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 !