基于光栅干涉仪的暗场成像的Cramér-Rao下界.pdf

IntroductionOver the past decades,dark-field imaging using agrating interferometer has showed great promise indiverse fields1,as it is compatible with conventionalsources,including X-ray tube sources2and neutrons3,anddoesnot requirehightemporal coherence4.Especially,dark-field imaging can provide informationabout the objects microstructures on a scale below thespatial resolution of the grating interferometer,and alsoenables visualization of spatially-resolved small-anglescattering properties5-6.In recent years,X-ray dark-fieldimaging with a grating interferometer has shown toprovide significant benefits for several applications,including but not limited to:mammography7,pulmonaryimaging8,materials analysis9,security screening10andfood sciences11.Meanwhile,neutron dark-field imagingusing a Talbot-Lau interferometer also has shown greatpotentialinnon-destructivetestingofmetallicmaterials12,research on magnetic materials13-14,and soon,owing to the unique neutron features of highpenetration and magnetic moment.In grating interferometry,several methods have beendeveloped for quantitative signal retrieval from measuredintensities15-20.Among them,the phase stepping(PS)technique is routinely used as the standard approach19,21.The noise performance of the 3 signals retrieved with PStechniquehavebeentheoreticallyanalyzedandexperimentally validated22-25.However,previous studiesdemonstrated that the noise variances of amplitude andphase signals obtained with a grating interferometer werealways higher than Cramr-Rao lower bound(CRLB)26.Numerical results implied a sub-optimality of PStechnique for refraction and dark-field imaging in termsof noise variance27.Their results also suggested thenecessity of developing advanced retrieval algorithms tofurther reduce the noise variance and improve doseefficiency.Recently,reference28discussed the noisestandard deviation and algorithm efficiency of phaseshifting interferometry theoretically and numerically usingCRLB.Although their studies were presented in terms ofthe optical pathlength,the obtained results were alsoapplicable to grating-based refraction imaging,with somemathematical transforms.However,there still lacks of acomprehensive evaluation on the algorithm performancewith respect to dark-field imaging using a gratinginterferometer.Herein the performance of the PS technique for dark-field retrieval using CRLB is evaluated.AnalyticalCramr-Rao lower bound of dark-field imaging using a grating interferometerLIU Bo,CHEN Zihan,GU Yao,CHEN Heng,WANG ZhiliDepartment of Optical Engineering,School of Physics,Hefei University of Technology,Hefei 230009,ChinaAbstract:In grating-based phase contrast imaging,the phase stepping technique is commonly utilized for data acquisition andsignal retrieval from acquired intensity data.However,the algorithm efficiency with respect to the dark-field retrieval has yet tobe sufficiently evaluated.Herein the algorithm efficiency of dark-field retrieval based on Cramr-Rao lower bound is evaluated.The theoretical analysis and numerical results demonstrates that fully efficient algorithm is currently available only for 3-step phasestepping technique,and other techniques with more phase steps are all sub-optimal.Quantitatively,the dependence of the algorithmefficiency on the phase step number and the visibility is investigated.It is shown that the phase stepping technique can nearlyapproach its theoretical optimal efficiency in the case of a low visibility.With a phase step greater than 5,the algorithm efficiencyis only 77.4%in the case of a high visibility.The study can provide some reference for signal-to-noise ratio improvement andpotential dose optimization in X-ray and neutron grating-based dark-field imaging.Keywords:dark-field imaging;grating interferometer;Cramr-Rao lower bound;visibilityReceived:2023-03-24Supported by the National Natural Science Foundation of China(U1532113,11475170,11905041),and the Fundamental Research Funds for the CentralUniversities(PA2020GDKC0024,JZ2022HGTB0244)Leading author:LIU Bo,Master,Research Direction:X-ray phasecontrast imaging,E-mail:Corresponding author:WANG Zhili,Doctor,Professor,ResearchDirection:X-ray phase contrast imaging,E-mail:DOI:10.3969/j.issn.1005-202X.2023.09.008第40卷第9期2023年 9月中国医学物理学杂志Chinese Journal of Medical PhysicsVol.40 No.9September 2023Medical imaging physics-1105expressions of CRLB were derived for 3-step and 4-stepPStechniques,respectively.Throughnumericalsimulations,we discussed the dependence of the CRLBandalgorithmefficiencyonvariousexperimentalparameters,including the mean intensity,the phase stepnumber and the visibility.The presented results can beuseful for advanced algorithm development and furthernoise reduction in grating-based dark-field imaging.1 Theory and methods1.1 Signal retrieval by PS techniqueIn grating interferometer,the PS technique iscommonly used as the standard approach for dataacquisition and retrieval of multi-contrast signals.In dataacquisition,one of the gratings is translated laterally overits period with a total ofNphase step,while intensitymeasurements are performed at each phase step.For eachdetector pixel,the measured intensity oscillationIkfor aphase stepkcan be expressed as2:Ik=I01+Vcos()+2kN()1 k N(1)whereI0,V,anddenote the mean intensity,the visibility,and the phase of the intensity oscillation,respectively,andNis the total number of phase step.For notation brevity,the spatial dependence of all terms has been omitted in Eq.