Biophotonics lecture11. January 2012
Today:
- Correct sampling in microscopy- Deconvolution techniques
Correct Sampling
What is SAMPLING?
Intensity [a.u.]
2 3 4 5 6 X [µm]1
Aliasing … suppose it is a sine-wave
Intensity [a.u.]
2 3 4 5 6
There are many sine-waves, SAMPLED with the same measurements.Which is the correct one?
Intensity [a.u.]
2 3 4 5 6 X [µm]
When sampling at the frequency of the signal, a zero-frequency is recorded!
Intensity [a.u.]
2 3 4 5 6 X [µm]
Intensity [a.u.]
2 3 4 5 6 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:
Inte
nsity
Spatial Coordinate
Inte
nsity
Spatial Coordinate
OTF
Aliasing in Fourier-spaceFourier-transform of Image
Inte
nsity
Aliased Frequencies
½ SamplingFrequency
Cut-off frequency=½ Nyquist Rate
SamplingFrequency
NyquistRate
Pixel sensitivityIntensity [a.u.]
2 3 4 5 6 X [µm]1
Convolution of pixel form factor with sample
Multiplication in Fourier-space
Reduced sensitivity at high spatial frequency
Optical Transfer Function
|kx,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 samplingReciprocal d-Sampling GridReal-space sampling:
Multiplied in real spacewith band-limited information
Regular samplingReciprocal d-Sampling GridReal-space sampling:
Widefield SamplingIn-Plane sampling distance
Axial sampling distance
obj
emxy NAd
4max,
)cos(1)sin(
2max,obj
obj
obj
emz NAd
Confocal SamplingIn-Plane sampling distance (very small pinhole)
else use widefield equation
Axial sampling distance
)cos(1)sin(
2max,obj
obj
obj
effz NAd
emex
eff
11
1
obj
effxy NAd
4max,
Confocal OTFs
WF
1 AU
0.3 AU
in-plane, in-focus OTF1.4 NA Objective
WF Limit
Hexagonal sampling
Advantage: ~17%+ less ‚almost empty‘ information collected+ less readout-noiseapproximation in confocal
Reciprocal d-Sampling GridReal-space sampling:
Multiplied in real spacewith band-limited information
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
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
MaximumLikelihood
Deconvolution
Fluorescence imaging
Sample: S(r)Point Spread Function, PSF: h(r)Ideal Image:
(Convolution operator )⨂
But: noisy image M(r) = N(M(r)) = E(r) + n(r) Poisson Noise
Naïve approach to deconvolution ?
Problems:Fourier space: , Frequencies , for which
Fourier space:
Noise amplification for low
Poisson distribution
Probability p for measuring M photons when expectation value is E photons:
Image: http://en.wikipedia.org
Poisson probability in images
Probability p for measuring image M with pixel values M(r) when expectation image E with expectation pixel values E(r):
(Probabilities multiply)
Or even:
Our goal:
For a given measurement image M, find the most likely sample distribution S.
We can calculate: and
But…
Bayes rule:
But rather: The prior(requires prior knowledge; can imply contraints, e.g. positivity)
Constant normalisation factor
Nevertheless:
Maximum likelihood deconvolution triesto maximise rather than (uniform prior).
The approach:
Take the negative natural logartihm and minimise.
Constant, therefore obsolete
Minimise with respect to S(r‘):
With:
Iterative minimisation:
Simple “steepest gradient” search:
Minimise function F(x) iteratively: with small
Applied to log-likelihood function:
With:
Richardson-Lucy iterative minimisation:
Richardson-Lucy:
Steepest gradient
Richardson Lucy (fix point iteration)
Has positivity constraint!
Richardson-Lucy:
Start with initial guess:
Problem with algorithm:
- Very slow- Not stable
MATLAB demonstration
Information & Photon noise
VirtualMicroscopy
Only Noise?
FT
NO!
10 Photons / Pixel
Band Extrapolation?
Object
Mean Error Energy
Mean EnergyRelative Energy Regain
With Photon Noise
Is this always possible?
White Noise Object
Is this always possible?
Unfortunately NOT !