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Biophotonics lecture 9. November 2011

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

ingrid-stevenson
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Biophotonics lecture 9. November 2011

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  1. Biophotonics lecture9. 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. Fourier-transformation • Plane Waves are simple points in reciprocal space • A lens performs a Fourier-transformbetween its Foci Fourier-transformation & Optics

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

  7. Fourier Transform

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

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

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

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

  12. real  imag. k k0 Examples real imag. x

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

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

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

  16. real imag. -k0 k0 Examples real x k

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

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

  19. Multiplication witha „spiral“ Real Space Fourier Space Theorems (Shifting) shift by Dx

  20. Convolution Real Space Fourier Space Theorems Multiplication

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

  22. Convolution ?

  23. The Running Wave

  24. ky ky kx kx Constructing images from waves CorrespondingSine-Wave SpatialFrequency AccumulatedFrequencies SumofWaves

  25. Constructing images from waves CorrespondingSine-Wave SpatialFrequency AccumulatedFrequencies SumofWaves

  26. Fourier-space& Optics

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

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

  29. Real Space (PSF) Reciprocal Space (ATF) Lens Cover Glass ky y x kx Focus z kz Oil Intensity in Focus (PSF)

  30. McCutchen generalised aperture Ewald sphere

  31. IFT Amplitude indicated by brightness Phase indicated by color

  32. Amplitude Intensity

  33. 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)

  34. IFT |.|2 square ? ? FT

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

  36. Region of Support kx,y kz Convolution: Drawing with a Brush

  37. kx,y kz Optical Transfer Function (OTF)

  38. Missing cone

  39. 2nsin(a)/l n sin(a)/l n/l kx,y kx,y a  = kz kz n (1-cos(a))/l n (1-cos(a))/ l Widefield OTF support

  40. Top view Missing cone

  41. Image = Sample  PSF FT(Image) = FT(Sample) * FT(PSF) contrast ky Optical Transfer Function 1 |kx,y| kx 0 |kx,y| [1/m] Cut-off limit A microscope is a Fourier-filter!

  42. kz kz ky kx kx 1 kx Image = Sample  PSF FT(Image) = FT(Sample) * FT(PSF) Real space Fourier domain Fourier domain Fourier Filtering suppresshigh spatialfrequencies DFT 0

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