1 Spatial Frequency or How I learned to love the Fourier Transform Jean Baptiste Joseph Fourier

2 Spatial Frequency 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 (1/f) 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)

7 What is spatial frequency? 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)  ) 2 cos(x    ) x g(x) cos(0.25  ) = cos(0.50  ) = cos(0.75  ) = cos(1.00  ) = cos(1.25  ) = … cos(1.50  ) = cos(1.75  ) = cos(2.00  ) = cos(2.25  ) =

9 What is spatial frequency? g(x) = A cos(x  2  f +  ) x g(x)

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

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

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

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

14 What is spatial frequency? g i (x) = A i cos(x  2  i/N +  i ), i = 0,1,2,3,…,N/2-1 i=N/2-1i=0 If N equals the number of pixel in a line, then... Lowest frequency Highest frequency

15 What is spatial frequency? g i (x) = A i cos(x  2  i/N +  i ), i = 0,1,2,3,…,N/2-1 i=N/2-1i=0 If N equals the number of pixel in a line, then... Lowest frequency Highest frequency

16 What is spatial frequency? g i (x) = A i cos(x  2  i/N +  i ), i = 0,1,2,3,…,N/2-1 i=N/2i=0 If N equals the number of pixel in a line, then... Lowest frequency Too high Redundant frequency

17 What is spatial frequency?

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20 What is spatial frequency? + + = Low frequency Medium frequency High frequency

21 What is spatial frequency? + + =

22 g(x)g(x) i=1 i=2 i=3 i=4 i=5 i= x

23 g(x)g(x) i=1 i=2 i=3 i=4 i=5 i= x

24 g(x)g(x) g(x)g(x) 64 terms 10 terms

25 g(x)g(x) i=1 i=2 i=3 i=4 i=5 i= x Fourier Decomposition of a step function (64 terms)

26 g(x)g(x) i=1 i=2 i=3 i=4 i=5 i= x Fourier Decomposition of a step function (11 terms)

27 What is spatial frequency? Any function of x (any shape) that can be represented by g(x) can also be represented by the summation of cosine functions

28 What is spatial frequency? + + = Low frequency Medium frequency High frequency

29 What is spatial frequency? 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 Domain Frequency Domain

30 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

31 Spatial Frequency or How I learned to love the Fourier Transform Jean Baptiste Joseph Fourier