(1).Subsequently,we can readily retrieve the meanintensityI0,the phaseand the visibilityVbyI0=1Nk=1NIk(2)=-tan-1k=1NIksin()2kNk=1NIkcos()2kN(3)V=2k=1NIksin()2kN2+k=1NIkcos()2kN2k=1NIk(4)In order to quantitatively retrieve the transmission,refraction,and dark-field signals,one needs to comparemeasurements with a sample in the beam,to referencemeasurements without sample,and thereby deduce thelocal changes of the intensity oscillation induced by thesample.In the following,the superscriptssandrwillconsistently denote the values measured with the samplein place and as a reference without.The transmissionsignalTis calculated by the negative logarithm of theratio of the mean intensity with sampleIs0and withoutIr0:T=-ln()Is0Ir0=()x,y,z dz(5)Where()x,y,zdenotes the samples linear attenuationcoefficient.The refraction signalRcan be obtained by thedifference of the sample phasesand the reference phaser:R=s-r(6)The dark-field signalDis given by the localreduction of the visibility of the intensity oscillation:D=VsVr(7)Where theVsdenotes the samples visibility and theVrdenotes the references visibility.These 3 signals arecalculated on a pixel-by-pixel basis,and normallydisplayed in the form of images.1.2 CRLB of dark-field signalThe CRLB gives the lowest possible noise varianceof the estimated parameters from a set of noisymeasurements.In the following,we will use CRLB as ametric to evaluate the noise variance of the dark-fieldsignal retrieved by the PS technique.The study alsoconsiders the Poisson distributed photon-counting noise,the most dominant noise source in real experiments,which has been validated for both X-ray and neutronimaging previously18,22,29-30.For photon-counting detectors,the measured datafollows the Poisson distribution.Under this noise model,the joint probabilityPof acquired intensities at all phasesteps is given byP()I0,V,=k=1N()IkIkIk!exp()-Ik(8)whereIkdenotes the measured intensity for a phase stepk,andIkdenotes the corresponding mean value.Secondly,the log-likelihood functionLshould bederived.In our case,it isL()I0,V,=lnP()I0,V,=k=1NIklnIk-Ik-lnIk!(9)Then taking the first partial derivatives of the log-likelihood functionLwith respect toI0,Vand:LI0=k=1NIk-IkIk1+Vcos()+2kN(10)LV=k=1NIk-IkIkI0cos()+2kN(11)L=k=1NIk-IkIkI0Vsin()+2kN(12)Thirdly,the Fisher information matrixJwith respecttoI0,Vandis calculated with Eq.(13).中国医学物理学杂志第40卷-1106J=-E 2LI20E2LI0VE2LI0E2LVI0E2LV2E2LVE2LI0E2LVE2L2(13)whereE denotes the expectation value with respect tothe given statistical model.For our Poisson distributioncase,by use of Eqs.(10)-(13),we can obtain the resultingFisher information matrix:J=k=1N1-Ik 2IkI202IkI0V2IkI02IkI0V2IkV22IkV2IkI02IkV2Ik2(14)where2IkI20=1+Vcos()+2kN2(15)2IkV2=I20cos2()+2kN(16)2Ik2=-I20V2sin()+2kN(17)2IkI0V=I01+Vcos()+2kNcos()+2kN(18)2IkI0=-I0V1+Vcos()+2kNsin()+2kN(19)2IkV=-I20Vcos()+2kNsin()+2kN(20)And then,by inverting the Fisher information matrix,the CRLB ofI0,V,can be obtained from the diagonalelement.For the visibility signalV,its CRLB is given bythe second diagonal element in the inverse of Fisherinformation matrix.Therefore,one yields the CRLB of thevisibility signal:CRLBV=J-122(21)Finally,by use of Eqs.(7)and(21),and errorpropagation formula29,we can obtain CRLB of theretrieved dark-field signal:CRLBD=()DVr2CRLBVr+()1Vr2CRLBVs(22)1.3 Algorithm efficiencyFor PS technique,the noise variance of the retrievedvisibility and dark-field signals have been calculatedthrough error propagation formula in previous studies22-23.A direct comparison between CRLB and the calculatednoise variance can reveal the efficiency of PS technique.The algorithm efficiency is defined as the ratio of CRLBto the calculated noise variance of dark-field signal:=()CRLBD2D 100%(23)where the noise variance2Dcan be found in reference23.The algorithm efficiency also indicates whether thereexists any potential for algorithm improvement.In thefollowing,the algorithm performance of several commonPS techniques in the dark-field retrieval will be evaluated.On the basis of general expressions derived above,thetheoretical formulaeof CRLBandthealgorithmefficiency of common PS techniques are presented anddiscussed.1.3.1 3-step PS technique The 3-step PS technique usethe least number of intensity measurements to retrieve thetransmission,refraction and dark-field signals quantitatively18-19.Computing the Fisher information matrix andfinding the inverse matrix will lead to CRLB.With trivialmathematical computations,we can obtain CRLB of theretrieved visibility by 3-step PS technique:CRLB3V=2-V2+Vcos()33I0(24)By use of Eqs.