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压缩感知问题统计建模及应用边昂;张建州【摘要】In this paper, a novel statistical model is proposed to describe the Gaussian noisy compressed sensing problem with Gaussian random measurement matrix, and under the statistical framework, some hypothesis tests are used to analyse the performance of the weighted median regression estimate compressive sensing signal reconstruction with an iterative hard threshold under the l0- regularized constraint. The χ2 test based computation sequence is proposed to improve the performance of its coordination descent computation sequence, and F test based data adaptive stopping criterion is presented to take the place of its manual stopping conditions of the maximal number of iterations and the lower bound of the residual energy. Practical performance of the proposal is evaluated via numerical experiments.%对高斯噪声下的高斯随机观测矩阵压缩感知问题建立了新的统计模型，并在该统计模型的基础上，引入相应的统计检验方法对l 0范式约束下的硬阈值加权中值回归重建算法进行分析。
基于贝叶斯压缩感知的复数稀疏信号恢复方法 王伟;唐伟民;王犇;雷舒杰 【摘 要】An effective Sparse Bayesian Learning algorithm exploiting Complex sparse Temporal correlation (CTSBL) is proposed in this paper, which is used to recover sparse complex signal. By exploiting the fact that the real and imaginary components of a complex value share the same sparsity pattern, it can improve the sparsity of the estimated signal. A multitask sparse signal recovery issue is transformed to a block sparse signal recovery issue of a single measurement by taking full advantage of the internal structure information among the multiple measurement vector signals. The experiments show that the proposed algorithm CTSBL achieves better recovery performance compared with the existing Complex MultiTask Bayesian Compressive Sensing (CMTBCS) algorithm and BOMP algorithm.%该文利用复数稀疏信号的时域相互关系提出一种新的稀疏贝叶斯算法(CTSBL)。该算法利用复数信号的实部与虚部分量具有相同的稀疏结构的特点，提升估计信号的稀疏程度。同时将多个测量信号间的内部结构信息引入到了信号恢复中，使原始的多测量稀疏信号恢复问题转变为单测量块稀疏信号恢复问题，使恢复性能得到了提升。理论分析和仿真结果证明，提出的 CTSBL 算法相较于目前的针对复数信号的多测量矢量贝叶斯压缩感知(CMTBCS)算法和块正交匹配追踪算法(BOMP)在估计精度上具有更好的性能。
基于小波树和互补分解的CS-MRI重建算法PEI Ying;ZHU Jin-xiu;YANG Yu-chen;WU Wen-xia【摘要】针对压缩感知(CS)核磁共振成像(MRI)重建算法中全变分(TV)正则项会导致图像细节丢失的问题,引入互补分解模型,结合小波树结构稀疏(简称小波树),提出一种基于小波树和互补分解的CS-MRI重建算法.利用互补分解将图像分成平滑分量和残差分量两个部分,并将平滑分量用于TV正则项,残差分量用于e1范数,可避免TV正则项在滤除噪声的同时滤除过多的细节信息;利用小波树结构稀疏可进一步补充小波稀疏等先验信息,减少测量值或提高信噪比.针对目标函数中存在平滑和残差两个未知分量,将目标函数分解为相应的两个子问题交替最小化进行求解.实验结果表明,与基于小波树的WaTMRI和基于TV的TVCMRI、FCSA等重建算法相比,其能在滤除噪声的同时有效改善MRI图像的细节信息.【期刊名称】《计算机技术与发展》【年(卷),期】2018(028)012【总页数】5页(P152-156)【关键词】核磁共振成像;压缩感知;互补分解;小波树结构稀疏(小波树);目标函数;重建算法【作者】PEI Ying;ZHU Jin-xiu;YANG Yu-chen;WU Wen-xia【作者单位】;;;【正文语种】中文【中图分类】TP301.60 引言核磁共振成像(magnetic resonance imaging，MRI)是医学成像，应用广泛。
