Introduction Inverse Problems

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1、IntroductiontoInverseProblemsGuillaumeBal1January29,20121ColumbiaUniversity,NewYorkNY,10027;gb2030@columbia.eduiiContents1WhatconstitutesanInverseProblem11.1ElementsofanInverseProblem(IP)....................11.1.1InjectivityandstabilityoftheMeasurementOperator.....

2、..11.1.2Noise",Modeling,andPriorInformation.............21.1.3Numericalsimulations........................41.2ExamplesofMeasurementOperator.....................61.3IPandModeling.ApplicationtoMRI...................81.4InverseProblemsandSmoothing:Hilbertscale.......

3、.......121.4.1Fouriertransformsandwell-posedness...............121.4.2Hilbertscaleanddegreesofill-posedness..............132IntegralGeometry.Radontransforms192.1TransmissionTomography..........................192.2TwodimensionalRadontransform.....................21

4、2.3ThreedimensionalRadontransform.....................282.4AttenuatedRadonTransform........................292.4.1SinglePhotonEmissionComputedTomography..........302.4.2RiemannHilbertproblem......................312.4.3InversionoftheAttenuatedRadonTransform........

5、...322.4.4Step(i):The@problem,anellipticequation............332.4.5Step(ii):jumpconditions......................352.4.6Step(iii):reconstructionformulas..................373IntegralGeometry.GeneralizedRayTransform393.1GeneralizedRayTransform:Settingintwodimensions..

6、.........403.1.1Familyofcurves............................403.1.2GeneralizedRayTransform......................403.1.3AdjointoperatorandrescaledNormaloperator..........413.2OscillatoryintegralsandFourierIntegralOperators............433.2.1Symbols,phases,andoscillato

7、ryintegrals..............433.2.2Parameterizedoscillatoryintegrals.................453.2.3DenitionofFourierIntegralOperators...............463.3Pseudo-dierentialoperatorsandGRT...................473.3.1Absenceofsingularitiesawayfromthediagonalx=y.......483.3.2Chan

8、geofvariablesandphase(xy)
..............493.3.3Choiceofaparametrix.........................50iiiivCONTENTS3.3.4Proofofsmoothingbyonederivative.