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)




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

								
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