(22)and(24),we can yield thecorresponding CRLB of the retrieved dark-field signalusing 3-step PS technique:CRLB3D=D231Ir02()Vr2-1+cos()3rVr+1Is02()Vs2-1+cos()3sVs(25)Previous studies has reported the noise variance of3-step PS technique for retrieved dark-field signals23,andthe expression is actually identical to Eq.(25),whichmeans that if extrinsic noise can be eliminated,the 3-stepPS technique can reach the theoretically optimal noiseefficiency.In other words,this algorithm is fully efficient(i.e.,=100%)according to Eq.(23).A similar result hasbeen observed for CRLB of the retrieved refractionsignal28,31.It is worth noting that fully efficient第9期刘波,等.基于光栅干涉仪的暗场成像的Cramr-Rao下界-1107algorithms are currently available only for 3-step PStechnique,and other PS techniques with more phase stepsare all sub-optimal,as discussed in the following.1.3.2 4-step PS technique Similar to the case of 3-stepPS technique,by calculating the diagonal elements of theinverse of the Fisher information matrix,we can obtainCRLB of the retrieved visibility by 4-step PS technique:CRLB4V=4-3V2-V2cos48I0(26)By use of Eqs.(22)and(26),we can yield the CRLBof the retrieved dark-field signal:CRLB4V=D241Ir02()Vr2-1-1+cos()4r2+1Is02()Vs2-1-1+cos()4s2(27)Previous studies reported the noise variance of theretrieved dark-field signal by 4-step PS technique as23:2D=D241Ir02()Vr2-1+1Is02()Vs2-1(28)Different from the CRLB in Eq.(27),the noisevariance is independent of the phasessandr.By use ofEqs.(27)and(28),we can calculate the algorithmefficiency of 4-step PS technique as:4=2()Vr2-1-1+cos()4r2+Ir0Is02()Vs2-1-1+cos()4s22()Vr2-1+Ir0Is02()Vs2-1100%(29)As revealed by Eq.(29),the algorithm efficiency4is generally lower than 100%,which indicates that 4-stepPS technique is sub-optimal in terms of the noise of theretrieved dark-field signal.Only at specific isolatedlocations determined byr=()2m+1 4ands=()2n+1 4,withm,nboth being integers,thealgorithm efficiency reaches 100%,meaning that the noisevariance is equal to the CRLB at these locations.1.3.3 N-step PS technique()N 5In order to effectivelyremove the effect of higher orders in obtaining Eq.(1),alarger integer is selected forN21.Since analyticalexpression of CRLB of the dark-field signal retrieved byN-step PS technique is difficult to be simplified,we willcompute these bounds numerically in the following.The noise variance of the dark-field signal retrievedby N-step PS technique was23:2D=D2N1Ir02()Vr2-1+1Is02()Vs2-1(30)For further analysis,the algorithm performance ofN-step PS technique will evaluated by calculating theCRLBnumericallyandcomparingitwiththecorresponding noise variance in Eq.(30).Especially,thedependences of the algorithm efficiency on the meanintensity,the phase step number and the visibility areinvestigated systematically.2 Results and discussionsThe developed models of CRLB are validated bynumerical calculations,and some quantitative insights intothe algorithm efficiency is further provided.In order toobtain results independent of the sample,we furtherrestricted the analysis to a sample-free region within theretrieved dark-field image32.Note that this choice iscommonly made in experimental measurements18,23.Therefore,the mean intensity,the visibility,and the phaseare equal for the reference and sample measurements,andthe noise is determined by calculating the variance overa background region of80 80pixels.As revealed by Eqs.(24)and(27),the CRLB ofretrieved dark-field signals exhibits a periodic oscillationas a function of the phasessandr.Similar results wereobtained for CRLB of refraction signals28.By contrast,the calculated noise variance is independent of the phasedistribution,with a phase step number greater than 3.Therefore,we investigated the dependence of CRLB ofdark-field signals on the phase distribution.For thispurpose,we calculated the ratio of the oscillationamplitude to the corresponding average as a function ofthe phase step number for different representativevisibilities(Figure 1).As expected,the ratio decreasesmonotonically when the phase step number increases,indicating a larger phase step number is effective toremove the effect of phase distribution.With a highvisibility of 0.74,the ratios are 23%and 10%for 4-stepand 5-step PS techniques,respectively,which means thatthe substantial effect of the phase distribution on CRLBcan not be neglected.With a phase step number greaterthan 5,the ratio decreases to less than 5%.The phasedistribution is considered to have a negligible contribution中国医学物理学杂志第40卷-1108to CRLB.Then we can reasonably take the average valueof CRLB to evaluate the algorithm efficiency.With alower visibility(0.30 and 0.50),the ratio is always below10%,meaning that the phase distribution has aninsignificant effect on CRLB.Based on results shown inFigure 1,for an impartial and convenient comparison,theaverage value of CRLB will be used for algorithmefficiency evaluations in the following.The de