1. Chapter 7: The Fourier Transform 7.1 Introduction
The Fourier transform allows us to perform tasks that would be impossible to perform any other way
It is more efficient to use the Fourier transform than a spatial filter for a large filter
The Fourier transform also allows us to isolate and process particular image frequencies

2. 7.2 Background

3. FIGURE 7.2
A periodic function may be written as the sum of sines and cosines of varying amplitudes and frequencies

4. 7.2 Background Fourier series
These are the equations for the Fourier series expansion of f (x), and they can be expressed in complex form
Fourier series

5. 7.2 Background
If the function is nonperiodic, we can obtain similar results by letting T → ∞, in which case
Fourier transform pair

6. 7.3 The One-Dimensional Discrete Fourier Transform

7. 7.3 The One-Dimensional Discrete Fourier Transform
Definition of the One-Dimensional DFT
This definition can be expressed as a matrix multiplication
where F is an N × N matrix defined by

8. 7.3 The One-Dimensional Discrete Fourier Transform
Given N, we shall define
e.g. suppose f = [1, 2, 3, 4] so that N = 4. Then

9. 7.3 The One-Dimensional Discrete Fourier Transform

10. 7.3 The One-Dimensional Discrete Fourier Transform
THE INVERSE DFT
If you compare Equation (7.3) with Equation 7.2 you will see that there are really only two differences:
There is no scaling factor 1/N
The sign inside the exponential function has been changed to positive.
The index of the sum is u, instead of x

11. 7.3 The One-Dimensional Discrete Fourier Transform

12. 7.3 The One-Dimensional Discrete Fourier Transform

13. 7.4 Properties of the One-Dimensional DFT
LINEARITY
This is a direct consequence of the definition of the DFT as a matrix product
Suppose f and g are two vectors of equal length, and p and q are scalars, with h = pf + qg
If F, G, and H are the DFT’s of f, g, and h, respectively, we have
SHIFTING
Suppose we multiply each element xn of a vector x by (−1)n. In other words, we change the sign of every second element
Let the resulting vector be denoted x’. The DFT X’ of x’ is equal to the DFT X of x with the swapping of the left and right halves

14. 7.4 Properties of the One-Dimensional DFT
e.g.

15. 7.4 Properties of the One-Dimensional DFT
Notice that the first four elements of X are the last four elements of X1 and vice versa

16. 7.4 Properties of the One-Dimensional DFT
SCALING
where k is a scalar and F= f
If you make the function wider in the x-direction, it's spectrum will become smaller in the x-direction, and vice versa
Amplitude will also be changed F
CONJUGATE SYMMETRY
CONVOLUTION

17. 7.4 Properties of the One-Dimensional DFT
THE FAST FOURIER TRANSFORM 2n

18. 7.5 The Two-Dimensional DFT
The 2-D Fourier transform rewrites the original matrix in terms of sums of corrugations

19. 7.5.1 Some Properties of the Two-Dimensional Fourier Transform
SIMILARITY
THE DFT AS A SPATIAL FILTER
SEPARABILITY

20. 7.5.1 Some Properties of the Two-Dimensional Fourier Transform
LINEARITY
THE CONVOLUTION THEOREM
Suppose we wish to convolve an image M with a spatial filter S
Pad S with zeroes so that it is the same size as M; denote this padded result by S’
Form the DFTs of both M and S’ to obtain (M)and (S’)
Form the element-by-element product of these two transforms:
Take the inverse transform of the result:
Put simply, the convolution theorem states or

21. 7.5.1 Some Properties of the Two-Dimensional Fourier Transform
THE DC COEFFICIENT
SHIFTING
DC coefficient
DC coefficient

22. 7.5.1 Some Properties of the Two-Dimensional Fourier Transform
CONJUGATE SYMMETRY
DISPLAYING YRANSFORMS
fft, which takes the DFT of a
vector,
ifft, which takes the inverse DFT of a vector,
fft2, which takes the DFT of a matrix,
ifft2, which takes the inverse DFT of a matrix, and
fftshift, which shifts a transform

23. 7.6 Fourier Transforms in MATLAB
e.g.
Note that the DC coefficient is indeed the sum of all the matrix values

24. 7.6 Fourier Transforms in MATLAB
e.g.

25. 7.6 Fourier Transforms in MATLAB
e.g.

26. 7.7 Fourier Transforms of Images

27. FIGURE 7.10

28. FIGURE 7.11

29. FIGURE 7.12

30. FIGURE 7.13
EXAMPLE 7.7.2

31. FIGURE 7.14
EXAMPLE 7.7.3

32. FIGURE 7.15
EXAMPLE 7.7.4

33. 7.7 Fourier Transforms of Images

34. 7.8 Filtering in the Frequency Domain
Ideal Filtering
LOW-PASS FILTERING

35. FIGURE 7.16

36. FIGURE 7.17
D = 15

37. 7.8 Filtering in the Frequency Domain
>> cfl = cf.*b
>> cfli = ifft2(cfl);
>> figure, fftshow(cfli, ’abs’)

38. FIGURE 7.18
D = 5
D = 30

39. 7.8 Filtering in the Frequency Domain
HIGH-PASS FILTERING

40. FIGURE 7.19

41. FIGURE 7.20

42. 7.8.2 Butterworth Filtering
Ideal filtering simply cuts off the Fourier transform at some distance from the center
It has the disadvantage of introducing unwanted artifacts (ringing) into the result
One way of avoiding these artifacts is to use as a filter matrix, a circle with a cutoff that is less sharp

43. FIGURE 7.21

44. FIGURE 7.22 & 7.23

45. FIGURE 7.24

46. FIGURE 7.25

47. FIGURE 7.26 >> bl = lbutter(c,15,1); >> cfbl = cf.*bl;
>> figure, fftshow(cfbl, ’log’);
>> cfbli = ifft2(cfbl);
>> figure, fftshow(cfbli, ’abs’)

48. FIGURE 7.27

49. 7.8.3 Gaussian Filtering
A wider function, with a large standard deviation, will have a low maximum

50. FIGURE 7.28

51. FIGURE 7.29

52. 7.9 Homomorphic Filtering
where f(x, y) is intensity, i(x, y) is the illumination and r(x, y) is the reflectance

53. 7.9 Homomorphic Filtering

54. FIGURE 7.32 function res=homfilt(im,cutoff,order,lowgain,highgain)
% HOMFILT(IMAGE,FILTER) applies homomorphic filtering
% to the image IMAGE
% with the given parameters
u=im2uint8(im/256);
u(find(u==0))=1;
l=log(double(u));
ft=fftshift(fft2(l));
f=hb_butter(im,cutoff,order,lowgain,highgain);
b=f.*ft;
ib=abs(ifft2(b));
res=exp(ib);

55. FIGURE 7.33 >>i=imread(‘newborn.tif’);
>>r=[1:256]’*ones(1,256);
>>x=double(i).*( *sin((r-32)/16));
>>imshow(i);figure;imshow(x/256);

56. FIGURE 7.34
>>xh=homfilt(x,10,2,0.5,2);
>>imshow(xh/16);

57. FIGURE 7.35 >> a=imread('arch.tif'); >> figure;imshow(a);
>> a1=a(:,:,1);
>> figure;imshow(a1);
>> a2=double(a1);
>> ah=homfilt(a2,128,2,0.5,2);
>> figure;imshow(ah/14);