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					                                     Reciprocal Space
                                     Fourier Transforms

   Outline
                 Introduction to reciprocal space
                 Fourier transformation
                 Some simple functions
                • Area and zero frequency components
                • 2- dimensions
                 Separable
                 Central slice theorem
                 Spatial frequencies
                 Filtering
                 Modulation Transfer Function




22.56 - lecture 3, Fourier imaging
                                     Reciprocal Space




                   real space                     reciprocal space




22.56 - lecture 3, Fourier imaging
                                                       Reciprocal Space
    In[1]:=   face   , 0 , 0 , 0 , .2 , .4 , .7 , .9 , 1.2 , 2. , 2.5 , 3.1 , 3.25 , 3.2 , 3.1 , 2.9 , 2.9 , 3 , 3.1 , 2.8 ,
                        0
                 2.9 , 2.8 , 2.7 , 2.6 , 2.8 , 2.9 , 3 , 2.9 , 2.7 , 2.3 , 2.1 , 1.7 , 1.4 , 1.2 , 1 , .8 , .8 , .7 ,
                 .7 , .65 , .6 , .5 , .4 , .3 , .2 , .1 , 0 , 0 , 0 , 0 , 0 ;

   In[33]:=   ListPlot     face , PlotJoine d      True ,   Axes   False   , PlotStyle        Thicknes s   0.01 




                                                                           20

                                                                           15

                                                                           10

                                                                            5


                                                                                      50         100      150      200       250
                                                                           -5

                                                                          -10

In[3]:=   periodic        Join                                     0 ;
                                 face , face , face , face , face ,  

In[5]:=   f  Fourier      periodic   ;

In[6]:=   ListPlot       RotateLeft        , 128 , PlotJoined
                                           Re f                           True ,   PlotRange       All ,
          PlotStyle          Thick nes s  0.01 



 22.56 - lecture 3, Fourier imaging
                                             Reciprocal Space

                                                                        20

                                                                        15

                                                                        10

                                                                         5


                                                                              50   100   150    200   250
                                                                        -5

                                                                       -10
                                                                                               real



                                                                       7.5

  ListPlot   RotateLe ft    Im , 128 , PlotJoine d
                                  f                         True ,     5
   PlotRange      All ,                                              2.5
  PlotStyle      Thicknes s  0.01 
                                                                              50   100   150    200   250
                                                                       -2.5

                                                                        -5

                                                                       -7.5              imaginary
22.56 - lecture 3, Fourier imaging
                                               Reconstruction
 filter     :  Table    n, 1 , If   256  n, 1 , 0 ,  , 1 , 256 
           n_            If i            i                      i           ;
  expand   :  ListPlot
             n_                      Re Fourier  filter
                                Take               f         , 54 ,
                                                               n
        PlotJoined    True , PlotRange      All , Axes   False , PlotStyle     Thicknes s   0.01 ,
    Dis playFunction       Identity  



                                                     8 Fourier components

                                                     16


                                                     32


                                                     64


                                                     128
22.56 - lecture 3, Fourier imaging
                                          Fourier Transforms

   For a complete story see:
                  Brigham “Fast Fourier Transform”
   Here we want to cover the practical aspects of Fourier Transforms.

   Define the Fourier Transform as:
                                                     
                                 G(k)  g(x)      g(x)eikx dx
                                                    

   There are slight variations on this definition (factors of π and the
   sign in the exponent), we will revisit these latter, i=√-1.

   Also recall that
                                     eikx  cos(kx)  isin(kx)

22.56 - lecture 3, Fourier imaging
                                          Reciprocal variables

   k is a wave-number and has units that are reciprocal to x:
           x -> cm
           k -> 2π/cm
   So while x describes a position in space, k describes a spatial
   modulation.
   Reciprocal variables are also called conjugate variables.
                                                  1
                                               0.75
                                                0.5
                                               0.25

           -4                        -2       -0.25              2                 4
                                               -0.5
                                              -0.75
                                                 -1
                                                  1
                                               0.75
                                                0.5
                                               0.25

           -4                        -2       -0.25              2                 4
                                               -0.5
                                              -0.75
                                                 -1

                                                                              2
                                                                     , k 
                                                                              
   Another pair of conjugate variables are time and angular frequency.
22.56 - lecture 3, Fourier imaging
                        Conditions for the Fourier Transform to Exist

   The sufficient condition for the Fourier transform to exist is that the
   function g(x) is square integrable,

                                     
                                              2
                                         g(x) dx  
                                     

   g(x) may be singular or discontinuous and still have a well defined
   Fourier transform.




