mindad@stanford.edu;yuxinh@stanfor.edu;hylu@stanfor.edu. 1

SeparationofMRmultibandimagesusingcomplexindependentcomponentanalysis

CS229FinalReportYuxinHu,MindaDeng,HaiyuLu

1DepartmentofElectricalEngineering,StanfordUniversity,2DepartmentofAppliedPhysics,StanfordUniversity,3DepartmentofPhysics,StanfordUniversity.

IntroductionMagnetic resonance imaging (MRI)hasbecomeapowerfulmedicalimagingmodalitywidelyusedinclinicalpractice.However,oneoftheproblemsofMRIisthateachimagerequiresalongscantime,especiallyfor3Dimaging.Apopularsolutiontothisproblemis“controlledaliasing inparallel imagingresults in higher acceleration” (CAIPIRINHA)1 inwhich the images of different slices are acquiredsimultaneously.Therefore,theresultingimageisamixtureoftherealspaceimages.Atraditionalwayof separating images from CAIPIRINHA is parallelimaging but this method requires additionalinformation and time. In this paper, we appliedcomplex-valued independent component analysis(ICA) to reconstruct images, which requires noadditionalinformation.Becausethelinearmixturematrixispixeldependent,imageswereseparatedinto blocks. To resolve the order and amplitudeambiguities, the adjacent blocks were partiallyoverlapped. Finally, the separated blocks wererecombinedusingaregion-growinglikemethod.

RelatedWork

One kind of MRI reconstruction methods isimplementedinthefrequencydomain(alsoknownas the k-space)2,3. Since themixtureof images inthe spatial domain is under-sampled in k-space,separating mixed images is equivalent torecovering the under-sampled points in k-space.Each pixel in k-space is approximately a linearcombination of its adjacent points and theseweightingscanbeappliedtothewholek-space.Toestimate these weightings, some fully sampledtrainingdatainthek-spaceisnecessary,whichwilltake extra time. Other calibration-lessreconstruction methods that require no extratrainingdataalsoexist4,5,6,butthesemethodsworkwell only when the center of k-space (low-frequency component) is sampled with highdensity. Therefore, we want to develop a real

calibration-less method with no need of anytrainingdatausingcomplex-valuedICA.

Complex-valued signals frequently appear invariousapplicationsuchascommunications,radar,and bio imaging. The independent componentanalysis (ICA)hasbeenoneof themostcommonand successful approaches to the blind sourceseparation problem, under the assumption ofstatisticallyindependenceinthesourcesignals.

MostoftheICAalgorithmsexploitoneofthetwofollowing properties: non-Gaussian or sampledependence7,8,FastICA9,Infomax10,andRADICAL11work well on non-Gaussian data but neglect thesample dependence. The second-order blindidentification algorithm (SOBI)12, the efficientalgorithmforblindseparationusingtimestructure(TDSEP)13,andtheweights-adjustedsecond-orderblind separation (WASOBI)14 exploit sampledependenceinthedata,howeverwithnoabilitytohandlingthenon-Gaussianity.

Inourpaper,weusethemutual informationrateas the cost function, which takes both the non-Gaussianity and sample independence intoaccount15.

DatasetandfeaturesMRItakesdatainthefrequencydomain(k-space)

Figure1 K-space and real imagesofone sliceofbraincombinedfromseveralrecorders.

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through phase and frequency encoding and thenreal space images can be constructed from theFourier transform of the data. In the dataacquisition process, multiple coils are used andthey have different sensitivities due to theirrelativepositions to theobjectbeing scanned. InMRI, these coils are like the recorders in thecocktailpartyproblem,andthesensitivitiesarethesameasthelinearmixturematrix𝐴.However,thesensitivities over the scanning area are pixeldependentforeachcoil.AsforCAIPIRINHA,twoormoredifferentslicesinreal space are scanned by each coil in the sametimesothatthescantimeisremarkablydecreased.However,postprocessingisnecessarytoseparatedifferentslicesafterscanning.In CAPIRINHA, to make the decomposition ofimageseasier,eachsliceisgivenadifferentshiftasshowninFigure2.Ourgoalofthisprojectisthentoseparatetherealspaceimagesofeachslicefromthe mixture we get from CAIPIRINHA. Our MRIscanningdatasetswerecollectedona3TMR750scanner (GE Healthcare. Waukesha, WI) with acommerciallyavailable32-channeltorsocoilfromthe Magnetic Resonance Systems ResearchLaboratoryatStanfordUniversity,withacquisitionmatrix size = 256×256, FOV = 240mm × 240mm,slicethickness=4mm.

