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Several necessary constraints on the filter lengths and symmetric features of wavelet frames are investigated starting at the constructed paraunitary matrix. The matrix is a symmetric extension of the polyphase matrix corresponding to 3\u2010band tight wavelet frames. Further, the parameterizations of 3\u2010band tight wavelet frames with 3<jats:italic>N<\/jats:italic> + 1 filter lengths are established. Examples of framelets with symmetry\/antisymmetry and Sobolev exponent are computed by appropriately choosing the parameters in the scheme.<\/jats:p>","DOI":"10.1155\/2012\/907175","type":"journal-article","created":{"date-parts":[[2012,10,3]],"date-time":"2012-10-03T21:07:06Z","timestamp":1349298426000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Symmetry Feature and Construction for the 3\u2010Band Tight Framelets with Prescribed Properties"],"prefix":"10.1155","volume":"2012","author":[{"given":"Jianjun","family":"Sun","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bin","family":"Huang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaodong","family":"Chen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lihong","family":"Cui","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2012,10,3]]},"reference":[{"key":"e_1_2_7_1_2","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780195094237.003.0002"},{"key":"e_1_2_7_2_2","doi-asserted-by":"publisher","DOI":"10.1109\/78.771037"},{"key":"e_1_2_7_3_2","doi-asserted-by":"publisher","DOI":"10.1109\/78.258087"},{"key":"e_1_2_7_4_2","doi-asserted-by":"publisher","DOI":"10.1006\/acha.2000.0301"},{"key":"e_1_2_7_5_2","doi-asserted-by":"publisher","DOI":"10.1006\/acha.2000.0332"},{"key":"e_1_2_7_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/S1063-5203(02)00510-9"},{"key":"e_1_2_7_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/S1063-5203(02)00511-0"},{"key":"e_1_2_7_8_2","doi-asserted-by":"publisher","DOI":"10.1023\/A:1021318804341"},{"key":"e_1_2_7_9_2","doi-asserted-by":"publisher","DOI":"10.1023\/A:1021339707805"},{"key":"e_1_2_7_10_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00365-002-0522-1"},{"key":"e_1_2_7_11_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2004.05.003"},{"key":"e_1_2_7_12_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2004.09.001"},{"key":"e_1_2_7_13_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2006.10.003"},{"key":"e_1_2_7_14_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2007.10.002"},{"key":"e_1_2_7_15_2","doi-asserted-by":"crossref","first-page":"551","DOI":"10.4310\/CMS.2011.v9.n2.a10","article-title":"Frame based segmentation for medical images","volume":"9","author":"Dong B.","year":"2011","journal-title":"Communications in Mathematical Sciences"},{"key":"e_1_2_7_16_2","doi-asserted-by":"publisher","DOI":"10.1006\/jfan.1996.3079"},{"key":"e_1_2_7_17_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02648888"},{"key":"e_1_2_7_18_2","doi-asserted-by":"publisher","DOI":"10.1109\/78.258088"},{"key":"e_1_2_7_19_2","doi-asserted-by":"publisher","DOI":"10.1023\/A:1018965102118"},{"key":"e_1_2_7_20_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2008.09.017"},{"key":"e_1_2_7_21_2","doi-asserted-by":"publisher","DOI":"10.1007\/s12190-008-0091-y"},{"key":"e_1_2_7_22_2","unstructured":"CuiL. 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