{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:42:50Z","timestamp":1740134570341,"version":"3.37.3"},"reference-count":22,"publisher":"Wiley","license":[{"start":{"date-parts":[[2013,1,1]],"date-time":"2013-01-01T00:00:00Z","timestamp":1356998400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2013]]},"abstract":"In order to achieve a mutiscale representation and texture extraction for textured image, a hierarchical(<\/mml:mo>B<\/mml:mi>V<\/mml:mi>,<\/mml:mo>G<\/mml:mi>p<\/mml:mi><\/mml:msub>,<\/mml:mo>L<\/mml:mi>2<\/mml:mn><\/mml:msup><\/mml:mrow>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>decomposition model is proposed in this paper. We firstly introduce the proposed model which is obtained by replacing the fixed scale parameter of the original(<\/mml:mo>B<\/mml:mi>V<\/mml:mi>,<\/mml:mo>G<\/mml:mi>p<\/mml:mi><\/mml:msub>,<\/mml:mo>L<\/mml:mi>2<\/mml:mn><\/mml:msup><\/mml:mrow>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>decomposition with a varying sequence. And then, the existence and convergence of the hierarchical decomposition are proved. Furthermore, we show the nontrivial property of this hierarchical decomposition. Finally, we introduce a simple numerical method for the hierarchical decomposition, which utilizes gradient decent for energy minimization and finite difference for the associated gradient flow equations. Numerical results show that the proposed hierarchical(<\/mml:mo>B<\/mml:mi>V<\/mml:mi>,<\/mml:mo>G<\/mml:mi>p<\/mml:mi><\/mml:msub>,<\/mml:mo>L<\/mml:mi>2<\/mml:mn><\/mml:msup><\/mml:mrow>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>decomposition is very appropriate for multiscale representation and texture extraction of textured image.<\/jats:p>","DOI":"10.1155\/2013\/107120","type":"journal-article","created":{"date-parts":[[2013,9,5]],"date-time":"2013-09-05T17:01:31Z","timestamp":1378400491000},"page":"1-14","source":"Crossref","is-referenced-by-count":1,"title":["Multiscale Image Representation and Texture Extraction Using Hierarchical Variational Decomposition"],"prefix":"10.1155","volume":"2013","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9140-4745","authenticated-orcid":true,"given":"Liming","family":"Tang","sequence":"first","affiliation":[{"name":"College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China"},{"name":"College of Mathematics and Physics, Chongqing University of Science and Technology, Chongqing 401331, China"}]},{"given":"Chuanjiang","family":"He","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China"}]}],"member":"311","reference":[{"year":"2001","key":"1"},{"issue":"1","key":"2","first-page":"14","volume":"50","year":"2003","journal-title":"Notices of the American Mathematics Society"},{"issue":"1\u20134","key":"3","first-page":"259","volume":"60","year":"1992","journal-title":"Physica D"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1137\/S1540345902416247"},{"issue":"4","key":"5","first-page":"554","volume":"2","year":"2003","journal-title":"Multiscale Modeling and Simulation"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1023\/A:1025384832106"},{"key":"7","first-page":"143","volume":"4","year":"2005","journal-title":"Applied Mathematics Research Express"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1137\/040610052"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.3969\/j.issn.1004-4132.2011.02.026"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1109\/83.661182"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2009.08.036"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2004.12.004"},{"issue":"2","key":"13","doi-asserted-by":"crossref","first-page":"281","DOI":"10.4310\/CMS.2008.v6.n2.a2","volume":"6","year":"2008","journal-title":"Communications in Mathematical Sciences"},{"issue":"1-2","key":"14","first-page":"7","volume":"20","year":"2004","journal-title":"Journal of Mathematical Imaging and Vision"},{"year":"2001","key":"15"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2005.860340"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1080\/00207160903134263"},{"year":"1992","key":"18"},{"issue":"2","key":"19","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/s002110050258","volume":"76","year":"1997","journal-title":"Numerische Mathematik"},{"key":"20","doi-asserted-by":"publisher","DOI":"10.1007\/s00245-004-0812-z"},{"key":"21","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-005-4783-8"},{"key":"22","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-010-0251-1"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2013\/107120.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2013\/107120.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2013\/107120.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,21]],"date-time":"2017-06-21T17:24:16Z","timestamp":1498065856000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/jam\/2013\/107120\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013]]},"references-count":22,"alternative-id":["107120","107120"],"URL":"https:\/\/doi.org\/10.1155\/2013\/107120","relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"type":"print","value":"1110-757X"},{"type":"electronic","value":"1687-0042"}],"subject":[],"published":{"date-parts":[[2013]]}}}