ebook img

Mixed one-bit compressive sensing with applications to overexposure correction for CT reconstruction PDF

0.47 MB·
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Mixed one-bit compressive sensing with applications to overexposure correction for CT reconstruction

1 Mixed One-bit Compressive Sensing with Application to Overexposure Correction for CT Reconstruction Xiaolin Huang, Member, IEEE, Yan Xia, Lei Shi, Yixing Huang, Ming Yan, Joachim Hornegger, and Andreas Maier, Member, IEEE 7 1 Abstract—When a measurement falls outsidethequantization ment can be described as 0 or measurable range, it becomes saturated and cannot be 2 used in classical reconstruction methods. For example, in C- p =max s−,min u⊤x+ε,s+ , (1) n armangiographysystems,whichprovideprojectionradiography, i n (cid:8) i (cid:9)o a fluoroscopy,digitalsubtractionangiography,andarewidelyused where x ∈ Rd is the signal to be estimated, u ∈ Rd is a J for medical diagnoses and interventions, the limited dynamic i range of C-arm flat detectors leads to overexposure in some sensing vector, ε stands for the noise, and s− and s+ are the 3 projections during an acquisition, such as imaging relatively lower and upper saturation thresholds, respectively. When an ] thin body parts (e.g., the knee). Aiming at overexposure cor- observationpi equalss+ or s−, we saythatthismeasurement V rection for computed tomography (CT) reconstruction, we in is saturated. The saturation phenomenon may happen when this paper propose a mixed one-bit compressive sensing (M1bit- C the detector has a narrow range [1], [2], [3]. In this paper,we CS) to acquire information from both regular and saturated . measurements. This method is inspired by the recent progress discusshowtorecoverxfromasensingsystemwithsaturation s c on one-bit compressive sensing, which deals with only sign and then apply the proposed method to correct overexposure [ observations. Its successful applications imply that information for C-arm computed tomography (C-arm CT). 1 cqaurarliietyd.bFyorsatthueraptreodpmoseeadsuMre1mbietn-CtsSismuosdeeful,latloteirmnpartionvgedreirceocvteiorny C-armCTischaracterizedwithhighspatialresolution,large v methods of multipliers is developed and an iterative saturation field of view, and 3D imaging capability. It has become a 4 detection scheme is established. Then we evaluate M1bit-CS on valuable tool for medical diagnoses, therapy guidance, and 9 one-dimensional signal recovery tasks. In some experiments, the interventions.Thekey technologythatenablesthese advances 6 performanceoftheproposedalgorithmsonmixedmeasurements is the flat-panel detector, which provides good soft-tissue 0 isalmostthesameasrecoveryonunsaturatedoneswiththesame 0 amountofmeasurements.Finally,weapplytheproposedmethod imaging performance, distortion-free images as well as high 1. to overexposure correction for CT reconstruction on a phantom detective quantum efficiency [4]. However, although typical 0 and a simulated clinical image. The results are promising, as C-arm flat-panel detectors have a dynamic range of 14-bit 7 the typical streaking artifacts and capping artifacts introduced that is sufficient for conventional fluoroscopy, it may not 1 by saturated projection data are effectively reduced, yielding be high enough to eliminate overexposure in all projections : significant error reduction compared with existing algorithms acquired during a 3D acquisition, such as imaging relatively v based on extrapolation. i thin body parts (e.g., the knee). If no counter-measures are X IndexTerms—one-bitcompressivesensing,CTreconstruction, carried out, strong direct radiation may hit certain detector r overexposure correction, ADMM regions in some views (the intensity range of the traveled X- a rays are greater than the detector’s inherent detectable range) I. INTRODUCTION and thus overexposethem. Consequently,reconstruction from IN a real-world measuring system, the measurable range overexposedprojectionstypicallyleadstoincorrectHounsfield unit (HU) values, streaking artifacts, and loss of clear outer could be limited and the final output for a linear measure- boundaries. Moreover, HU values of a homogeneous object tend to become smaller when they are far away from the This work is partially supported by Alexander von Humboldt Foundation object center, resulting in so-called capping artifacts. Since andNationalNaturalScienceFoundationofChina(NSFC,61603248).L.Shi alltheseoverexposureartifactsimpairthefinalvisualizationof issupportedbytheNSFC(11571078),theJointResearchFundbyNSFCand Research Grants Council of Hong Kong (11461161006 and CityU 104012) lowcontrastobjects,itisofpracticalsignificancetodevelopan and“ZhuoXue”program ofFudanUniversity. appropriate overexposure correction scheme on the saturated X. Huang is with Institute of Image Processing and Pattern Recogni- images to compensate for these artifacts. tion, Shanghai Jiao Tong University, Shanghai, P.R. China. Y. Xia is with Department of Radiology, Stanford University, CA, USA. L. Shi is with To give an intuitive impression of saturation phenomenon SchoolofMathematicalSciences,FudanUniversity,Shanghai,P.R.China.Y. happens in CT scan, a knee phantom is designed in Fig.1(a). Huang, J.Hornegger, andA.Maier arewithPattern Recognition Labofthe Its360◦ projectiondatawithoutandwithsaturationareshown Friedrich-Alexander-Universita¨t Erlangen-Nu¨rnberg, Erlangen, Germany. M. Yan(correspondingauthor)iswiththeDepartmentofComputationalMathe- in Fig.1(c) and 1(d), respectively. If there is no saturation, matics, Science and Engineering, Michigan State University, MI, USA. (e- then the object can be reconstructedwell by classical analyti- mails: [email protected], [email protected], [email protected], cal algorithms, such as the standard filtered back projection [email protected], [email protected], [email protected], an- [email protected].) (FBP, [5]) or iterative reconstruction