{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,30]],"date-time":"2025-12-30T15:38:29Z","timestamp":1767109109873,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,2,23]],"date-time":"2021-02-23T00:00:00Z","timestamp":1614038400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61802212 and 61872203"],"award-info":[{"award-number":["61802212 and 61872203"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007129","name":"Natural Science Foundation of Shandong Province","doi-asserted-by":"publisher","award":["ZR2019BF017"],"award-info":[{"award-number":["ZR2019BF017"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100018532","name":"Major Scientific and Technological Innovation Project of Shandong Province","doi-asserted-by":"publisher","award":["2019JZZY010127, 2019JZZY010132 and 2019JZZY010201"],"award-info":[{"award-number":["2019JZZY010127, 2019JZZY010132 and 2019JZZY010201"]}],"id":[{"id":"10.13039\/501100018532","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Continuous orthogonal moments, for which continuous functions are used as kernel functions, are invariant to rotation and scaling, and they have been greatly developed over the recent years. Among continuous orthogonal moments, polar harmonic Fourier moments (PHFMs) have superior performance and strong image description ability. In order to improve the performance of PHFMs in noise resistance and image reconstruction, PHFMs, which can only take integer numbers, are extended to fractional-order polar harmonic Fourier moments (FrPHFMs) in this paper. Firstly, the radial polynomials of integer-order PHFMs are modified to obtain fractional-order radial polynomials, and FrPHFMs are constructed based on the fractional-order radial polynomials; subsequently, the strong reconstruction ability, orthogonality, and geometric invariance of the proposed FrPHFMs are proven; and, finally, the performance of the proposed FrPHFMs is compared with that of integer-order PHFMs, fractional-order radial harmonic Fourier moments (FrRHFMs), fractional-order polar harmonic transforms (FrPHTs), and fractional-order Zernike moments (FrZMs). The experimental results show that the FrPHFMs constructed in this paper are superior to integer-order PHFMs and other fractional-order continuous orthogonal moments in terms of performance in image reconstruction and object recognition, as well as that the proposed FrPHFMs have strong image description ability and good stability.<\/jats:p>","DOI":"10.3390\/s21041544","type":"journal-article","created":{"date-parts":[[2021,2,23]],"date-time":"2021-02-23T20:19:36Z","timestamp":1614111576000},"page":"1544","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Invariant Image Representation Using Novel Fractional-Order Polar Harmonic Fourier Moments"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3742-5614","authenticated-orcid":false,"given":"Chunpeng","family":"Wang","sequence":"first","affiliation":[{"name":"School of Computer Science and Technology (School of Cyber Security), Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China"},{"name":"Shandong Provincial Key Laboratory of Computer Networks, Shandong Computer Science Center (National Supercomputer Center in Jinan), Qilu University of Technology (Shandong Academy of Sciences), Jinan 250014, China"},{"name":"Shandong Provincial Key Laboratory for Distributed Computer Software Novel Technology, Jinan 250358, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hongling","family":"Gao","sequence":"additional","affiliation":[{"name":"School of Computer Science and Technology (School of Cyber Security), Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China"},{"name":"Shandong Provincial Key Laboratory of Computer Networks, Shandong Computer Science Center (National Supercomputer Center in Jinan), Qilu University of Technology (Shandong Academy of Sciences), Jinan 250014, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Meihong","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Computer Science and Technology (School of Cyber Security), Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China"},{"name":"Shandong Provincial Key Laboratory of Computer Networks, Shandong Computer Science Center (National Supercomputer Center in Jinan), Qilu University of Technology (Shandong Academy of Sciences), Jinan 250014, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jian","family":"Li","sequence":"additional","affiliation":[{"name":"School of Computer Science and Technology (School of Cyber Security), Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China"},{"name":"Shandong Provincial Key Laboratory of Computer Networks, Shandong Computer Science Center (National Supercomputer Center in Jinan), Qilu University of Technology (Shandong Academy of Sciences), Jinan 250014, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bin","family":"Ma","sequence":"additional","affiliation":[{"name":"School of Computer Science and Technology (School of Cyber Security), Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China"},{"name":"Shandong Provincial Key Laboratory of Computer Networks, Shandong Computer Science Center (National Supercomputer Center in Jinan), Qilu University of Technology (Shandong Academy of Sciences), Jinan 250014, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qixian","family":"Hao","sequence":"additional","affiliation":[{"name":"School of Computer Science and Technology (School of Cyber Security), Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China"},{"name":"Shandong Provincial Key Laboratory of Computer Networks, Shandong Computer Science Center (National Supercomputer Center in Jinan), Qilu University of Technology (Shandong Academy of Sciences), Jinan 250014, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,2,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1016\/j.sigpro.2018.11.011","article-title":"Color medical image lossless watermarking using chaotic system and accurate quaternion polar harmonic transforms","volume":"157","author":"Xia","year":"2019","journal-title":"Signal Process."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1914","DOI":"10.1109\/TIFS.2016.2566261","article-title":"A Reversible Data Hiding Scheme Based on Code Division Multiplexing","volume":"11","author":"Ma","year":"2016","journal-title":"IEEE Trans. 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