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FourierSlicePhotography

RenNgStanfordUniversityConventionalPhotographLightFieldPhotographyCapturethelightfieldinsidethecamerabodyHand-HeldLightFieldCameraMediumformatdigitalcameraCamerain-use16megapixelsensorMicrolensarrayLightFieldinaSingleExposureLightFieldinaSingleExposureLightFieldInsidetheCameraBodyDigitalRefocusingDigitalRefocusingQuestionsAboutDigitalRefocusing

Whatisthecomputationalcomplexity?

Arethereefficientalgorithms?Whatarethelimitsonrefocusing?

Howfarcanwemovethefocalplane?Overview

FourierSlicePhotographyTheoremFourierRefocusingAlgorithmTheoreticalLimitsofRefocusingPreviousWorkIntegralphotographyLippmann1908,Ives1930Lotsofvariants,especiallyin3DTV

Okoshi1976,Javidi&Okano2002Closestvariantisplenopticcamera

Adelson&Wang1992FourieranalysisoflightfieldsChaietal.2000RefocusingfromlightfieldsIsaksenetal.2000,Stewartetal.2003FourierSlicePhotographyTheoremIntheFourierdomain,aphotographisa

2Dsliceinthe4Dlightfield.

Photographsfocusedatdifferentdepthscorrespondto2Dslicesatdifferenttrajectories.DigitalRefocusingbyRay-TracingLensSensoruxDigitalRefocusingbyRay-TracingLensSensoruImaginaryfilmxDigitalRefocusingbyRay-TracingLensSensoruxImaginaryfilmDigitalRefocusingbyRay-TracingLensSensoruImaginaryfilmxDigitalRefocusingbyRay-TracingLensSensoruxImaginaryfilmRefocusingasIntegralProjectionLensSensoruxImaginaryfilmxuRefocusingasIntegralProjectionLensSensoruxImaginaryfilmxuRefocusingasIntegralProjectionLensSensoruxxuImaginaryfilmRefocusingasIntegralProjectionLensSensoruxxuImaginaryfilmClassicalFourierSliceTheorem2DFourier

Transform1DFourier

TransformIntegral

Projection

Slicing2DFourier

TransformClassicalFourierSliceTheorem1DFourier

TransformIntegral

Projection

SlicingClassicalFourierSliceTheorem2DFourier

Transform1DFourier

TransformIntegral

Projection

SlicingFourierDomainClassicalFourierSliceTheoremIntegral

Projection

SlicingSpatialDomainSpatialDomainClassicalFourierSliceTheoremIntegral

Projection

SlicingFourierDomainFourierSlicePhotographyTheoremIntegral

Projection

SlicingFourierDomainSpatialDomainFourierSlicePhotographyTheorem4DFourier

TransformIntegral

Projection

SlicingFourierSlicePhotographyTheorem4DFourier

Transform2DFourier

TransformIntegral

Projection

SlicingFourierSlicePhotographyTheorem4DFourier

Transform2DFourier

TransformIntegral

Projection

SlicingFourierSlicePhotographyTheorem4DFourier

Transform2DFourier

TransformIntegral

Projection

SlicingPhotographicImagingEquationsSpatial-DomainIntegralProjectionFourier-DomainSlicingPhotographicImagingEquationsSpatial-DomainIntegralProjectionFourier-DomainSlicingPhotographicImagingEquationsSpatial-DomainIntegralProjectionFourier-DomainSlicingTheoremLimitationsFilmparalleltolensEverydaycamera,notviewcameraAperturefullyopenClosingaperturerequiresspatialmaskOverview

FourierSlicePhotographyTheoremFourierRefocusingAlgorithmTheoreticalLimitsofRefocusingExistingRefocusingAlgorithmsExistingrefocusingalgorithmsareexpensiveO(N4) wherelightfieldhas

NsamplesineachdimensionAllarevariantsonintegralprojectionIsaksen etal. 2000Vaish etal. 2004Levoy etal. 2004Ng etal. 2005RefocusinginSpatialDomain4DFourier

Transform2DFourier

TransformIntegral

Projection

SlicingRefocusinginFourierDomain4DFourier

TransformInverse

2DFourier

TransformIntegral

Projection

SlicingRefocusinginFourierDomain4DFourier

TransformInverse

2DFourier

TransformIntegral

Projection

SlicingAsymptoticPerformanceFourier-domainslicingalgorithmPre-process:O(N4logN)Refocusing:O(N2logN)Spatial-domainintegrationalgorithmRefocusing:O(N4)ResamplingFilterChoiceTrianglefilter(quadrilinear)Kaiser-Besselfilter

(width2.5)Goldstandard(spatialintegration)Overview

FourierSlicePhotographyTheoremFourierRefocusingAlgorithmTheoreticalLimitsofRefocusingProblemStatementAssumealightfieldcamerawithAnf/AlensNxNpixelsundereachmicrolensIfwecomputerefocusedphotographs

fromtheselightfields,overwhatrange

canwemovethefocalplane?AnalyticalassumptionAssumeband-limitedlightfieldsBand-LimitedAnalysisBand-LimitedAnalysisLightfieldshot

withcameraBand-widthof

measuredlightfieldBand-LimitedAnalysisBand-LimitedAnalysisBand-LimitedAnalysisPhotographicImagingEquationsSpatial-DomainIntegralProjectionFourier-DomainSlicingResultsofBand-LimitedAnalysisAssumealightfieldcamerawithAnf/AlensNxNpixelsundereachmicrolensFromitslightfieldswecanRefocusexactlywithin

depthoffieldofanf/(AN)lensInourprototypecameraLensisf/412x12pixelsundereachmicrolensTheoreticallyrefocuswithin

depthoffieldofanf/48lensLightFieldPhotoGalleryStanfordQuadRodin’sBurghersofCalaisPalaceofFineArts,SanFranciscoPalaceofFineArts,SanFranciscoWaitingtoRaceStartoftheRaceSummaryofMainContributions

Formaltheoremaboutrelationship

betweenlightfieldsandphotographsComputationalapplicationgivesasymptoticallyfastrefocusingalgorithmTheoreticalapplicationgives

analyticsolutionfor

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