part 2: fourier transforms - welcome to the course · part 2: fourier transforms key to...
TRANSCRIPT
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Part 2: Fourier transforms
Key to understanding NMR, X-ray crystallography,
and all forms of microscopy
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Sine waves
y(t) = A sin(wt + p)y(x) = A sin(kx + p)
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To completely specify a sine wave, you need its (1) direction, (2) wavelength or frequency, (3) amplitude, and (4) phase shift
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y = sin(x)
y = sin(2x)
y = sin(3x)
y = sin(x) - 2.3sin(2x) + 1.8sin(3x)
Adding sine waves
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Taking sine wave sums apart(a Fourier “decomposition”, or “transform”)
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Fourier transforms in music and hearing
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Fourier decompositions may not be exact - depends on how many terms you use
(“resolution”)
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The mathematical details
The balance between sine and cosine terms can be equivalently introduced by giving each sine term a “phase”
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One-dimensional sine waves and their sums Concept check questions:
• What four parameters define a sine wave?
• What is the difference between a temporal and a spatial frequency?
• What in essence is a “Fourier transform”?
• How can the amplitude of each Fourier component of a waveform be found?
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x
y
0 1 2 3 4 5 6 7 8 9
0.3 0.4 1.3 3.4 4.5 4.2 2.8 2.4 1.4 0.1
“analog”
“digital”
12
34
0
Analog versus digital images
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Fourier transform
A0 P0 A1 P1 A2 P2 A3 P3 A4 P4 A5 P5
10 numbers
10 numbers + 2(5 amps & phases + “DC”
component)
A0 = amplitude of “DC” componentA1 = amplitude of “fundamental” frequency (one wavelength across box)P1 = phase of “fundamental” frequency componentA2 = amplitude of first “harmonic” (two wavelengths across box)P2 = phase of first harmonicA3 = amplitude of second harmonicP3 = phase of second harmonic etc.A5 = amplitude of “Nyquist” frequency component
Consider a one-dimensional array:
0.3 0.4 1.3 3.4 4.5 4.2 2.8 2.4 1.4 0.1
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One dimensional functions and transforms (spectra)
Hecht, Fig. 11.13
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One dimensional functions and transforms (spectra)
Hecht, Fig. 11.13
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More complicated functions and their spectra
Hecht, Fig. 11.38
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One-dimensional reciprocal space Concept check questions:
• What is the difference between an “analog” and a “digital” image?
• What is the “fundamental” frequency? A “harmonic”? “Nyquist” frequency?
• What is “reciprocal” space? What are the axes?
• What does a plot of the Fourier transform of a function in reciprocal space tell you?
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In microscopy we deal with 2-D images and transforms
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Two-dimensional waves and images Concept check questions:
• What does a two-dimensional sine wave look like?
• What do the “Miller” indices “h” and “k” indicate?
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Fourier transform
xh
k
0
-1
-2
-3
-4
-5
5
4
3
2
1
0 1 2 3 4 5
A01 P01
A00 P00 A10 P10 A20 P20
A0,-5 P0,-5
A05 P05
A50 P50
A55 P55
A5,-5 P5,-5
A23 P23
N2 numbers
~N2 numbers
y
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. .
A simple 2-D image and transform (diffraction pattern)
x
yfx
fy
“Real” space : coordinates “Reciprocal” space : spatial “frequencies”
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.
.
Another simple 2-D image and transform
x
yfx
fy
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More complex 2-D images and transforms
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Briegel et al., PNAS 2009
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“Resolution”
Note here the “power” or intensity of each Fourier component is being plotted, not the phase, and for
any real image, the pattern is symmetric
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http://sharp.bu.edu/~slehar/fourier/fourier.html
FT
FTFT
FT
FT
“low pass” filter “high pass” filter
“band pass” filter
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Two-dimensional transforms and filters Concept check questions:
• In the Fourier transform of a real image, how much of reciprocal space (positive and negative values of “h” and “k”) is unique?
• If an image “I” is the sum of several component images, what is the relationship of its Fourier transform to the Fourier transforms of the component images?
• What part of a Fourier transform is not displayed in a power spectrum?
• What does the “resolution” of a particular pixel in reciprocal space refer to?
• What is a “low pass” filter? “High pass”? “Band pass”?
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In X-ray crystallography, 3-D microscopy, and 3-D NMR
we deal with 3-D images and transforms
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FT
h
k0
-1
-2
-3
-4
-5
5
4
3
2
1
0 1 2 3 4 5
A000 P000
A5,-5,0 P5,-5,0
l
A550 P550 A050 P050
-5
0
5N3 numbers
~N3 numbers
What does a 3-D FT look like?
A05,-5 P05,-5
A050 P050
A555 P555
A550 P550
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Three-dimensional waves and transforms Concept check questions:
• What does a three-dimensional sine wave look like?
• What does the third “Miller” index “l” represent?
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Convolution
=
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Zhu et al., JSB 2004
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Convolution and cross-correlation Concept check questions:
• What is a “convolution”?
• What is the “convolution theorem”?
• What is a “point spread function”?
• What does convolution have to do with the structure of crystals?
• What is “cross-correlation”?
• How might cross-correlations be used in cryo-EM?