Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine...
-
Upload
norman-bishop -
Category
Documents
-
view
214 -
download
2
Transcript of Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine...
![Page 1: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/1.jpg)
Automation
Mike MarshNational Center for Macromolecular Imaging
Baylor College of Medicine
Single-Particle Reconstructions and VisualizationEMAN Tutorial and Workshop
March 14, 2007
Fourier Transforms, Filtering and Convolution
![Page 2: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/2.jpg)
Fourier Transform is an invertible operator
Image Fourier Transform
v2 will display image or its transform
FT
![Page 3: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/3.jpg)
Fourier Transform is an invertible operator
Image
f(x,y)F(kx,ky)
x
y
0 Nx
Ny
{F(kx,ky)} = f(x,y)} {f(x,y)} = F(kx,ky)
Nx ⁄ 2
Ny ⁄ 2Fourier Transform
![Page 4: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/4.jpg)
Continuous Fourier Transform
dsesFxf
dxexfsF
xsi
xsi
2
2
)()(
)()(
dsesFxf
dxexfsF
ixs
ixs
)(2
1)(
)()(
dsesFxf
dxexfsF
ixs
ixs
)(2
1)(
)(2
1)(
f(x) = F(s)
Euler’s Formula
![Page 5: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/5.jpg)
Some Conventions
• Image Domain
• Forward Transform
• f(x,y,z)• g(x)• F
• Fourier Domain– Reciprocal space– Fourier Space– K-space– Frequency Space
• Reverse Transform, Inverse Transform
• F(kx,ky,kz)• G(s)• F
![Page 6: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/6.jpg)
Math Review - Periodic Functions
If there is some a, for a function f(x), such that
f(x) = f(x + na)
then function is periodic with the period a
0
a 2a 3a
![Page 7: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/7.jpg)
Math Review - Attributes of cosine wave
Amplitude
Phase
f(x) = cos (x)
f(x) = 5 cos (x)
f(x) = 5 cos (x + 3.14)
-5
-4
-3
-2
-1
0
1
2
3
4
5
-10 -5 0 5 10
-5
-3
-1
1
3
5
-10 -5 0 5 10
-5
-3
-1
1
3
5
-10 -5 0 5 10
![Page 8: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/8.jpg)
-5
-3
-1
1
3
5
-10 -5 0 5 10
-5
-4
-3
-2
-1
0
1
2
3
4
5
-10 -5 0 5 10
Math Review - Attributes of cosine wave
Amplitude
Phase
Frequency
f(x) = 5 cos (x)
f(x) = 5 cos (x + 3.14)
f(x) = 5 cos (3 x + 3.14)
-5
-3
-1
1
3
5
-10 -5 0 5 10
![Page 9: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/9.jpg)
Math Review - Attributes of cosine wave
f(x) = cos (x)
Amplitude, Frequency, Phase
-5
-3
-1
1
3
5
-10 -5 0 5 10
f(x) = A cos (kx + )
![Page 10: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/10.jpg)
Math Review - Complex numbers
• Real numbers:1-5.2
• Complex numbers4.2 + 3.7i9.4447 – 6.7i-5.2 (-5.2 + 0i)
1i
![Page 11: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/11.jpg)
Math Review - Complex numbers
• Complex numbers4.2 + 3.7i9.4447 – 6.7i-5.2 (-5.2 + 0i)
• General FormZ = a + biRe(Z) = a
Im(Z) = b
• AmplitudeA = | Z | = √(a2 + b2)
• Phase = Z = tan-1(b/a)
![Page 12: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/12.jpg)
Math Review – Complex Numbers
• Polar CoordinateZ = a + bi
• AmplitudeA = √(a2 + b2)
• Phase = tan-1(b/a)
a
b
A
![Page 13: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/13.jpg)
Math Review – Complex Numbers and Cosine Waves
• Cosine wave has three properties– Frequency– Amplitude– Phase
• Complex number has two properties– Amplitude– Wave
• Complex numbers to represent cosine waves at varying frequency– Frequency 1: Z1 = 5 +2i– Frequency 2: Z2 = -3 + 4i– Frequency 3: Z3 = 1.3 – 1.6i
![Page 14: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/14.jpg)
Fourier Analysis
Decompose f(x) into a series of cosine waves that when summed reconstruct f(x)
![Page 15: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/15.jpg)
Fourier Analysis in 1D. Audio signals
-5
-4
-3
-2
-1
0
1
2
3
4
5
0 200 400 600 800 1000 1200 1400
-5
-4
-3
-2
-1
0
1
2
3
4
5
0 200 400 600 800 1000 1200 1400
5 10 15(Hz)
5 10 15(Hz)
Amplitude OnlyAmplitude Only
![Page 16: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/16.jpg)
Fourier Analysis in 1D. Audio signals
-5
-4
-3
-2
-1
0
1
2
3
4
5
0 200 400 600 800 1000 1200 1400
5 10 15(Hz)
Your ear performs fourier analysis.
![Page 17: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/17.jpg)
Fourier Analysis in 1D. Spectrum Analyzer.
iTunes performs fourier analysis.
![Page 18: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/18.jpg)
Fourier Synthesis
Summing cosine waves reconstructs the original function
![Page 19: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/19.jpg)
Fourier Synthesis of Boxcar Function
Boxcar function
Periodic Boxcar
Can this function be reproduced with cosine waves?
