Biophotonics lecture 9. November 2011. Last time (Monday 7. November)  Review of Fourier Transforms (will be repeated in part today)  Contrast enhancing.

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Presentation transcript:

Biophotonics lecture 9. November 2011

Last time (Monday 7. November)  Review of Fourier Transforms (will be repeated in part today)  Contrast enhancing techniques in microscopy  Brightfield microscopy  Darkfield microscopy  Phase Constrast Microscopy  Polarisation Contrast Microscopy  Differential Interference Contrast (DIC) Microscopy

Today  Part 1: Review of Fourier Transforms  1D, 2D  Fourier filtering  Fourier transforms in microscopy: ATF, ASF, PSF, OTF  Part 2: Sampling theory

Fourier-transformation & Optics

Plane Waves are simple points in reciprocal space A lens performs a Fourier-transform between its Foci Fourier-transformation

Fourier-transformation & Optics Fourier- plane Object Image f f f f Laser

Fourier Transform

The Complex Plane real imaginary 1 i =  -1 ab  A

The Complex Wave real imaginary x Wavenumber: k [waves / m] x

Frequency space: k [1/m] x [m] Real space: Intensity Amplitude Excurse: Spatial Frequencies

: from: Even better approximation: Fourier Analysis : from: ~history/PictDisplay/Fourier.html

Examples x real imag. k k0k0  real imag.

Non-Periodic Examples (rect) x real k

Non-Periodic Examples (triang) x real k

Examples (comb function) x real k Inverse Scaling Law !

Examples xk k0k0 real imag. -k 0 real

Theorems (Real Valued) Function is Real Valued Real SpaceFourier Space Function is Self-Adjunct:

Theorems (Real + Symmetric) Function is Real Valued & Symmetric Real SpaceFourier Space Function is Real Valued & Symmetric

Theorems (Shifting) shift by  x Real SpaceFourier Space Multiplication with a „spiral“

Theorems Multiplication Real SpaceFourier Space Convolution

Theorems (Scaling) scaling by a Real SpaceFourier Space Inverse scaling 1/a

Convolution ?

The Running Wave

Constructing images from waves Sum of Waves Corresponding Sine-Wave kxkx kyky kxkx kyky Accumulated Frequencies Spatial Frequency

Constructing images from waves Sum of Waves Corresponding Sine-Wave Accumulated Frequencies Spatial Frequency

Fourier-space & Optics

Fourier-transformation & Optics Fourier- plane Object Image f f f f Laser Low Pass Filter

Fourier-transformation & Optics Fourier- plane Object Image f f f f Laser High Pass Filter

Intensity in Focus (PSF) Reciprocal Space (ATF) kxkx kzkz kyky Real Space (PSF) x z y Lens Focus Oil Cover Glass

Ewald sphere McCutchen generalised aperture

IFT Amplitude indicated by brightness Phase indicated by color

AmplitudeIntensity

Point spread function (PSF) The image generated by a single point source in the sample. A sample consisting of many points has to be “repainted” using the PSF as a brush.  Convolution ! Image = Sample  PSF FT(Image) = FT(Sample) * FT(PSF)

IFT FT |.| 2 square ? ?

I(x) = |A(x)| 2 = A(x) · A(x) * I(k) = A(k)  A(-k) OTF CTF ~~ ~ * Fourier Transform Intensity in Focus (PSF), Epifluorescent PSF ?

Convolution: Drawing with a Brush k x,y kzkz Region of Support

Optical Transfer Function (OTF) k x,y kzkz

Missing cone

Widefield OTF support  kzkz k x,y n/ n sin   k x,y kzkz = 2nsin  n (1-cos  n (1-cos  

Missing cone Top view

Optical Transfer Function kxkxkxkx kykykyky |k x,y | |k x,y | [1/m] contrast Cut-off limit 0 1 A microscope is a Fourier-filter! Image = Sample  PSFFT(Image) = FT(Sample) * FT(PSF)

Fourier Filtering kxkxkxkx kykykyky Fourier domain Real space Fourier domain DFT suppress high spatial frequencies kxkxkxkx kzkzkzkz kzkzkzkz 0 1 kxkxkxkx Image = Sample  PSFFT(Image) = FT(Sample) * FT(PSF)