transforms using quaternion functions as an illustrative example.<\/jats:p>","DOI":"10.3390\/axioms13060402","type":"journal-article","created":{"date-parts":[[2024,6,14]],"date-time":"2024-06-14T10:42:34Z","timestamp":1718361754000},"page":"402","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Weighted Convolution for Quaternion Linear Canonical Cosine Transform and Its Application"],"prefix":"10.3390","volume":"13","author":[{"given":"Rongbo","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mathematics and Computer Science, Yanan University, Yanan 716000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7336-0814","authenticated-orcid":false,"given":"Qiang","family":"Feng","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science, Yanan University, Yanan 716000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1772","DOI":"10.1063\/1.1665805","article-title":"Linear canonical transformations and their unitary representations","volume":"12","author":"Moshinsky","year":"1971","journal-title":"J. Math. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1049\/iet-spr.2015.0028","article-title":"Convolution and correlation theorems for the two-dimensional linear canonical transform and its applications","volume":"10","author":"Feng","year":"2016","journal-title":"IET Signal Process."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"6173","DOI":"10.1007\/s00034-021-01759-w","article-title":"A Novel Sub-Nyquist FRI Sampling and Reconstruction Method in Linear Canonical Transform Domain","volume":"40","author":"Xin","year":"2021","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1529","DOI":"10.1109\/LCOMM.2020.2988947","article-title":"Jittered Sampling in Linear Canonical Domain","volume":"24","author":"Zhang","year":"2020","journal-title":"IEEE Commun. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1364\/JOSAA.482872","article-title":"Fast numerical calculation of the offset linear canonical transform","volume":"40","author":"Chen","year":"2023","journal-title":"J. Opt. Soc. Am. A\u2014Opt. Image Sci. Vis."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2237","DOI":"10.1109\/TSP.2019.2903031","article-title":"Discrete Linear Canonical Transform Based on Hyperdifferential Operators","volume":"67","author":"Bartan","year":"2019","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Urynbassarova, D., and Teali, A.A. (2023). Convolution, Correlation, and uncertainty principles for the quaternion offset linear canonical transform. Mathematics, 11.","DOI":"10.3390\/math11092201"},{"key":"ref_8","first-page":"159","article-title":"Multidimensional linear canonical transform and convolution","volume":"37","author":"Kundu","year":"2022","journal-title":"J. Ramanujan Math. Soc."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"6009","DOI":"10.1109\/TSP.2019.2951191","article-title":"Convolution and Multichannel Sampling for the Offset Linear Canonical Transform and Their Applications","volume":"67","author":"Wei","year":"2019","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1007\/s43036-021-00164-z","article-title":"Multiresolution analysis for linear canonical S transform","volume":"6","author":"Bhat","year":"2021","journal-title":"Adv. Oper. Theory"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/j.sigpro.2019.07.008","article-title":"Weighted Heisenberg-Pauli-Weyl uncertainty principles for the linear canonical transform","volume":"165","author":"Feng","year":"2019","journal-title":"Signal Process."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Bruni, V., Cardinali, M.L., and Vitulano, D. (2022). An MDL-Based Wavelet Scattering Features Selection for Signal Classification. Axioms, 11.","DOI":"10.3390\/axioms11080376"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"109385","DOI":"10.1016\/j.sigpro.2024.109385","article-title":"Generalized spectrum analysis of Chirp Cyclostationary signals associate with linear canonical transform","volume":"218","author":"Miao","year":"2024","journal-title":"Signal Process."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1186\/s13634-021-00753-3","article-title":"On a new Wigner-Ville distribution associated with linear canonical transform","volume":"2021","author":"Xin","year":"2021","journal-title":"EURASIP J. Adv. Signal Process."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Li, Z.W., and Gao, W.B. (2023). Inequalities for the Windowed Linear Canonical Transform of Complex Functions. Axioms, 12.","DOI":"10.3390\/axioms12060554"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Bahri, M., and Karim, S.A.A. (2022). Novel Uncertainty Principles Concerning Linear Canonical Wavelet Transform. Mathematics, 10.","DOI":"10.3390\/math10193502"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Li, Y.M., Jiang, M.J., Wei, D., and Deng, Y. (2024). Novel Image Encryption Algorithm Based on Compressive Sensing and a Two-Dimensional Linear Canonical Transform. Fractal Fract., 8.","DOI":"10.3390\/fractalfract8020092"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"129262","DOI":"10.1016\/j.optcom.2023.129262","article-title":"Optical image encryption based on linear canonical transform with sparse representation","volume":"533","author":"Qasim","year":"2023","journal-title":"Opt. Commun."