Digital Image Processing COSC 6380/4393

Transcription

1 Digital Image Processing COSC 6380/4393 Lecture 10 Feb 14 th, 2019 Pranav Mantini Slides from Dr. Shishir K Shah and S. Narasimhan

2 Time and Frequency example : g(t) = sin(2π f t) + (1/3)sin(2π (3f) t)

3 Time and Frequency example : g(t) = sin(2pi f t) + (1/3)sin(2pi (3f) t) = +

4 Frequency Spectra example : g(t) = sin(2pi f t) + (1/3)sin(2pi (3f) t) = +

5 Periodic Function Sum of sine and cosine waves: f t

6 Periodic Function Sum of sine and cosine waves: f t = a 0 + a 1 cos t + a 2 cos 2t + b 1 sin t + b 2 sin 2t +

7 Recap 0 2π sin mt dt= 0 2π 0 cos mt dt= 0 2π න sin mt cos nt dt = π sin mt sin nt dt =? ( m! = n) 0 2π sin mt sin nt dt =? (m = n) 0 2π cos mt cos nt dt =? ( m! = n) 0 2π cos mt cos nt dt =? (m = n)

8 Periodic Function f t = a 0 + a 1 cos t + a 2 cos 2t + b 1 sin t + b 2 sin 2t + Sum of sine and cosine waves: a 0 = 1 2π න 0 a n = 1 π න 0 2π 2π f(t)dt f t cos nt dt b n = 1 π න 0 2π f t sin nt dt

9 Periodic Function 2π 4π 6π

10 Periodic Function 2π 4π 6π Sum of sine and cosine waves: a 0 = 1/2 a n = 0 Cosine Frequency spectra 0 if n is even b n = ቐ 2 if n is odd nπ

11 Periodic Function 2π 4π 6π Sum of sine and cosine waves: a 0 = 1/2 a n = 0 0 if n is even b n = ቐ 2 if n is odd nπ a 0 Sin Frequency spectra 1/2

12 Periodic Function 2π 4π 6π Sum of sine and cosine waves: a 0 = 1/2 a n = 0 0 if n is even b n = ቐ 2 if n is odd nπ b 1 Sin Frequency spectra 2/π

13 Periodic Function 2π 4π 6π Sum of sine and cosine waves: a 0 = 1/2 a n = 0 0 if n is even b n = ቐ 2 if n is odd nπ Sin Frequency spectra b 2

14 Periodic Function 2π 4π 6π Sum of sine and cosine waves: a 0 = 1/2 a n = 0 Sin Frequency spectra 0 if n is even b n = ቐ 2 if n is odd nπ b 3 2/3π

15 Periodic Function 2π 4π 6π Sum of sine and cosine waves: a 0 = 1/2 a n = 0 Sin Frequency spectra 0 if n is even b n = ቐ 2 if n is odd nπ b 5 2/5π

16 Frequency Spectra

17 Frequency Spectra 2/3π 2/π Sin(t) 2/3π Sin(3t) = + =

18 Frequency Spectra 2/5π b 5 2/5π Sin(5t) = + =

19 Frequency Spectra 2/7π Sin(7t) = + =

20 Frequency Spectra = + =

21 Frequency Spectra = + =

22 Frequency spectra f t = a 0 + a 1 cos t + a 2 cos 2t + b 1 sin t + b 2 sin 2t +

23 Frequency spectra f t = a 0 + a 1 cos t + a 2 cos 2t + b 1 sin t + b 2 sin 2t + Cos frequencies Sin frequencies a0 a1 a2 a3 a4 a5 a6 a7 2 0 b0 b1 b2 b3 b4 b5 b6 b7

24 Frequency spectra f t = a 0 + a 1 cos t + a 2 cos 2t + b 1 sin t + b 2 sin 2t + Cos frequencies Sin frequencies a0 a1 a2 a3 a4 a5 a6 a7 2 0 b0 b1 b2 b3 b4 b5 b6 b7 (a 0, b 0 ) Corresponds to frequency 0 (a 1, b 1 ) Corresponds to frequency 1. (a n, b n ) Corresponds to frequency n Combine them for compact representation

25 Complex Form of Fourier Series Compact Form easier to represent and integrate

26 Complex Form of Fourier Series Compact Form easier to represent and integrate f t = a 0 + a 1 cos t + a 2 cos 2t + b 1 sin t + b 2 sin 2t +

27 Complex Form of Fourier Series Compact Form easier to represent and integrate f t = a 0 + a 1 cos t + a 2 cos 2t + b 1 sin t + b 2 sin 2t + f t = a 0 + n=1 a n cos(nt) + b n sin(nt) n=1

28 Complex Form of Fourier Series Compact Form easier to represent and integrate f t = a 0 + a 1 cos t + a 2 cos 2t + b 1 sin t + b 2 sin 2t + f t = a 0 + n=1 a n cos(nt) + n=1 b n sin(nt) Euler s Rule: e 1 nt = cos nt + 1 sin nt, e 1 nt = cos nt 1 sin nt

29 Complex Form of Fourier Series Compact Form easier to represent and integrate f t = a 0 + a 1 cos t + a 2 cos 2t + b 1 sin t + b 2 sin 2t + f t = a 0 + n=1 a n cos(nt) + n=1 b n sin(nt) Euler s Rule: e 1 nt = cos nt + 1 sin nt, e 1 nt = cos nt 1 sin nt cos nt = 1 2 e 1 nt + e 1nt sin nt = 1 2 (e 1 nt e 1 nt )

30 Complex Form of Fourier Series Compact Form easier to represent and integrate f t = a 0 + a 1 cos t + a 2 cos 2t + b 1 sin t + b 2 sin 2t + f t = a 0 + f t = a 0 + n=1 n=1 a n cos(nt) + n=1 b n sin(nt) a n 1 2 e 1nt + e 1nt + b n 1 2 (e 1nt e 1nt ) cos nt = 1 2 e 1 nt + e 1nt sin nt = 1 2 (e 1 nt e 1 nt )

31 Complex Form of Fourier Series Compact Form easier to represent and integrate f t = a 0 + a 1 cos t + a 2 cos 2t + b 1 sin t + b 2 sin 2t + f t = a 0 + f t = a 0 + n=1 f t = a n=1 n=1 a n cos(nt) + n=1 b n sin(nt) a n 1 2 e 1nt + e 1nt + b n 1 2 (e 1nt e 1nt ) (a n 1b n )e 1nt + (a n + 1b n ) e 1nt

32 Complex Form of Fourier Series Compact Form easier to represent and integrate f t = a 0 + a 1 cos t + a 2 cos 2t + b 1 sin t + b 2 sin 2t + f t = a 0 + f t = a 0 + n=1 f t = a n=1 n=1 a n cos(nt) + n=1 b n sin(nt) a n 1 2 e 1nt + e 1nt + b n 1 2 (e 1nt e 1nt ) (a n 1b n )e 1nt + (a n + 1b n ) e 1nt

33 Fourier Transform We want to understand the frequency n of our signal. So, let s reparametrize the signal by x instead of t: f(x) Fourier Transform c n c n = 1 2 (a n 1b n ) n 1

34 Periodic Function f t = a 0 + a 1 cos t + a 2 cos 2t + b 1 sin t + b 2 sin 2t + Sum of sine and cosine waves: a 0 = 1 2π න 0 a n = 1 π න 0 2π 2π f(t)dt f t cos nt dt b n = 1 π න 0 2π f t sin nt dt

35 Fourier Transform We want to understand the frequency n of our signal. So, let s reparametrize the signal by x instead of t: f(x) Fourier Transform c n c n = 1 2 (a n 1b n ) n 1 c n = 1 2π 2π න f x cos nx dx 1 2π 0 2π 1 න f x 0 sin nx dx

36 Fourier Transform We want to understand the frequency n of our signal. So, let s reparametrize the signal by x instead of t: f(x) Fourier Transform c n c n = 1 2π න 0 c n = a n jb n 2π f x cos nx dx 1 = 1 2π න 0 2π 2π j න 0 2π f x f x cos nx 1 sin nx dx sin nx dx

37 Fourier Transform We want to understand the frequency n of our signal. So, let s reparametrize the signal by x instead of t: f(x) Fourier Transform c n c n = 1 2π න 0 c n = a n jb n 2π f x cos nx dx 1 = 1 2π න 0 2π 2π j න 0 2π f x f x cos nx 1 sin nx dx sin nx dx Note: e 1nx = cos nx 1sin nx

