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Signal Process."],"abstract":"Abstract<\/jats:title>\n \n In this paper, we consider the frame-based block sparse signal recovery via a\n \n \n $$l_2\/l_1$$<\/jats:tex-math>\n \n \n \n l<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \/<\/mml:mo>\n \n l<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -synthesis method. A new kind of null space property based on the given dictionary\n D<\/jats:italic>\n (block\n D<\/jats:italic>\n -NSP) is proposed. It is proved that sensing matrices satisfying the block\n D<\/jats:italic>\n -NSP is not just a sufficient and necessary condition for the\n \n \n $$l_2\/l_1$$<\/jats:tex-math>\n \n \n \n l<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \/<\/mml:mo>\n \n l<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -synthesis method to exactly recover signals that are block sparse in frame\n D<\/jats:italic>\n , but also a sufficient and necessary condition for the\n \n \n $$l_2\/l_1$$<\/jats:tex-math>\n \n \n \n l<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \/<\/mml:mo>\n \n l<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -synthesis to stably recover signals which are block-compressible in frame\n D<\/jats:italic>\n . To the best of our knowledge, this new property is the first sufficient and necessary condition for successful signal recovery via the\n \n \n $$l_2\/l_1$$<\/jats:tex-math>\n \n \n \n l<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \/<\/mml:mo>\n \n l<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -synthesis. In addition, we also characterize the theoretical performance of recovering signals via the\n \n \n $$l_2\/l_1$$<\/jats:tex-math>\n \n \n \n l<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \/<\/mml:mo>\n \n l<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -synthesis in the case of the measurements are disturbed.\n <\/jats:p>","DOI":"10.1186\/s13634-024-01175-7","type":"journal-article","created":{"date-parts":[[2024,7,16]],"date-time":"2024-07-16T05:03:00Z","timestamp":1721106180000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Frame-based block sparse compressed sensing via $$l_2\/l_1$$-synthesis"],"prefix":"10.1186","volume":"2024","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5647-3278","authenticated-orcid":false,"given":"Fengong","family":"Wu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Penghong","family":"Zhong","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huasong","family":"Xiao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chunmei","family":"Miao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,7,16]]},"reference":[{"key":"1175_CR1","doi-asserted-by":"publisher","first-page":"282","DOI":"10.1016\/j.acha.2011.12.002","volume":"33","author":"A Aldroubi","year":"2012","unstructured":"A. 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