1. Biophotonics lecture 9. November 2011

2. 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

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

4. Fourier-transformation & Optics

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

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

7. Fourier Transform

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

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

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

11. : from: http://members.nbci.com/imehlmir/ Even better approximation: Fourier Analysis : from: http://www-groups.dcs.st-and.ac.uk/ ~history/PictDisplay/Fourier.html

12. Examples x real imag. k k0k0  real imag.

13. Non-Periodic Examples (rect) x real k

14. Non-Periodic Examples (triang) x real k

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

16. Examples xk k0k0 real imag. -k 0 real

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

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

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

20. Theorems Multiplication Real SpaceFourier Space Convolution

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

22. Convolution ?

23. The Running Wave

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

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

26. Fourier-space & Optics

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

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

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

30. Ewald sphere McCutchen generalised aperture

31. IFT Amplitude indicated by brightness Phase indicated by color

32. AmplitudeIntensity

33. 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)

34. IFT FT |.| 2 square ? ?

35. 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 ?

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

37. Optical Transfer Function (OTF) k x,y kzkz

38. Missing cone

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

40. Missing cone Top view

41. 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)

42. 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)