1. Biophotonics lecture 11. January 2012

2. Today: -Correct sampling in microscopy -Deconvolution techniques

3. Correct Sampling

4. What is SAMPLING? Intensity [a.u.] 2 3456 X [µm] 1

5. Aliasing … suppose it is a sine-wave Intensity [a.u.] 2 3456 There are many sine-waves, SAMPLED with the same measurements. Which is the correct one?

6. Intensity [a.u.] 2 3456 X [µm] When sampling at the frequency of the signal, a zero-frequency is recorded!

7. Intensity [a.u.] 2 3456 X [µm]

8. Intensity [a.u.] 2 3456 X [µm] Problem: too high frequencies will be aliased, they will seemingly become lower frequencies

9. But … high frequencies are not transmitted well. Object: Microscope Image: Intensity Spatial Coordinate Intensity Spatial Coordinate OTF

10. Aliasing in Fourier-space Fourier-transform of Image Intensity Aliased Frequencies ½ Sampling Frequency Cut-off frequency =½ Nyquist Rate Sampling Frequency Nyquist Rate

11. Pixel sensitivity Intensity [a.u.] 2 3456 X [µm] 1 Convolution of pixel form factor with sample  Multiplication in Fourier-space  Reduced sensitivity at high spatial frequency

12. Optical Transfer Function |k x,y | [1/m] contrast Cut-off limit 0 1 rectangle form-factor OTF sampled

13. Consequences of high sampling Confocal: high Zoom  more bleaching? No! if laser is dimmed or scan-speed adjusted  bad signal to noise ratio? Yes, but photon positions are only measured more accurately  binning still possible  high SNR. Readout noise is a problem at high spatial sampling (CCD)

14. Optimal Sampling?

15. Regular sampling Reciprocal  -Sampling Grid Real-space sampling: Multiplied in real space with band-limited information

16. Regular sampling Reciprocal  -Sampling Grid Real-space sampling:

17. Widefield Sampling  In-Plane sampling distance  Axial sampling distance

18. Confocal Sampling  In-Plane sampling distance (very small pinhole) else use widefield equation  Axial sampling distance

19. Confocal OTFs WF 1 AU 0.3 AU in-plane, in-focus OTF 1.4 NA Objective WF Limit

20. Hexagonal sampling Advantage: ~17% + less ‚almost empty‘ information collected + less readout-noise approximation in confocal Reciprocal d-Sampling Grid Real-space sampling: Multiplied in real space with band-limited information

21. 63× 1.4 NA Oil Objective (n=1.516), excitation at 488 nm, emission at 520 nm  l eff = 251.75 nm, a = 67.44 deg widefield in-plane: d xy < 92.8 nm  maximal CCD pixelsize: 63×92.8 = 5.85 µm confocal in-plane:d xy < 54.9 nm widefield axial: d z < 278.2 nm confocal axial: d z < 134.6 nm Fluorescence Sampling Example

22. OTF is not zero but very small (e.g. confocal in-plane frequency) Object possesses no higher frequencies You are only interested in certain frequencies (e.g. in counting cells, serious under-sampling is acceptable) Reasons for undersampling

23. If you need high resolution or need to detect small samples  sample your image correctly along all dimensions Sampling Summary

24. Maximum Likelihood Deconvolution

27. Image: http://en.wikipedia.org

30. The prior (requires prior knowledge; can imply contraints, e.g. positivity) Constant normalisation factor

31. Constant, therefore obsolete

37. MATLAB demonstration

38. Information & Photon noise Virtual Microscopy Only Noise? FT NO! 10 Photons / Pixel

39. Band Extrapolation? Object Mean Error Energy Mean Energy Relative Energy Regain

40. With Photon Noise

41. Is this always possible? White Noise Object

42. Is this always possible? Unfortunately NOT !