1. Digtial Image Processing, Spring 2006 1 ECES 682 Digital Image Processing Oleh Tretiak ECE Department Drexel University

2. Digtial Image Processing, Spring 20062 About the Course Homework 2 due today Midterm exam next week  Covers first three homeworks  90 minutes (second half of class)

3. Digtial Image Processing, Spring 20063 Last Week’s Lecture Image Enhancement in the Spatial Domain  Gray level transformations  Histogram processing  Arithmetic/Logic operations  Spatial filtering  Smoothing  Sharpening Matlab image processing  Image datatypes  Image display

4. Digtial Image Processing, Spring 20064 This Week’s Lecture Chapter 4, Image enhancement in the frequency domain  Fourier transform and the frequency domain  Filtering with Fourier methods  Spatial vs. Fourier filtering  Smoothing filters  Sharpening filters  Laplacian  Unsharp masking, homomorphic filtering  Funny stuff with the FFT  Convolution and correlation

5. Digtial Image Processing, Spring 20065 Mr. Joseph Fourier To analyze a heat transient problem, Fourier proposed to express an arbitrary function by the formula

6. Digtial Image Processing, Spring 20066 Fourier Methods Continuous time, real function, finite interval Sine/cosine Fourier series Continuous time, complex function, finite interval Fourier series, complex exponentials Discrete time, complex function, infinite interval Fourier transform, finite interval in frequency Discrete time, complex function, finite interval Discrete Fourier transform (DFT) Two dimensional complex function, infinite intervals 2-D Fourier transform Two dimensional complex function, polar coordinates Fourier-Bessel transform, angular harmonics

7. Digtial Image Processing, Spring 20067 FT and FFT We normally deal with low-pass functions centered at the origin f(x) F(u)  Space range -X/2 < x < X/2  Frequency range -W< u <W Natural coordinates for DFT are f n  Space range 0 ≤ n < N  Frequency range 0 ≤ k < N

8. Digtial Image Processing, Spring 20068 DFT Example

9. Digtial Image Processing, Spring 20069 2D FT Example

10. Digtial Image Processing, Spring 200610 Another Example

11. Digtial Image Processing, Spring 2006 11 Examples of 2DFT a b c a b c Image Fourier transform

12. Digtial Image Processing, Spring 200612 Two-Dimensional Systems We would like to have a system model for vision. h x(u,v) y(u,v) Input: Image Output: Our mind’s perception

13. Digtial Image Processing, Spring 200613 ‘Typical’ Visual Spatial Response

14. low contrast high contrast

15. Digtial Image Processing, Spring 2006 15 Objective value (intensity) Subjective (perceived) value Mach Bands

16. Digtial Image Processing, Spring 2006 16 The circles have the same objective intensity.

17. Digtial Image Processing, Spring 2006 17

19. Digtial Image Processing, Spring 200619 How to Filter 1.Multiply image by (-1) x+y Image dimensions MxN 2.Compute F(u, v) DFT DC at M/2, N/2. F(u, v) complex valued 3.Multiply F(u, v) by H(u, v) DC for H(u, v) at M/2, N/2. 4.Compute inverse DFT of result in (3) 5.Take real part of result in (4) 6.Multiply result in (5) by (-1) x+y

20. Digtial Image Processing, Spring 200620 Notch Filter

21. Digtial Image Processing, Spring 200621 Fourier Low- and High-Pass Filters

22. Digtial Image Processing, Spring 200622 High-Boost Filter

23. Digtial Image Processing, Spring 200623 Space and Frequency Filters

24. Digtial Image Processing, Spring 200624 Radial Low-Pass Filter

25. Digtial Image Processing, Spring 200625 Power Distribution

26. Digtial Image Processing, Spring 200626 Power Removal (a) Original image, (b) 8% power removal, (c) 5.4% power removal, (d) 4.3%, (e) 2%, (f) 0.5%. Radii are 5, 15, 30, 80, and 230. Max frequency is 250

27. Digtial Image Processing, Spring 200627 Ideal vs. Butterworth

28. Digtial Image Processing, Spring 200628 Ideal vs. Gaussian

29. Digtial Image Processing, Spring 200629 ‘Morphological’ Filtering

30. Digtial Image Processing, Spring 200630 Sharpening Filters

31. Digtial Image Processing, Spring 200631 Sharpening: Ideal vs. Butterworth

32. Digtial Image Processing, Spring 200632 Sharpening: Ideal vs. Gaussian

33. Digtial Image Processing, Spring 200633 Laplacian in the Frequency Domain

34. Digtial Image Processing, Spring 200634 Homomorphic Filtering

35. Digtial Image Processing, Spring 200635 Correlation and Finding Things

36. Digtial Image Processing, Spring 200636 More About the Fourier Transform Shift Linearity Scaling Rotation Seperability Forward and inverse Padding and wraparound

37. Digtial Image Processing, Spring 200637 Wraparound: Example

38. Digtial Image Processing, Spring 200638 Summary Fourier methods in image processing  Filtering  Other Filtering  Space domain N 2 image, M 2 filter  Cost = cN 2 M 2  Fourier domain  Cost = kN 2 logN Other  Spectral estimation

39. Digtial Image Processing, Spring 200639 References on the FT Ron Bracewell, The Fourier Transform and its Applications, McGraw-Hill, 2000 About Josef Fourier  www-groups.dcs.st-and.ac.uk (University of Saint Andrews MacTutor history of mathematics web site). The image on the right is from that site.