methods, such as the 2 simultaneous algebraic reconstruction technique (SART, [6]). In contrast to the existing saturation correction methods When there is saturation in the sinogram data, e.g. Fig.1(d), thatrequireprior-knowledge,in this paperwe proposea data- CT reconstruction from the saturated data is quite hard. For drivenmethodto findx directlyfrommeasurementsobserved example, directly applying FBP on the saturated data in by (1). The key is to acquire information from the saturated Fig.1(d) outputs Fig.1(b), which is far from satisfactory. measurements, which do not provide analog information but docontainone-bitinformation:whenameasurementisupper- saturated (lower-saturated), i.e., p = s+(s−), we know that i u⊤x+ε is greater (smaller) than or equal to s+ (s−). i Dataminingfrominequalitiesistheoreticallypossiblewhen x is sparse, which hasbeen verifiedby the recentprogresson one-bit compressive sensing (1bit-CS). The concept of 1bit- CS was firstly introduced by [12] to recover sparse signals from measurements containing only one-bit information, i.e., (a) (b) sign measurements. It is rooted in compressive sensing (CS, [13],[14],[15]),andtherearesomeinterestingdiscussionson both theory and algorithms; see, e.g., [16] [17] [18] [19] [20] [21]. Unlike the pure 1bit-CS, (1) has analog measurements as well. Therefore,we refer to the methoddealing with (1) as mixed one-bit compressive sensing (M1bit-CS). There is few (c) study on this topic. To the best of our knowledge,it has been considered only by [22], [23]. But they put major emphasis on quantization error and assumed that there was no noise on the saturation measurements. In this paper, we are going to recover sparse signals from noise-corruptedmeasurementswithsaturation.Themajorcon- (d) tributions include: Fig. 1. (a) A knee phantom; (b) reconstructed by FBP from saturated • propose M1bit-CS models; projections shown by (d); (c) full projection data; (d) saturated projection data(themeasurable rangeκ=0.5pmax;seeSectionV-Afordetails). • establish the corresponding alternating direction method of multipliers (ADMM); Due to the loss of information, signal recovery from satu- • design an iterative saturation detection scheme; ratedmeasurementsarenotstraightforward.InCTreconstruc- • evaluate M1bit-CS on one-dimensional signal recovery tion, primarily targeted to correct the overexposure artifacts problems; forkneeimaging,anoptimization-basedmultipleshapefitting • apply M1bit-CS on overexpose correction for C-arm CT approach was suggested to extrapolate the missing data using and achieve very promising performance. a pair of cylinder shapes that are fitted in the sinogram Before discussing M1bit-CS, we introduce some notations domain [7]. The parameters describing the cylinders are es- to describe (1) that is a linear sensing system with saturation. timated from the overexposed data by minimizing the least The dimension of the signal is denoted by d, the number squareerror.However,themethodwouldfacechallengeswhen of total measurements by m, and the number of saturated it comes to complicated shapes that cannot be modeled with measurementsby n. A boolean vector Ψ is used to indicate a a compositionof cylindricalor elliptical shapes. Furthermore, measurement is saturated or not. Mathematically, additional hardware, such as a shaped pre-filteration (bowtie) can be deployed to correct the saturation during the data Ψ =1 ⇔ u⊤x+ε>s+ or u⊤x+ε<s−. i i i acquisition [8]. However, that method requires the imaged Furthermore,forsaturatedmeasurements,i.e.,Ψ =1,weuse object to be exactly centered in the rotation center and the i y ∈ {−1,+1} to distinguish the upper-saturation (y = 1) use of a bowtie filter also appears not flexible enough for the i i and lower-saturation (y =−1): applications of C-arm CT systems. Another way to prevent i saturation at the object is to apply modeling clay wrapped y (u⊤x+ε−s )≥0, ∀i:Ψ =1, (2) i i i i aroundthe imagedobjectas an additionalabsorberduringthe data acquisition [9], [10]. However, the method is not ideal where s = s+,∀i : y = 1 and s = s−,∀i : y = −1. Then i i i i since the modeling clay may cause great discomfort for the our task of signal recovery from (1) is to find a sparse signal patient. Alternatively,a Kinect camera was used to obtain the x such that u⊤x is close to p when Ψ = 0 and coincides i i i objectdepthinformation,whichcanbecombinedintoaC-arm with the saturation inequality (2) when Ψ =1. i system for correcting overexposured projections [11]. More Therestofthispaperisorganizedasfollows.InSectionII, specifically,the Kinectdepthdata isused to findthe pointsof M1bit-CSmodelsareestablished.Fastalgorithmsaredesigned intersection between the X-ray beam path and object surface. inSectionIII.SectionIVevaluatestheproposedalgorithmsin Overexposed pixels on the projection images are corrected numericalexperiments.SectionVappliestheproposedmethod by extrapolating the absorption along the corresponding line to overexposure correction for C-arm CT reconstruction. In integrals. Section VI, a conclusion is given to end this paper. 