22.56 - lecture 3, Fourier imaging
                                     The Fourier transform is complex

   The Fourier transform G(k) and the original function g(x) are both in
   general complex.
                                       g(x)  Gr (k) iGi (k)
   The Fourier transform can be written as,

                g(x)  G(k)  A(k)ei(k )
                         2
                A  G  Gr  Gi2
                A  amplitude spectrum, or magnitude spectrum
                  phase spectrum
                            2
                A2  G 2  Gr  Gi2  power spectrum


22.56 - lecture 3, Fourier imaging
                             The Fourier transform when g(x) is real

   The Fourier transform G(k) has a particularly simple form when g(x)
   is purely real               
                       Gr (k)   g(x)cos(kx)dx
                                                
                                                
                                     Gi (k)     g(x)sin(kx)dx
                                                
   So the real part of the Fourier transform reports on the even part of
   g(x) and the imaginary part on the odd part of g(x).




22.56 - lecture 3, Fourier imaging
                           The Fourier transform of a delta function

   The Fourier transform of a delta function should help to convince
   you that the Fourier transform is quite general (since we can build
   functions from delta functions).
                                                  
                                      x      x eikx dx
                                                  
   The delta function picks out the zero frequency value,

                                        x  eik 0  1
                                       x  1




                                          x                          k
22.56 - lecture 3, Fourier imaging
                               The Fourier transform of a delta function

   So it take all spatial frequencies to create a delta function.
 In[1]:=   delta    , x_   Sum
                    n_            Cos  x ,   n , n, 1 
                                       k      k,           ;

 In[7]:=   Plot  delta  , x ,  ,  3 , 3  PlotRange
                          1         x         ,              All ,
           PlotStyle      Thicknes s 0.01 



                                                                            60



                                                                            40
                            QuickTime™ an d a
                         Anima tion d ecompressor
                      are need ed to see this picture.                      20




                                                                 -10   -5        5   10




22.56 - lecture 3, Fourier imaging
                                          The Fourier transform

   The fact that the Fourier transform of a delta function exists shows
   that the FT is complete.

   The basis set of functions (sin and cos) are also orthogonal.

                                     
                                      cos(k1x)cos(k2 x)dx   (k1  k2 )
                                     
   So think of the Fourier transform as picking out the unique spectrum
   of coefficients (weights) of the sines and cosines.




22.56 - lecture 3, Fourier imaging
                       The Fourier transform of the TopHat Function

   Define the TopHat function as,
                                                 x  1
                                                 1;
                                         g(x)  
                                                 x  1
                                                 0;

                                                   
   The Fourier transform is,              G(k)     g(x)eikx dx
                                                   
   which reduces to,
                                     1
                                                     sin(k)
                            G(k)  2  cos(kx)dx  2         2sinc(k)
                                     0                  k




22.56 - lecture 3, Fourier imaging
                       The Fourier transform of the TopHat Function

   For the TopHat function                         x  1
                                                   1;
                                           g(x)  
                                                   x  1
                                                   0;

                                                        1
   The Fourier transform is, G(k)  2  cos(kx)dx  2 sin(k)  2sinc(k)
                                      0                  k
                                                   2


                                                 1.5


                                                   1


                                                 0.5



                                     -20   -10              10   20

                                                 -0.5




22.56 - lecture 3, Fourier imaging
                   The Fourier reconstruction of the TopHat Function

                                 2


                               1.5


                                 1


                               0.5



      -20        -10                         10         20

                               -0.5




                    In[33]:=   square  , x_  : 
                                       n_
                                2  Sum Sin  k   Cos  x ,  , 1 , n 
                                         2     k           k      k         ;

                    In[36]:=   Table  Plot square  , x ,  ,  2 , 2 
                                                       n          x         ,
                                 PlotRange      All , PlotStyle     Thick nes s   0.01 ,
                                 , 0 , 128 
                                 n           




22.56 - lecture 3, Fourier imaging
                          The Fourier transform of a cosine Function

   Define the cosine function as,

                                      g(x)  cos(k0 x)

   where k0 is the wave-number of the original function.
   The Fourier transform is,         
                             G(k)   cos(k0 x)eikx dx
   which reduces to,                
                            
               G(k)         cos(k0 x)cos(kx)dx   (k  k0 )  (k  k0 )
                           
   cosine is real and even, and so the Fourier transform is also real and
   even. Two delta functions since we can not tell the sign of the
   spatial frequency.

