Fourier-transformation & Optics

Biophotonics lecture9. November 2011Last time (Monday 7. November)Review of Fourier Transforms (will be repeated in part today)Contrast enhancing techniques in microscopyBrightfield microscopyDarkfield microscopyPhase Constrast MicroscopyPolarisation Contrast MicroscopyDifferential Interference Contrast (DIC) MicroscopyTodayPart 1: Review of Fourier Transforms1D, 2DFourier filteringFourier transforms in microscopy: ATF, ASF, PSF, OTFPart 2: Sampling theoryFourier-transformation & OpticsFourier-transformation & OpticsPlane Waves are simple points in reciprocal space A lens performs a Fourier-transformbetween its FociFourier-transformationFourier-transformation & OpticsFourier-planeObjectImageffffLaserFourier Transform

The Complex Planerealimaginary1-1i = -1ab

AThe Complex WaverealimaginaryxWavenumber: k [waves / m]

xFrequency space:k [1/m]x [m]Real space:IntensityAmplitudeExcurse: Spatial Frequenciesfrom: http://members.nbci.com/imehlmir/

Even better approximation:Fourier Analysis

from: http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Fourier.htmlExamples

x

realimag.kk0realimag.

Non-Periodic Examples (rect)

xreal

kreal

Non-Periodic Examples (triang)xreal

kreal

Examples (comb function)xreal

kreal

Inverse Scaling Law !Examples

x

kk0realimag.

-k0realTheorems (Real Valued)

Function isReal ValuedReal SpaceFourier SpaceFunction isSelf-Adjunct:

Theorems (Real + Symmetric)

Function isReal Valued &SymmetricReal SpaceFourier SpaceFunction isReal Valued &SymmetricTheorems (Shifting)

shift by DxReal SpaceFourier Space

Multiplication witha spiral

Theorems

MultiplicationReal SpaceFourier Space

Convolution

Theorems (Scaling)scaling by aReal SpaceFourier Space

Inverse scaling 1/a

Convolution?The Running Wave

Constructing images from wavesSum of WavesCorrespondingSine-WavekxkykxkyAccumulatedFrequenciesSpatialFrequency24

Constructing images from wavesSum of WavesCorrespondingSine-WaveAccumulatedFrequenciesSpatialFrequency25Fourier-space & Optics

Fourier-transformation & OpticsFourier-planeObjectImageffffLaser

Low Pass Filter

Fourier-transformation & OpticsFourier-planeObjectImageffffLaser

High Pass Filter

Intensity in Focus (PSF)Reciprocal Space (ATF)kxkzkyReal Space (PSF)xzyLensFocusOilCover Glass

Ewald sphereMcCutchengeneralisedaperture

IFTAmplitude indicated by brightnessPhase indicated by color

AmplitudeIntensity

Point spread function (PSF)

The image generated by a single pointsource in the sample.

A sample consisting of many points hasto be repainted using the PSF as abrush.

Convolution !Image = Sample PSF

FT(Image) = FT(Sample) * FT(PSF)

IFTFT|.|2

square??I(x) = |A(x)|2 = A(x) A(x)* I(k) = A(k) A(-k) OTFCTF~~~ *Fourier TransformIntensity in Focus (PSF), Epifluorescent PSF?Convolution: Drawing with a Brushkx,ykzRegion of SupportOptical Transfer Function (OTF)kx,ykz

Missing coneWidefield OTF supportakzkx,yn/ln sin(a)/lkx,ykz=2nsin(a)/l n (1-cos(a)) /ln (1-cos(a)) / l

Missing cone

Top viewOptical Transfer Functionkxky|kx,y||kx,y| [1/m]contrastCut-off limit01A microscope is a Fourier-filter!

Image = Sample PSFFT(Image) = FT(Sample) * FT(PSF)

Fourier Filtering

kxkyFourier domainReal spaceFourier domainDFTsuppresshigh spatialfrequencieskxkzkz01kxImage = Sample PSFFT(Image) = FT(Sample) * FT(PSF)