60-520 Presentation Image Filters

1. 60-520 PresentationImage Filters School of Computer Science University of Windsor November 2003

2. Outline • Introduction • Spatial Filtering • Smoothing • Sharpening • Frequency-Domain Filtering • Low pass • High pass • Summary Image Filters

3. Introduction • Filtering is the process of replacing a pixel with a value based on some operations or functions. • The operations/functions used on the original image are called filters. • or masks, kernels, templates, windows… Image Filters

4. Introduction • In digital image processing, filters are usually used to • suppress the high frequencies in an image • i.e., smoothing the image • suppress the low frequencies in an image • i.e., enhancing or detecting edges in the image Image Filters

5. Introduction • Image filters fall into two categories: • Spatial domain • Filters are based on direct manipulation of pixels on an image plane. • Frequency domain • Filters are based on modifying the Fourier transform (FT) of an image. Image Filters

6. Spatial Filters • The general processes can be denoted by the expression: • f(x,y) is the input image • g(x,y) is the processed image • T is an operator on f, defined over some neighborhood of (x,y) Image Filters

7. Spatial Filters • The principal approach in defining a neighborhood about a point (x,y) • use a subimage area centered at (x,y) • shapes of the neighborhood • circle • square • rectangular Image Filters

8. x (x-1,y-1) (x,y-1) (x+1,y-1) (x,y) (x-1,y) (x,y) (x+1,y) Image f(x,y) (x-1,y+1) (x,y+1) (x+1,y+1) y Spatial Filters Example: 3×3 neighborhood about a point (x,y) in an image Image Filters

9. x w(-1,-1) w(0,-1) w(1,-1) Mask w(-1,0) w(0,0) w(1,0) Image f(x,y) w(-1,1) w(0,1) w(1,1) y f(x-1,y-1) f(x,y-1) f(x+1,y-1) Mask coefficients f(x-1,y) f(x,y) f(x+1,y) f(x-1,y+1) f(x,y+1) f(x+1,y+1) Pixels under mask Image Filters

10. Spatial Filters – linear filters • For linear spatial filtering, the result, R, at a point (x,y) is R=w(-1,-1)f(x-1,y-1) + w(0,-1)f(x,y-1) + …+ w(0,0)f(x,y) +… + w(0,1)f(x,y+1) + w(1,1)f(x+1,y+1) Image Filters

11. Spatial Filters – convolution • In general, linear filtering of an image is given by the expression: • The image f is of size M×N • The filter mask is of size m×n m=2a+1, n=2b+1 Image Filters

12. Spatial Filters – smoothing • Smoothing filters are used for blurring and for noise reduction. • Smoothing, linear spatial filter • average filters • reduce “sharp” transitions • side effect Image Filters

13. 1 1 1 1 1 1 1 1 1 Spatial Filters – smoothing, linear • Mean filters • example: Gaussian noise Original 5×5 mean filter 3×3 mean filter Image Filters

14. 1 1 1 1 1 1 1 1 1 Spatial Filters – smoothing, linear • Mean filters • example: Salt and pepper 5×5 mean filter 3×3 mean filter Image Filters

15. 1 2 1 2 4 2 1 2 1 Spatial Filters – smoothing, linear • Weighted average filters • example: • general expression: Image Filters

16. Spatial Filters – smoothing, nonlinear • Order-statistic filters • nonlinear spatial filters • order/rank the pixels contained in the image area encompassed by the filter Image Filters

17. Spatial Filters – smoothing, nonlinear • Median filters • replace a pixel value with the median of its neighboring pixel values • example: Neighborhood values: 15, 19, 20, 23, 24, 25, 26, 27, 50 Median value: 24 Image Filters

18. Spatial Filters – smoothing, nonlinear • Median filters • have excellent noise-reduction capabilities V.S. Gaussian noise removed By 3×3 median filter Gaussian noise removed by 3×3 mean filter Image Filters

19. Spatial Filters – smoothing, nonlinear • Median filters • are particularly effective in salt & pepper V.S. Salt & pepper removed By 3×3 median filter Salt & pepper removed by 3×3 mean filter Image Filters

