Transformations

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Transformations
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Fourier transformation
forward inverse
f(t) cos(2?5t) cos(2? 10t) cos(2?
20t) cos(2? 50t)
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Spectrum, phase
In general F(u) is a complex function   F(u)
R(u) j I(u) F(u) ej ?(u)  F(u)  
 ? ( R2(u) I2(u) ) the Fourier spectrum of
f(x) ? (u) tan-1 ( I(u) / R(u) ) the phase
angle of f(x)
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2 D Fourrier
F(u,v)    ? ? f(x,y) e-j2? (uxvy) dx
dyf(x,y)     ? ? F(u,v) ej2? (uxvy)du dv
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Convolution

• c(x) f(x) ? g(x) ? f(?)g(x-?) d ?
• C(u) F(u)G(u)
• Point spread function of a lens
• Light on ideal point (?,?) spread over pixels
  (x,y) according h(x,?,y,? )
• p(x,y) ? ? w (?,?) h(x,?,y,? ) d ? d ?
• Linear h(x,?,y,? ) h(x-?,y-? )
• p(x,y) ? ? w (?,?) h(x-?,y-? ) d ? d ?
  w ? h
• P(u,v) W(u,v)H(u,v)

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Discrete Fourier Transformation
In 2-D the DFT becomes Fu,v 1/MN x0?M-1 
y0 ? N-1 fx,y e -j2? (xu / M yv /
N)fx,y    u0?M-1  v0?N-1  Fu,v
e j2? (xu / M yv / N)
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Fast Fourier Transformation

• To calculate Fu for u0,1...N-1 it takes NN
  multiplications and N(N-1) summations of complex
  numbers (e... in a table).
• The complexity of a DFT is therefore proportional
  to N2.
• Transform 1 DFT of N terms into 2 DFTs of N/2
  terms.
• We can apply this recursively and reach a
  complexity of N log2N.
• special purpose hardware chips wirh parallel
  processing

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Use in CT
g ? (x')   ? f(x',y') dy'    x' x cos ?   y
sin ?      y' -x sin ?  y cos ?
FT( g ? (x')) F(u cos ?, u sin ?).
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Other transformations

• DFT example of whole class of transformations
• T(u) x0 ? N-1 f(x) g(x,u)    with g the
  forward transformation kernelf(x)  u0 ? N-1
  T(u) h(x,u)   with h the inverse transformation
  kernel
• Discrete Cosine cos( (2x1)u? / 2N) , JPEG, MPEG
• T(u,v) x0 ? N-1 y0 ? N-1  f(x,y)
  g(x,y,u,v)f(x,y)  u0 ? N-1 v0 ? N-1 T(u,v)
  h(x,y,u,v)
• g(x,y,u,v)   g1(x,u) g2(y,v) separable 2D
  N 1D

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Continuous wavelets
Mexican-hat ?(x) c (1-x2) exp(-x2/2) the second
derivative of a Gaussian
Construction of the Morlet wavelet as a sinus
modeled by a Gaussian function
set of wavelet basis functions ?s,t(x) ?s,t(x)
?( (x-t) / s) / ?s, s gt 0 the scale and  t the
translation The CWT of f(x) is then Wf(s,t)
ltf, ?s,tgt ? f(x) ?s,t(x) dx f(x) (1 / C? ) 
? ?  Wf(s,t) ?s,t(x) dt ds/s2
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Continuous wavelet transform
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Time frequency tilings
In the discrete wavelet transform one works with
factors 2 Also here there is a Fast Wavelet
Transformation
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Example 3 scale 2D FWT
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Example