3 II. MIXED ONE-BIT COMPRESSIVE SENSING where γ >0. This modification is denoted as M1bit-CS with A. (One-bit) Compressive Sensing ℓ2-normregularization(M1bit-CSR)andtheoriginalone(5)is Aclassicalcompressivesensing(CS)modelforunsaturated M1bit-CS with ℓ2-norm constraint (M1bit-CSC). When kxk2 is known or can be estimated accurately, one can choose data,i.e.,Ψ =0,∀i,canbeformulatedasthefollowingwell- i M1bit-CSC, otherwise, M1bit-CSR is applicable. known model: m For the analog part, we use the least squares loss for L1 xm∈iRnd µkxk1+ 21 L1(u⊤i x−pi). (3) ianlstohiaspppalipcearb.lIet.sFroorbuthstemsaotduirfiatciaotniopnasr,ts,udcihreacstlyHiunbheerriltoisnsg, tahree Xi=1 ideaof 1bit-CSleadsto the linearloss. Butherewe havealso The loss function in the above formulation could be the analogmeasurementsthatcanprovidemagnitudeinformation, least squares loss. The related theory and algorithms have anditmakesusingthenaturalhingelossforsigninconsistency been insightfully investigated in the last decade; see [24] and possible. The linear and the hinge loss are covered by the references therein. pinball loss defined below: If we only have sign measurements, which could be re- garded as an extreme quantization process, or equivalently t, t≥0, s+ = 0 and s− being very close to zero in (1)1, we obtain Lτ(t)=(cid:26) −τt, t<0, (7) theso-calledone-bitcompressivesensing(1bit-CS,[12][16]). Many 1bit-CS models can be expressed as follows: where τ describes the slope in the negativepart: when τ =0, it is the hinge loss and when τ =−1, it reduces to the linear m min µkxk1+ L2(yi(u⊤i x)) (4) loss. The pinball loss has been applied for both regression x∈Rd ([25] [26]) and classification ([27] [28]). For M1bit-CS, it is Xi=1 possible to take any value from −1 to 0 for τ. The best one s.t. kxk2 =c, is problem-dependent and we will investigate its choice in wherecisagivenconstant.Therearetwodifferencesbetween numerical experiments. (3) and (4). First, a different loss function L2 is used for In M1bit-CS model, the parameter µ controls the signal the inconsistency of the sign. Second, because multiplying a sparsity,andit existsalsoin regularCS models.Thisparame- signal by a positive constant does not change the signs of ter can be selected according to prior-knowledgeor by cross- linear measurements, the ℓ2 norm constraint (c=1 is always validation.λisatrade-offbetweentheanalogandthesaturated assumed) is needed. measurements. Heuristically, we set it as λ = m such that 100n In recent works on 1bit-CS ([19] [20] [21]), the nonconvex λissmallerwhentherearemoresaturatedmeasurements.For constraint kxk2 =c is relaxed to kxk2 ≤c for computational M1bit-CSR (6), there is an additional parameter γ which is efficiency.Butthisrelaxationmakesitinconvenienttouse the used to adjust the norm of the reconstructed signal. In 1bit- hinge loss, which is a natural choice for sign inconsistency: CS, the requirementon the signal normis crucial. For M1bit- minimizing mi=1max{0,yi(u⊤i x)} in a norm-ball kxk2 ≤c CS, it becomes less important since there are also analog leads to a triPvial and unreasonable solution x=0. Therefore, measurements but still it helps the signal recovery. in convex 1bit-CS models, L2 usually takes the linear loss −y (u⊤x), which could be explained as to pursue high i i correlation between y and u⊤x. C. Iterative Saturation Detection i i In the sensing system (1), if the observation is s+ or s−, B. Mixed One-bit Compressive Sensing we know that it is saturated and use it in M1bit-CS as a To consider both analog measurements and one-bit infor- one-bit measurement. But in some systems, u⊤x itself has i mation, we propose the following mixed one-bit compressive a lower/upper bound and then it is interesting and useful to sensing (M1bit-CS): distinguishwhetherameasurementreachesthebounds.Inthis paper, we focus on the lower-saturation and assume that the min µkxk1+ L1(u⊤i x−pi) lowerboundiszero.Discussionsonupper-saturationandother x∈Rd i:XΨi=0 bounding values are similar. +λ L2(yi(u⊤i x−si)) (5) With prior-knowledgeon non-negativenessof the measure- i:XΨi=1 ments, we have s.t. kxk2 ≤c. p =s− ⇔ 0≤u⊤x≤s−. i i Instead of putting the ℓ2 norm in the constraint, we can also place it in the objective function and come to the following For example, in a CT scan system, u⊤x is related to the i problem, integral of the tissue density along a X-ray. It is certainly γ non-negative and u⊤x = 0 means that the corresponding xm∈iRnd µkxk1+ 2kxk22 (6) X-ray hits the detecitor directly without crossing any tissue. + L1(u⊤i x−pi)+λ L2(yi(u⊤i x−si)), IΨf th=ose0,opbse=rva0tioannsdwuisteh tuh⊤iemx a=s 0anaarleogfomuneads,uwreemceanntssient i:ΨXi=0 i:ΨXi=1 i i M1bit-CS. They will be helpful, especially for reconstructing 1Weassumethatthenegative measurements areupperboundedbys−. the boundary. 4 The detection criterion is: and updates the dual variable (α,β). Updating (e,z) can be 1, 0<u⊤x≤s−, decoupledintoupdatingzandeindependently.Wechoosethe Ψi =(cid:26) 0, u⊤x=i 0. updating order (e,z,x,α,β) in our ADMM implementation. i 1) e-subproblem:we have to solve Accordingly, we can use the reconstructed result to update Ψ and then to solve M1bit-CS with the new guess of Ψ, min λ L (e )+ α e −s +u⊤x e τ i i i i i iteratively. Note that incorrectly detecting a saturated mea- i:ΨXi=1 i:XΨi=1 (cid:0) (cid:1) surementas a zero one is more harmfulthan regardinga zero +θ1 e −s +u⊤x 2, measurementasasaturatedone.Therefore,theinitialguessis 2 i i i Ψ =1forallp =s−.Foralower-saturatedandnon-negative i:XΨi=1(cid:0) (cid:1) i i sensingsystem,theiterativesaturationdetection(ISD)scheme of which the optimal solution is analytically given by is 1d)escsreitbseadtubrealtoiown:indicator Ψ :=1,if p ≤s−; ei =Sτ si−u⊤i x−αi/θ1,λ/θ1 , ∀i:Ψi =1. i i (cid:0) (cid:1) 2) signal recovery with Ψ, denote the result as x˜; Here Sτ is the shrinkage operator for the pinball loss: 3) calculate the measurements of x˜, i.e., q˜i :=u⊤i x˜; t−ρ, if t≥ρ, 0, if p >s−; Sτ(t,ρ)= 0, if t∈(τρ,ρ), 4) update Ψi := 0, if pii ≤s− and q˜i ≤s−/10;  t−τρ, if t≤τρ.  1, if p ≤s− and q˜ >s−/10; 2) z-subproblem: we have to solve i i 5) set pi =0 fori:pi ≤s− and Ψi =0; mzin ιc(z)+β⊤z+ θ22 kz−xk22, 6) go to 2) until there is no change on Ψ. of which the analytical solution is III. ALTERNATINGDIRECTION METHOD OFMULTIPLIERS z=Pc(x−β/θ2). In this paper, we use the least squares loss and the pinball Here Pc is the projection of a vector onto the ℓ2 ball lossforM1bit-CS(5)and(6).Thentheyareconvexproblems ktk2 ≤c: and many existing methods can be applied. We tried several methods including primal-dual algorithms [29] and different P (t)= t, if ktk2 ≤c, Alternating Direction Methods of Multipliers (ADMM, [30]) c (cid:26) c t/ktk2, if ktk2 >c. implementations [31]. In this paper, we introduce an imple- 3) x-subproblem: we have to solve mentation of ADMM, which is efficient in our experiments, 1 but of course is not necessarily optimal for other tasks. mxin µkxk1+ 2 (u⊤i x−pi)2+(Uα+β)⊤x InordertoapplyADMM,wereformulate(5)byintroducing i:XΨi=0 two auxiliaryvariablese andz, andthe equivalentproblemis + θ1 e −s +u⊤x 2+ θ2 kz−xk2. 