1 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?

Aliasing … suppose it is a sine-wave … maybe we can know! Object: Microscope Image: Intensity Spatial Coordinate Intensity Spatial Coordinate

Aliasing in Fourier-space Fourier-transform of Image Intensity Aliased Frequencies ½ Sampling Frequency ½ Nyquist Frequencies

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

Optical Transfer Function |k x,y | [1/m] contrast Cut-off limit 0 1 rectange form-factor specimen 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)

9 Optimal Sampling?

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

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; 3D: ABA, ABC stacking Reciprocal  -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  eff = nm,  = 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) 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

Detector generates high-frequency noise? Detector generates high-frequency noise?  Measure this noise (e.g. dark exposure and 2D FFT)  Avoid aliasing by sampling above this noise frequency. Traps and Pitfalls

FFT of dark CCD exposure (2 µs)

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