# Automation by LFnO319

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```									Automation
Fourier Transforms,
Filtering and Convolution

Mike Marsh
National Center for Macromolecular Imaging
Baylor College of Medicine

Single-Particle Reconstructions and Visualization
EMAN Tutorial and Workshop
March 14, 2007
Fourier Transform is an invertible operator

Image                            Fourier Transform

FT

v2 will display image or its transform
Fourier Transform is an invertible operator

Image                                Fourier Transform
Ny ⁄ 2
Ny

y                                              F(kx,ky)
f(x,y)                                           Nx ⁄ 2

0                   Nx
x

{F(kx,ky)} = f(x,y)}            {f(x,y)}     = F(kx,ky)
Continuous Fourier Transform


F (s)             f ( x)e i 2xs dx


f(x) = F(s)              f ( x)   F ( s )e i 2xs ds



F ( s )   f ( x)e ixs dx


1            
f ( x) 
2        

F ( s )eixs ds

1           
F ( s) 
2      

f ( x)e ixs dx

Euler’s Formula                        1           
f ( x) 
2      

F ( s )e ixs ds
Some Conventions

• Image Domain        • Fourier Domain
–   Reciprocal space
–   Fourier Space
–   K-space
–   Frequency Space

• Reverse Transform,
• Forward Transform     Inverse Transform

• f(x,y,z)            • F(kx,ky,kz)
• g(x)                • G(s)
• F                   • F
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

a           2a       3a
0
Math Review - Attributes of cosine wave
5

3

f(x) = cos (x)   -10   -5
1

-1 0    5   10
-3

-5

5
4
3

Amplitude           f(x) = 5 cos (x)                2
1
0
-10   -5    -1 0   5   10
-2
-3
-4
-5

5

Phase       f(x) = 5 cos (x + 3.14)                3

1

-10   -5   -1 0    5   10
-3

-5
Math Review - Attributes of cosine wave
5
4
3
2

Amplitude            f(x) = 5 cos (x)   -10   -5
1
0
-1 0   5   10
-2
-3
-4
-5

5

3
Phase        f(x) = 5 cos (x + 3.14)               1

-10   -5   -1 0   5   10
-3

-5

5

Frequency f(x) = 5 cos (3 x + 3.14)                3

1

-10   -5   -1 0   5   10
-3

-5
Math Review - Attributes of cosine wave
5

3

f(x) = cos (x)              1

-10   -5   -1 0   5   10
-3

-5

f(x) = A cos (kx + )

Amplitude, Frequency, Phase
Math Review - Complex numbers

• Real numbers:
1
-5.2


• Complex numbers
4.2 + 3.7i
9.4447 – 6.7i        i  1
-5.2 (-5.2 + 0i)
Math Review - Complex numbers

• Complex numbers        • Amplitude
4.2 + 3.7i               A = | Z | = √(a2 + b2)
9.4447 – 6.7i
-5.2 (-5.2 + 0i)       • Phase
 =  Z = tan-1(b/a)
• General Form
Z = a + bi
Re(Z) = a
Im(Z) = b
Math Review – Complex Numbers

• Polar Coordinate         
Z = a + bi
A           b

• Amplitude
A=   √(a2   +   b2 )
a           

• Phase
 = tan-1(b/a)
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
Fourier Analysis

Decompose f(x) into a series of cosine
waves that when summed reconstruct
f(x)
Fourier Analysis in 1D. Audio signals
5
4
3                                                    Amplitude Only
2
1
0
-1 0     200    400    600   800   1000    1200   1400
-2
-3                                                       5    10    15
-4
-5                                                           (Hz)

5
4
3
2
1
0
-1 0     200    400    600    800   1000   1200    1400
-2
-3
5    10    15
-4
-5
(Hz)
Fourier Analysis in 1D. Audio signals
5
4
3
2
1
0
-1 0    200   400   600   800   1000   1200   1400
-2
-3
5    10    15
-4
-5
(Hz)

Fourier Analysis in 1D. Spectrum Analyzer.

iTunes performs fourier analysis.
Fourier Synthesis

Summing cosine waves reconstructs the
original function
Fourier Synthesis of Boxcar Function

Boxcar function

Periodic Boxcar

Can this function be reproduced with cosine waves?
k=1. One cycle per period

A1·cos(2kx + 1)
k=1

 A ·cos(2kx +  )
1

k             k
k=1
k=2. Two cycles per period

A2·cos(2kx + 2)
k=2

 A ·cos(2kx +  )
2

k             k
k=1
k=3. Three cycles per period

A3·cos(2kx + 3)
k=3

 A ·cos(2kx +  )
3

k             k
k=1
Fourier Synthesis. N Cycles

A3·cos(2kx + 3)
k=3

 A ·cos(2kx +  )
N

k             k
k=1
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 iris
128x128 pixels
Fourier Synthesis of a 2D Function

F(2,3)
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
• Low frequency terms
– Close to origin in Fourier Space
– Changes with great spatial extent (like ice

• High frequency terms
– Closer to edge in Fourier Space
– Necessary to represent edges or high-
resolution features
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
Cutoff Low-pass Filter

Image is blurred
Sharp features are lost
Ringing artifacts

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0   0.2   0.4   0.6   0.8   1
Butterworth Low-pass Filter

Flat in the pass-band
Zero in the stop-band
No ringing
Gaussian Low-pass Filter

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0     0.2   0.4   0.6   0.8   1
Butterworth High-pass Filter

• Note the loss of solid
densities
How the filter looks in 2D

unprocessed
bandpass

lowpass

highpass
Filtering with EMAN2

LowPass Filters
filtered=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 Filters
filtered=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 Filters
filtered=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})
Convolution

Convolution of some function f(x) with
some kernel g(x)
Continuous

Discrete

*       =
Convolution in 2D

x                     x x

    =

x x
x                  x x x x
x xx x
    =      x x
Convolution Theorem

fg=        {FG}

f=          FG
G

Convolution in image domain
Is equivalent to multiplication in fourier domain
Contrast Theory

Power spectrum

PS = F2(s) CTF2(s) Env2(s) + N2(s)
Incoherant average of transform

obs(x) = f(x)  psf(x)  env(x) + n(x)
observed image                                            noise
f(x) for true particle        envelope function
Lowpass Filtering by Convolution

fg=
1

{FG}            0.9
0.8
0.7
0.6
0.5

• Camera shake                0.4
0.3

• Crystallographic B-factor   0.2
0.1
0
0   0.2   0.4   0.6   0.8   1
Review
Fourier Transform is invertible operator   Fourier Filters
Math Review
Low-pass
Periodic functions                         High-pass
Amplitude, Phase and                  Band-pass
Frequency
Band-stop
Complex number
Amplitude and Phase
Convolution Theorem
Fourier Analysis (Forward Transform)
Decomposition of periodic signal            Deconvolute by Division in
into cosine waves                             Fourier Space

Fourier Synthesis (Inverse Transform)
Summation of cosine waves into          All Fourier Filters can be
multi-frequency waveform                    expressed as real-space
Convolution Kernels
Fourier Transforms in 1D, 2D, 3D, ND

Image Analysis                             Lens does Foureir transforms
Image (real-valued)                     Diffraction Microscopy
Transform (complex-valued,
amplitude plot)

• Wikipedia
• Mathworld
• The Fourier Transform and its
Applications. Ronald Bracewell
Lens Performs Fourier Transform
Gibbs Ringing

• 5 waves

• 25 waves

• 125 waves

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