2 i i i 2 2 xm,ei,nz µkxk1+ 21i:XΨi=0(u⊤i x−pi)2 Itisessentialliy:XΨai=qu1a(cid:0)draticprogramm(cid:1)ingplusanℓ1item: s.t. ei+=λyii:(XΨui⊤i=x1L−τ(sei)i),∀+iι:cΨ(zi)=1, (8) mxin µ+kxθk11+ 21i:ΨPei=−0(us⊤i+xu−⊤pxi)+2+αiθ222(cid:13)(cid:13)(cid:13).x−z+ θβ2(9(cid:13)(cid:13)(cid:13))22 z=x, 2 i:ΨPi=1(cid:16) i i i θ1(cid:17) where ιc(z) is an indicatorfunctionthatreturns0 if kzk2 ≤c It does not have analytical solutions. We approximately and +∞ otherwise. Note here e is m-dimensionalfor consis- solve this problem using FISTA [32]. tency but only the n components with Ψ =1 have effect. The ADMM for M1bit-CSC (5) is summarized in Al- i The corresponding augmented Lagrangian with the dual gorithm 1. When the x-subproblem is exactly solved, the variables (the Lagrange multipliers: α, β) is convergence of Algorithm 1 follows the classical analysis of ADMM;see,e.g.,[30].Otherwise,theconvergenceresultsfor L (x,e,z;α,β) A inexact ADMM in [33] and [34] can be applied. = µkxk1+ 21 (u⊤i x−pi)2+λ Lτ(ei)+ιc(z) CSWRe(6c)a,nanedsttahbelisahlgoAriDthMmMisisnimtihlearstaomAelgwoariythmfor1Mex1cbeipt-t i:ΨXi=0 i:ΨXi=1 that the update of z becomes solving the following problem: + α e −s +u⊤x + θ1 e −s +u⊤x 2 i:ΨXi=1 i(cid:0) i i i (cid:1) 2 i:ΨXi=1(cid:0) i i i (cid:1) mzin γ2kzk22+β⊤z+ θ22 kz−xk22, +β⊤(z−x)+ θ2 kz−xk2, of which the optimal solution is analytically given by 2 2 where θ1 > 0 and θ2 > 0. In every iteration, the ADMM z= θ2x−β. (10) updates(e,z)andxin a Gauss-Seidelstyle withfixed(α,β) θ2+γ 5 Algorithm 1: ADMM for M1bit-CSC (5) A. Parameters Selection Input The parameter τ in the pinball loss is important to charac- ui ∈Rd,Ψi ∈{0,1},i=1,2,...,m;pi ∈R,∀i: terize the M1bit-CS model. As discussed previously, τ =−1 Ψi =0;yi ∈{−1,+1},si∈R,∀i:Ψ=1; give is for 1bit-CS, τ = 0 is for classification problems. To parameters µ,λ numerically investigate the performanceof differentτ values, Initialize the following situations are considered: x,z,β :=0d, e,α:=0m, θ1,θ2 >0 1) signalsparsityK =100,numberofmeasurementsm= repeat 500, number of saturated observations n = 100, noise ei :=Sτ si−u⊤i x−αi/θ1,λ/θ1 ,∀i:Ψi =1 level sn =20; z:=Pc((cid:0)x−β/θ2) (cid:1) 2) K,m,sn are as the same as 1), but n=50; update x by solving (9) 3) K,m,s are as the same as 1), but n=200; n α:=α+θ1(e−s+U⊤x) 4) K,m,n are as the same as 1), but sn =10. β :=β+θ2(z−x) WethenapplyAlgorithm1torecoverthesignalandreportthe until stopping criteria is satisfied; averageSNRover100trialsinFig.2.Fromtheresult,onecan observe that a small and negative τ performs well. The trend is that when the saturation ratio m/n is large, the absolute value of the best τ is also large, which coincides with the IV. EXPERIMENTALVALIDATION experiments reported in 1bit-CS literatures: τ = −1 is better than τ =0 in 1bit-CS where m=n. In the rest of this paper, Inthissection,weevaluatetheproposedalgorithmsonone- we always use τ =− n . Additional tuning can improve the dimensional sparse signal recovery problems. The parameters 5m performance but requires more computational burden. will be discussed numerically, and then the algorithms will be compared with two existing methods: i) simply dropping those saturated measurements and using the remaining mea- 35 surements to recover the sparse signal by lasso; ii) Robust 30 Dequantized Compressive Sensing (RDCS, [23]), which is 25 designed for both saturation and quantization error. When R20 there is no quantization error, RDCS takes the following SN15 formulation: 10 5 1 xm∈iRnd µkxk1+ 2i:XΨi=0(u⊤i x−pi)2 0-1 -0.8 -0.6 -0τ.4 -0.2 0 0.2 s.t. yi(u⊤i x−si)≥0, ∀i:Ψi =1. (11) Fig. 2. Reconstructed SNRs for different τ values. CASE 1): blue solid curve; CASE2):reddashedcurve; CASE3):black dottedcurve; CASE4): greendot-dashedcurve. It assumes that there is no noise in the saturated measure- ments. Thus model (11) is not suitable when there is noise ForM1bit-CSR(6),thereisanadditionalparameterγwhich in measurements before quantization. In fact, model (11) is is used to adjustthe normof the reconstructedsignal.In 1bit- equivalent to model (5) with τ =0, λ= +∞, and c = +∞. CS, the requirementon the signal normis crucial. For M1bit- The ADMM algorithms for lasso and RDCS can be found CS, it becomes less important since there are also analog in [30] and [23], respectively. All the experiments are done measurements but still it helps the signal recovery. We in withMatlab2014bonWindows7withCorei5-3.10GHzand the following consider different γ values for d = 1000,K = 8.0 GB RAM. 100,m = 500,n = 400, and s = 10. The average SNR n Let us consider an experiment with d = 1000. Denote over 100 trials for different γ values are displayed by the the true signal as x¯, of which the non-zero components are blue solid curve in Fig.3, which also shows the norm of the generated firstly following the standard Gaussian distribution reconstructedsignalsbythereddashedcurve.Asexpected,the and then normalizedsuch that kx¯k2 =1. The sensing vectors highest SNR is obtained when kx˜k2 is near one. We in this are drawn from the standard Gaussian distribution. There paperuseγ =10−4.Notethatwhentherearefewersaturation is Gaussian noise in the measurements. The noise level is data, the influence of the norm constraint is weaken. In that measured by the ratio of the summed squared magnitude of case, kxk2 could be simply regardedas a regularizationterm. noise-free measurements to that of noise and denoted by s . n For the saturation part, we choose the saturation levels such B. Synthetic Experiments that there are n/2 upper saturated measurements and n/2 In this subsection, we compare M1bit-CS with lasso and lower saturated ones. Suppose that the reconstructed signal is RDCS on synthetic data. The sparsity parameter λ is tuned x˜, its quality is measured by the signal-to-noise ratio (SNR) based on lasso, and the same value is then applied to other in dB: methods.We first set d=1000,K =300,m=500,s =10 n and consider the recovery performance with different satura- SNR(x¯,x˜)=10log kx¯k2/kx¯−x˜k2 . 