目前MRI应用的关键在于快速成像。
Nyquist采样定理需两倍带宽，不符合实际应用[1]，现常用方法是在压缩感知(compressed sensing，CS)[2-3]框架下，从欠采样k空间中重建MRI数据，能有效减少采样时间，达到快速成像的目的。
关于CS-MRI重建算法的研究有很多。
例如，Lusting等[4]利用全变分(total variation，TV)和小波构建目标函数，提出共轭梯度算法(CG)，但重建时间有待提高；FISTA算法[5]通过计算更合适的起始点以加快收敛速度；TVCMRI[6]、RecPF[7]和FCSA[8]算法利用算子分割、变量分割思想求解联合正则算子，提高重建速度和质量；文献[9]利用图像在频域的特性优化测量矩阵并提出迭代加权算法，提高了重建精度；文献[2,10]利用MR图像低秩特性进行奇异值分解，但过程过于复杂；文献[11-12]提出结构稀疏理论，图像不仅在小波域有稀疏性，其小波稀疏系数也有特定的四叉树结构，在此基础上文献[13]提出YALL1算法，利用小波树结构稀疏代替小波稀疏构建目标函数，以提高图像稀疏度；文献[14]提出WaTMRI算法，联合小波树结构稀疏和小波稀疏分别拥有的结构稀疏和稀疏性，联合改善图像质量；Park等[15]提出互补分解，将完整图像分为平滑和残差两个分量，仅将平滑分量用于TV，残差分量用于1范数，以解决全变分导致的细节过平滑问题；文献[16]利用贪婪算法提高重建速度，但需要确定图像稀疏度，缺乏实际性；文献[17]基于p范数构建重建算法，计算较为复杂。
点云匹配和ICP算法概述Iterative Closest Point (ICP) is an algorithm employed to minimize the difference between two clouds of points.点云匹配分类法（1）•全局匹配算法 Globe•局部匹配算法LocalSalvi, J. (2007). "A review of recent range image registration methods with accuracy evaluation." Image and Vision Computing 25(5): 578-596.Mellado, N. and D. Aiger (2014). "SUPER 4PCS Fast Global Point cloud Registration via Smart Indexing."点云匹配分类法（2）•基于点的匹配•基于特征的匹配•点特征•VPF•FHPF•…•基于线特征•"Algorithms for Matching 3D Line Sets."•"Line segment-based approach for accuracy assessment of MLS point clouds in urban areas.“•Poreba, M. and F. Goulette (2015). "A robust linear feature-based procedure for automated registration of point clouds." Sensors (Basel) 15(1): 1435-1457.Coarse to fine registration粗-精过程粗配的⽬的：提供刚体变换初始估计Salvi, J., et al. (2007).改进ICP算法Besl, P. J. and N. D. Mckay (1992). "A Method for Registration of 3-D Shapes." IEEE Transactions on Pattern Analysis and Machine Intelligence 14(2): 239-256.Siegwart, R., et al. (2015). "A Review of Point Cloud Registration Algorithms for Mobile Robotics." Foundations and Trends in Robotics.•加快搜索效率•K-D树•Voronoi图•不同的距离量测⽅式•点到点•点到线 PLICP•Censi, A. (2008). "An ICP variant using a point-to-line metric." IEEE International Conference on Robotics & Automation. IEEE,: 19-25.•CSM（Canonical Scan Matcher）源码•点到⾯•Low, K.-L. (2004).•⾯到⾯ GICPICP算法求解•Closed Form•SVD•Unit Quaternions单位四元数•The ICP error function minimization via orthonormal matrices•Dual Quaternions•数值解法•LM算法（Levenberg-Marquardt algorithm）•Jerbić, B., et al. (2015). "Robot Assisted 3D Point Cloud Object Registration." Procedia Engineering 100: 847-852.•点到⾯线性最⼩⼆乘法•Low, K.-L. (2004). "Linear Least-Squares Optimization for Point-to-Plane ICP Surface Registration."