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Biophotonics lecture 11. January 2012

Biophotonics lecture 11. January 2012. Today: Correct sampling in microscopy Deconvolution techniques. Correct Sampling. Intensity [a.u.]. What is SAMPLING?. X [µm]. 1. 2. 3. 4. 5. 6. Intensity [a.u.]. 2. 3. 4. 5. 6.

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Biophotonics lecture 11. January 2012

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  1. Biophotonics lecture11. January 2012

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

  3. Correct Sampling

  4. Intensity [a.u.] What is SAMPLING? X [µm] 1 2 3 4 5 6

  5. Intensity [a.u.] 2 3 4 5 6 There are many sine-waves, SAMPLED with the same measurements.Which is the correct one? Aliasing … suppose it is a sine-wave

  6. Intensity [a.u.] 2 3 4 5 6 When sampling at the frequency of the signal, a zero-frequency is recorded! X [µm]

  7. Intensity [a.u.] 2 3 4 5 6 X [µm]

  8. Intensity [a.u.] 2 3 4 5 6 Problem:too high frequencies will be aliased, they will seemingly become lower frequencies X [µm]

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

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

  11. Convolution of pixel form factor with sample  Multiplication in Fourier-space  Reduced sensitivity at high spatial frequency Intensity [a.u.] Pixel sensitivity X [µm] 1 2 3 4 5 6

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

  13. Confocal: high Zoom morebleaching? Consequences of high sampling No! iflaserisdimmedor scan-speed adjusted badsignaltonoiseratio? Yes, but photonpositionsareonlymeasuredmoreaccuratelybinning still possiblehigh SNR. Readoutnoiseis a problemat high spatialsampling (CCD)

  14. Optimal Sampling?

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

  16. Reciprocald-Sampling Grid Real-space sampling: Regular sampling

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

  18. In-Plane sampling distance (very small pinhole) • elseusewidefieldequation •  Axial sampling distance Confocal Sampling

  19. in-plane, in-focus OTF1.4 NA Objective Confocal OTFs WF Limit 1 AU 0.3 AU WF

  20. Reciprocal d-Sampling Grid Multiplied in real spacewith band-limited information Real-space sampling: Hexagonal sampling Advantage: ~17%+ less ‚almostempty‘ informationcollected+ lessreadout-noiseapproximation in confocal

  21. 63× 1.4 NA Oil Objective (n=1.516), excitationat 488 nm, emissionat 520 nm leff = 251.75 nm, a = 67.44 deg widefield in-plane: dxy < 92.8 nm  maximal CCD pixelsize: 63×92.8 = 5.85 µm confocal in-plane: dxy < 54.9 nm widefield axial: dz < 278.2 nm confocal axial: dz < 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. MaximumLikelihoodDeconvolution

  25. Fluorescence imagingSample: S(r)Point SpreadFunction, PSF: h(r)Ideal Image: (Convolution operator ⨂)But: noisy image M(r) = N(M(r)) = E(r) + n(r)  Poisson Noise

  26. Naïve approach to deconvolution ?Problems:Fourier space: , Frequencies , for whichFourier space: Noise amplificationforlow

  27. Poisson distributionProbability p formeasuring M photonswhenexpectationvalueis E photons: Image: http://en.wikipedia.org

  28. Poisson probability in images Probability p formeasuringimage M withpixelvalues M(r) whenexpectationimage E withexpectationpixelvalues E(r): (Probabilities multiply) Or even: 

  29. Our goal: For a givenmeasurementimage M, find themostlikely sample distribution S. We can calculate: and But…

  30. Bayes rule: But rather: The prior(requirespriorknowledge; canimplycontraints, e.g. positivity) Constant normalisationfactor

  31. Nevertheless: Maximum likelihooddeconvolutiontriestomaximiseratherthan (uniform prior). The approach: Take thenegative naturallogartihmandminimise. Constant, therefore obsolete

  32. Minimisewithrespectto S(r‘): With:

  33. Iterative minimisation: Simple “steepest gradient” search: Minimise function F(x) iteratively: with small Applied to log-likelihood function: With:

  34. Richardson-Lucy iterative minimisation:

  35. Richardson-Lucy: Steepestgradient Richardson Lucy (fix point iteration) Has positivity constraint!

  36. Richardson-Lucy: Start withinitialguess: Problem withalgorithm: Veryslow Not stable

  37. MATLAB demonstration

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

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

  40. With Photon Noise

  41. White Noise Object Is this always possible?

  42. Is this always possible? Unfortunately NOT !

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