10 2 2 tionratiosn/mvaryingfrom0%to40%.The averageresults (cid:0) (cid:1) 6 and if the computational time requirement is not critical, it is 1188 1.3 1166 1.2 worthy to use M1bit-CS to improve the recovery accuracy. 1144 1.1 1122 1 35 0.8 SNRSNR11024680246800 000000......456789x˜kk2 SNR(dB)1122305050 lRMMaDs11sCbboiiStt--CCSSCR time(s)000000......234567 lRMMaDs11sCbboiiStt--CCSSCR --66 --55..55 --55 --44..55 --44 --33..55 --33 --22..55 --22 --11..55 --11 log10(γ) 5 0.1 040n0um600be80r0 o10f00m12e00as1u400re1m600en18t00s2000 0400num600be80r0 o10f00m12e00as1u400re1m600en18t00s2000 Fig.3. Reconstruction performance fordifferent γ values.SNR:bluesolid curve;Theℓ2 normofthereconstructed signal:reddashedcurve. (a) (b) Fig.5. Recoveryperformanceoflasso(blackdotted),RDCS(bluedashed), M1bit-CSC (red solid line with triangle), and M1bit-CSR (red solid) for over 100 trials are displayed in Fig.4(a). When the saturation different numbersofmeasurements.(a)SNR;(b)computational time. ratio is low, there are plenty of analog measurements, then lasso can reconstruct the signal well. When n/m increases, lasso’srecoveryqualitydramaticallydecreases.Saturatedmea- V. OVEREXPOSURE CORRECTION FORCT surements are helpful to improve the recovery performance, RECONSTRUCTION which is the reason why RDCS, M1bit-CSC, and M1bit-CSR have significantly better results than lasso. Among the three A. Problem Formulation algorithms, M1bit-CSC performs the best. Its advantage over Nowwe areatthestagetoapplytheproposedM1bit-CSto RDCSisduetothefactsthatnoiseonsaturatedmeasurements CTreconstructiontocorrectoverexposure.InaCTsystem,the are suppressed and the signal’s norm information is useful objectdensityimagecanberepresentedinitsdiscreteformas for recovery. If the ℓ2 norm of the true signal is unknown a vector x, and the acquiredprojectiondata as q. When there or cannot be well estimated, then we use M1bit-CSR, whose is no saturation, the X-ray transform of the object x can be performance is slightly worse than M1bit-CSC. Similar per- written as formance could be observed in Fig.4(b), which is the average q=U⊤x, (12) recoveryresultsover100trailsfordifferentsparselevelswith a fixed saturation ratio n/m=20%. where U is the system matrix. Suppose that the intensity of one X-ray before entering the object is I0 and the intensity 7 40 departing the object is I1, then Lambert-Beer’s law tells us: lasso B)56 RMMD11bbCiiStt--CCSSCR B)233505 lRMMaDs11sbbCoiiStt--CCSSCR I1 =I0e−q. d4 d ( (20 Due to the physical limits of the detector or the analog-to- R3 R SN2 SN1105 digital converter, there is a maximal intensity Imax and any intensity exceeding this threshold cannot be distinguished. In 1 5 00 0.s04at0.u08r0a.12t i o0.1n6 0.r20a t 0i.2o4 0n.28 / 0m.32 0.36 0.4 nu05m0 be10r0 of150non200-zer25o0 c3o00mp3o50nen400ts other words, we cannot observe the analog value when (a) (b) q <log(I0/Imax). Fig.4. Recoveryperformanceoflasso(blackdotted),RDCS(bluedashed), In practice, the maximal and minimal detectable value of the M1bit-CSC (redsolid line with triangle), and M1bit-CSR (redsolid) for(a) differentsaturationratiosand(b)differentnumbersofnon-zerocomponents. detectorcan be adjustedat eachprojectionangleforexposure control but the measurable dynamic range of the detector is WethenkeepthesignalsparsityasK =300,thesaturation fixed and denoted by κ = log(Iβ,max/Iβ,min), where Iβ,max ratio as n/m = 20%, and vary the number of measurements and Iβ,min are the maximal and minimal intensities that are m from 350 to 2000. The average SNRs over 100 trials are measured by the detector at projection angle β. At each reportedinFig.5(a).Wecanobservetheeffectoftheproposed projection, Iβ,min is determined by the object, and Iβ,max algorithmsinminingknowledgefromsaturatedmeasurements. is adjusted correspondingly afterwards by the fixed κ as Forexample,M1bit-CSCwith800measurements(640analog Iβ,max =Iβ,mineκ. Then we can choose a dynamic threshold measurements and 160 saturated ones) has a similar SNR sβ = log(I0/Iβ,max) = log(I0/Iβ,min)−κ = pβ,max −κ, as lasso with 1000 measurements (800 analog measurements wherepβ,max isthemaximumprojectionvalueaftertheminus are used and the left 200 ones are dropped). It means that log transform at projection β. Thus, the observation p is saturatedmeasurementshavealmostthesamepowerasanalog a truncation of q, i.e., p = max{s ,q }, where s takes i i i i ones in recovering the sparse signal in that case and our the value of s with β being the angle containing the i- β algorithms are able to explore all the power with a longer th projection. Since the observation stands for the integral computational time. Generally, the computational time of of the tissue intensity, it should be non-negative. In fact, M1bit-CS is about 10 times of that of lasso, as reported in there are amount of X-rays that hit the detector directly, Fig.5(b).Wethinkthatwhentherearesaturatedmeasurements i.e., the corresponding measurements is zero. Thus, when a 7 measurementgetsaturated,peopleusuallysetitsvalueaszero. Therefore, the observations in CT reconstruction are q , if q >s , p = i i i i (cid:26) 0, if qi ≤si. (a) When overexposure happens, traditional CT reconstruction methods are not suitable, as shown by Fig.1(b). To correct overexposure, the researchers typically considered extrapola- tiontocompensatethemissinginformation;seee.g.,[35][36] [37].Amongthem,themethodgivenby[38]thatassumesthe (b) missing data as line integrals of a partial water cylinder is