Image analysis I
C9940 3-Dimensional Transmission Electron Microscopy
S1007 Doing structural biology with the electron microscope
March 21, 2016
Outline
Image analysis I
Fourier transforms
Why do we care?
Theory
Examples in 1D
Examples in 2D
Digitization
Fourier filtration
Contrast transfer function
Resolution
Fourier transforms
Outline
Image analysis I
Fourier transforms
Why do we care?
Theory
Examples in 1D
Examples in 2D
Digitization
Fourier filtration
Contrast transfer function
Resolution
A quiz
http://www.microscopy.ethz.ch
A quiz
http://electron6.phys.utk.edu
http://www.microscopy.ethz.ch
http://electron6.phys.utk.edu
Relationship between imaging and diffraction
The only difference between microscopy
and diffraction is that, in microscopy,
you can focus the scattered radiation
into an image.
How do X-ray microscopes work?
http://ssrl.slac.stanford.eduhttp://www.microscopy.ethz.ch
How do X-ray microscopes work?
http://ssrl.slac.stanford.edu
Fabrizio...Barrett, 1999, Nature
Best resolution: ~20nm
Outline
Image analysis I
Fourier transforms
Relationship between imaging and diffraction
Theory
Examples in 1D
Examples in 2D
Digitization
Fourier filtration
Contrast transfer function
Resolution
Relevance of Fourier transforms to EM
Fourier transform ~ diffraction pattern
see John Rodenburg's site, http://rodenburg.org

Fourier series
A Fourier series is an expansion of a
periodic function f(x) in terms of an infinite
sum of sines and cosines
Fourier transforms: Exponential form
f: function which we are transforming (1D)
x: axis coordinate
i: √-1
k: spatial frequency
F(k): Fourier coefficient at frequency k
Fourier transforms: Exponential form
Euler's Formula:
ba +i
Fourier transforms: Sines + cosines
(NOTE: This isn't the same a & b from the previous slide.)
Fourier transforms: Definition
ba +i
a
b
Amplitude, A:
Phase, Ф:
Ф
A
Fourier coefficients, discrete functions
sine and cosine are orthogonal.
The inner product of the two is zero.
A function with the same frequency but
with an offset will have some components
of both sine and cosine.
That is, a and b will be non-zero.
A function with a different frequency will
have coefficients of zero.
http://cnx.org
The higher the
spatial frequencies
(i.e., higher
resolution) that are
included, the more
faithful the
representation of
the original function
will be.
Some properties

As n increases, so does the spatial frequency,
i.e., the “resolution.”
 For example, sin(2x) oscillates faster than sin(x)

Computation of a Fourier transform is a
completely reversible operation.
 There is no loss of information.

Fourier terms (or coefficients) have amplitude and
phase.