问题•观测误差•部分重叠•离群点Outlier、噪声（经常是错误点或者异常点）•不满⾜⼀⼀对应的条件解决⽅法•剔除 Rejection•PCL类库中采⽤•权重⽅法•稳健⽅法Bergström, P. and O. Edlund (2014). "Robust registration of point sets using iteratively reweighted least squares."H. Pottmann, S. Leopoldseder, and M. Hofer. Simultaneous registration of multiple views of a 3D object. ISPRS Archives 34/3A (2002), 265-270.Andreas Nüchter(2008).3D Robotic Mapping-The Simultaneous Localization and Mapping Problem with Six Degrees of Freedom标准ICP标准ICP算法是最早提出的基于点-点距离的算法，另外⼀种是基于点-⾯的算法，由chen提出，好多⽂献所说的恶Chen's Method。
ITERATIVELYREWEIGHTEDALGORITHMSFORCOMPRESSIVESENSINGRickChartrandLosAlamosNationalLaboratoryrickc@lanl.govWotaoYin∗RiceUniversitywotao.yin@rice.edu
ABSTRACTThetheoryofcompressivesensinghasshownthatsparsesignalscanbereconstructedexactlyfrommanyfewermea-surementsthantraditionallybelievednecessary.In[1],itwasshownempiricallythatusingpminimizationwithp<1can
dosowithfewermeasurementsthanwithp=1.Inthispaperweconsidertheuseofiterativelyreweightedalgorithmsforcomputinglocalminimaofthenonconvexproblem.Inpar-ticular,aparticularregularizationstrategyisfoundtogreatlyimprovetheabilityofareweightedleast-squaresalgorithmtorecoversparsesignals,withexactrecoverybeingobservedforsignalsthataremuchlesssparsethanrequiredbyanunreg-ularizedversion(suchasFOCUSS,[2]).Improvementsarealsoobservedforthereweighted-1approachof[3].
IndexTerms—Compressivesensing,signalreconstruc-tion,nonconvexoptimization,iterativelyreweightedleastsquares,1minimization.
1.INTRODUCTIONRecentpapers[4,5]haveintroducedtheconceptknownascompressivesensing(amongotherrelatedterms).Thebasicprincipleisthatsparseorcompressiblesignalscanberecon-structedfromasurprisinglysmallnumberoflinearmeasure-ments,providedthatthemeasurementssatisfyanincoherenceproperty(see,e.g.,[6]foranexplanationofincoherence).Suchmeasurementscanthenberegardedasacompressionoftheoriginalsignal,whichcanberecoveredifitissufficientlycompressible.Afewofthemanypotentialapplicationsaremedicalimagereconstruction[7],imageacquisition[8],andsensornetworks[9].Thefirstalgorithmpresentedinthiscontextisknownasbasispursuit[10].LetΦbeanM×Nmeasurementma-trix,andΦx=bthevectorofMmeasurementsofanN-dimensionalsignalx.Thereconstructedsignalu∗isthemin-
imizerofthe1norm,subjecttothedata:
minuu1,subjecttoΦu=b.(1)
∗Thesecondauthorwassupportedbyaninternalfacultyresearchgrant
fromtheDeanofEngineeringatRiceUniversity.HewouldliketothankElaineHaleandYinZhangforvaluablehelpwithcoding.
AremarkableresultofCand`esandTao[11]isthatif,forexample,therowsofΦarerandomlychosen,Gaussiandis-tributedvectors,thereisaconstantCsuchthatifthesupportofxhassizeKandM≥CKlog(N/K),thenthesolutionto(1)willbeexactlyu∗=xwithoverwhelmingprobability.
