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1 Image Transforms Fourier Transform Basic idea 1

2 Image Transforms Fourier transform theory Let f(x) be a continuous function of a real variable x. The Fourier transform of f(x) is F ( u) f ( x)exp[ j2πux] = dx Given F(u), f(x) can be obtained by using the inverse Fourier transform f ( x) F( u)exp[ j2πux] = du 2

3 Image Transforms Fourier transform theory The Fourier transform F(u) is general complex F ( u) = R( u) + ji( u) It is often convenient to write it in the form ( 2 2 R ( u) + I ( u) ) j ( u) F( u) = 2 exp 1 [ ] ( ) iφ ( u ) φ = F u e 3

4 Image Transforms Fourier transform theory Magnitude and Phase ( u) = ( R ( u) I ( u) ) 2 F + φ 2 2 ( u) = ( R ( u) I ( u) ) P + ( u) = tan 1 I R ( u) ( ) u Fourier Spectrum of f(x) Power Spectrum (spectrum density function) of f(x) Phase angle 4

5 Image Transforms Fourier transform theory Frequency ( u) = ( R ( u) I ( u) ) 2 F + φ ( u) = tan 1 I R ( u) ( ) u u is called the frequency variable Euler s formula exp [ j2πux] = cos 2πux j sin 2πux 5

6 Image Transforms Fourier transform theory Intuitive interpretation An infinite sum of sine and cosine terms, each u determines the frequency of its corresponding sine cosine pair F f ( u) f ( x)exp[ j2πux] = dx ( x) F( u)exp[ j2πux] = du 6

7 Image Transforms Fourier transform 7

8 Image Transforms Fourier transform When W become smaller, what will happen to the spectrum? 8

9 Image Transforms Discrete Fourier transform Continuous function f(x) is discretized into a sequence { f ( x ) f ( x + x), f ( x + 2 x),, f ( x + ( N 1) x) } 0, 0 0 L 0 by taking N samples x units apart 9

10 Image Transforms Discrete Fourier transform pair of the sampled function F N 1 1 ( u) f ( x + x x) for = N x= u = 0 0 0,1,2,..., exp N 1 f j2πux N 1 ( x) F( u) for N = u= 0 x j2πux exp N = 0,1,2,..., N 1 10

11 Image Transforms Fourier transform of unit impulse function 0 t 0 t 0 δ( t) = and δ( t) dt = 1 t = 0 11

12 Image Transforms Fourier transform of unit impulse function F[ δ ( x)] δ = jux e = 1 =0 jux = ( x) e dx δ(x) F x F(ju) 1 0 x 0 u 12

13 Image Transforms Fourier transform of unit impulse train Here t = x and ω = u 13

14 Convolution Convolution The convolution of two functions f(x) and g(x), denote f(x)*g(x) f ( x) g( x) = f ( a) g( x a) da 14

15 Convolution Convolution An example 15

16 Convolution Convolution and Spatial Filtering f(x,y) w(x,y) f(x,y)*w(x,y) 16

17 Convolution Convolution theorem f ( x) g( x) F( u) G( u) f ( x) g( x) F( u) G( u) 17

18 Sampling Sampling f(t) F(u) FT s(t) -w w S(u) t FT 1/ t 1/ t s(t)f(t) 1/ t S(u)*F(u) FT 18

19 Sampling Sampling 1/ t t FT -w w G(u) f(t) -w w G(u)[S(u)*F(u)]= F(u)] FT 19

20 Sampling Theorem Bandwidth, Sample Rate, and Nyquist Theorem 1/ t The sampling rate (Nyquist rate) must be at least two times the bandwidth of a bandlimited signal -w w G(u) t 2w -w w G(u)[S(u)*F(u)]= F(u)] 20

21 Aliasing Over- and under-sampling Anti-aliasing filtering 21

22 Aliasing Consider an image with 512 alternating vertical black and white stripes. (You may not even be able to see the alternating stripes because of poor screen resolution. But take my word for it, they are there.) Source: 22

23 Aliasing The image is created by sampling an image with 512 alternating values of black (gray = 0) and white (gray = 255). Starting in row 0, 512 samples of the image are taken. For each successive row, 1 fewer sample is taken from row 0, (i.e. for row 1, take 511 samples, for row 2, take 510 samples,... for row 511, take 1 sample). The whole row is then reconstructed from the samples by pixel replication. The result is a colossal aliasing pattern. Source: 23

24 More examples Aliasing 24

25 More examples Aliasing 25

26 More examples Aliasing 26

27 Image Transforms 2D Fourier Transform (Fourier Transform of Images) F ( u, v) f ( x, y)exp[ j2π ( ux + vy) ] = dxdy ( x, y) F( u, v)exp[ j2π ( ux + vy ] f = ) dudv 27

28 Image Transforms 2D Fourier Transform (Fourier Fourier Transform of Images) Spectrum 2 2 of f(x) 1 ( u, v) = ( R ( u, v) I ( u, v) ) 2 F ( u, v) = ( R ( u, v) I ( u, v) ) P + φ ( u, v) = tan 1 I R ( u, v) ( ) u, v Power Spectrum (spectrum density function) of f(x) Phase angle 28

29 29 Image Transforms 2D Discrete Fourier Transform (Fourier Transform of Digital Images) ( ) ( ) 1 0,1,2,..., 1 0,1,2,..., 2 exp, 1, = = = = = N v M u for N vy M ux j y y x x x x f MN v u F M x N y π ( ) ( ) 1 0,1,2,..., 1 0,1,2,..., 2 exp,, = = + = = = N y M x for N vy M ux j v v u u F y x f M u N v π

30 Frequency Domain Processing What does frequency mean in an image? 30

31 Frequency Domain Processing What does frequency mean in an image? 31

32 Frequency Domain Processing What does frequency mean in an image? 32

33 Frequency Domain Processing What does frequency mean in an image? High frequency components fast changing/sharp features Low frequency components slow changing/smooth features 33

34 Frequency Domain Processing The foundation of frequency domain techniques is the convolution theorem f ( x, y) g( x, y) F( u, v) G( u, v) 34

35 Frequency Domain Processing H(u, v) is called the transfer function 35

36 Frequency Domain Processing Typical lowpass filters and their transfer functions 36

37 Frequency Domain Processing Typical lowpass filters and their transfer functions 37

38 Frequency Domain Processing Example 38

39 Frequency Domain Processing Example 39

40 Frequency Domain Processing Typical lowpass filters and their transfer functions 40

41 Frequency Domain Processing Example 41

42 Frequency Domain Processing Typical lowpass filters and their transfer functions 42

43 Frequency Domain Processing Example 43

44 Frequency Domain Processing Example 44

45 Frequency Domain Processing Example 45

46 Frequency Domain Processing Typical highpass filters and their transfer functions 46

47 Frequency Domain Processing Typical highpass filters and their transfer functions 47

48 Frequency Domain Processing Typical highpass filters and their transfer functions 48

49 Frequency Domain Processing Examples 49

50 Frequency Domain Processing Examples 50

51 Frequency Domain Processing Examples 51

52 Frequency Domain Processing More examples 52

53 Frequency Domain Processing Examples 53

54 Frequency Domain Processing Examples 54

55 Frequency Domain Processing Spatial vs frequency domain 55

56 Frequency Domain Processing Spatial vs frequency domain 56

57 Frequency Domain Processing Examples 57

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