20. Spatial Filters – smoothing, nonlinear • Max filters • maximum of neighboring pixel values • useful for finding the brightest points in an image • Min filters • minimum of neighboring pixel values • useful for finding the darkest points in an image Image Filters

21. Spatial Filters – sharpening • Principal objective • highlight fine detail in an image • enhance detail that has been blurred • Sharpening can be accomplished by spatial differentiation Image Filters

22. Spatial Filters – sharpening • For one dimensional function f(x) • first order derivative • second order derivative Image Filters

23. 5 5 4 3 2 1 0 0 0 6 0 0 0 0 1 3 1 0 0 0 0 7 7 7 7 -1 -1 -1 -1 -1 0 0 6 -6 0 0 0 1 2 -2 -1 0 0 0 7 0 0 0 -1 0 0 0 0 1 0 6 -12 6 0 0 1 1 -4 1 1 0 0 7 -7 0 0 Spatial Filters – sharpening • A sample (a) a scan line (b) image strip (c) first derivative (d) second derivative (a) (b) (c) (d) Image Filters

24. Spatial Filters – sharpening • The Laplacian • second derivative of a two dimensional function f(x,y) = [f(x+1,y)+f(x-1,y)+f(x,y+1)+f(x,y-1)] -4f(x,y) Image Filters

25. -1 1 0 2 1 1 0 -1 1 1 1 2 -4 -8 -4 2 1 1 -1 1 0 2 1 1 1 -1 0 Spatial Filters – sharpening • The Laplacian • use a convolution mask to approximate Image Filters

26. Spatial Filters – sharpening • The Laplacian • example: Image Filters

27. Spatial Filters – sharpening • The Laplacian • example: Image Filters

28. Frequency Filters – Fourier transform • Fourier transform (FT) • decompose an image into its sine and cosine components • transform real space images into Fourier or frequency space images • In a frequency space image, each point represents a particular frequency contained in the real domain image. Image Filters

29. Frequency Filters – Fourier transform • Discrete Fourier transform (DFT) • Inverse DFT Image Filters

30. Frequency Filters – Fourier transform • example: FT (log) Image Filters

31. DFT Filter function Inverse DFT F(u,v) H(u,v)F(u,v) f(x,y) Input image g(x,y) Processed image Frequency Filters • Basic steps for filtering in the frequency domain Image Filters

32. Frequency Filters • Frequencies in an image correspond to the rate of change in pixel values • High frequencies • rapid changes of gray level values • Low frequencies • slow changes of gray level values Image Filters

33. Frequency Filters • Lowpass filters • attenuate high frequencies while “passing” low frequencies • Highpass filters • attenuate low frequencies while “passing” high frequencies Image Filters

34. Frequency Filters – lowpass filters • Ideal lowpass filters (ILPF) Image Filters

35. Frequency Filters – lowpass filters • Butterworth lowpass filters (BLPF) Image Filters

36. Frequency Filters – lowpass filters • Gaussian lowpass filters (GLPF) Image Filters

37. Frequency Filters – highpass filters • Highpass filters • Ideal higpass filters (IHPF) • Butterworth highpass filters (BHPF) • Gaussian highpass filters (GHPF) Image Filters

38. Image Filters

39. Frequency Filters – bandpass filters • Bandpass filters • attenuate very low frequencies and very high frequencies • enhance edges while reducing the noise at the same time Image Filters

40. Frequency Filters • Examples: (lowpass filters) ILPF with cut-off frequency of 1/3 ILPF with cut-off frequency of 1/2 BLPF with cut-off frequency of 1/3 BLPF with cut-off frequency of 1/2 Gaussian noise Original Image Filters

41. Frequency Filters • Examples: (highpass filters) Image Filters

42. Frequency Filters • Relationship and comparison with spatial filters • spatial filtering • frequency filtering Image Filters

43. Frequency Filters • Comparison with spatial filters • more computational efficient • more intuitive Image Filters

44. Summary • Filtering is the operation of applying a transform on an image in order to enhance it. • Filtering techniques can be subdivided into two types • Spatial domain filtering • Frequency domain filtering Image Filters

45. Summary • Filtering techniques are very useful in image analysis and processing • Noise removal • Edge detection Image Filters

46. The end Thank you & Questions ? Image Filters