We first neglected the pixel dependence of thesensitivityofeachcoilandtestedouralgorithmontwo simulation data sets, which were artificiallycombinedwithtwodifferentweightingsandshifts.By implementing the complex-valued ICA, it is

reduced to the cocktail party problem with twospeakersandtwomicrophones.Next,weworkedon the realMRIdatawithpixeldependenceoneachcoil. Inotherwords,wecanno longer use a constant linearmixturematrix𝐴forthewholeimagetobuildtheICAmodel.Methodsandalgorithms

Linearmixturemodel

Let 𝑁 statistically independent complex-valuedsources 𝑠 𝑡 = 𝑠& 𝑡 , … , 𝑠) 𝑡 * be mixedthrough an 𝑁×𝑁 nonsingular mixing matrix𝐴)×) so that we obtain the mixtures 𝑥 𝑡 =𝑥& 𝑡 , … , 𝑥) 𝑡 * = 𝐴 ∗ 𝑠(𝑡) , 1 ≤ 𝑡 ≤ 𝑇 , wherethet is thediscretetime index.Themixturescanbe separated by constructing a demixing matrix𝑊)×) as 𝑦 𝑡 = 𝑊 ∗ 𝑥(𝑡) , where 𝑦 𝑡 =𝑦& 𝑡 , … , 𝑦) 𝑡 * istheseparation.

Costfunction

A natural choice for the independent separation𝑦 𝑡 is the mutual information rate15, whichprovidesageneralframeworktohandlebothnon-Gaussianityandsample-dependence,amongthe𝑁randomvariables𝑦6 𝑡 ,𝑖 = 1, … , 𝑁:

𝑀𝐼𝑅(𝑦&, … , 𝑦)) = 𝐻< 𝑦6 − log |det(𝑊𝑊E)| − 𝐻<(𝑥))

6F&

where 𝐻< 𝑦6 is the entropy rate of the 𝑖𝑡ℎseparatedsourcedefinedas

𝐻< 𝑦6 = lim*→K

𝐻 𝑦6𝑇

= lim*→K

− 𝑃 𝑦6M log(𝑃 𝑦6M )*MF&

𝑇

where𝐻 𝑦6 is theentropyof the vector𝑦6 , 𝑦6M is𝑦6’s𝑛𝑡ℎelementand𝑇is𝑦’sdimension. Since the𝐻<(𝑥)is independentof𝑊, thecostfunctioncanbewrittenas:

𝐽<(𝑊) = 𝐻< 𝑦6 − 2log |det(𝑊)|)

6F&

The role of log|det(𝑊)| is a regularization term.Since the entropy rate𝐻< 𝑦6 is not scale invariant,i.e.,𝐻< 𝑦6 ≠ 𝐻< 𝑎𝑦6 for𝑎 ≠ 1 , then without theregularization term the cost function can be

Figure2AtypicalCAIPIRINHAimageC)combinedfromtwodifferentslicesofabdomenA)andB)withshift.

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minimizedbyscalingalone.

GradientdescendVSNewton’salgorithm

It’s possible to use gradient descend rule tominimize the cost function𝐽<(𝑊) with respect tothedemixingmatrix𝑊directly.

𝜕𝜕𝑊

𝐽<(𝑊) =𝜕𝜕𝑦6

𝐻< 𝑦6𝜕𝑦6𝜕𝑊

− 2WUV)

6F&

To achieve faster convergence, we implement adecoupling procedure to simplify the problem intominimizing 𝐽<(𝑊) with respect to each of its rowvectors𝑊6.Thedecouplingprocedurenotonlyavoidsthe complicated matrix optimization problem, butalso make efficient Newton algorithm becometractableafterdecoupling.ForaNewtonupdate,theHessiancanbecomputedusing15

𝜕W

𝜕𝑊6𝜕𝑊6𝐽<(𝑊6) =

𝜕W

𝜕𝑦6W𝐻< 𝑦6 𝑥𝑥* −

𝐻6𝑊6

*ℎ6W

)

6F&

where 𝐻6 = 𝐼 − 𝑊X* 𝑊X𝑊X

* U&𝑊X with 𝑊X =

𝑊&,𝑊W, … ,𝑊6U&,𝑊6Y&, … ,𝑊) * and ℎ6 is a unitlength vector that is perpendicular to all the rowvectorsof𝑊except𝑊6.

Pixeldependentlinearmixturematrix

Toapplytheabovecomplex-valuedICAmethodinseparating mixed MRI images due to under-sampling,weneed tomake further assumptions.For a real-worldMRI scanner, the linearmixturematrix𝐴ispixeldependent.Therefore,wecannotdirectly implement complex-valued ICA assuming𝐴isconstantforallpixelsintheimage.However,𝐴 varies relatively slowly as a function of pixelposition. This allows us to group close pixelstogether intoblocks andassume the same linearmixturematrixforallthepixelsinthesameblock.After obtaining thedemixingmatrixwe then cancombine the blocks together to reconstruct theunmixed images. However, the ICA algorithmsuffers from the order/amplitude/phaseambiguities.Thatis,theoutputmaydifferfromits“true” value by a complex constant and outputordermayvary.Tosolvetheseintrinsicambiguities,themixedimagesneedtobedividedintopartiallyoverlappedblocks.Theoverlappedregion isused

to sort theoutputs. To illustratehow the sortingand correction work, we denote the overlappedregions fromblock 1 and block 2 as vectors𝑎, 𝑏.Thenwedefinehowclosetwovectorsarebythe

normalizedinnerproduct [\]

[ ]andusethisasthe

sorting criterion. After matching the order, wecorrect the amplitude and phase through aweightedaverage:𝑏^_<<`^a`b =

[c]c𝑤66 𝑏,where𝑤6 =

[c[cc.