efficient. It is based on FBP algorithm and in the following, we refer this method as FBP-WCE (FBP with water cylinder extrapolation).ItsreconstructionresultisdisplayedinFig.6(a). One can see significant improvement from the standard FBP method, of which the result is given by Fig.1(b). (c) Fig.7. Saturation indicator Ψforsaturated datashowninFig.1(d):(a)the B. Knee Phantom trueΨ;(b)indicatordetectedbyM1bit-CSR-ISD;(c)differencebetweenthe trueandthedetectedsaturationindicator(blue:falsedetection;green:missing The overexposurephenomenonis a kind of one-sided satu- detection). ration.IfΨisknown,wecandirectlyapplyM1bit-CSR(6)to dooverexposurecorrection(wedonotknowtheℓ2normofthe image and M1bit-CSC is not suitable). Note that the sparsity times,thedetectionprocessistime-consuming.Utilizingprior- for CT reconstruction is on the gradient domain, i.e., total knowledgemay reduce the detection time, which could be an variation(TV)isusedastheregularizationterm;see[39][40] interesting study in the future. [41] for TV regularization CT reconstruction.When applying M1bit-CS on CT reconstruction, we do not use FISTA but a 120 TVminimizationalgorithmtosolvethex-subproblem(9).The 100 reconstructionimageisshowninFig.6(b),whichpreliminarily U)80 H shows that acquiring information from saturated projections (60 E largely helps CT reconstruction. S40 M30 R20 10 0 5ISD10iter15ation20 25 30 (a) (b) Fig.8. (a)ReconstructedresultofM1bit-CS-ISDfromsaturatedprojections showninFig.1(d);(b)RMSE(HU)indifferent ISDiterations. (a) (b) C. Simulated Clinical Data Finally, we apply the proposed method on a clinical head Fig. 6. Reconstructed results from saturated projection Fig.1(d) (a) FBP- WCE;(b)M1bit-CSR. dataset. The data are acquired with a Siemens Artis zee an- giographicC-arm system (Siemens Healthcare GmbH, Forch- In practice, when p = 0, we do not know whether this heim, Germany).In this experiment,we chooseone slice of a i projection ray is saturated or does not hit the object. Thus, 3D clinical head dataset as the ground truth image (Fig.9(a)) the ISD scheme proposed in Section II-C should be used and reproject it to simulate the acquired sinogram data in together with M1bit-CSR (abbreviated as M1bit-CSR-ISD). thefan-beamsystemwiththe followingtrajectoryparameters: Again consider the projections shown in Fig.1(d). Fig.7(b) the source-to-isocenter distance is 750 mm and isocenter-to- displaysthe detectedΨ. The differencebetweenthe idealand detectordistanceis450mm.Theangularstepis 1degreeand thedetectedΨisshowninFig.7(c),wherethefalsedetections the totalscanrangeis 360degrees.The equal-spaceddetector (zero measurements that are detected as saturated ones) are length is 620 mm with the pixel length 1 mm. markedinblueandmissingdetection(saturatedmeasurements The full projections are shown in Fig.10(a). When the that are regarded as zero ones) are in green. detection range κ is limited, there could be saturation for The final reconstruction image is given by Fig.8, which the projection. In Fig.10(b) and Fig.10(c), observations for shows that a small number of incorrect detections can be κ = 0.6pmax and κ = 0.4pmax are displayed, respectively. tolerated by M1bit-CSR (6) and the reconstruction result is Ourtaskistorecovertheimagefromthesaturatedprojections still satisfactory. As plotted in Fig.8(b), root of mean square via M1bit-CS-ISD. The results are compared with FBP and error(RMSE,inHU)isdecreasingwiththeincreasingofISD SART, two standard CT reconduction frameworks. For FBP, iterations. Since ISD requires to solving M1bit-CSR multiple we apply the modification given by [38] that utilizes water 8 cylinderextrapolationtoremedymissingprojectionscausedby information from the saturated data. As shown in Fig.9(d), truncation or overexposure.For SART, one can remove those most of outer boundaries are nicely restored and streaking saturated projectionswhen they are found, for which the ISD artifacts are effectively eliminated. can be used as well. We denote this method as SART-ISD, of Next, we artificially add Gaussian noise on the projections whichthedetectionschemeisasthesameasM1bit-CSR-ISD and the standard deviation of the noise σ = 0.1. The recon- but SART is used as the reconstruction method. structed images of FBP-WCE (with a smooth filter), SART- ISD,andM1bit-CSR-ISDare displayedin Fig.11. Comparing theimagesinFig.9(d)and11(d),wecanfindthattheproposed method is not sensitive to additive noise, which is rooted in the loss functions of (5) and (6). If the projections contain outliers, then some robust losses can be applied. (a) (b) (a) (b) (c) (d) Fig. 9. Reconstruction results for the clinical data (κ = 0.6pmax): (a) groundtruth; (b)FBP-WCE;(c)SART-ISD;(d)M1bit-CSR-ISD. (c) (d) Fig. 11. Reconstruction results from the noise-corrupted and saturated projections (κ = 0.6pmax and σ = 0.1): (a) ground truth; (b) FBP-WCE; (c)SART-ISD;(d)M1bit-CSR-ISD. (a) Whenthedetectablerangeissmallersuchthatκ=0.4pmax, saturation becomes worse. Then the performance of FBP and SART, even with water cylinder extrapolation and the ISD scheme, dramatically drops. In practice, this heavy saturation rarely happens and is out of the scope of FBP-WCE. But (b) even with this heavy saturation, the proposed M1bit-CSR- ISD can output a good result. The reconstructed images and enlargements of a region are illustrated in Fig.12 and Fig.13, respectively. In this case, both FBP-WCE and SART- ISD fail to restore the clear outer boundaries of the patient, while M1bit-CSR-ISD is still able to achieve this in a more (c) accurate manner. We further report the saturation detection result in Fig.14, from which one can observe that most of the Fig. 10. Projections of Fig.9(a): (a) full projection data; (b) saturated saturations have been properly detected. projection data with κ = 0.6pmax; (c) saturated projection data with κ=0.4pmax. In Table I, we list the RMSE (in HU) for FBP, FBP-WCE, SART-ISD, M1bit-CSR