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Spatial Frequencies

Spatial Frequencies . 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. What is spatial frequency?.

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Spatial Frequencies

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  1. 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? g(x) = A cos(x) g(x) 2 A x

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

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

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

  8. What is spatial frequency? Let us take arbitrary g(x) x g(x) 0.00 2 cos(0.25) = 0.707106... 0.25 2 cos(0.50) = 0.0 0.50 2 cos(0.75) = -0.707106... 0.75 2 cos(1.00) = -1.0 1.00 2 cos(1.25) = -0.707106… 1.25 2 cos(1.50) = 0 1.50 2 cos(1.75) = 0.707106... 1.75 2 cos(2.00) = 1.0 2.00 2 cos(2.25) = 0.707106... 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) We calculate discrete values of g(x) for various values of x We substitute values of A, f and 

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

  10. What is spatial frequency? g(x) = A cos(x  2f + ) gi(x) = Ai 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) i=1 i=2 i=3 i=4 i=5 i=63 x 0 127 Example of periodic function created by summing standard trivial functions

  14. g(x) i=1 i=2 i=3 i=4 i=5 i=10 x 0 127 Example of periodic function created by summing standard trivial functions

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

  16. Fourier Decomposition of a step function (64 terms) g(x) i=1 i=2 i=3 i=4 i=5 Example of periodic function created by summing standard trivial functions x i=63 0 127

  17. Fourier Decomposition of a step function (11 terms) g(x) i=1 i=2 i=3 Example of periodic function created by summing standard trivial functions i=4 i=5 i=10 x 0 63

  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 Spatial Domain gi(x) = 1.3, 2.1, 1.4, 5.7, …., i=0,1,2…N-1 N pieces of information Frequency Domain N pieces of information N/2 amplitudes (Ai, i=0,1,…,N/2-1) and N/2 phases (i, i=0,1,…,N/2-1) and

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

  21. EXAMPLE

  22. Substitute values 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 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 complex Complex conjugate • |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

  26. 1-D DFT and IDFT Equal time intervals • 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 frequency intervals n = 0, 1, 2,….., N-1 k = 0, 1, 2,….., N-1

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