{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:58:38Z","timestamp":1760241518387,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2018,4,8]],"date-time":"2018-04-08T00:00:00Z","timestamp":1523145600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we will construct a new multilevel system in the Fourier domain using the harmonic wavelet. The main advantages of harmonic wavelet are that its frequency spectrum is confined exactly to an octave band, and its simple definition just as Haar wavelet. The constructed multilevel system has the circular shape, which forms a partition of the frequency domain by shifting and scaling the basic wavelet functions. To possess the circular shape, a new type of sampling grid, the circular-polar grid (CPG), is defined and also the corresponding modified Fourier transform. The CPG consists of equal space along rays, where different rays are equally angled. The main difference between the classic polar grid and CPG is the even sampling on polar coordinates. Another obvious difference is that the modified Fourier transform has a circular shape in the frequency domain while the polar transform has a square shape. The proposed sampling grid and the new defined Fourier transform constitute a completely Fourier transform system, more importantly, the harmonic wavelet based multilevel system defined on the proposed sampling grid is more suitable for the distribution of general images in the Fourier domain.<\/jats:p>","DOI":"10.3390\/sym10040101","type":"journal-article","created":{"date-parts":[[2018,4,11]],"date-time":"2018-04-11T12:16:50Z","timestamp":1523449010000},"page":"101","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Framework for Circular Multilevel Systems in the Frequency Domain"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4655-8571","authenticated-orcid":false,"given":"Guomin","family":"Sun","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 610054, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jinsong","family":"Leng","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 610054, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7504-0424","authenticated-orcid":false,"given":"Carlo","family":"Cattani","sequence":"additional","affiliation":[{"name":"Engineering School, DEIM, University of Tuscia, 01100 Viterbo, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,4,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"674","DOI":"10.1109\/34.192463","article-title":"A Theory for Multiresolution Signal Decomposition: The Wavelet Representation","volume":"11","author":"Mallat","year":"1989","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2317","DOI":"10.1155\/2011\/231754","article-title":"A Study on Conjugate Quadrature Filters","volume":"2011","author":"Leng","year":"2011","journal-title":"EURASIP J. Adv. Signal Process."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1663","DOI":"10.1016\/j.camwa.2006.05.005","article-title":"Construction and properties of multiwavelet packets with arbitrary scale and the related algorithms of decomposition and reconstruction","volume":"51","author":"Leng","year":"2006","journal-title":"Comput. Math. Appl."},{"key":"ref_4","unstructured":"Leng, J., Huang, T., and Fu, Y. (2008, January 12\u201315). Construction of bivariate nonseparable compactly supported biorthogonal wavelets. Proceedings of the International Conference on Machine Learning and Cybernetics IEEE, Kunming, China."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"764","DOI":"10.1137\/060650283","article-title":"A Framework for Discrete Integral Transformations I-The Pseudopolar Fourier Transform","volume":"30","author":"Averbuch","year":"2007","journal-title":"Siam J. Sci. Comput."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1137\/1033097","article-title":"The Fractional Fourier Transform and Applications","volume":"33","author":"Bailey","year":"1991","journal-title":"SIAM Rev."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1109\/TAU.1969.1162034","article-title":"Chirp Z-transform algorithm","volume":"17","author":"Rabiner","year":"1969","journal-title":"IEEE Trans. Audio Electroacoust."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1002\/cpa.10116","article-title":"New tight frames of curvelets and optimal representations of objects with smooth singularities","volume":"57","author":"Candes","year":"2004","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2091","DOI":"10.1109\/TIP.2005.859376","article-title":"The contourlet transform: an efficient directional multiresolution image representation","volume":"14","author":"Do","year":"2005","journal-title":"IEEE Trans. Image Process."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1291","DOI":"10.1137\/110854497","article-title":"ShearLab: A Rational Design of a Digital Parabolic Scaling Algorithm","volume":"5","author":"Kutyniok","year":"2011","journal-title":"Siam J. Imaging Sci."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Kutyniok, G. (2012). Shearlets. Applied and Numerical Harmonic Analysis, Springer.","DOI":"10.1007\/978-0-8176-8316-0"},{"key":"ref_12","first-page":"203","article-title":"Harmonic Wavelet Analysis","volume":"443","author":"Newland","year":"1993","journal-title":"Proc. R. Soc. Math. Phys. Eng. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1191","DOI":"10.1016\/j.camwa.2005.07.001","article-title":"Harmonic wavelets towards solution of nonlinear PDE","volume":"50","author":"Cattani","year":"2005","journal-title":"Comput. Math. Appl."},{"key":"ref_14","unstructured":"Cattani, C. (July, January 29). Fractals Based on Harmonic Wavelets. Proceedings of the International Conference on Computational Science and Its Applications, Seoul, South Korea."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1007\/s11235-009-9208-3","article-title":"Harmonic wavelet approximation of random, fractal and high frequency signals","volume":"43","author":"Cattani","year":"2010","journal-title":"Telecommun. Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1007\/BF01456927","article-title":"Zur Theorie der orthogonalen Funktionensysteme","volume":"71","author":"Haar","year":"1911","journal-title":"Math. Ann."},{"key":"ref_17","first-page":"532","article-title":"A Shannon-Runge-Kutta-Gill Method for Convection-Diffusion Equations","volume":"46","author":"Duan","year":"2013","journal-title":"Math. Probl. Eng."},{"key":"ref_18","first-page":"562","article-title":"Framelet","volume":"21","author":"Cai","year":"2012","journal-title":"IEEE Trans. Image Process."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/4\/101\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:59:59Z","timestamp":1760194799000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/4\/101"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,4,8]]},"references-count":18,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2018,4]]}},"alternative-id":["sym10040101"],"URL":"https:\/\/doi.org\/10.3390\/sym10040101","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2018,4,8]]}}}