with ISD and with an idealsaturation For κ = 0.6pmax, the reconstruction results of FBP-WCE matrix. Compared with classical FBP, FBP-WCE and SART- andSART-ISDaregiveninFig.9(b)and9(c),respectively.As ISD havesignificantimprovements.Butthe schemesare basi- shown before, the traditional FBP method cannot handle the callydifferent:FBP-WCEapplieswatercylinderextrapolation saturated data. With water cylinder extrapolation, the recon- as the complements of the missing projections, and SART- struction quality has been improved but loss of clear patient ISD is to iteratively detect and drop incorrect projections. boundaries still happens. The overall performance of SART- The proposed M1bit-CSR can adequately use the information ISD is slightly better than FBP-WCE but capping artifact carried by overexposed measurements and then outputs good can be identified at the object border. Further improvement reconstruction results. The gap between M1bit-CSR-ISD and is obtained using the proposed M1Bit-CSR-ISD to acquire M1bit-CSRwiththeidealΨisnotverybig,whichontheone 9 handshowsthe accuracyofthe proposedISDmethod,andon the other hand confirms the robustness of M1bit-CSR against a part of incorrect detection. TABLEI ROOTOFMEANSQUAREERROR(HU)OFRECONSTRUCTIONRESULTS FBP FBP SART M1bit-CSR M1bit-CSR data WCE ISD ISD (idealΨ) kneephantom (a) (b) (κ=0.5pmax) 182.4 97.93 54.91 8.547 5.461 head (κ=0.6pmax) 43.22 35.34 31.97 11.62 6.930 headwithnoise (κ=0.6pmax) 43.84 36.71 34.22 11.72 7.001 head (κ=0.4pmax) 132.2 87.93 70.63 14.81 12.39 In M1bit-CS models, the saturation indicator Ψ is a key input.Inthispaper,wedesignedasimplebutefficientiterative methodtodetectΨ.Furtherimprovementmaycomefromthe (c) (d) use of prior-knowledge and reasonable assumptions on tissue structures. At least, those prior-knowledge provides a good Fig. 12. Reconstruction results for the clinical data (κ = 0.4pmax): (a) ground truth (yellow rectangle is enlarged in Fig.13); (b) FBP-WCE; (c) initial guess. An accurate initial guess can reduce both the SART-ISD;(d)M1bit-CSR-ISD. reconstruction error and the detection time. VI. CONCLUSION Aiming at signal reconstruction when saturation happens, we established mixed one-bit compressive sensing that could be regarded as a bridge between the regular and one-bit CS. We developedthe correspondingADMM and an iterative sat- uration detection method. They are then successfully applied (a) (b) (c) (d) to overexposure correction for C-arm CT reconstruction. The Fig.13. Anenlargementofaregionoftheclinicaldata,whichismarkedby improvement from the existing methods is quite significant. yellowrectangleinFig.12(a):(a)groundtruth;(b)FBP-WCE;(c)SART-ISD; Generally, the reconstruction performance of the proposed (d)M1bit-CSR-ISD. method is satisfactory even in the presence of severe detector saturation, showing the benefit of our method on considering saturated projections. CT scanning that has overexposureis a typical example of sensing systems with saturation. The promising performance of M1bit-CS implies its potential use on other tasks with measurementsbeyondthereliablesensingregion.Wecaneven takeadvantageofsaturationphenomenaforspecific purposes. For example, one can sample signs for a part of the measure- (a) mentstoreducetheexpenseoftheanalog-to-digitalconversion without significantly sacrificing the recovery accuracy. In low-dose computed tomography, when we largely reduce the radiationdosetothelevelsuchthatsomeprojectionsarebelow the threshold, then those projection values become unreliable (b) but they do provide one-bit information. Then M1bit-CS can also be applied to improve the performance. ACKNOWLEDGMENT The authors would like to thank Prof. Ji Liu from the (c) University of Rochester for sharing the code of RDCS. Fig.14. Saturationindicatorfortheclinicalphantomwithκ=0.4pmax.(a) thetruesaturationindicator;(b)ΨdetectedbyM1bit-CSR-ISD;(c)difference REFERENCES between the true and the detected saturation indicator (blue: false detection; green:missingdetection). [1] J. S. Feng, “Sensor saturation effects on NLTS measurements [of magnetoresistive sensors],” IEEE Transactions on Magnetics, vol. 34, no.4,pp.1952–1954,1998. 10 [2] Y. Fang, Y. Zhang, N. Qi, and X. Dong, “AM-AFM system analysis [23] J. Liu and S. J. Wright, “Robust dequantized compressive sensing,” and output feedback control design with sensor saturation,” IEEE AppliedandComputationalHarmonicAnalysis,vol.37,no.2,pp.325– Transactions onNanotechnology, vol.12,no.2,pp.190–202,2013. 346,2014. [3] E.dosSantos,G.Cardoso,P.Farias,andA.deMorais,“CTsaturation [24] Y.C. EldarandG.Kutyniok, CompressedSensing: Theory andAppli- detectionbasedonthedistancebetweenconsecutivepointsintheplans cations. CambridgeUniversity Press,2012. formedbythesecondarycurrentsamplesandtheirdifference-functions,” [25] A.ChristmannandI.Steinwart,“HowSVMscanestimatequantilesand IEEETransactionsonPowerDelivery,vol.28,no.1,pp.29–37,2013. the median,” in Advances in Neural Information Processing Systems, [4] N.Strobel, O.Meissner, J.Boese, T.Brunner, B.HeiGl, M.Hoheisel, 2007,pp.305–312. G. Lauritsch, M. NaGel, M. Pfister, E.-P. Ru¨hrnschopf, B. Scholz, [26] I.Steinwart andA.Christmann, “Estimating conditional quantiles with B.Schreiber, M.Spahn,M.Zellerhoff, andK.Klingenbeck-Regn, “3D the help of the pinball loss,” Bernoulli, vol. 17, no. 1, pp. 211–225, imagingwithflat-detectorC-armsystems,”inMultisliceCT. Springer, 2011. 2009,pp.33–51. [27] X. Huang, L. Shi, and J. A.K. Suykens, “Support vector machine [5] L.Feldkamp, L.Davis,andJ.Kress,“Practical cone-beam algorithm,” classifierwithpinballloss,”IEEETransactionsonPatternAnalysisand JournaloftheOpticalSocietyofAmericaA,vol.1,no.6,pp.612–619, Machine Intelligence, vol.36,no.5,pp.984–997,2014. 