Biophotonics lecture 11. January 2012

Today: -Correct sampling in microscopy -Deconvolution techniques

Correct Sampling

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

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

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

Intensity [a.u.] X [µm]

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

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

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

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

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

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)

Optimal Sampling?

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

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

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

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

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

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

63× 1.4 NA Oil Objective (n=1.516), excitation at 488 nm, emission at 520 nm  l eff = nm, a = 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 < nm confocal axial: d z < nm Fluorescence Sampling Example

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

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

Maximum Likelihood Deconvolution

Image:

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

Constant, therefore obsolete

MATLAB demonstration

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

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

With Photon Noise

Is this always possible? White Noise Object

Is this always possible? Unfortunately NOT !