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1661","DOI":"10.1109\/TSP.2002.1011207","article-title":"Fractional cosine, sine, and Hartley transforms","volume":"50","author":"Pei","year":"2002","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2641","DOI":"10.3934\/math.2024130","article-title":"The discrete convolution for fractional cosine-sine series and its application in convolution equations","volume":"9","author":"Wang","year":"2024","journal-title":"AIMS Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"e12170","DOI":"10.1049\/sil2.12170","article-title":"Fractional Fourier cosine and sine Laplace weighted convolution and its application","volume":"17","author":"Xiang","year":"2023","journal-title":"IET Signal Process."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"8057","DOI":"10.1007\/s11042-018-6595-z","article-title":"Fractional quaternion cosine transform and its application in color image copy-move forgery detection","volume":"78","author":"Chen","year":"2019","journal-title":"Multimed. Tools Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3651","DOI":"10.1002\/mma.4251","article-title":"Convolution theorem for fractional cosine-sine transform and its application","volume":"40","author":"Feng","year":"2017","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1198","DOI":"10.1109\/78.923302","article-title":"The discrete fractional cosine and sine transforms","volume":"49","author":"Pei","year":"2001","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_25","unstructured":"Hamilton, W.R. (1866). Elements of Quaternions, Longmans Green."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Kantor, I.L., and Solodovnikov, A.S. (1989). Hypercomplex Number: An Elementary Introduction to Algebras, Springer.","DOI":"10.1007\/978-1-4612-3650-4"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"606","DOI":"10.1007\/s10851-013-0430-y","article-title":"Convolution products for hypercomplex Fourier transforms","volume":"48","author":"Bujack","year":"2014","journal-title":"J. Math. Imaging Vis."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1007\/s11868-019-00283-5","article-title":"Uncertainty principle for the two sided quaternion windowed Fourier transform","volume":"11","author":"Brahim","year":"2020","journal-title":"J. Pseudo-Differ. Oper."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2511","DOI":"10.1016\/j.sigpro.2008.04.012","article-title":"Fractional quaternion Fourier transform, convolution and correlation","volume":"88","author":"Xu","year":"2008","journal-title":"Signal Process."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"485","DOI":"10.23919\/cje.2021.00.225","article-title":"Convolution theorem associated with the QWFRFT","volume":"32","author":"Mei","year":"2023","journal-title":"Chin. J. Electron."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1062979","DOI":"10.1155\/2019\/1062979","article-title":"Two-dimensional quaternion linear canonical transform: Properties, convolution, correlation and uncertainty principle","volume":"2019","author":"Bahri","year":"2019","journal-title":"J. Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"108743","DOI":"10.1016\/j.sigpro.2022.108743","article-title":"Convolution theorem associated with quaternion linear canonical transforms and applications","volume":"201","author":"Hu","year":"2023","journal-title":"Signal Process."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1007\/s00006-020-1042-4","article-title":"Quaternion windowed linear canonical transform of two-dimensional signals","volume":"30","author":"Gao","year":"2020","journal-title":"Adv. Appl. Clifford Algebr."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Yang, H.H., Feng, Q., Wang, X.X., Urynbassarova, D., and Teali, A.A. (2024). Reduced Biquaternion Windowed Linear Canonical Transform: Properties and Applications. Mathematics, 12.","DOI":"10.3390\/math12050743"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1007\/s00034-022-02127-y","article-title":"Uncertainty Principles for Wigner-Ville Distribution Associated with the Quaternion Offset Linear Canonical Transform","volume":"42","author":"Urynbassarova","year":"2023","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"786","DOI":"10.1515\/dema-2022-0175","article-title":"Wigner-Ville distribution and ambiguity function associated with the quaternion offset linear canonical transform","volume":"55","author":"Bhat","year":"2022","journal-title":"Demonstr. Math."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1109\/97.664179","article-title":"A convolution and product theorem for the fractional Fourier transform","volume":"5","author":"Zayed","year":"1998","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Bhandari, A., and Zayed, A. (2018). Convolution and product theorems for the special affine Fourier transform. Frontiers in Orthogonal Polynomials and q-Series, World Scientific Publishers.","DOI":"10.1142\/9789813228887_0006"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Nussbaumer, H.J. (1981). Fast Fourier Transform and Convolution Algorithms, Springer.","DOI":"10.1007\/978-3-662-00551-4"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/6\/402\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:59:01Z","timestamp":1760108341000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/6\/402"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,14]]},"references-count":39,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["axioms13060402"],"URL":"https:\/\/doi.org\/10.3390\/axioms13060402","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,6,14]]}}}