22.56 - lecture 3, Fourier imaging
                            The Fourier transform of a sine Function

   Define the sine function as,

                                      g(x)  sin(k0 x)

   where k0 is the wave-number of the original function.
   The Fourier transform is,         
                             G(k)   sin(k0 x)eikx dx
   which reduces to,                
                             
              G(k)  i  sin(k0 x)sin(kx)dx  i (k  k0 )  (k  k0 )
                            
   sine is real and odd, and so the Fourier transform is imaginary and
   odd. Two delta functions since we can not tell the sign of the spatial
   frequency.

22.56 - lecture 3, Fourier imaging
                                     Telling the sense of rotation

      Looking at a cosine or sine alone one can not tell the sense of
      rotation (only the frequency) but if you have both then the sign
      is measurable.




22.56 - lecture 3, Fourier imaging
                                     Symmetry

Even/odd
      if g(x) = g(-x), then G(k) = G(-k)
      if g(x) = -g(-x), then G(k) = -G(-k)
Conjugate symmetry
      if g(x) is purely real and even, then G(k) is purely real.
      if g(x) is purely real and odd, then G(k) is purely imaginary.
      if g(x) is purely imaginary and even, then G(k) is purely imaginary.
      if g(x) is purely imaginary and odd, then G(k) is purely real.




22.56 - lecture 3, Fourier imaging
                          The Fourier transform of the sign function

   The sign function is important in filtering applications, it is defined
   as,                             1; x  0
                          sgn(x)  
                                    1;
                                    x  0

   The FT is calculated by expanding about the origin,

                                                  2
                                     sgn x   i
                                                   k




22.56 - lecture 3, Fourier imaging
                     The Fourier transform of the Heaviside function

   The Heaviside (or step) function can be explored using the result of
   the sign function            1
                        (x)  1 sgn( x)
                                2
   The FT is then,
                           
                            1                
                                                1       1        
               x    1 sgn( x)    sgn( x)
                           
                            2                
                                                2       2        


                                                         i
                                     x    (k) 
                                                         k



22.56 - lecture 3, Fourier imaging
                                         The shift theorem

   Consider the conjugate pair,
                                                    gx   G(k)

   what is the FT of                 g x  a 

                                               
                             g x  a      g(x  a)eikx dx
                                               
   rewrite as,                  
                                g(x  a)eik (xa)eika d(x  a)
                                
   The new term is not a function of x,

                      gx  a  eikaG(k)
   so you pick up a frequency dependent phase shift.

22.56 - lecture 3, Fourier imaging
                                                                The shift theorem
 In[18]:=   d :  Table   Table Cos  x  n 2 Pi 16 ,  , 0 , 15 
                                       2                    n          ,
                  ,  10 , 10 , 20 63 
                  x                      ;

 In[35]:=   f  Table RotateLe ft   Fourier       
                                                d  n  , 32 ,  , 1 , 16 
                                                                    n          ;

 In[19]:=   ListPlot3D   , 
                         d   PlotRange           
                                               1 , 1  Mesh   False
                                                         ,                 


                                                                            2
                                                                                                                  15
                                                                            0

                                                                            -2
                                                                                                             10


                                                                                    20
                                                                                                         5
    1                                                                                    40

   0.5                                                                                              60
      0                                               15
   -0.5
     -1                                          10


                 20
                                            5
                          40

                                     60
                                                           2

                                                                                                   15
                                                           0

                                                           -2
                                                                                              10


                                                                   20
                                                                                         5
                                                                           40

22.56 - lecture 3, Fourier imaging                                                  60
                                          The similarity theorem

   Consider the conjugate pair, gx   G(k)


   what is the FT of                      g ax 

                                                                       k
                                                           1           i ax
                                                ikx
            g ax                g(ax)e          dx      g(ax)e a d(ax)
                                                         a 

                                                    1 k
                                        g ax   G( )
                                                    a a
   so the Fourier transform scales inversely with the scaling of g(x).