38 Fourier Transform We want to understand the frequency n of our signal. So, let s reparametrize the signal by x instead of t: f(x) Fourier Transform c n c n = 1 2π න 0 c n = a n jb n 2π f x cos nx dx 1 = 1 2π න 0 2π 2π j න 0 2π f x f x cos nx 1 sin nx dx sin nx dx Note: e 1nx = cos nx jsin nx F n = 1 2π 2π න f x e 1nx dx 0

39 Fourier Transform We want to understand the frequency u of our signal. So, let s reparametrize the signal by x instead of t: f(x) Fourier Transform F u F u = න F u = c u f x e 1ux dx Spatial Domain (x) Frequency Domain (u) (Frequency Spectrum F(u))

40 Inverse Fourier Transform (IFT) Frequency Domain (u) Spatial Domain (x) f x = න = න c u e 1ut du F(u) e 1ut du f t = a 0 + a n cos(nt) + b n sin(nt) n=1 n=1

41 Discrete Fourier Transform Spatial Domain (x) Fourier Transform Discrete Fourier Transform F u = න Frequency Domain (u) F u = f x e 1ux x= f x e 1ux dx Frequency Domain (u) Inverse Fourier Transform Inverse Discrete Fourier Transform f(x) = න f x = Spatial Domain (x) F u e 1ux du u= F u e 1ux e 1x = cosx e 1x = cosx + 1sinx 1sinx

42 From 1D 2D One dimension (x) frequency (u) Two dimensions (i, j) Frequencies along (I,j) (u,v)

43 Sinusoidal Images sin(i + j)(u = 1, v = 1) sin(i + 0.5j)(u = 1, v = 0.5) sin(0.5i + 0.5j) u = v = 0.5 Waveform Contour plots From Wolframalpha

44 From 1D 2D Two dimensions (i, j) Frequencies along (I,j) (u,v) Can be decomposed into 2D sin(ui + vj) and cos(ui + vj) waves with frequencies (u,v)

45 Discrete Fourier Transform Spatial Domain (x) Fourier Transform Discrete Fourier Transform F u = න Frequency Domain (u) F u = f x e 1ux x= f x e 1ux dx Frequency Domain (u) Inverse Fourier Transform Inverse Discrete Fourier Transform f(x) = න f x = Spatial Domain (x) F u e 1ux du u= F u e 1ux e 1x = cosx e 1x = cosx + 1sinx 1sinx

46 2D Discrete Fourier Transform Spatial Domain (i,j) Fourier Transform Discrete Fourier Transform Frequency Domain (u,v) F u, v = න න F u, v = f i, j e 1(ui+vj) x= y= f i, j e 1(ui+vj) di dj Frequency Domain (u,v) Inverse Fourier Transform Inverse Discrete Fourier Transform f(i, j) = න f i, j = Spatial Domain (i,j) න u= v= F u, v e 1(ui+vj) du dv F u, v e 1(ui+vj)

47 Images as 2D waves Are Images 2D Waves? Continuous or discrete? Are they periodic? Can we apply DFT on images?

48 Sinusoidal Images We shall make frequent discussion in this module of the frequency content of an image. First consider images having the simplest frequency content. A digital sine image I is an image having elements and a digital cosine image has elements where u and v are integer frequencies in the i- and j- directions (measured in cycles/image; notice division by N). 48

49 Sinusoidal Images The radial frequency (how fast the image oscillates in its direction of propagation) is The angle of the wave (relative to the i-axis) is 49

50 2D Discrete Fourier Transform Spatial Domain (i,j) Fourier Transform Discrete Fourier Transform Frequency Domain (u,v) F u, v = න න F u, v = f i, j e 1(ui+vj) x= y= f i, j e 1(ui+vj) di dj Frequency Domain (u,v) Inverse Fourier Transform Inverse Discrete Fourier Transform f(i, j) = න f i, j = Spatial Domain (i,j) න u= v= F u, v e 1(ui+vj) du dv F u, v e 1(ui+vj)

51 2D Discrete Fourier Transform If I is an image of size N then Sin image Cos image Let ሚI be the DFT of the I N 1 N 1 ሚI u, v = I i, j e 12π N (ui+vj) i=0 j=0 F u, v = f i, j e 1(ui+vj) x= y=

52 2D Inverse Discrete Fourier Transform Let ሚI be the DFT of the I N 1 I i, j = u=0 N 1 v=0 ሚI u, v e 12π N (ui+vj) f i, j = F u, v e 1(ui+vj) u= v=