Thewhole process of our project can be dividedintofivesteps.Step 1: Compress the data using principalcomponentanalysissowehavethesamenumberofrecordersasspeakers(slices)16.Step 2: Fourier transform raw data to getmixedrealspaceimages.Step 3: Divide images into multiple partiallyoverlapped blocks as described above. For eachblock,theimagesaremixedduetounder-sampling.Step4:Applycomplex-valuedICAtoalltheblocks.

a) b)

c) d)

e) f)

Figure3a)andb)arethemixturesignal.c)andd)showsthe independent component separated by the mutualinformation cost function. e) and f) shows theindependent component separated by the mutualinformationratecostfunction.

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Step 5: Apply order, amplitude and phasecorrections to the unmixed blocks and thenrecombinethemblockbyblocktoreconstructtheunmixedimages.Results

In the first part of our result, we simulated ICAseparationresultsbyusingmutualinformationandusingmutualinformationrateasthecostfunctiontoshowhowmutual informationratecanexploitsampledependence.Weassumed that the linearmixturematrix𝐴isconstantoverentireMRIimage,whichisusuallynottrueinpractice.

Thesimulationresultsareshowninfigure3.e)andf) indicates that themutual informationratecostfunction successfully separates two independentcomponents without artificial features whichappearsinc)andd).

In thissimulation,weknowthereal independent

component. Therefore, we calculated the rootmeansquareofthedifferencesbetweenseparatedIC with the real images to compare the mutualinformation and mutual information rate. Weobservedoneorderofmagnitudeenhancementinmutualinformationratecostfunction.

Till this point, we can draw the conclusion thatbecauseofthesampledependenceinMRIdataitisnecessarytoimplementthemutualinformationrateasthecostfunction.

In the second part of our result, we applied thiscomplex-valuedICAmethodtorealcoildata,firstwithout dividing them into blocks. As shown infigure 4 a)-d), clearly, the edges are not wellseparated, especially for the right half of theimages.

We then divided images into blocks before weapplythesamealgorithm.As shown in figure 4 e)-h), detailed features arebetter separated and resolved. Also, edges aremore clear. Comparing the resulting separatedimages, we concluded that for most part of theunmixedimages,groupingpixelsintoblocksbeforeapplying complex-valued ICA did improve theoverallperformance.

Also, as shown in figure 5, dividing into blocksgreatlyreducestherootmeansquareerroroftheseparated images compared to rescanned “true”images by 25%. The variance in the root meansquare error also decreases, indicating that

MIascostfunction MIRascostfunction

1 0.122474487 0.023043437

2 0.119582607 0.020124612

3 0.187082869 0.020832667

4 0.154919334 0.021610183

5 0.170293864 0.022022716

6 0.126491106 0.022847319

7 0.154919334 0.022203603

8 0.187082869 0.020469489

9 0.192353841 0.021142375

10 0.161245155 0.022405357

Table1.Tendifferentdatasetsseparatedbymutualinformation cost function and mutual informationrate function. Root mean square differences(𝑅𝑀𝑆(𝑟𝑒𝑎𝑙𝐼𝐶 − 𝑠𝑒𝑝𝑎𝑟𝑎𝑡𝑒𝑑𝐼𝐶))arelisted.

Figure4a)andb)showthemixedimagesoftwocoils;c)andd)showtheseparatedimages.

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complex-valued ICA with divided blocks is morerobust.

Conclusionandfuturework

In this project, we applied cICA algorithm toseparatemixedMRimagesduetounder-sampling.Resultsfromsimulatedmixedimagesshowedthatmutual information rateworksbetteras thecostfunction, since it does not require independentdata sets. Also, we showed that dividing originalmixed images into blocks and then applyingcomplex-valued ICA improved the performancebecause it accounts for the fact that the mixingmatrices are pixel dependent. Unmixed imageswere successfully reconstructed from blocks byregiongrowingmethodanddetailedfeatureswerewellseparatedandresolved.In traditional reconstruction methods, the linear

relationship between sampling points in thefrequency domain is assumed and learned fromthe training data. We can apply more complexmodelstoreconstructmoredetails.Inaddition,inthiscomplexICAalgorithm,wecanaddsomeotherconstraints from some properties ofMR images,e.g. sparsity and locally low rank5,17, to achievefaster convergence and better reconstructionresults.Reference

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Figure 4 e) and f) show the separated blocks withoutorder,amplitudeandphasecorrections;g)andh)showtherecombinedimagesbytheregion-growingmethod.

Figure5AverageRMSerrorwithout/withdividingintoblocks

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