![Page 20: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/20.jpg)
k=1. One cycle per period
A1·cos(2kx + 1)k=1
Ak·cos(2kx + k)k=1
1
![Page 21: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/21.jpg)
k=2. Two cycles per period
A2·cos(2kx + 2)k=2
Ak·cos(2kx + k)k=1
2
![Page 22: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/22.jpg)
k=3. Three cycles per period
A3·cos(2kx + 3)k=3
Ak·cos(2kx + k)k=1
3
![Page 23: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/23.jpg)
Ak·cos(2kx + k)N
Fourier Synthesis. N Cycles
A3·cos(2kx + 3)k=3
k=1
![Page 24: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/24.jpg)
Fourier Synthesis of a 2D Function
An image is two dimensional data.
Intensities as a function of x,y
White pixels represent the highest intensities.
Greyscale image of iris128x128 pixels
![Page 25: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/25.jpg)
Fourier Synthesis of a 2D Function
F(2,3)F(2,3)
![Page 26: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/26.jpg)
Fourier Filters
• Change the image by changing which frequencies of cosine waves go into the image
• Represented by 1D spectral profile
• 2D Profile is rotationally symmetrized 1D profile
![Page 27: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/27.jpg)
• Low frequency terms– Close to origin in Fourier Space– Changes with great spatial extent (like ice
gradient), or particle size
• High frequency terms– Closer to edge in Fourier Space– Necessary to represent edges or high-
resolution features
![Page 28: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/28.jpg)
Frequency-based Filters
• Low-pass Filter (blurs) – Restricts data to low-frequency components
• High-pass Filter (sharpens) – Restricts data to high-frequency-componenets
• Band-pass Filter– Restrict data to a band of frequencies
• Band-stop Filter– Suppress a certain band of frequencies
![Page 29: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/29.jpg)
Cutoff Low-pass Filter
Image is blurred
Sharp features are lost
Ringing artifacts
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
![Page 30: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/30.jpg)
Butterworth Low-pass Filter
Flat in the pass-band
Zero in the stop-band
No ringing
![Page 31: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/31.jpg)
Gaussian Low-pass Filter
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
![Page 32: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/32.jpg)
Butterworth High-pass Filter
• Note the loss of solid densities
![Page 33: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/33.jpg)
How the filter looks in 2D
unprocessed
lowpass
highpass
bandpass
![Page 34: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/34.jpg)
Filtering with EMAN2
LowPass Filtersfiltered=image.process(‘filter.lowpass.guass’, {‘sigma’:0.10})
filtered=image.process(‘filter.lowpass.butterworth’, {‘low_cutoff_frequency’:0.10, ‘high_cutoff_frequency’:0.35})
filtered=image.process(‘filter.lowpass.tanh’, {‘cutoff_frequency’:0.10, ‘falloff’:0.2})
HighPass Filtersfiltered=image.process(‘filter.highpass.guass’, {‘sigma’:0.10})
filtered=image.process(‘filter.highpass.butterworth’, {‘low_cutoff_frequency’:0.10, ‘high_cutoff_frequency’:0.35})
filtered=image.process(‘filter.highpass.tanh’, {‘cutoff_frequency’:0.10, ‘falloff’:0.2})
BandPass Filtersfiltered=image.process(‘filter.bandpass.guass’, {‘center’:0.2,‘sigma’:0.10})
filtered=image.process(‘filter.bandpass.butterworth’, {‘low_cutoff_frequency’:0.10, ‘high_cutoff_frequency’:0.35})
filtered=image.process(‘filter.bandpass.tanh’, {‘cutoff_frequency’:0.10, ‘falloff’:0.2})
![Page 35: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/35.jpg)
Convolution
Convolution of some function f(x) with some kernel g(x)
* =
Continuous
Discrete
![Page 36: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/36.jpg)
x x
Convolution in 2D
x
x x
=
x
x x
=x x
x xx xx x
![Page 37: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/37.jpg)
Microscope Point-Spread-Function is Convolution
![Page 38: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/38.jpg)
Convolution Theorem
f g = {FG}
f = FG
G
Convolution in image domainIs equivalent to multiplication in fourier domain
![Page 39: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/39.jpg)
Contrast Theory
observed imagef(x) for true particle
point-spread function
envelope functionnoise
obs(x) = f(x) psf(x) env(x) + n(x)
Incoherant average of transform
F2(s) CTF2(s) Env2(s) + N2(s)
Power spectrum
PS =
![Page 40: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/40.jpg)
Lowpass Filtering by Convolution
f g = {FG}
• Camera shake• Crystallographic B-factor
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
![Page 41: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/41.jpg)
Review
Fourier Transform is invertible operator
Math ReviewPeriodic functions
Amplitude, Phase and Frequency
Complex numberAmplitude and Phase
Fourier Analysis (Forward Transform)Decomposition of periodic signal into cosine waves
Fourier Synthesis (Inverse Transform)Summation of cosine waves into multi-frequency waveform
Fourier Transforms in 1D, 2D, 3D, ND
Image AnalysisImage (real-valued)Transform (complex-valued,
amplitude plot)
Fourier FiltersLow-pass High-pass Band-passBand-stop
Convolution TheoremDeconvolute by Division in
Fourier Space
All Fourier Filters can be expressed as real-space Convolution Kernels
Lens does Foureir transforms Diffraction Microscopy
![Page 42: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/42.jpg)
Further Reading
• Wikipedia
• Mathworld
• The Fourier Transform and its Applications. Ronald Bracewell
![Page 43: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/43.jpg)
Lens Performs Fourier Transform
![Page 44: Automation Mike Marsh National Center for Macromolecular Imaging Baylor College of Medicine Single-Particle Reconstructions and Visualization EMAN Tutorial.](https://reader036.fdocuments.net/reader036/viewer/2022070409/56649e995503460f94b9ca23/html5/thumbnails/44.jpg)
Gibbs Ringing
• 5 waves
• 25 waves
• 125 waves