1. Transformations Theo Schouten

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

3. 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) Theo Schouten

4. 2 D Fourrier F(u,v) =     f(x,y) e-j2(ux+vy) dx dyf(x,y) =      F(u,v) e+j2(ux+vy)du dv Theo Schouten

5. 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) Theo Schouten

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

7. 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 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 Theo Schouten

8. 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 ). Theo Schouten

9. 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 kernelf(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-1y=0  N-1  f(x,y) g(x,y,u,v)f(x,y)  = u=0  N-1v=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 Theo Schouten

10. 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 > 0 the scale and  t the translation • The CWT of f(x) is then: • Wf(s,t) = <f, s,t> =  f(x) s,t(x) dx • f(x) = (1 / C )   Wf(s,t) s,t(x) dt ds/s2 Theo Schouten

11. Continuous wavelet transform Theo Schouten

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

13. Example 3 scale 2D FWT Theo Schouten

14. Example Theo Schouten