2007Theo Schouten1 Transformations

2007Theo Schouten2 Fourier transformation forward inverse f(t) = cos(2*  *5*t) + cos(2*  *10*t) + cos(2*  *20*t) + cos(2*  *50*t)

2007Theo Schouten3 Spectrum, phase In general F(u) is a complex function: F(u) = R(u) + j I(u) = | F(u) | e j  (u) | F(u) | =  ( R 2 (u) + I 2 (u) ) : the Fourier spectrum of f(x)  (u) = tan -1 ( I(u) / R(u) ) : the phase angle of f(x)

2007Theo Schouten4 2 D Fourrier F(u,v) =   f(x,y) e -j2  (ux+vy) dx dy f(x,y) =   F(u,v) e +j2  (ux+vy) du dv

2007Theo Schouten5 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)

2007Theo Schouten6 Discrete Fourier Transformation In 2-D the DFT becomes: F[u,v] = 1/MN x=0  M-1 y=0  N-1 f[x,y] e -j2  (xu / M + yv / N) f[x,y] = u=0  M-1 v=0  N-1 F[u,v] e +j2  (xu / M + yv / N)

2007Theo Schouten7 Fast Fourier Transformation To calculate F[u] for u=0,1...N-1 it takes N*N multiplications and N*(N-1) summations of complex numbers (e... in a table). The complexity of a DFT is therefore proportional to N 2. Transform 1 DFT of N terms into 2 DFTs of N/2 terms. We can apply this recursively and reach a complexity of N log 2 N. special purpose hardware chips wirh parallel processing

2007Theo Schouten8 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  ).

2007Theo Schouten9 Other transformations DFT example of whole class of transformations T(u) = x=0  N-1 f(x) g(x,u) with g the forward transformation kernel f(x) = u=0  N-1 T(u) h(x,u) with h the inverse transformation kernel Discrete Cosine: cos( (2x+1)u  / 2N), JPEG, MPEG T(u,v) = x=0  N-1 y=0  N-1 f(x,y) g(x,y,u,v) f(x,y) = u=0  N-1 v=0  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

2007Theo Schouten10 Continuous wavelets Mexican-hat  (x)= c (1-x 2 ) exp(-x 2 /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 > 0 the scale and t the translation The CWT of f(x) is then: W f (s,t) = =  f(x)  s,t (x) dx f(x) = (1 / C  )   W f (s,t)  s,t (x) dt ds/s 2

2007Theo Schouten11 Continuous wavelet transform

2007Theo Schouten12 Time frequency tilings In the discrete wavelet transform one works with factors 2 Also here there is a Fast Wavelet Transformation

2007Theo Schouten13 Example 3 scale 2D FWT

2007Theo Schouten14 Example