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ECES 682 Digital Image Processing

ECES 682 Digital Image Processing. Oleh Tretiak ECE Department Drexel University. About the Course. Homework 2 due today Midterm exam next week Covers first three homeworks 90 minutes (second half of class). Last Week’s Lecture. Image Enhancement in the Spatial Domain

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ECES 682 Digital Image Processing

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

  2. About the Course • Homework 2 due today • Midterm exam next week • Covers first three homeworks • 90 minutes (second half of class) Digtial Image Processing, Spring 2006

  3. 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 Digtial Image Processing, Spring 2006

  4. 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 Digtial Image Processing, Spring 2006

  5. Mr. Joseph Fourier • To analyze a heat transient problem, Fourier proposed to express an arbitrary function by the formula Digtial Image Processing, Spring 2006

  6. Fourier Methods Digtial Image Processing, Spring 2006

  7. 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 fn • Space range 0 ≤ n< N • Frequency range 0 ≤ k < N Digtial Image Processing, Spring 2006

  8. DFT Example Digtial Image Processing, Spring 2006

  9. 2D FT Example Digtial Image Processing, Spring 2006

  10. Another Example Digtial Image Processing, Spring 2006

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

  12. x(u,v) y(u,v) h Two-Dimensional Systems • We would like to have a system model for vision. • Input: Image • Output: Our mind’s perception Digtial Image Processing, Spring 2006

  13. ‘Typical’ Visual Spatial Response Digtial Image Processing, Spring 2006

  14. low contrast high contrast

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

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

  17. Digtial Image Processing, Spring 2006

  18. How to Filter • Multiply image by (-1)x+y Image dimensions MxN • Compute F(u, v) DFT DC at M/2, N/2. F(u, v) complex valued • Multiply F(u, v) by H(u, v) DC for H(u, v) at M/2, N/2. • Compute inverse DFT of result in (3) • Take real part of result in (4) • Multiply result in (5) by (-1)x+y Digtial Image Processing, Spring 2006

  19. Notch Filter Digtial Image Processing, Spring 2006

  20. Fourier Low- and High-Pass Filters Digtial Image Processing, Spring 2006

  21. High-Boost Filter Digtial Image Processing, Spring 2006

  22. Space and Frequency Filters Digtial Image Processing, Spring 2006

  23. Radial Low-Pass Filter Digtial Image Processing, Spring 2006

  24. Power Distribution Digtial Image Processing, Spring 2006

  25. 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 Digtial Image Processing, Spring 2006

  26. Ideal vs. Butterworth Digtial Image Processing, Spring 2006

  27. Ideal vs. Gaussian Digtial Image Processing, Spring 2006

  28. ‘Morphological’ Filtering Digtial Image Processing, Spring 2006

  29. Sharpening Filters Digtial Image Processing, Spring 2006

  30. Sharpening: Ideal vs. Butterworth Digtial Image Processing, Spring 2006

  31. Sharpening: Ideal vs. Gaussian Digtial Image Processing, Spring 2006

  32. Laplacian in the Frequency Domain Digtial Image Processing, Spring 2006

  33. Homomorphic Filtering Digtial Image Processing, Spring 2006

  34. Correlation and Finding Things Digtial Image Processing, Spring 2006

  35. More About the Fourier Transform • Shift • Linearity • Scaling • Rotation • Seperability • Forward and inverse • Padding and wraparound Digtial Image Processing, Spring 2006

  36. Wraparound: Example Digtial Image Processing, Spring 2006

  37. Summary • Fourier methods in image processing • Filtering • Other • Filtering • Space domain N2 image, M2 filter • Cost = cN2M2 • Fourier domain • Cost = kN2logN • Other • Spectral estimation Digtial Image Processing, Spring 2006

  38. 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. Digtial Image Processing, Spring 2006

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