Correct Sampling 1 What is SAMPLING Intensity a

Correct Sampling 1

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

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? 2 3 4 5 6

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

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

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

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

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? 9

Regular sampling Real-space sampling: Reciprocal d-Sampling Grid 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 in-plane, in-focus OTF 1. 4 NA Objective WF Limit 1 AU WF 0. 3 AU

Hexagonal sampling Real-space sampling: Reciprocal d-Sampling Grid Multiplied in real space with band-limited information Advantage: ~17% + less ‚almost empty‘ information collected + less readout-noise approximation in confocal; 3 D: ABA, ABC stacking

Fluorescence Sampling Example 63× 1. 4 NA Oil Objective (n=1. 516), excitation at 488 nm, emission at 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

Reasons for Undersampling 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)

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

FFT of dark CCD exposure (2 µs)

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