1984. [28] Y. Xu, Z. Yang, and X. Pan, “A novel twin support-vector machine withpinballloss,”IEEETransactionsonNeuralNetworksandLearning [6] A.H.AndersenandA.C.Kak,“Simultaneousalgebraicreconstruction Systems,2016. technique (SART): A superior implementation of the ART algorithm,” [29] A. Chambolle and T. Pock, “A first-order primal-dual algorithm for Ultrasonic Imaging,vol.6,no.1,pp.81–94,1984. convexproblemswithapplicationstoimaging,”JournalofMathematical [7] A. Preuhs, M. Berger, Y. Xia, A. Maier, J. Hornegger, and ImagingandVision,vol.40,no.1,pp.120–145,2011. R. Fahrig, “Over-exposure correction in CT using optimization- [30] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed based multiple cylinder fitting,” in Bildverarbeitung fu¨r die optimizationandstatisticallearningviathealternatingdirectionmethod Medizin 2015, H. Handels, Ed., 2015, pp. 35–40. [Online]. Available: of multipliers,” Foundations and Trends in Machine Learning, vol. 3, https://www5.informatik.uni-erlangen.de/Forschung/Publikationen/2015/Preuhs15-OCI.pdf no.1,pp.1–122,2011. [8] N. Mail, D. Moseley, J. Siewerdsen, and D. Jaffray, “The influence [31] M.YanandW.Yin,“Selfequivalenceofthealternatingdirectionmethod ofbowtie filtration oncone-beam CTimagequality,” Medical Physics, of multipliers,” in Spllitiing Methods in Communication and Imaging, vol.36,no.1,pp.22–32,2009. Science and Engineering, R. Glowinski, S. Osher, and W. Yin, Eds. [9] J.-H. Choi, R. Fahrig, A. Keil, T. F. Besier, S. Pal, E. J. McWalter, Springer-Verlag, 2016. G. S. Beaupre´, and A. Maier, “Fiducial marker-based correction for [32] A.BeckandM.Teboulle,“Afastiterativeshrinkage-thresholding algo- involuntary motion in weight-bearing C-arm CT scanning of knees rithmforlinearinverseproblems,”SIAMJournalonImagingSciences, (PartI):Numericalmodel-basedoptimization,”MedicalPhysics,vol.40, vol.2,no.1,pp.183–202,2009. no.9,p.091905,2013. [33] M.K.Ng,F.Wang,andX.Yuan,“Inexactalternatingdirectionmethods [10] J.-H. Choi, A. Maier, A. Keil, S. Pal, E. J. McWalter, G. S. Beaupre´, for image recovery,” SIAM Journal on Scientific Computing, vol. 33, G. E. Gold, and R. Fahrig, “Fiducial marker-based correction for no.4,pp.1643–1668,2011. involuntarymotioninweight-bearingC-armCTscanningofknees(Part [34] L.Shen and S. Pan, “Inexact indefinite proximal ADMMs for2-block II):Experiment,” Medical physics,vol.41,no.6,p.061902,2014. separable convex programs and applications to 4-block DNNSDPs,” [11] J.Rausch, A.Maier, R.Fahrig,J.-H.Choi,W.Hinshaw,F.Schebesch, arXivpreprintarXiv:1505.04519, 2015. S.Haase,J.Wasza,J.Hornegger,andC.Riess,“Kinect-basedcorrection [35] B.Ohnesorge,T.Flohr,K.Schwarz,J.P.Heiken,andJ.P.Bae,“Efficient of overexposure artifacts in knee imaging with C-Arm CT systems,” correction for CT image artifacts caused by objects extending outside International JournalofBiomedical Imaging,vol.2016,2016. thescanfieldofview,”MedicalPhysics,vol.27,no.1,pp.39–46,2000. [12] P.T.BoufounosandR.G.Baraniuk,“1-bitcompressivesensing,”inthe [36] A.Maier,B.Scholz,andF.Dennerlein,“Optimization-based extrapola- 42ndAnnualConferenceonInformationSciences andSystems. IEEE, tionfortruncation correction,” in2ndCTMeeting, 2012,pp.390–394. 2008,pp.16–21. [37] Y.Xia,H.Hofmann,F.Dennerlein,C.Schwemmer,S.Bauer,G.Chin- [13] E. J. Cande`s and T. Tao, “Decoding by linear programming,” IEEE talapani, P.Chinnadurai, J.Hornegger,andA.Maier,“Towardsclinical Transactions on Information Theory, vol. 51, no. 12, pp. 4203–4215, applicationofaLaplaceoperator-basedregionofinterestreconstruction 2005. algorithm in C-arm CT,” IEEE Transactions on Medical Imaging, [14] D. L.Donoho, “Compressed sensing,” IEEETransactions onInforma- vol.33,pp.593–606,2014. tionTheory,vol.52,no.4,pp.1289–1306, 2006. [38] J.Hsieh,E.Chao,J.Thibault,B.Grekowicz,A.Horst,S.McOlash,and [15] E. J. Cande`s, J. Romberg, and T. Tao, “Robust uncertainty principles: T. J. Myers, “A novel reconstruction algorithm to extend the CT scan Exactsignalreconstruction fromhighlyincompletefrequency informa- field-of-view,” Medical Physics,vol.31,no.9,pp.2385–2391,2004. tion,” IEEE Transactions on Information Theory, vol. 52, no. 2, pp. [39] H.YuandG.Wang,“Asoft-thresholdfilteringapproachforreconstruc- 489–509,2006. tion from a limited number of projections,” Physics in Medicine and Biology, vol.55,no.13,p.3905,2010. [16] J.N.Laska,Z.Wen,W.Yin,andR.G.Baraniuk,“Trust,butverify:Fast [40] L.Ritschl, F.Bergner, C.Fleischmann, and M.Kachelrieß, “Improved and accurate signal recovery from 1-bit compressive measurements,” total variation-based CTimagereconstruction applied toclinical data,” IEEE Transactions on Signal Processing, vol. 59, no. 11, pp. 5289– PhysicsinMedicine andBiology, vol.56,no.6,p.1545,2011. 5301,2011. [41] Z.Chen,X.Jin,L.Li,andG.Wang,“Alimited-angleCTreconstruction [17] M. Yan, Y. Yang, and S. Osher, “Robust 1-bit compressive sensing methodbasedonanisotropicTVminimization,”PhysicsinMedicineand usingadaptiveoutlierpursuit,”IEEETransactionsonSignalProcessing, Biology, vol.58,no.7,p.2119,2013. vol.60,no.7,pp.3868–3875, 2012. [18] L.Jacques,J.N.Laska,P.T.Boufounos,andR.G.Baraniuk,“Robust1- bitcompressivesensingviabinarystableembeddingsofsparsevectors,” IEEE Transactions on Information Theory, vol. 59, no. 4, pp. 2082– 2102,2013. [19] Y. Plan and R. Vershynin, “Robust 1-bit compressed sensing and sparse logistic regression: A convex programming approach,” IEEE TransactionsonInformationTheory,vol.59,no.1,pp.482–494,2013. [20] L. Zhang, J. Yi, and R. Jin, “Efficient algorithms for robust one- bit compressive sensing,” in Proceedings of the 31st International Conference onMachineLearning, 2014,pp.820–828. [21] R. Zhu and Q. Gu, “Towards a lower sample complexity for robust one-bit compressed sensing,” in Proceedings ofthe 32nd International Conference onMachineLearning, 2015,pp.739–747. [22] J. N. Laska, P. T. Boufounos, M. A. Davenport, and R. G. Baraniuk, “Democracy in action: Quantization, saturation, and compressive sens- ing,”AppliedandComputationalHarmonicAnalysis,vol.31,no.3,pp. 429–443,2011.

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.