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Biophotonics lecture 9. November 2011. Last time (Monday 7. November). Review of Fourier Transforms (will be repeated in part today) Contrast enhancing techniques in microscopy Brightfield microscopy Darkfield microscopy Phase Constrast Microscopy Polarisation Contrast Microscopy

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Presentation Transcript
last time monday 7 november
Last time (Monday 7. November)
  • Review of Fourier Transforms (will be repeated in part today)
  • Contrast enhancing techniques in microscopy
      • Brightfield microscopy
      • Darkfield microscopy
      • Phase Constrast Microscopy
      • Polarisation Contrast Microscopy
      • Differential Interference Contrast (DIC) Microscopy
today
Today
  • Part 1: Review of Fourier Transforms
        • 1D, 2D
        • Fourier filtering
        • Fourier transforms in microscopy: ATF, ASF, PSF, OTF
  • Part 2: Sampling theory
fourier transformation optics1

Fourier-transformation

  • Plane Waves are simple points in reciprocal space
  • A lens performs a Fourier-transformbetween its Foci
Fourier-transformation & Optics
fourier transformation optics2

f

f

f

f

Laser

Object

Fourier-plane

Image

Fourier-transformation & Optics
the complex plane

imaginary

b

i = -1

A

real

a

-1

1

The Complex Plane
frequency space

Excurse: Spatial Frequencies

Real space:

Frequency space:

Intensity

x [m]

Amplitude

k [1/m]

even better approximation

from:http://www-groups.dcs.st-and.ac.uk/

~history/PictDisplay/Fourier.html

Even better approximation:

from:http://members.nbci.com/imehlmir/

Fourier Analysis

examples

real

imag.

k

k0

Examples

real

imag.

x

examples comb function

real

k

Examples (comb function)

real

x

Inverse Scaling Law !

examples1

real

imag.

-k0

k0

Examples

real

x

k

theorems real valued

Function is

Self-Adjunct:

Real Space

Fourier Space

Theorems (Real Valued)

Functionis

Real Valued

theorems real symmetric

Real Space

Fourier Space

Theorems (Real + Symmetric)

Function is

Real Valued &

Symmetric

Function is

Real Valued &

Symmetric

theorems

Convolution

Real Space

Fourier Space

Theorems

Multiplication

theorems scaling

Inverse scaling 1/a

Real Space

Fourier Space

Theorems (Scaling)

scaling by a

constructing images from waves

ky

ky

kx

kx

Constructing images from waves

CorrespondingSine-Wave

SpatialFrequency

AccumulatedFrequencies

SumofWaves

constructing images from waves1
Constructing images from waves

CorrespondingSine-Wave

SpatialFrequency

AccumulatedFrequencies

SumofWaves

fourier transformation optics3

f

f

f

f

Laser

Object

Fourier-plane

Image

Fourier-transformation & Optics

Low Pass Filter

fourier transformation optics4

f

f

f

f

Laser

Object

Fourier-plane

Image

Fourier-transformation & Optics

High Pass Filter

intensity in focus psf

Real Space (PSF)

Reciprocal Space (ATF)

Lens

Cover Glass

ky

y

x

kx

Focus

z

kz

Oil

Intensity in Focus (PSF)
slide30

McCutchen

generalised

aperture

Ewald sphere

slide31

IFT

Amplitude indicated by brightness

Phase indicated by color

slide32

Amplitude

Intensity

slide33

Point spread function (PSF)

  • The image generated by a single pointsource in the sample.
  • A sample consisting of many points hasto be “repainted” using the PSF as abrush.
  • Convolution !
  • Image = Sample  PSF
  • FT(Image) = FT(Sample) * FT(PSF)
slide34

IFT

|.|2

square

?

?

FT

slide35

Intensity in Focus (PSF), Epifluorescent PSF

Fourier Transform

~ *

~

~

I(k) = A(k) A(-k)

OTF

CTF

I(x) = |A(x)|2 = A(x) · A(x)*

?

widefield otf support

2nsin(a)/l

n sin(a)/l

n/l

kx,y

kx,y

a

=

kz

kz

n (1-cos(a))/l

n (1-cos(a))/ l

Widefield OTF support
slide40

Top view

Missing cone

o ptical t ransfer f unction

Image = Sample  PSF FT(Image) = FT(Sample) * FT(PSF)

contrast

ky

Optical Transfer Function

1

|kx,y|

kx

0

|kx,y| [1/m]

Cut-off limit

A microscope is a Fourier-filter!

fourier filtering

kz

kz

ky

kx

kx

1

kx

Image = Sample  PSF FT(Image) = FT(Sample) * FT(PSF)

Real space

Fourier domain

Fourier domain

Fourier Filtering

suppresshigh spatialfrequencies

DFT

0