TherequiredCdependsonthedesiredprobabilityofsuccess,whichinanycasetendstooneasN→∞.DonohoandTan-ner[12]havecomputedsharpreconstructionthresholdsforGaussianmeasurements,sothatforanychoiceofsparsityKandsignalsizeN,therequirednumberofmeasurementsMfor(1)torecoverxcanbedeterminedprecisely.VariantsoftheseresultshaveincludedΦbeingarandomFouriersubmatrix,orhavingvalues±1/√
Nwithequalprob-
ability.Moregeneralmatricesareconsideredin[6,13].Also,xcanbesparsewithrespecttoanybasis,withureplacedwithΨuforsuitableunitaryΨ.Afamilyofiterativegreedyalgorithms[14,15,16]havebeenshowntoenjoyasimilarexactreconstructionproperty,generallywithlesscomputationalcomplexity.However,thesealgorithmsrequiremoremeasurementsforexactreconstruc-tionthanthebasispursuitmethod.Intheotherdirection,itwasshownin[1]thatanonconvexvariantofbasispursuitwillproduceexactreconstructionwithfewermeasurements.Specifically,the1normisreplaced
withthepnorm,where0actuallyanorm,thoughd(x,y)=x−yppisametric):
minuupp,subjecttoΦu=b.(2)
Thatfewermeasurementsarerequiredforexactreconstruc-tionthanwhenp=1wasdemonstratedbynumericalex-perimentsin[1],withrandomandnonrandomFouriermea-surements.Atheoremwasalsoprovenintermsofthere-strictedisometryconstantsofΦ,generalizingaresultof[17]toshowthataconditionsufficientfor(2)torecoverxexactlyisweakerforsmallerp.Morerecently,forthecaseofrandom,Gaussianmeasurements,theaboveconditionofCand`esandTaohasbeenshown(usingdifferentmethods)[18]togener-alizeto
M≥C1(p)K+pC2
(p)Klog(N/K),(3)
whereC1,C2aredeterminedexplicitly,andareboundedin
p.Thus,thedependenceofthesufficientnumberofmea-
38691-4244-1484-9/08/$25.00 ©2008 IEEEICASSP 2008surementsMonthesignalsizeNdecreasesasp→0.Theconstantsarenotsharp,however.2.ALGORITHMSFORNONCONVEXCOMPRESSIVESENSINGEarlypapersconsideringiterativelyreweightedleastsquares(IRLS)approachesinclude[19,20].IRLSforsolving(2)hasmostlybeenconsideredforp≥1.Thecaseofp<1wasstudiedbyRaoandKreutz-Delgado[2].Theapproachistoreplacethepobjectivefunctionin(2)byaweighted2norm,minuNi=1wiu2i,subjecttoΦu=b,(4)wheretheweightsarecomputedfromthepreviousiterateu(n−1),sothattheobjectivein(4)isafirst-orderapproxi-mationtothepobjective:wi=|u(n−1)i|p−2.Thesolutionof(4)canbegivenexplicitly,givingthenextiterateu(n):u(n)=QnΦTΦQnΦT)−1b,(5)whereQnisthediagonalmatrixwithentries1/wi=|u(n−1)i|2−p.ThiscomesfromsolvingtheEuler-Lagrangeequationof(4),usingtheconstrainttosolvefortheLagrangemultipliers,thensubstitutingthisvalueintothesolution.Theresultisafixed-pointiterationforsolvingtheEuler-Lagrangeequationof(2).Inthispaper,weareconsidering0≤p≤1.Thecaseofp=0correspondstoanobjectiveoflog(|ui|2).Giventhatp−2willbenegative,theweightswiareundefinedwheneveru(n−1)i=0.Acommonapproachfordealingwiththisissueistoregularizetheoptimizationproblem,byincorporatingasmall>0.Forexample,belowweconsiderthedampingapproach:wi=(u(n−1)i)2+p/2−1.(6)Asobservedin[2],theiteration(5)dependsonlyon1/wi,whichifdefineddirectlyas|u(n−1)i|2−piswelldefinedforanyvalueofu(n−1)i.Inprinciple,noregularizationisneeded.Formanysystems,however,thematrixbeinginvertedin(5)istooill-conditionedtoallowaccuratecomputationofu(n).InSection3,weperformexperimentsthatshowthatthestrategyofusingarelativelylargein(6),thenrepeatingtheprocessofdecreasingafterconvergenceandrepeatingtheiteration(5),dramaticallyimprovestheabilityofIRLStore-coversparsesignals.Exactrecoverybecomespossiblewithmanyfewermeasurements,orwithsignalsthataremuchlesssparse.This-regularizationstrategywasusedeffectivelywithaprojectedgradientalgorithmin[1].Itmustbenotedthat(2)isanonconvexoptimizationprob-lemwhenp<1,andallofthealgorithmsconsideredhereareonlydesignedtoproducelocalminima.However,thefactthatinpracticeweareabletorecoversignalsexactly,com-binedwiththeoreticalresults[1,18]thatgivecircumstancesinwhich(2)hasaunique,globalminimizerthatisexactlyu∗=x,stronglysuggeststhatthecomputedlocalminimiz-