The diffraction pattern is the physical
manifestation of the Fourier transform
– Phase information is lost in a diffraction pattern.
– An image contains both phase and amplitude information.
Some simple 1D transforms:
a 1D lattice
Some simple 1D transforms:
a box
http://cnx.org
Later, you will learn that multiplying a step function is bad,
because of these ripples in Fourier space.
Fourier transforms: plot of a Gaussian
Xx
f(x) F(X)
Some simple 1D transforms:
a sharp point (Dirac delta function)
http://en.labs.wikimedia.org/wiki/Basic_Physics_of_Nuclear_Medicine/Fourier_Methods
Some simple 2D Fourier transforms:
a row of points
Some simple 2D Fourier transforms:
a series of lines
Some simple 2D Fourier transforms:
a sharp disc
Some simple 2D Fourier transforms:
a 2D lattice
Some simple 2D Fourier transforms:
a 2D lattice
Some simple 2D Fourier transforms: a helix
Outline
Image analysis I
Fourier transforms
Relationship between imaging and diffraction
Theory
Examples in 1D
Examples in 2D
Digitization
Fourier filtration
Contrast transfer function
Resolution
Digitization in 2D
Digitization in 1D: Sampling
Digitization: Is our sampling good enough?
Here, our sampling is good enough.
Digitization in 1D: Bad sampling
What's the best resolution we can get
from a given sampling rate?
A 4-pixel “image”
1 2 3 4
In other words, what is the most rapid oscillation we can detect?
What's the best resolution we can get
from a given sampling rate?
In other words, what is the most rapid oscillation we can detect?
ANSWER: Alternating light and dark pixels.
A 4-pixel “image”
The period of this finest oscillation is 2 pixels.
The spatial frequency of this oscillation is 0.5 px-1
.
The finest detectable oscillation is what is known as “Nyquist frequency.”
The edge of the Fourier transform corresponds to Nyquist frequency.
origin
spatial frequencyspatial frequency
Nyquist frequency
Nyquist frequency
The period of this finest oscillation is 2 pixels.
The spatial frequency of this oscillation is 0.5 px-1
.
The finest detectable oscillation is what is known as “Nyquist frequency.”
The edge of the Fourier transform corresponds to Nyquist frequency.
What do we mean by pixel size?
http://www.en.wikipedia.org
Typical magnification: 50,000X
Typical detector element: 15μm
(pixel size on the camera scale)
Pixel size on the specimen scale:
15 x 10-6
m/px / 50000 =
3.0 x 10-10
m/px = 3.0 Å/px
In other words,
the best resolution we
can achieve (or, the
finest oscillation we
can detect) at 3.0 Å/px
is 6.0 Å.
It will be worse due to interpolation,
so to be safe, a pixel should be 3X
smaller than your target resolution.
What do we mean by spatial frequency?
origin
spatial frequency
From Wikipedia
Fourier
filtration
From Wikipedia PowerspectrumProfile
A “high-pass”
filter
A “low-pass” filter
Contrast transfer function
Why do we defocus?
1.0
0.1
Typical amplitude contrast is estimated a 0.08-0.12
(minus noise)
water
macromolecule
Instead of amplitude contrast,
we'll use phase contrast.
Phase contrast in light microscopy
Bright-field image Phase-contrast image
http://www.microbehunter.com
In EM, even with defocus, the contrast is poor.
Signal-to-noise ratio for cryoEM typically given to be between 0.07 and 0.10.
E. coli 70S ribosomes, field width ~1440Å.
unfiltered filtered
Optical path
Specimen
Back focal plane
Image plane
At focus, all we would see is amplitude contrast.
Specimen
Back focal plane
Image plane
Optical path with defocus
Perfect focus
Image plane
Focal plane
O
A B
OA path of unscattered beam
OB path of scattered beam
The length OA is
also the amount of defocus Δf
Optical path with defocus
What is the path difference between the scattered and unscattered beams?
Path difference as a function of Δf
OB = OA/cos(a)
Expressed in the number of wavelengths λ
Phase difference is the sine
O
A B
a
Some typical values
OA = Δf = 10,000 Å
λ = 0.02 Å
a < 0.01
A more precise formulation of the CTF can be found in
Erickson & Klug A (1970). Philosophical Transactions of the Royal Society B. 261:105.
Proper form the CTF
where:
Cs: spherical aberration
k: spatial frequency (resolution)
How does the CTF affect an image?
original
original
combined
Still a zero present
Outline
Image analysis I
Fourier transforms
Relationship between imaging and diffraction
Theory
Examples in 1D
Examples in 2D
Digitization
Fourier filtration
Contrast transfer function
Resolution
How do we evaluate the quality of a
reconstruction?
Now, how do we compare the two half-set
reconstructions?
images
“odd” reconstruction
“even” reconstruction
We split the data set into halves and compare them.
Fourier Shell Correlation (FSC)
Properties:
- Fourier terms have amplitude + phase.
- Correlation values range from -1 to +1.
- Noise should give an average of 0.
- The comparison is done as a function of spatial frequency (or “resolution”)
Reconstruction1
Reconstruction2
term 1 term 2
Fourier Shell Correlation;
A better example
It is controversial what single number to use to describe this curve,
but a common practice is to report the value where the FSC=0.5
as the nominal resolution.
Fourier Shell Correlation;
A better example
The FSC is not a foolproof metric.
You can “fool” your data, or be fooled, into an artifactually good FSC.