Chapter 3: The Image, its Mathematical and Physical Background

1. Chapter 3: The Image, its Mathematical and Physical Background 3.1. Overview 3.2. Linear Integral Transforms 3.3. Images as Stochastic Processes 3.4. Image Formation Physics

2. 3.1.2. Dirac Delta Function Heaviside function: 3.1. Overview Pulse:

3. Impulse: Dirac Delta Function: 2D delta function:

4. Sampling: Image function:

5. Show Proof: 4

6. by Hospital’s rule 3-5

7. 3.1.3. Convolution

8. 。Discrete case: Wraparound If N < A+B,

9. In practice, 5-8

10. 5-9

11. Properties: Show Proof:

12. 2-D convolution: Discrete: 3-11

13. Correlation: ( )

14. Transform: → domain 1 domain 2 e.g., Fourier transform Frequency domain Spatial domain → → Advantages: implicit explicit properties Examples: Fourier, wavelet, cosine, radon, slant, Haar, Hough, Paley-Walsh transforms, …..…. 3.2. Linear Integral Transforms 3.2.3. Fourier Transform

15. Fourier series -- A periodic (T) function f(x) can be written as where In complex form where and 3-14

16. Since 3-15

17. 3-16

18. In continuous case, 3-17

19. Fourier transform (FT) 1-D: aggregates the sinuous component of f.

20. Complex spectrum Phase spectrum Amplitude spectrum Power spectrum Properties: 1. 2. (Parseval’s theory)

21. Examples: 3-20

22. 3-21

23. Discrete Fourier transform Input signal: 3-22

24. FFT: Successive Doubling Method Then Let Assume . Let N = 2M. ∴ 3-23

25. Let 3-26

26. --------- (B) Consider 3-26

27. 3-26

28. ○ Analysis: The Fourier sequence F(u), u = 0 , … , N-1 of f(x), x = 0 , … , N-1can be formed from sequences u = 0 , …… , M-1 Recursively divide F(u) and F(u+M); eventually, each contains one element F(w), i.e., w = 0, and F(w) = f(x). 3-27

29. ○ Example: Input sequence { f(0), f(1), ……, f(7) } Computing needs { f(0), f(2), f(4), f(6) } Computing needs { f(1), f(3), f(5), f(7) } 3-28

30. {f(0),f(2),f(4),f(6)}  {f(0),f(4)}, {f(2),f(6)} {f(1),f(3),f(5),f(7)}  {f(1),f(5)}, {f(3),(7)} {f(0),f(4)}  {f(0)}, {f(4)}, {f(2),f(6)}  {f(2)}, {f(6)} {f(1),f(5)}  {f(1)}, {f(5)}, {f(3),f(7)}  {f(3)}, {f(7)} Reorder the input sequence into {f(0), f(4), f(2), f(6), f(1), f(5), f(3),f(7)} ＊Bit-reversal rule 3-29

31. 3-30

32. ○ Inverse FFT ← Given ← compute into FFT. The output is i. Input ii. Taking the complex conjugate and multiplying by N , yields the f(x) 3-31

33. ○ 2-D: Discrete Fourier transform 3-32

34. 3-33

35. Properties: Linearity Periodicity Conjugate symmetry 3-34

36. Shifting

37. 。 Examples: 1-D signal 2-D image f F F’ 3-36

38. Rotation Polar coordinates: Correlation theorem Convolution theorem 3-37

40. ○ Any real function can always be decomposed into its even and odd parts, i.e., e.g.,

41. ○ Convolution involving impulse function ※ Copyf(x) at the location of each impulse 3-40

42. 3.2.5. Sampling theory Continuous function Dense sampling Sparse sampling Objective: looking at the question of how many sampling should be taken so that no information is lost in the sampling process

43. Spatial domain Frequency domain f(x): band-limited function Sampling function Sampling 3-42

44. Whittaker-Shannon sampling theory 1-D: 2-D: Interpolation Reconstruction Formula 3-43

45. 3.2.7. Wavelet transform Fourier spectrum provides all the frequencies present in a signal but does not tell where they are present. Windowed Fourier transformsuffers from the dilemma: Small range – poor frequency resolution Large range – poor localization Wavelet: wave that is only nonzero in a small region Wave Wavelet

46. Types of wavelets: Haar: Morlet: Mexican hat:DOG, LOG

47. ○ Operations on wavelet: (a) Dilation: i) Squashing ii) Expanding (b) Translation: i) Shift to the right ii) Shift to the left (c) Magnitude change: i) Amplification ii) Minification 3-46

48. Any function can be expressed as a sum of wavelets of the form 3-47

49. Wavelet transform:decomposes a function into a set of wavelets where : wavelets : mother wavelet New variables: scale translation Inverse wavelet transform:synthesize a function from wavelets coefficients

50. Discrete wavelet transform: Approximation coefficients (cA) : scaling functions Detail coefficients (cD) : wavelet functions Inverse discrete wavelet transform: