1. Spatial Frequencies Spatial Frequencies

2. Why are Spatial Frequencies important? Efficient data representation Provides a means for modeling and removing noise Physical processes are often best described in “frequency domain” Provides a powerful means of image analysis

3. What is spatial frequency? Instead of describing a function (i.e., a shape) by a series of positions It is described by a series of cosines

4. What is spatial frequency? A g(x) = A cos(x) 22 x g(x)

5. What is spatial frequency? Period (L) Wavelength ( ) Frequency f=(1/ ) Amplitude (A) Magnitude (A) A cos(x  2  /L) g(x) = A cos(x  2  / ) A cos(x  2  f) x g(x)

6. What is spatial frequency? A g(x) = A cos(x  2  f) x g(x) (1/f) period

7. But what if cosine is shifted in phase? g(x) = A cos(x  2  f +  ) x g(x) 

8. What is spatial frequency? g(x) = A cos(x  2  f +  ) A=2 m f = 0.5 m -1 = 0.25  = 45  g(x) = 2 cos(x  2  (0.5) + 0.25  ) 2 cos(x   + 0.25  ) x g(x) 0.002 cos(0.25  ) = 0.707106... 0.252 cos(0.50  ) = 0.0 0.502 cos(0.75  ) = -0.707106... 0.752 cos(1.00  ) = -1.0 1.002 cos(1.25  ) = -0.707106… 1.252 cos(1.50  ) = 0 1.502 cos(1.75  ) = 0.707106... 1.752 cos(2.00  ) = 1.0 2.002 cos(2.25  ) = 0.707106... Let us take arbitrary g(x) We substitute values of A, f and  We calculate discrete values of g(x) for various values of x

9. What is spatial frequency? g(x) = A cos(x  2  f +  ) x g(x) We calculate discrete values of g(x) for various values of x

10. What is spatial frequency? g(x) = A cos(x  2  f +  ) g i (x) = A i cos(x  2  i/N +  i ), i = 0,1,2,3,…,N/2-1

11. We try to approximate a periodic function with standard trivial (orthogonal, base) functions + + = Low frequency Medium frequency High frequency

12. We add values from component functions point by point + + =

13. g(x)g(x) i=1 i=2 i=3 i=4 i=5 i=63 0127 x Example of periodic function created by summing standard trivial functions

14. g(x)g(x) i=1 i=2 i=3 i=4 i=5 i=10 0127 x Example of periodic function created by summing standard trivial functions

15. g(x)g(x) g(x)g(x) 64 terms 10 terms Example of periodic function created by summing standard trivial functions

16. g(x)g(x) i=1 i=2 i=3 i=4 i=5 i=63 0127 x Fourier Decomposition of a step function (64 terms) Example of periodic function created by summing standard trivial functions

17. g(x)g(x) i=1 i=2 i=3 i=4 i=5 i=10 063 x Fourier Decomposition of a step function (11 terms) Example of periodic function created by summing standard trivial functions

18. Main concept – summation of base functions Any function of x (any shape) that can be represented by g(x) can also be represented by the summation of cosine functions Observe two numbers for every i

19. Information is not lost when we change the domain g i (x) = 1.3, 2.1, 1.4, 5.7, …., i=0,1,2…N-1 N pieces of information N/2 amplitudes (A i, i=0,1,…,N/2-1) and N/2 phases (  i, i=0,1,…,N/2-1) and Spatial Spatial Domain Frequency Domain

20. What is spatial frequency? g i (x) Are equivalent They contain the same amount of information and The sequence of amplitudes squared is the SPECTRUM Information is not lost when we change the domain

21. EXAMPLE

22. A cos(x  2  i/N) frequency (f) = i/N wavelength (p) = N/I N=512 i f p 0 0 infinite 1 1/512 512 16 1/32 32 256 1/2 2 Substitute values Assuming N we get this table which relates frequency and wavelength of component functions

23. More examples to give you some intuition ….

24. Fourier Transform Notation g(x) denotes an spatial domain function of real numbers –(1.2, 0.0), (2.1, 0.0), (3.1,0.0), … G() denotes the Fourier transform G() is a symmetric complex function (-3.1,0.0), (4.1, -2.1), (-3.1, 2.1), …(1.2,0.0) …, (-3.1,-2.1), (4.1, 2.1), (-3.1,0.0) G[g(x)] = G(f) is the Fourier transform of g(x) G -1 () denotes the inverse Fourier transform G -1 (G(f)) = g(x)

25. Power Spectrum and Phase Spectrum |G(f)| 2 = G(f)  G(f)* is the power spectrum of G(f) –(-3.1,0.0), (4.1, -2.1), (-3.1, 2.1), … (1.2,0.0),…, (-3.1,-2.1), (4.1, 2.1) –9.61, 21.22, 14.02, …, 1.44,…, 14.02, 21.22 tan -1 [Im(G(f))/Re(G(f))] is the phase spectrum of G(f) –0.0, -27.12, 145.89, …, 0.0, -145.89, 27.12 complex Complex conjugate

26. 1-D DFT and IDFT 1-D DFT and IDFT Discrete Domains –Discrete Time: k = 0, 1, 2, 3, …………, N-1 –Discrete Frequency:n = 0, 1, 2, 3, …………, N-1 Discrete Fourier Transform Inverse DFT Equal time intervals Equal frequency intervals n = 0, 1, 2,….., N-1 k = 0, 1, 2,….., N-1