Automation

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)









Your ear performs fourier analysis.

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

gradient), or particle size





• 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

Microscope Point-Spread-Function is Convolution

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

point-spread 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)

Further Reading





• Wikipedia

• Mathworld

• The Fourier Transform and its

Applications. Ronald Bracewell

Lens Performs Fourier Transform

Gibbs Ringing









• 5 waves







• 25 waves







• 125 waves