文档名：信号重构中l1l2极小化的等价模型理论研究
摘要：非凸l1-l2极小化方法是信号重构中最有效、最重要的模型之一,为研究信号恢复中极小化l1-l2重构的条件,从极小化l1-l2的定义出发,结合零空间的性质,基于对原优化模型中的目标函数进行松弛并利用相干性函数,推导出与原优化问题等价的新目标函数.结合信号重构中限制等距性的充分性条件,得到了极小化l1-l2完全重构的充分性条件,并证明了在噪声情形下两类条件的等价性.
Abstract：Summary:Signalreconstructionisadataprocessingtechnologybasedonpartialsignaltorecoverthewholesignal.Itiswidelyusedinopticalcommunication,imageprocessing,datacompressionandotherfields.Themainproblemistofindthesparsesolutionofagroupofunderdeterminedlinearornonlinearequations.Whenusingthetraditionalsignalreconstructionmethodtotransformtheoriginalproblemintoanunderdeterminedsystemofequations,itneedstoaddsomerestrictiveconditionssuchassparsity,soastoachievesignalreconstruction.Inordertosolvetheshortcomingsofthetraditionalsignalreconstructionmodel,suchaslowaccuracy,slowefficiency,andexcessivedependenceonsignalsparsity,itisofgreatsignificancetofurtherstudythenewsignalreconstructionmodel.Amongthem,therecentlyproposednonconvexl1-l2minimizationnormmethodisoneofthemosteffectiveandimportantmodelsinsignalreconstruction.Inthispaper,aclassofimportantnonconvexl1-l2minimazationnormmodelsproposedinrecentyearsarestudied,andal1-l2minimazationnormsignalreconstructionmodelwithequivalentsignificanceisproposed,andthesignalreconstructiontheoryisanalyzedtheoretically.Basedonthedefinitionofl1-l2minimazationnormmodelandthenatureofzerospace,theobjectivefunctionintheoriginaloptimizationmodelisrelaxed,andthenewobjectivefunctionequivalenttotheoriginaloptimizationproblemisderivedbyusingthecoherencefunction.Twoequivalentmathematicalmodelsofl1-l2minimazationnormareestablished,andthefeasibilityofsignalreconstructionisstudiedintheory.Proposition1:Suppose(A∈Rm×n(m<n))isameasurementmatrix,ifthenon-zerovectorAx=bsatisfies(G(x,h)=||h||1-(||x+h||2-||x||2)﹥0)Thenthesignalx∈Rncanbereconstructedthroughl1-l2minimizationnorminAx=b,where,h≠0and(h∈ker(A)={z∈Rn:Az=0}).Proposition2:SupposeA∈Rm×n(m<n)isameasurementmatrix,ifitexistst∈(0,1],sothatthenon-zerovectorx∈Rnsatisfies(G(t;x,h)=t||h||1-(||x+h||2-||x||2)﹥0)Thenthesignalx∈Rncanbereconstructedthroughl1-l2minimizationnorminAx=b,where,h≠0and(h∈ker(A)={z∈Rn:Az=0}).Inordertoexplorethesufficientconditionsforsignalrecoveryusingl1-l2minimazationnorm,thearticlebasedonproposition1andproposition2,researchstherelationshipbetweensignalreconstructionandn-dimensionalinnerproductandcoefficientinzerospace.Combiningwiththenecessaryandsufficientconditionsofsignalreconstructioninthegeneralformofthel1-l2minimazationnormmodel,thesufficientconditionsofsignalcompletereconstructionareestablishedforthel1-l2minimazationnormmodelintheequivalentcase.Theeffectsofparametersandzerospaceonsignalreconstructionareanalyzed,andtwokindsofrobustconditionsthatcanguaranteesignalreconstructioninthecaseofnonoiseareproved.Theorem1:AssumingthemeasurementmatrixA∈Rm×n(m<n),thesignalx∈RninAx=bcanbereconstructedbyl1-l2minimizationnorm,asufficientconditionexistst∈(0,1],sothat(||h||1-/t||x||2﹥0)Where,(h∈ker(A)={z∈Rn:Az=0}),||h2||=1.Theorem2:SupposeA∈Rm×n(m<n)isameasurementmatrix.Ifthereisarealnumberλi,suchthat(||H||1-[x,H]/||x||2﹥0)Thenthesignalx∈RninAx=bcanbereconstructedbyl1-l2minimizationnorm,where,(H=λ1(h)1+λ2(h)2+…+λj(h)j,(h)i∈ker(A)={z∈Rn:Az=0})Theconclusionshowsthatwhencertainconditionsaresatisfied,thesignalcanbeeffectivelyreconstructedbytheequivalentl1-l2minimazationnormmodel.
作者：谢挺 刘昊 张艺萍 张望哲Author：XIETing LIUHao ZHANGYiping ZHANGWangzhe
作者单位：重庆理工大学理学院,重庆400054
刊名：重庆理工大学学报 PKU
Journal：JournalofChongqingInstituteofTechnology
年，卷(期)：2024, 38(9)
分类号：O242.26
关键词：压缩感知 信号重构 零空间
Keywords：compressedsensing signalreconstruction nullspace
机标分类号：O141.4O224TP301.6
在线出版日期：2024年7月11日
基金项目：重庆市自然科学基金，重庆市技术创新与应用发展专项重点项目信号重构中l1-l2极小化的等价模型理论研究[
期刊论文] 重庆理工大学学报--2024, 38(9)谢挺 刘昊 张艺萍 张望哲非凸l1-l2极小化方法是信号重构中最有效、最重要的模型之一,为研究信号恢复中极小化l1-l2重构的条件,从极小化l1-l2的定义出发,结合零空间的性质,基于对原优化模型中的目标函数进行松弛并利用相干性函数,推导出与原优化...参考文献和引证文献
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信号重构中l1-l2极小化的等价模型理论研究 Research on the equivalent model of l1-l2 minimization in signal reconstruction
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