22.56 - lecture 3, Fourier imaging
                                              The similarity theorem
  In[3]:=   square  Table 
               Table    64  n , 0 , If   64  n , 0 , 1 ,  , 1 , 128 
                       If x                 x                      x           ,
                , 1 , 16 
                n          ;

 In[13]:=   f  Table 
               RotateLe ft   Fourier   RotateLe ft    square    
                                                                   n  , 64 , 64 ,
                , 1 , 16 
                n          ;




         1                                                          3
       0.8                                                          2
       0.6                                                  15                                           15
        0.4                                                         1
        0.2                                                             0
           0                                           10                                           10


                      50                         5                              50              5

                                  100                                                     100




22.56 - lecture 3, Fourier imaging
                                     The similarity theorem
                                                                              8
                                                                                         1
                                                                              6
                                                                                         2a
                                                                              4


                                                                              2


                                                       -3      -2        -1          1        2        3


                        -a           a
                                                                              -2

                                                                                                  1
                                                                                                  a

      1
    0.8
    0.6                                           15   3
     0.4
     0.2                                               2
        0                                    10                                                        15
                                                       1
                                                           0
                 50                                                                               10
                                         5

                             100
                                                                    50                    5

                                                                               100
22.56 - lecture 3, Fourier imaging
                                          Rayleigh’s theorem

   Also called the energy theorem,

                                                    
                                             2               2
                                        g(x) dx       G(k) d(k)
                                                   
   The amount of energy (the weight) of the spectrum is not changed
   by looking at it in reciprocal space.

   In other words, you can make the same measurement in either real or
   reciprocal space.




22.56 - lecture 3, Fourier imaging
                                     The zero frequency point

   Also weight of the zero frequency point corresponds to the total
   integrated area of the function g(x)
                                                                 
                    g(x) k0        g(x)eikx dx             g(x)dx
                                                      k0       




22.56 - lecture 3, Fourier imaging
                                     The Inverse Fourier Transform

   Given a function in reciprocal space G(k) we can return to direct
   space by the inverse FT,

                                              1 
                          g(x)  1G(k)       G(k)eikx dk
                                             2 
                                                      
   To show this, recall that                 G(k)     g(x)eikx dx
                                                      

                   1                  1        
                       G(k)eikx 'dk   dx g(x)  eik (x'x )dk
                  2                2      
                                                                    2 ( x' x)
                                                           g( x')


22.56 - lecture 3, Fourier imaging
                              The Fourier transform in 2 dimensions

   The Fourier transform can act in any number of dimensions,

                                         
                                               ik y y ik x x
                   g(x, y)x,y    g(x, y)e       e        dxdy
                                   

   It is separable
                              g(x, y)x,y  g(x, y)x g(x, y)y


   and the order does not matter.




22.56 - lecture 3, Fourier imaging
                                         Central Slice Theorem

   The equivalence of the zero-frequency rule in 2D is the central slice
   theorem.
                                                                ik y y ik x x
               g(x, y)x,y                          g(x, y)e        e        dxdy
                                     k x 0        
   or                                                                                     k x 0

                                                       ik y       
                   g(x, y)x,y                     e   y
                                                                dy  g(x, y)dx
                                        k x 0                               k x 0

   So a slice of the 2-D FT that passes through the origin corresponds
   to the 1 D FT of the projection in real space.




22.56 - lecture 3, Fourier imaging
                                     Filtering

   We can change the information content in the image by manipulating
   the information in reciprocal space.




   Weighting function in k-space.



22.56 - lecture 3, Fourier imaging
                                     Filtering

   We can also emphasis the high frequency components.




   Weighting function in k-space.




22.56 - lecture 3, Fourier imaging
                                     Modulation transfer function

                           i(x, y)  o(x, y)  PSF(x, y)  noise


                        I(k x , k y )  O(k x , k y ) MTF(kx , k y ) noise




22.56 - lecture 3, Fourier imaging

				
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